(* Title: HOL/Algebra/AbelCoset.thy Id: $Id: AbelCoset.thy,v 1.7 2008/03/17 21:34:23 wenzelm Exp $ Author: Stephan Hohe, TU Muenchen *) theory AbelCoset imports Coset Ring begin section {* More Lifting from Groups to Abelian Groups *} subsection {* Definitions *} text {* Hiding @{text "<+>"} from @{theory Sum_Type} until I come up with better syntax here *} hide const Plus constdefs (structure G) a_r_coset :: "[_, 'a set, 'a] => 'a set" (infixl "+>\<index>" 60) "a_r_coset G ≡ r_coset (|carrier = carrier G, mult = add G, one = zero G|))," a_l_coset :: "[_, 'a, 'a set] => 'a set" (infixl "<+\<index>" 60) "a_l_coset G ≡ l_coset (|carrier = carrier G, mult = add G, one = zero G|))," A_RCOSETS :: "[_, 'a set] => ('a set)set" ("a'_rcosets\<index> _" [81] 80) "A_RCOSETS G H ≡ RCOSETS (|carrier = carrier G, mult = add G, one = zero G|)), H" set_add :: "[_, 'a set ,'a set] => 'a set" (infixl "<+>\<index>" 60) "set_add G ≡ set_mult (|carrier = carrier G, mult = add G, one = zero G|))," A_SET_INV :: "[_,'a set] => 'a set" ("a'_set'_inv\<index> _" [81] 80) "A_SET_INV G H ≡ SET_INV (|carrier = carrier G, mult = add G, one = zero G|)), H" constdefs (structure G) a_r_congruent :: "[('a,'b)ring_scheme, 'a set] => ('a*'a)set" ("racong\<index> _") "a_r_congruent G ≡ r_congruent (|carrier = carrier G, mult = add G, one = zero G|))," constdefs A_FactGroup :: "[('a,'b) ring_scheme, 'a set] => ('a set) monoid" (infixl "A'_Mod" 65) --{*Actually defined for groups rather than monoids*} "A_FactGroup G H ≡ FactGroup (|carrier = carrier G, mult = add G, one = zero G|)), H" constdefs a_kernel :: "('a, 'm) ring_scheme => ('b, 'n) ring_scheme => ('a => 'b) => 'a set" --{*the kernel of a homomorphism (additive)*} "a_kernel G H h ≡ kernel (|carrier = carrier G, mult = add G, one = zero G|)), (|carrier = carrier H, mult = add H, one = zero H|)), h" locale abelian_group_hom = abelian_group G + abelian_group H + var h + assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |) (| carrier = carrier H, mult = add H, one = zero H |) h" lemmas a_r_coset_defs = a_r_coset_def r_coset_def lemma a_r_coset_def': includes struct G shows "H +> a ≡ \<Union>h∈H. {h ⊕ a}" unfolding a_r_coset_defs by simp lemmas a_l_coset_defs = a_l_coset_def l_coset_def lemma a_l_coset_def': includes struct G shows "a <+ H ≡ \<Union>h∈H. {a ⊕ h}" unfolding a_l_coset_defs by simp lemmas A_RCOSETS_defs = A_RCOSETS_def RCOSETS_def lemma A_RCOSETS_def': includes struct G shows "a_rcosets H ≡ \<Union>a∈carrier G. {H +> a}" unfolding A_RCOSETS_defs by (fold a_r_coset_def, simp) lemmas set_add_defs = set_add_def set_mult_def lemma set_add_def': includes struct G shows "H <+> K ≡ \<Union>h∈H. \<Union>k∈K. {h ⊕ k}" unfolding set_add_defs by simp lemmas A_SET_INV_defs = A_SET_INV_def SET_INV_def lemma A_SET_INV_def': includes struct G shows "a_set_inv H ≡ \<Union>h∈H. {\<ominus> h}" unfolding A_SET_INV_defs by (fold a_inv_def) subsection {* Cosets *} lemma (in abelian_group) a_coset_add_assoc: "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> (M +> g) +> h = M +> (g ⊕ h)" by (rule group.coset_mult_assoc [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_coset_add_zero [simp]: "M ⊆ carrier G ==> M +> \<zero> = M" by (rule group.coset_mult_one [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_coset_add_inv1: "[| M +> (x ⊕ (\<ominus> y)) = M; x ∈ carrier G ; y ∈ carrier G; M ⊆ carrier G |] ==> M +> x = M +> y" by (rule group.coset_mult_inv1 [OF a_group, folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) lemma (in abelian_group) a_coset_add_inv2: "[| M +> x = M +> y; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |] ==> M +> (x ⊕ (\<ominus> y)) = M" by (rule group.coset_mult_inv2 [OF a_group, folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) lemma (in abelian_group) a_coset_join1: "[| H +> x = H; x ∈ carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x ∈ H" by (rule group.coset_join1 [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_solve_equation: "[|subgroup H (|carrier = carrier G, mult = add G, one = zero G|); x ∈ H; y ∈ H|] ==> ∃h∈H. y = h ⊕ x" by (rule group.solve_equation [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_repr_independence: "[|y ∈ H +> x; x ∈ carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|)), |] ==> H +> x = H +> y" by (rule group.repr_independence [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_coset_join2: "[|x ∈ carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|)),; x∈H|] ==> H +> x = H" by (rule group.coset_join2 [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_monoid) a_r_coset_subset_G: "[| H ⊆ carrier G; x ∈ carrier G |] ==> H +> x ⊆ carrier G" by (rule monoid.r_coset_subset_G [OF a_monoid, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_rcosI: "[| h ∈ H; H ⊆ carrier G; x ∈ carrier G|] ==> h ⊕ x ∈ H +> x" by (rule group.rcosI [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_rcosetsI: "[|H ⊆ carrier G; x ∈ carrier G|] ==> H +> x ∈ a_rcosets H" by (rule group.rcosetsI [OF a_group, folded a_r_coset_def A_RCOSETS_def, simplified monoid_record_simps]) text{*Really needed?*} lemma (in abelian_group) a_transpose_inv: "[| x ⊕ y = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |] ==> (\<ominus> x) ⊕ z = y" by (rule group.transpose_inv [OF a_group, folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) (* --"duplicate" lemma (in abelian_group) a_rcos_self: "[| x ∈ carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|)), |] ==> x ∈ H +> x" by (rule group.rcos_self [OF a_group, folded a_r_coset_def, simplified monoid_record_simps]) *) subsection {* Subgroups *} locale additive_subgroup = var H + struct G + assumes a_subgroup: "subgroup H (|carrier = carrier G, mult = add G, one = zero G|))," lemma (in additive_subgroup) is_additive_subgroup: shows "additive_subgroup H G" by (rule additive_subgroup_axioms) lemma additive_subgroupI: includes struct G assumes a_subgroup: "subgroup H (|carrier = carrier G, mult = add G, one = zero G|))," shows "additive_subgroup H G" by (rule additive_subgroup.intro) (rule a_subgroup) lemma (in additive_subgroup) a_subset: "H ⊆ carrier G" by (rule subgroup.subset[OF a_subgroup, simplified monoid_record_simps]) lemma (in additive_subgroup) a_closed [intro, simp]: "[|x ∈ H; y ∈ H|] ==> x ⊕ y ∈ H" by (rule subgroup.m_closed[OF a_subgroup, simplified monoid_record_simps]) lemma (in additive_subgroup) zero_closed [simp]: "\<zero> ∈ H" by (rule subgroup.one_closed[OF a_subgroup, simplified monoid_record_simps]) lemma (in additive_subgroup) a_inv_closed [intro,simp]: "x ∈ H ==> \<ominus> x ∈ H" by (rule subgroup.m_inv_closed[OF a_subgroup, folded a_inv_def, simplified