192 lines
7.8 KiB
Text
192 lines
7.8 KiB
Text
import algebra.group_theory ..move_to_lib eq2
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open pi pointed algebra group eq equiv is_trunc trunc
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namespace group
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-- definition pmap_mul [constructor] {A B : Type*} [inf_pgroup B] (f g : A →* B) : A →* B :=
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-- pmap.mk (λa, f a * g a) (ap011 mul (respect_pt f) (respect_pt g) ⬝ !one_mul)
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-- definition pmap_inv [constructor] {A B : Type*} [inf_pgroup B] (f : A →* B) : A →* B :=
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-- pmap.mk (λa, (f a)⁻¹) (ap inv (respect_pt f) ⬝ !one_inv)
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definition pmap_mul [constructor] {A B : Type*} (f g : A →* Ω B) : A →* Ω B :=
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pmap.mk (λa, f a ⬝ g a) (respect_pt f ◾ respect_pt g ⬝ !idp_con)
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definition pmap_inv [constructor] {A B : Type*} (f : A →* Ω B) : A →* Ω B :=
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pmap.mk (λa, (f a)⁻¹ᵖ) (respect_pt f)⁻²
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definition inf_group_pmap [constructor] [instance] (A B : Type*) : inf_group (A →* Ω B) :=
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begin
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fapply inf_group.mk,
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{ exact pmap_mul },
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{ intro f g h, fapply pmap_eq,
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{ intro a, exact con.assoc (f a) (g a) (h a) },
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{ rexact eq_of_square (con2_assoc (respect_pt f) (respect_pt g) (respect_pt h)) }},
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{ apply pconst },
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{ intros f, fapply pmap_eq,
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{ intro a, exact one_mul (f a) },
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{ esimp, apply eq_of_square, refine _ ⬝vp !ap_id, apply natural_square_tr }},
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{ intros f, fapply pmap_eq,
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{ intro a, exact mul_one (f a) },
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{ reflexivity }},
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{ exact pmap_inv },
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{ intro f, fapply pmap_eq,
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{ intro a, exact con.left_inv (f a) },
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{ exact !con_left_inv_idp⁻¹ }},
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end
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definition group_trunc_pmap [constructor] [instance] (A B : Type*) : group (trunc 0 (A →* Ω B)) :=
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!trunc_group
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definition Group_trunc_pmap [reducible] [constructor] (A B : Type*) : Group :=
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Group.mk (trunc 0 (A →* Ω (Ω B))) _
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definition Group_trunc_pmap_homomorphism [constructor] {A A' B : Type*} (f : A' →* A) :
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Group_trunc_pmap A B →g Group_trunc_pmap A' B :=
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begin
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fapply homomorphism.mk,
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{ apply trunc_functor, intro g, exact g ∘* f},
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{ intro g h, induction g with g, induction h with h, apply ap tr,
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fapply pmap_eq,
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{ intro a, reflexivity },
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{ refine _ ⬝ !idp_con⁻¹,
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refine whisker_right _ !ap_con_fn ⬝ _, apply con2_con_con2 }}
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end
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definition Group_trunc_pmap_isomorphism [constructor] {A A' B : Type*} (f : A' ≃* A) :
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Group_trunc_pmap A B ≃g Group_trunc_pmap A' B :=
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begin
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apply isomorphism.mk (Group_trunc_pmap_homomorphism f),
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apply @is_equiv_trunc_functor,
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exact to_is_equiv (pequiv_ppcompose_right f),
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end
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definition Group_trunc_pmap_pid [constructor] {A B : Type*} (f : Group_trunc_pmap A B) :
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Group_trunc_pmap_homomorphism (pid A) f = f :=
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begin
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induction f with f, apply ap tr, apply eq_of_phomotopy, apply pcompose_pid
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end
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definition Group_trunc_pmap_pconst [constructor] {A A' B : Type*} (f : Group_trunc_pmap A B) :
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Group_trunc_pmap_homomorphism (pconst A' A) f = 1 :=
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begin
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induction f with f, apply ap tr, apply eq_of_phomotopy, apply pcompose_pconst
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end
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definition Group_trunc_pmap_pcompose [constructor] {A A' A'' B : Type*} (f : A' →* A) (f' : A'' →* A')
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(g : Group_trunc_pmap A B) : Group_trunc_pmap_homomorphism (f ∘* f') g =
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Group_trunc_pmap_homomorphism f' (Group_trunc_pmap_homomorphism f g) :=
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begin
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induction g with g, apply ap tr, apply eq_of_phomotopy, exact !passoc⁻¹*
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end
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definition Group_trunc_pmap_phomotopy [constructor] {A A' B : Type*} {f f' : A' →* A} (p : f ~* f') :
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@Group_trunc_pmap_homomorphism _ _ B f ~ Group_trunc_pmap_homomorphism f' :=
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begin
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intro f, induction f, exact ap tr (eq_of_phomotopy (pwhisker_left a p))
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end
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definition ab_inf_group_pmap [constructor] [instance] (A B : Type*) : ab_inf_group (A →* Ω (Ω B)) :=
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⦃ab_inf_group, inf_group_pmap A (Ω B), mul_comm :=
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begin
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intro f g, fapply pmap_eq,
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{ intro a, exact eckmann_hilton (f a) (g a) },
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{ rexact eq_of_square (eckmann_hilton_con2 (respect_pt f) (respect_pt g)) }
