src/HOL/Algebra/Bij.thy

 (*  Title:      HOL/Algebra/Bij
     ID:         $Id$
     Author:     Florian Kammueller, with new proofs by L C Paulson
 *)
 
-
-header{*Bijections of a Set, Permutation Groups, Automorphism Groups*} 
+header {* Bijections of a Set, Permutation Groups, Automorphism Groups *}
 
 theory Bij = Group:
 
 constdefs
-  Bij :: "'a set => (('a => 'a)set)" 
+  Bij :: "'a set => ('a => 'a) set"
     --{*Only extensional functions, since otherwise we get too many.*}
-    "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
+  "Bij S == extensional S \<inter> {f. f`S = S & inj_on f S}"
 
-   BijGroup ::  "'a set => (('a => 'a) monoid)"
-   "BijGroup S == (| carrier = Bij S, 
-		     mult  = %g: Bij S. %f: Bij S. compose S g f,
-		     one = %x: S. x |)"
+  BijGroup :: "'a set => ('a => 'a) monoid"
+  "BijGroup S ==
+    (| carrier = Bij S,
+      mult = %g: Bij S. %f: Bij S. compose S g f,
+      one = %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)
+  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)
+  by (auto simp add: Bij_def Pi_def)
 
 lemma Bij_imp_apply: "f \<in> Bij S ==> f ` S = S"
-by (simp add: Bij_def)
+  by (simp add: Bij_def)
 
 lemma Bij_imp_inj_on: "f \<in> Bij S ==> inj_on f S"
-by (simp add: Bij_def)
+  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)
+  by (simp add: Bij_def)
 
 
-subsection{*Bijections Form a Group*}
+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 (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 (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
 
 lemma Bij_compose_restrict_eq:
      "f \<in> Bij S ==> compose S (restrict (Inv S f) S) f = (\<lambda>x\<in>S. x)"
 apply (rule compose_Inv_id)
  apply (simp add: Bij_imp_inj_on)
-apply (simp add: Bij_imp_apply) 
+apply (simp add: Bij_imp_apply)
 done
 
 theorem group_BijGroup: "group (BijGroup S)"
-apply (simp add: BijGroup_def) 
+apply (simp add: BijGroup_def)
 apply (rule groupI)
     apply (simp add: compose_Bij)
    apply (simp add: id_Bij)
   apply (simp add: compose_Bij)
   apply (blast intro: compose_assoc [symmetric] Bij_imp_funcset)
  apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
-apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij) 
+apply (blast intro: Bij_compose_restrict_eq restrict_Inv_Bij)
 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) 
+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 |]        
+ 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 (simp add: Bij_def)
 apply (subgoal_tac "EX x':S. EX y':S. x = h x' & y = h y'", clarify)
  apply (simp add: eq [symmetric] Inv_f_f funcset_mem [THEN funcset_mem], blast)
 done
 
 constdefs
- auto :: "('a,'b) monoid_scheme => ('a => 'a)set"
+  auto :: "('a, 'b) monoid_scheme => ('a => 'a) set"
   "auto G == hom G G \<inter> Bij (carrier G)"
 
-  AutoGroup :: "[('a,'c) monoid_scheme] => ('a=>'a) monoid"
+  AutoGroup :: "('a, 'c) monoid_scheme => ('a => 'a) monoid"
   "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 group.axioms id_Bij) 
+  by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
 
 lemma mult_funcset: "group G ==> mult G \<in> carrier G -> carrier G -> carrier G"
   by (simp add:  Pi_I group.axioms)
 
 lemma restrict_Inv_hom:

 
 lemma inv_BijGroup:
      "f \<in> Bij S ==> m_inv (BijGroup S) f = (%x: S. (Inv S f) x)"
 apply (rule group.inv_equality)
 apply (rule group_BijGroup)
-apply (simp_all add: BijGroup_def restrict_Inv_Bij Bij_compose_restrict_eq)  
+apply (simp_all add: BijGroup_def restrict_Inv_Bij Bij_compose_restrict_eq)
 done
 
 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 (rule group.subgroupI)
+    apply (rule group_BijGroup)
+   apply (force simp add: auto_def BijGroup_def)
+  apply (blast intro: dest: id_in_auto)
  apply (simp del: restrict_apply
-	     add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom) 
+             add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
 apply (simp add: BijGroup_def auto_def Bij_imp_funcset compose_hom compose_Bij)
 done
 
 theorem AutoGroup: "group G ==> group (AutoGroup G)"
-apply (simp add: AutoGroup_def) 
+apply (simp add: AutoGroup_def)
 apply (rule Group.subgroup.groupI)
-apply (erule subgroup_auto)  
-apply (insert Bij.group_BijGroup [of "carrier G"]) 
-apply (simp_all add: group_def) 
+apply (erule subgroup_auto)
+apply (insert Bij.group_BijGroup [of "carrier G"])
+apply (simp_all add: group_def)
 done
 
 end
-