--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Algebra/Bij.thy Fri May 02 10:25:42 2003 +0200
@@ -0,0 +1,148 @@
+(* 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*}
+
+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. 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 |)"
+
+
+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)
+
+
+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, 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
+
+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)
+done
+
+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 : 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'", 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 intro: dest: id_in_auto)
+ apply (simp del: restrict_apply
+ 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 (rule Group.subgroup.groupI)
+apply (erule subgroup_auto)
+apply (insert Bij.group_BijGroup [of "carrier G"])
+apply (simp_all add: group_def)
+done
+
+end
+