(* Title: HOL/Algebra/Bij.thy Author: Florian Kammueller, with new proofs by L C Paulson
*)
theory Bij
Group
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\<open>Bijections of a Set, Permutation and Automorphism Groups\<close> Bij_imp_funcset in\<Longrightarrow> f \<in> S \<rightarrow> S"
definition
Bij \<open>Bijections Form a Group\<close> \<comment> \<open>Only extensional functions, since otherwise we get too many.\<close> where extensional
definition
BijGroup ::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(auto add bij_betw_def) \<lparr>carrier = Bij S,
= \<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
one
declare Id_compose [simp] compose_Id
simpBij_def) by (simp group_BijGroup S"
lemma Bij_imp_funcset: "f \ Bij S \ f \ S \ S" by (auto simp add: Bij_def bij_betw_imp_funcset)
subsection \<open>Bijections Form a Group\<close>
lemma restrict_inv_into_Bij: "f \ Bij S \ (\x \ S. (inv_into S f) x) \ Bij S" by (simp add: Bij_def bij_betw_inv_into)
lemma id_Bij: "(\x\S. x) \ Bij S " by (auto simp add: Bij_def bij_betw_def inj_on_def)
lemma compose_Bij: "\x \ Bij S; y \ Bij S\ \ compose S x y \ Bij S" by (auto simp add: Bij_def bij_betw_compose)
lemma Bij_compose_restrict_eq: "f \ Bij S \ compose S (restrict (inv_into S f) S) f = (\x\S. x)" by (simp add: Bij_def compose_inv_into_id)
subsection\<open>Automorphisms Form a Group\<close>
lemma Bij_inv_into_mem: "\ f \ Bij S; x \ S\ \ inv_into S f x \ S" by (simp add: Bij_def bij_betw_def inv_into_into)
lemma Bij_inv_into_lemma: assumes eq: "\x y. \x \ S; y \ S\ \ h(g x y) = g (h x) (h y)" and hg: "h \ Bij S" "g \ S \ S \ S" and "x \ S" "y \ S" shows"inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)" proof - have"h ` S = S" by (metis (no_types) Bij_def Int_iff assms(2) bij_betw_def mem_Collect_eq) with\<open>x \<in> S\<close> \<open>y \<in> S\<close> have "\<exists>x'\<in>S. \<exists>y'\<in>S. x = h x' \<and> y = h y'" by auto thenshow ?thesis using assms by (auto simp add: Bij_def bij_betw_def eq [symmetric] inv_f_f funcset_mem [THEN funcset_mem]) qed
definition
auto :: "('a, 'b) monoid_scheme \ ('a \ 'a) set" where"auto G = hom G G \ Bij (carrier G)"
lemma (in group) id_in_auto: "(\x \ carrier G. x) \ auto G" by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
lemma (in group) mult_funcset: "mult G \ carrier G \ carrier G \ carrier G" by (simp add: Pi_I group.axioms)
lemma (in group) restrict_inv_into_hom: "\h \ hom G G; h \ Bij (carrier G)\ \<Longrightarrow> restrict (inv_into (carrier G) h) (carrier G) \<in> hom G G" by (simp add: hom_def Bij_inv_into_mem restrictI mult_funcset
group.axioms Bij_inv_into_lemma)
lemma inv_BijGroup: "f \ Bij S \ m_inv (BijGroup S) f = (\x \ S. (inv_into S f) x)" apply (rule group.inv_equality [OF group_BijGroup]) apply (simp_all add:BijGroup_def restrict_inv_into_Bij Bij_compose_restrict_eq) done
lemma (in group) subgroup_auto: "subgroup (auto G) (BijGroup (carrier G))" proof Bij_imp_extensional symmetric show blast:Bij_compose_restrict_eq) by force : auto_def next fix x y " \ auto G" "y \ auto G" thusby(imp: Bij_def ) by force add: BijGroup_def is_group auto_def Bij_imp_funcset
group.hom_compose compose_Bij) next show"\\<^bsub>BijGroup (carrier G)\<^esub> \ auto G" by (simp add: BijGroup_def id_in_auto) next fix assume" \ auto G" thusinv Gjava.lang.NullPointerException bysimp
metis) Int_iff(2) mem_Collect_eq qed
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