(* Title: HOL/Algebra/Bij.thy ID: $Id: Bij.thy,v 1.8 2006/08/03 12:58:12 ballarin Exp $ Author: Florian Kammueller, with new proofs by L C Paulson *) theory Bij imports Group begin section {* Bijections of a Set, Permutation Groups and Automorphism Groups *} constdefs Bij :: "'a set => ('a => 'a) set" --{*Only extensional functions, since otherwise we get too many.*} "Bij S ≡ extensional S ∩ {f. bij_betw f S S}" BijGroup :: "'a set => ('a => 'a) monoid" "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" by (simp add: Bij_def) 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_Bij: "f ∈ Bij S ==> (λx ∈ S. (Inv S f) x) ∈ Bij S" by (simp add: Bij_def bij_betw_Inv) 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 S f) S) f = (λx∈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 ∈ Bij S; x ∈ S|] ==> Inv S f x ∈ S" by (simp add: Bij_def bij_betw_def Inv_mem) lemma Bij_Inv_lemma: assumes eq: "!!x y. [|x ∈ S; y ∈ S|] ==> h(g x y) = g (h x) (h y)" shows "[|h ∈ Bij S; g ∈ S -> S -> S; x ∈ S; y ∈ 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 "∃x'∈S. ∃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 ∩ Bij (carrier G)" AutoGroup :: "('a, 'c) monoid_scheme => ('a => 'a) monoid" "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" by (simp add: Pi_I group.axioms) lemma (in group) restrict_Inv_hom: "[|h ∈ hom G G; h ∈ Bij (carrier G)|] ==> restrict (Inv (carrier G) h) (carrier G) ∈ hom G G" by (simp add: hom_def Bij_Inv_mem restrictI mult_funcset group.axioms Bij_Inv_lemma) lemma inv_BijGroup: "f ∈ Bij S ==> m_inv (BijGroup S) f = (λx ∈ 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 (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 "\<one>BijGroup (carrier G) ∈ auto G" by (simp add: BijGroup_def id_in_auto) next fix x assume "x ∈ auto G" thus "invBijGroup (carrier G) x ∈ auto G" by (simp del: restrict_apply add: inv_BijGroup auto_def restrict_Inv_Bij restrict_Inv_hom) qed theorem (in group) AutoGroup: "group (AutoGroup G)" by (simp add: AutoGroup_def subgroup.subgroup_is_group subgroup_auto group_BijGroup) end
lemma Bij_imp_extensional:
f ∈ Bij S ==> f ∈ extensional S
lemma Bij_imp_funcset:
f ∈ Bij S ==> f ∈ S -> S
lemma restrict_Inv_Bij:
f ∈ Bij S ==> restrict (Inv S f) S ∈ Bij S
lemma id_Bij:
(λx∈S. x) ∈ Bij S
lemma compose_Bij:
[| x ∈ Bij S; y ∈ Bij S |] ==> compose S x y ∈ Bij S
lemma Bij_compose_restrict_eq:
f ∈ Bij S ==> compose S (restrict (Inv S f) S) f = (λx∈S. x)
theorem group_BijGroup:
group (BijGroup S)
lemma Bij_Inv_mem:
[| f ∈ Bij S; x ∈ S |] ==> Inv S f x ∈ S
lemma Bij_Inv_lemma:
[| !!x y. [| x ∈ S; y ∈ S |] ==> h (g x y) = g (h x) (h y); h ∈ Bij S;
g ∈ S -> S -> S; x ∈ S; y ∈ S |]
==> Inv S h (g x y) = g (Inv S h x) (Inv S h y)
lemma id_in_auto:
(λx∈carrier G. x) ∈ auto G
lemma mult_funcset:
op ⊗ ∈ carrier G -> carrier G -> carrier G
lemma restrict_Inv_hom:
[| h ∈ hom G G; h ∈ Bij (carrier G) |]
==> restrict (Inv (carrier G) h) (carrier G) ∈ hom G G
lemma inv_BijGroup:
f ∈ Bij S ==> invBijGroup S f = restrict (Inv S f) S
lemma subgroup_auto:
subgroup (auto G) (BijGroup (carrier G))
theorem AutoGroup:
group (AutoGroup G)