Theory Bij

```(*  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 ⇒ ('a ⇒ 'a) set"
― ‹Only extensional functions, since otherwise we get too many.›
where "Bij S = extensional S ∩ {f. bij_betw f S S}"

definition
BijGroup :: "'a set ⇒ ('a ⇒ 'a) monoid"
where "BijGroup S =
⦇carrier = Bij S,
mult = λg ∈ Bij S. λf ∈ Bij S. compose S g f,
one = λx ∈ S. x⦈"

declare Id_compose [simp] compose_Id [simp]

lemma Bij_imp_extensional: "f ∈ Bij S ⟹ f ∈ extensional S"

lemma Bij_imp_funcset: "f ∈ Bij S ⟹ f ∈ S → S"
by (auto simp add: Bij_def bij_betw_imp_funcset)

subsection ‹Bijections Form a Group›

lemma restrict_inv_into_Bij: "f ∈ Bij S ⟹ (λx ∈ S. (inv_into S f) x) ∈ Bij S"

lemma id_Bij: "(λx∈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 = (λx∈S. x)"

theorem group_BijGroup: "group (BijGroup S)"
apply (rule groupI)
apply (auto simp: compose_Bij id_Bij Bij_imp_funcset Bij_imp_extensional compose_assoc [symmetric])
apply (blast intro: Bij_compose_restrict_eq restrict_inv_into_Bij)
done

subsection‹Automorphisms Form a Group›

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 ‹x ∈ S› ‹y ∈ S› have "∃x'∈S. ∃y'∈S. x = h x' ∧ y = h y'"
by auto
then show ?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)"

definition
AutoGroup :: "('a, 'c) monoid_scheme ⇒ ('a ⇒ 'a) monoid"
where "AutoGroup G = BijGroup (carrier G) ⦇carrier := auto G⦈"

lemma (in group) id_in_auto: "(λx ∈ 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"

lemma (in group) restrict_inv_into_hom:
"⟦h ∈ hom G G; h ∈ Bij (carrier G)⟧
⟹ restrict (inv_into (carrier G) h) (carrier G) ∈ 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 = (λx ∈ S. (inv_into S f) x)"
apply (rule group.inv_equality [OF group_BijGroup])
done

lemma (in group) subgroup_auto:
"subgroup (auto G) (BijGroup (carrier G))"
proof (rule subgroup.intro)
show "auto G ⊆ carrier (BijGroup (carrier G))"
by (force simp add: auto_def BijGroup_def)
next
fix x y
assume "x ∈ auto G" "y ∈ auto G"
thus "x ⊗⇘BijGroup (carrier G)⇙ y ∈ auto G"
by (force simp add: BijGroup_def is_group auto_def Bij_imp_funcset
group.hom_compose compose_Bij)
next
show "𝟭⇘BijGroup (carrier G)⇙ ∈ auto G" by (simp add:  BijGroup_def id_in_auto)
next
fix x
assume "x ∈ auto G"
thus "inv⇘BijGroup (carrier G)⇙ x ∈ auto G"
by (simp del: restrict_apply