monoid_record_simps]) subsection {* Normal additive subgroups *} subsubsection {* Definition of @{text "abelian_subgroup"} *} text {* Every subgroup of an @{text "abelian_group"} is normal *} locale abelian_subgroup = additive_subgroup H G + abelian_group G + assumes a_normal: "normal H (|carrier = carrier G, mult = add G, one = zero G|))," lemma (in abelian_subgroup) is_abelian_subgroup: shows "abelian_subgroup H G" by (rule abelian_subgroup_axioms) lemma abelian_subgroupI: assumes a_normal: "normal H (|carrier = carrier G, mult = add G, one = zero G|))," and a_comm: "!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊕G y = y ⊕G x" shows "abelian_subgroup H G" proof - interpret normal ["H" "(|carrier = carrier G, mult = add G, one = zero G|)),"] by (rule a_normal) show "abelian_subgroup H G" by (unfold_locales, simp add: a_comm) qed lemma abelian_subgroupI2: includes struct G assumes a_comm_group: "comm_group (|carrier = carrier G, mult = add G, one = zero G|))," and a_subgroup: "subgroup H (|carrier = carrier G, mult = add G, one = zero G|))," shows "abelian_subgroup H G" proof - interpret comm_group ["(|carrier = carrier G, mult = add G, one = zero G|)),"] by (rule a_comm_group) interpret subgroup ["H" "(|carrier = carrier G, mult = add G, one = zero G|)),"] by (rule a_subgroup) show "abelian_subgroup H G" apply unfold_locales proof (simp add: r_coset_def l_coset_def, clarsimp) fix x assume xcarr: "x ∈ carrier G" from a_subgroup have Hcarr: "H ⊆ carrier G" by (unfold subgroup_def, simp) from xcarr Hcarr show "(\<Union>h∈H. {h ⊕G x}) = (\<Union>h∈H. {x ⊕G h})" using m_comm[simplified] by fast qed qed lemma abelian_subgroupI3: includes struct G assumes asg: "additive_subgroup H G" and ag: "abelian_group G" shows "abelian_subgroup H G" apply (rule abelian_subgroupI2) apply (rule abelian_group.a_comm_group[OF ag]) apply (rule additive_subgroup.a_subgroup[OF asg]) done lemma (in abelian_subgroup) a_coset_eq: "(∀x ∈ carrier G. H +> x = x <+ H)" by (rule normal.coset_eq[OF a_normal, folded a_r_coset_def a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_inv_op_closed1: shows "[|x ∈ carrier G; h ∈ H|] ==> (\<ominus> x) ⊕ h ⊕ x ∈ H" by (rule normal.inv_op_closed1 [OF a_normal, folded a_inv_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_inv_op_closed2: shows "[|x ∈ carrier G; h ∈ H|] ==> x ⊕ h ⊕ (\<ominus> x) ∈ H" by (rule normal.inv_op_closed2 [OF a_normal, folded a_inv_def, simplified monoid_record_simps]) text{*Alternative characterization of normal subgroups*} lemma (in abelian_group) a_normal_inv_iff: "(N \<lhd> (|carrier = carrier G, mult = add G, one = zero G|)),) = (subgroup N (|carrier = carrier G, mult = add G, one = zero G|)), & (∀x ∈ carrier G. ∀h ∈ N. x ⊕ h ⊕ (\<ominus> x) ∈ N))" (is "_ = ?rhs") by (rule group.normal_inv_iff [OF a_group, folded a_inv_def, simplified monoid_record_simps]) lemma (in abelian_group) a_lcos_m_assoc: "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <+ (h <+ M) = (g ⊕ h) <+ M" by (rule group.lcos_m_assoc [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_lcos_mult_one: "M ⊆ carrier G ==> \<zero> <+ M = M" by (rule group.lcos_mult_one [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_l_coset_subset_G: "[| H ⊆ carrier G; x ∈ carrier G |] ==> x <+ H ⊆ carrier G" by (rule group.l_coset_subset_G [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_l_coset_swap: "[|y ∈ x <+ H; x ∈ carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|)),|] ==> x ∈ y <+ H" by (rule group.l_coset_swap [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_l_coset_carrier: "[| y ∈ x <+ H; x ∈ carrier G; subgroup H (|carrier = carrier G, mult = add G, one = zero G|)), |] ==> y ∈ carrier G" by (rule group.l_coset_carrier [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_l_repr_imp_subset: assumes y: "y ∈ x <+ H" and x: "x ∈ carrier G" and sb: "subgroup H (|carrier = carrier G, mult = add G, one = zero G|))," shows "y <+ H ⊆ x <+ H" apply (rule group.l_repr_imp_subset [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) apply (rule y) apply (rule x) apply (rule sb) done lemma (in abelian_group) a_l_repr_independence: assumes y: "y ∈ x <+ H" and x: "x ∈ carrier G" and sb: "subgroup H (|carrier = carrier G, mult = add G, one = zero G|))," shows "x <+ H = y <+ H" apply (rule group.l_repr_independence [OF a_group, folded a_l_coset_def, simplified monoid_record_simps]) apply (rule y) apply (rule x) apply (rule sb) done lemma (in abelian_group) setadd_subset_G: "[|H ⊆ carrier G; K ⊆ carrier G|] ==> H <+> K ⊆ carrier G" by (rule group.setmult_subset_G [OF a_group, folded set_add_def, simplified monoid_record_simps]) lemma (in abelian_group) subgroup_add_id: "subgroup H (|carrier = carrier G, mult = add G, one = zero G|)), ==> H <+> H = H" by (rule group.subgroup_mult_id [OF a_group, folded set_add_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcos_inv: assumes x: "x ∈ carrier G" shows "a_set_inv (H +> x) = H +> (\<ominus> x)" by (rule normal.rcos_inv [OF a_normal, folded a_r_coset_def a_inv_def A_SET_INV_def, simplified monoid_record_simps]) (rule x) lemma (in abelian_group) a_setmult_rcos_assoc: "[|H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G|] ==> H <+> (K +> x) = (H <+> K) +> x" by (rule group.setmult_rcos_assoc [OF a_group, folded set_add_def a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_group) a_rcos_assoc_lcos: "[|H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G|] ==> (H +> x) <+> K = H <+> (x <+ K)" by (rule group.rcos_assoc_lcos [OF a_group, folded set_add_def a_r_coset_def a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcos_sum: "[|x ∈ carrier G; y ∈ carrier G|] ==> (H +> x) <+> (H +> y) = H +> (x ⊕ y)" by (rule normal.rcos_sum [OF a_normal, folded set_add_def a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) rcosets_add_eq: "M ∈ a_rcosets H ==> H <+> M = M" -- {* generalizes @{text subgroup_mult_id} *} by (rule normal.rcosets_mult_eq [OF a_normal, folded set_add_def A_RCOSETS_def, simplified monoid_record_simps]) subsection {* Congruence Relation *} lemma (in abelian_subgroup) a_equiv_rcong: shows "equiv (carrier G) (racong H)" by (rule subgroup.equiv_rcong [OF a_subgroup a_group, folded a_r_congruent_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_l_coset_eq_rcong: assumes a: "a ∈ carrier G" shows "a <+ H = racong H `` {a}" by (rule subgroup.l_coset_eq_rcong [OF a_subgroup a_group, folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps]) (rule a) lemma (in abelian_subgroup) a_rcos_equation: shows "[|ha ⊕ a = h ⊕ b; a ∈ carrier G; b ∈ carrier G; h ∈ H; ha ∈ H; hb ∈ H|] ==> hb ⊕ a ∈ (\<Union>h∈H. {h ⊕ b})" by (rule group.rcos_equation [OF a_group