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end⦄
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definition ab_group_trunc_pmap [constructor] [instance] (A B : Type*) :
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ab_group (trunc 0 (A →* Ω (Ω B))) :=
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!trunc_ab_group
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definition AbGroup_trunc_pmap [reducible] [constructor] (A B : Type*) : AbGroup :=
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AbGroup.mk (trunc 0 (A →* Ω (Ω B))) _
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/- Group of functions whose codomain is a group -/
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definition group_pi [instance] [constructor] {A : Type} (P : A → Type) [Πa, group (P a)] : group (Πa, P a) :=
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begin
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fapply group.mk,
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{ apply is_trunc_pi },
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{ intro f g a, exact f a * g a },
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{ intros, apply eq_of_homotopy, intro a, apply mul.assoc },
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{ intro a, exact 1 },
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{ intros, apply eq_of_homotopy, intro a, apply one_mul },
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{ intros, apply eq_of_homotopy, intro a, apply mul_one },
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{ intro f a, exact (f a)⁻¹ },
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{ intros, apply eq_of_homotopy, intro a, apply mul.left_inv }
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end
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definition Group_pi [constructor] {A : Type} (P : A → Group) : Group :=
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Group.mk (Πa, P a) _
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/- we use superscript in the following notation, because otherwise we can never write something
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like `Πg h : G, _` anymore -/
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notation `Πᵍ` binders `, ` r:(scoped P, Group_pi P) := r
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definition Group_pi_intro [constructor] {A : Type} {G : Group} {P : A → Group} (f : Πa, G →g P a)
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: G →g Πᵍ a, P a :=
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begin
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fconstructor,
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{ intro g a, exact f a g },
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{ intro g h, apply eq_of_homotopy, intro a, exact respect_mul (f a) g h }
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end
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-- definition AbGroup_trunc_pmap_homomorphism [constructor] {A A' B : Type*} (f : A' →* A) :
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-- AbGroup_trunc_pmap A B →g AbGroup_trunc_pmap A' B :=
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-- Group_trunc_pmap_homomorphism f
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/- Group of functions whose codomain is a group -/
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-- definition group_arrow [instance] (A B : Type) [group B] : group (A → B) :=
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-- begin
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-- fapply group.mk,
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-- { apply is_trunc_arrow },
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-- { intro f g a, exact f a * g a },
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-- { intros, apply eq_of_homotopy, intro a, apply mul.assoc },
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-- { intro a, exact 1 },
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-- { intros, apply eq_of_homotopy, intro a, apply one_mul },
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-- { intros, apply eq_of_homotopy, intro a, apply mul_one },
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-- { intro f a, exact (f a)⁻¹ },
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-- { intros, apply eq_of_homotopy, intro a, apply mul.left_inv }
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-- end
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-- definition Group_arrow (A : Type) (G : Group) : Group :=
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-- Group.mk (A → G) _
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-- definition ab_group_arrow [instance] (A B : Type) [ab_group B] : ab_group (A → B) :=
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-- ⦃ab_group, group_arrow A B,
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-- mul_comm := by intros; apply eq_of_homotopy; intro a; apply mul.comm⦄
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-- definition AbGroup_arrow (A : Type) (G : AbGroup) : AbGroup :=
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-- AbGroup.mk (A → G) _
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-- definition pgroup_ppmap [instance] (A B : Type*) [pgroup B] : pgroup (ppmap A B) :=
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-- begin
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-- fapply pgroup.mk,
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-- { apply is_trunc_pmap },
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-- { intro f g, apply pmap.mk (λa, f a * g a),
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-- exact ap011 mul (respect_pt f) (respect_pt g) ⬝ !one_mul },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply mul.assoc },
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-- { intro f, apply pmap.mk (λa, (f a)⁻¹), apply inv_eq_one, apply respect_pt },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply one_mul },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply mul_one },
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-- { intros, apply pmap_eq_of_homotopy, intro a, apply mul.left_inv }
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-- end
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-- definition Group_pmap (A : Type*) (G : Group) : Group :=
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-- Group_of_pgroup (ppmap A (pType_of_Group G))
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-- definition AbGroup_pmap (A : Type*) (G : AbGroup) : AbGroup :=
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-- AbGroup.mk (A →* pType_of_Group G)
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-- ⦃ ab_group, Group.struct (Group_pmap A G),
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-- mul_comm := by intro f g; apply pmap_eq_of_homotopy; intro a; apply mul.comm ⦄
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-- definition Group_pmap_homomorphism [constructor] {A A' : Type*} (f : A' →* A) (G : AbGroup) :
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-- Group_pmap A G →g Group_pmap A' G :=
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-- begin
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-- fapply homomorphism.mk,
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-- { intro g, exact g ∘* f},
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-- { intro g h, apply pmap_eq_of_homotopy, intro a, reflexivity }
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-- end
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end group
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