src/HOL/Algebra/Bij.thy

more on bij_betw

(*  Title:      HOL/Algebra/Bij.thy
    ID:         $Id$
    Author:     Florian Kammueller, with new proofs by L C Paulson
*)

header {* Bijections of a Set, Permutation Groups, Automorphism Groups *}

theory Bij = Group:

constdefs
  Bij :: "'a set => ('a => 'a) set"
    --{*Only extensional functions, since otherwise we get too many.*}
  "Bij S == extensional S \<inter> {f. bij_betw f S 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 |)"


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 bij_betw_imp_funcset)


subsection {*Bijections Form a Group *}

lemma restrict_Inv_Bij: "f \<in> Bij S ==> (%x:S. (Inv S f) x) \<in> Bij S"
  by (simp add: Bij_def bij_betw_Inv)

lemma id_Bij: "(\<lambda>x\<in>S. x) \<in> Bij S "
  by (auto simp add: Bij_def bij_betw_def inj_on_def)

lemma compose_Bij: "[| x \<in> Bij S; y \<in> Bij S|] ==> compose S x y \<in> Bij S"
  by (auto simp add: Bij_def bij_betw_compose) 

lemma Bij_compose_restrict_eq:
     "f \<in> Bij S ==> compose S (restrict (Inv S f) S) f = (\<lambda>x\<in>S. x)"
  by (simp add: Bij_def compose_Inv_id)

theorem group_BijGroup: "group (BijGroup S)"
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)
done


subsection{*Automorphisms Form a Group*}

lemma Bij_Inv_mem: "[|  f \<in> Bij S;  x \<in> S |] ==> Inv S f x \<in> S"
by (simp add: Bij_def bij_betw_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 bij_betw_def)
apply (subgoal_tac "\<exists>x'\<in>S. \<exists>y'\<in>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 G == hom G G \<inter> Bij (carrier G)"

  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)

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:
      "[|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 mult_funcset
                group.axioms Bij_Inv_lemma)

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)
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 dest: id_in_auto)
 apply (simp del: restrict_apply
             add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom)
apply (auto simp add: BijGroup_def auto_def Bij_imp_funcset group.hom_compose
                      compose_Bij)
done

theorem AutoGroup: "group G ==> group (AutoGroup G)"
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)
done

end