a_subgroup, folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcos_disjoint: shows "[|a ∈ a_rcosets H; b ∈ a_rcosets H; a≠b|] ==> a ∩ b = {}" by (rule group.rcos_disjoint [OF a_group a_subgroup, folded A_RCOSETS_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcos_self: shows "x ∈ carrier G ==> x ∈ H +> x" by (rule group.rcos_self [OF a_group _ a_subgroup, folded a_r_coset_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcosets_part_G: shows "\<Union>(a_rcosets H) = carrier G" by (rule group.rcosets_part_G [OF a_group a_subgroup, folded A_RCOSETS_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_cosets_finite: "[|c ∈ a_rcosets H; H ⊆ carrier G; finite (carrier G)|] ==> finite c" by (rule group.cosets_finite [OF a_group, folded A_RCOSETS_def, simplified monoid_record_simps]) lemma (in abelian_group) a_card_cosets_equal: "[|c ∈ a_rcosets H; H ⊆ carrier G; finite(carrier G)|] ==> card c = card H" by (rule group.card_cosets_equal [OF a_group, folded A_RCOSETS_def, simplified monoid_record_simps]) lemma (in abelian_group) rcosets_subset_PowG: "additive_subgroup H G ==> a_rcosets H ⊆ Pow(carrier G)" by (rule group.rcosets_subset_PowG [OF a_group, folded A_RCOSETS_def, simplified monoid_record_simps], rule additive_subgroup.a_subgroup) theorem (in abelian_group) a_lagrange: "[|finite(carrier G); additive_subgroup H G|] ==> card(a_rcosets H) * card(H) = order(G)" by (rule group.lagrange [OF a_group, folded A_RCOSETS_def, simplified monoid_record_simps order_def, folded order_def]) (fast intro!: additive_subgroup.a_subgroup)+ subsection {* Factorization *} lemmas A_FactGroup_defs = A_FactGroup_def FactGroup_def lemma A_FactGroup_def': includes struct G shows "G A_Mod H ≡ (|carrier = a_rcosetsG H, mult = set_add G, one = H|))," unfolding A_FactGroup_defs by (fold A_RCOSETS_def set_add_def) lemma (in abelian_subgroup) a_setmult_closed: "[|K1 ∈ a_rcosets H; K2 ∈ a_rcosets H|] ==> K1 <+> K2 ∈ a_rcosets H" by (rule normal.setmult_closed [OF a_normal, folded A_RCOSETS_def set_add_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_setinv_closed: "K ∈ a_rcosets H ==> a_set_inv K ∈ a_rcosets H" by (rule normal.setinv_closed [OF a_normal, folded A_RCOSETS_def A_SET_INV_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcosets_assoc: "[|M1 ∈ a_rcosets H; M2 ∈ a_rcosets H; M3 ∈ a_rcosets H|] ==> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)" by (rule normal.rcosets_assoc [OF a_normal, folded A_RCOSETS_def set_add_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_subgroup_in_rcosets: "H ∈ a_rcosets H" by (rule subgroup.subgroup_in_rcosets [OF a_subgroup a_group, folded A_RCOSETS_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcosets_inv_mult_group_eq: "M ∈ a_rcosets H ==> a_set_inv M <+> M = H" by (rule normal.rcosets_inv_mult_group_eq [OF a_normal, folded A_RCOSETS_def A_SET_INV_def set_add_def, simplified monoid_record_simps]) theorem (in abelian_subgroup) a_factorgroup_is_group: "group (G A_Mod H)" by (rule normal.factorgroup_is_group [OF a_normal, folded A_FactGroup_def, simplified monoid_record_simps]) text {* Since the Factorization is based on an \emph{abelian} subgroup, is results in a commutative group *} theorem (in abelian_subgroup) a_factorgroup_is_comm_group: "comm_group (G A_Mod H)" apply (intro comm_group.intro comm_monoid.intro) prefer 3 apply (rule