more fundamental pred-to-set conversions for range and domain by means of inductive_set
(* Title: HOL/Algebra/Bij.thy
Author: Florian Kammueller, with new proofs by L C Paulson
*)
theory Bij
imports Group
begin
section {* Bijections of a Set, Permutation and Automorphism Groups *}
definition
Bij :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) set"
--{*Only extensional functions, since otherwise we get too many.*}
where "Bij S = extensional S \<inter> {f. bij_betw f S S}"
definition
BijGroup :: "'a set \<Rightarrow> ('a \<Rightarrow> 'a) monoid"
where "BijGroup S =
\<lparr>carrier = Bij S,
mult = \<lambda>g \<in> Bij S. \<lambda>f \<in> Bij S. compose S g f,
one = \<lambda>x \<in> S. x\<rparr>"
declare Id_compose [simp] compose_Id [simp]
lemma Bij_imp_extensional: "f \<in> Bij S \<Longrightarrow> f \<in> extensional S"
by (simp add: Bij_def)
lemma Bij_imp_funcset: "f \<in> Bij S \<Longrightarrow> f \<in> S \<rightarrow> S"
by (auto simp add: Bij_def bij_betw_imp_funcset)
subsection {*Bijections Form a Group *}
lemma restrict_inv_into_Bij: "f \<in> Bij S \<Longrightarrow> (\<lambda>x \<in> S. (inv_into S f) x) \<in> Bij S"
by (simp add: Bij_def bij_betw_inv_into)
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: "\<lbrakk>x \<in> Bij S; y \<in> Bij S\<rbrakk> \<Longrightarrow> 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 \<Longrightarrow> compose S (restrict (inv_into S f) S) f = (\<lambda>x\<in>S. x)"
by (simp add: Bij_def compose_inv_into_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] dest: Bij_imp_funcset)
apply (simp add: id_Bij Bij_imp_funcset Bij_imp_extensional, simp)
apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij)
done
subsection{*Automorphisms Form a Group*}
lemma Bij_inv_into_mem: "\<lbrakk> f \<in> Bij S; x \<in> S\<rbrakk> \<Longrightarrow> inv_into S f x \<in> S"
by (simp add: Bij_def bij_betw_def inv_into_into)
lemma Bij_inv_into_lemma:
assumes eq: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> h(g x y) = g (h x) (h y)"
shows "\<lbrakk>h \<in> Bij S; g \<in> S \<rightarrow> S \<rightarrow> S; x \<in> S; y \<in> S\<rbrakk>
\<Longrightarrow> inv_into S h (g x y) = g (inv_into S h x) (inv_into 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
definition
auto :: "('a, 'b) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) set"
where "auto G = hom G G \<inter> Bij (carrier G)"
definition
AutoGroup :: "('a, 'c) monoid_scheme \<Rightarrow> ('a \<Rightarrow> 'a) monoid"
where "AutoGroup G = BijGroup (carrier G) \<lparr>carrier := auto G\<rparr>"
lemma (in group) id_in_auto: "(\<lambda>x \<in> carrier G. x) \<in> auto G"
by (simp add: auto_def hom_def restrictI group.axioms id_Bij)
lemma (in group) mult_funcset: "mult G \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
by (simp add: Pi_I group.axioms)
lemma (in group) restrict_inv_into_hom:
"\<lbrakk>h \<in> hom G G; h \<in> Bij (carrier G)\<rbrakk>
\<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 \<in> Bij S \<Longrightarrow> m_inv (BijGroup S) f = (\<lambda>x \<in> S. (inv_into S f) x)"
apply (rule group.inv_equality)
apply (rule 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 (rule subgroup.intro)
show "auto G \<subseteq> carrier (BijGroup (carrier G))"
by (force simp add: auto_def BijGroup_def)
next
fix x y
assume "x \<in> auto G" "y \<in> auto G"
thus "x \<otimes>\<^bsub>BijGroup (carrier G)\<^esub> y \<in> auto G"
by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset
group.hom_compose compose_Bij)
next
show "\<one>\<^bsub>BijGroup (carrier G)\<^esub> \<in> auto G" by (simp add: BijGroup_def id_in_auto)
next
fix x
assume "x \<in> auto G"
thus "inv\<^bsub>BijGroup (carrier G)\<^esub> x \<in> auto G"
by (simp del: restrict_apply
add: inv_BijGroup auto_def restrict_inv_into_Bij restrict_inv_into_hom)
qed
theorem (in group) AutoGroup: "group (AutoGroup G)"
by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto
group_BijGroup)
end