a_factorgroup_is_group) apply (rule group.axioms[OF a_factorgroup_is_group]) apply (rule comm_monoid_axioms.intro) apply (unfold A_FactGroup_def FactGroup_def RCOSETS_def, fold set_add_def a_r_coset_def, clarsimp) apply (simp add: a_rcos_sum a_comm) done lemma add_A_FactGroup [simp]: "X ⊗(G A_Mod H) X' = X <+>G X'" by (simp add: A_FactGroup_def set_add_def) lemma (in abelian_subgroup) a_inv_FactGroup: "X ∈ carrier (G A_Mod H) ==> invG A_Mod H X = a_set_inv X" by (rule normal.inv_FactGroup [OF a_normal, folded A_FactGroup_def A_SET_INV_def, simplified monoid_record_simps]) text{*The coset map is a homomorphism from @{term G} to the quotient group @{term "G Mod H"}*} lemma (in abelian_subgroup) a_r_coset_hom_A_Mod: "(λa. H +> a) ∈ hom (|carrier = carrier G, mult = add G, one = zero G|)), (G A_Mod H)" by (rule normal.r_coset_hom_Mod [OF a_normal, folded A_FactGroup_def a_r_coset_def, simplified monoid_record_simps]) text {* The isomorphism theorems have been omitted from lifting, at least for now *} subsection{*The First Isomorphism Theorem*} text{*The quotient by the kernel of a homomorphism is isomorphic to the range of that homomorphism.*} lemmas a_kernel_defs = a_kernel_def kernel_def lemma a_kernel_def': "a_kernel R S h ≡ {x ∈ carrier R. h x = \<zero>S}" by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps]) subsection {* Homomorphisms *} lemma abelian_group_homI: includes abelian_group G includes abelian_group H assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |) (| carrier = carrier H, mult = add H, one = zero H |) h" shows "abelian_group_hom G H h" apply (intro abelian_group_hom.intro abelian_group_hom_axioms.intro) apply (rule G.abelian_group_axioms) apply (rule H.abelian_group_axioms) apply (rule a_group_hom) done lemma (in abelian_group_hom) is_abelian_group_hom: "abelian_group_hom G H h" by (unfold_locales) lemma (in abelian_group_hom) hom_add [simp]: "[| x : carrier G; y : carrier G |] ==> h (x ⊕G y) = h x ⊕H h y" by (rule group_hom.hom_mult[OF a_group_hom, simplified ring_record_simps]) lemma (in abelian_group_hom) hom_closed [simp]: "x ∈ carrier G ==> h x ∈ carrier H" by (rule group_hom.hom_closed[OF a_group_hom, simplified ring_record_simps]) lemma (in abelian_group_hom) zero_closed [simp]: "h \<zero> ∈ carrier H" by (rule group_hom.one_closed[OF a_group_hom, simplified ring_record_simps]) lemma (in abelian_group_hom) hom_zero [simp]: "h \<zero> = \<zero>H" by (rule group_hom.hom_one[OF a_group_hom, simplified ring_record_simps]) lemma (in abelian_group_hom) a_inv_closed [simp]: "x ∈ carrier G ==> h (\<ominus>x) ∈ carrier H" by (rule group_hom.inv_closed[OF a_group_hom, folded a_inv_def, simplified ring_record_simps]) lemma (in abelian_group_hom) hom_a_inv [simp]: "x ∈ carrier G ==> h (\<ominus>x) = \<ominus>H (h x)" by (rule group_hom.hom_inv[OF a_group_hom, folded a_inv_def, simplified ring_record_simps]) lemma (in abelian_group_hom) additive_subgroup_a_kernel: "additive_subgroup (a_kernel G H h) G" apply (rule additive_subgroup.intro) apply (rule group_hom.subgroup_kernel[OF a_group_hom, folded a_kernel_def, simplified ring_record_simps]) done text{*The kernel of a homomorphism is an abelian subgroup*} lemma (in abelian_group_hom) abelian_subgroup_a_kernel: "abelian_subgroup (a_kernel G H h) G" apply (rule abelian_subgroupI) apply (rule group_hom.normal_kernel[OF a_group_hom, folded a_kernel_def, simplified ring_record_simps]) apply (simp add: G.a_comm) done lemma (in abelian_group_hom) A_FactGroup_nonempty: assumes X: "X ∈ carrier (G A_Mod a_kernel G H h)" shows "X ≠ {}" by (rule group_hom.FactGroup_nonempty[OF a_group_hom, folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X) lemma (in abelian_group_hom) FactGroup_contents_mem: assumes X: "X ∈ carrier (G A_Mod (a_kernel G H h))" shows "contents (h`X) ∈ carrier H" by (rule group_hom.FactGroup_contents_mem[OF a_group_hom, folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X) lemma (in abelian_group_hom) A_FactGroup_hom: "(λX. contents (h`X)) ∈ hom (G A_Mod (a_kernel G H h)) (|carrier = carrier H, mult = add H, one = zero H|))," by (rule group_hom.FactGroup_hom[OF a_group_hom, folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) lemma (in abelian_group_hom) A_FactGroup_inj_on: "inj_on (λX. contents (h ` X)) (carrier (G A_Mod a_kernel G H h))" by (rule group_hom.FactGroup_inj_on[OF a_group_hom, folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) text{*If the homomorphism @{term h} is onto @{term H}, then so is the homomorphism from the quotient group*} lemma (in abelian_group_hom) A_FactGroup_onto: assumes h: "h ` carrier G = carrier H" shows "(λX. contents (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H" by (rule group_hom.FactGroup_onto[OF a_group_hom, folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule h) text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*} theorem (in abelian_group_hom) A_FactGroup_iso: "h ` carrier G = carrier H ==> (λX. contents (h`X)) ∈ (G A_Mod (a_kernel G H h)) ≅ (| carrier = carrier H, mult = add H, one = zero H |)" by (rule group_hom.FactGroup_iso[OF a_group_hom, folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) section {* Lemmas Lifted from CosetExt.thy *} text {* Not eveything from \texttt{CosetExt.thy} is lifted here. *} subsection {* General Lemmas from \texttt{AlgebraExt.thy} *} lemma (in additive_subgroup) a_Hcarr [simp]: assumes hH: "h ∈ H" shows "h ∈ carrier G" by (rule subgroup.mem_carrier [OF a_subgroup, simplified monoid_record_simps]) (rule hH) subsection {* Lemmas for Right Cosets *} lemma (in abelian_subgroup) a_elemrcos_carrier: assumes acarr: "a ∈ carrier G" and a': "a' ∈ H +> a" shows "a' ∈ carrier G" by (rule subgroup.elemrcos_carrier [OF a_subgroup a_group, folded a_r_coset_def, simplified monoid_record_simps]) (rule acarr, rule a') lemma (in abelian_subgroup) a_rcos_const: assumes hH: "h ∈ H" shows "H +> h = H" by (rule subgroup.rcos_const [OF a_subgroup a_group, folded a_r_coset_def, simplified monoid_record_simps]) (rule hH) lemma (in abelian_subgroup) a_rcos_module_imp: assumes xcarr: "x ∈ carrier G" and x'cos: "x' ∈ H +> x" shows "(x' ⊕ \<ominus>x) ∈ H" by (rule subgroup.rcos_module_imp [OF a_subgroup a_group, folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) (rule xcarr, rule x'cos) lemma (in abelian_subgroup) a_rcos_module_rev: assumes "x ∈ carrier G" "x' ∈ carrier G" and "(x' ⊕ \<ominus>x) ∈ H" shows "x' ∈ H +> x" using assms by (rule subgroup.rcos_module_rev [OF a_subgroup a_group, folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) lemma (in abelian_subgroup) a_rcos_module: assumes "x ∈ carrier G" "x' ∈ carrier G" shows "(x' ∈ H +> x) = (x' ⊕ \<ominus>x ∈ H)" using assms by (rule subgroup.rcos_module [OF a_subgroup a_group, folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) --"variant" lemma (in abelian_subgroup) a_rcos_module_minus: includes ring G assumes carr: "x ∈ carrier G" "x' ∈ carrier G" shows "(x' ∈ H +> x) = (x' \<ominus> x ∈ H)" proof - from carr have "(x' ∈ H +> x) = (x' ⊕ \<ominus>x ∈ H)" by (rule a_rcos_module) with carr show "(x' ∈ H +> x) = (x' \<ominus> x ∈ H)" by (simp add: minus_eq) qed lemma (in abelian_subgroup) a_repr_independence': assumes y: "y ∈ H +> x" and xcarr: "x ∈ carrier G" shows "H +> x = H +> y" apply (rule a_repr_independence) apply (rule y) apply (rule xcarr) apply (rule a_subgroup) done lemma (in abelian_subgroup) a_repr_independenceD: assumes ycarr: "y ∈ carrier G" and repr: "H +> x = H +> y" shows "y ∈ H +> x" by (rule group.repr_independenceD [OF a_group a_subgroup, folded a_r_coset_def, simplified monoid_record_simps]) (rule ycarr, rule repr) subsection {* Lemmas for the Set of Right Cosets *} lemma (in abelian_subgroup) a_rcosets_carrier: "X ∈ a_rcosets H ==> X ⊆ carrier G" by (rule subgroup.rcosets_carrier [OF a_subgroup a_group, folded A_RCOSETS_def, simplified monoid_record_simps]) subsection {* Addition of Subgroups *} lemma (in abelian_monoid) set_add_closed: assumes Acarr: "A ⊆ carrier G" and Bcarr: "B ⊆ carrier G" shows "A <+> B ⊆ carrier G" by (rule monoid.set_mult_closed [OF a_monoid, folded set_add_def, simplified monoid_record_simps]) (rule Acarr, rule Bcarr) lemma (in abelian_group) add_additive_subgroups: assumes subH: "additive_subgroup H G" and subK: "additive_subgroup K G" shows "additive_subgroup (H <+> K) G" apply (rule additive_subgroup.intro) apply (unfold set_add_def) apply (intro comm_group.mult_subgroups) apply (rule a_comm_group) apply (rule additive_subgroup.a_subgroup[OF subH]) apply (rule additive_subgroup.a_subgroup[OF subK]) done end
lemma a_r_coset_defs:
op +>G == op #>(| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
H #>G a == UN h:H. {h ⊗G a}
lemma a_r_coset_def':
H +>G a == UN h:H. {h ⊕G a}
lemma a_l_coset_defs:
op <+G == op <#(| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
a <#G H == UN h:H. {a ⊗G h}
lemma a_l_coset_def':
a <+G H == UN h:H. {a ⊕G h}
lemma A_RCOSETS_defs:
a_rcosetsG H == rcosets(| carrier = carrier G, mult = op ⊕G, one = \<zero>G |) H
rcosetsG H == UN a:carrier G. {H #>G a}
lemma A_RCOSETS_def':
a_rcosetsG H == UN a:carrier G. {H +>G a}
lemma set_add_defs:
op <+>G == op <#>(| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
H <#>G K == UN h:H. UN k:K. {h ⊗G k}
lemma set_add_def':
H <+>G K == UN h:H. UN k:K. {h ⊕G k}
lemma A_SET_INV_defs:
a_set_invG H == set_inv(| carrier = carrier G, mult = op ⊕G, one = \<zero>G |) H
set_invG H == UN h:H. {invG h}
lemma A_SET_INV_def':
a_set_invG H == UN h:H. {\<ominus>G h}
lemma a_coset_add_assoc:
[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> M +> g +> h = M +> g ⊕ h
lemma a_coset_add_zero:
M ⊆ carrier G ==> M +> \<zero> = M
lemma a_coset_add_inv1:
[| M +> x ⊕ \<ominus> y = M; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |]
==> M +> x = M +> y
lemma a_coset_add_inv2:
[| M +> x = M +> y; x ∈ carrier G; y ∈ carrier G; M ⊆ carrier G |]
==> M +> x ⊕ \<ominus> y = M
lemma a_coset_join1:
[| H +> x = H; x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) |]
==> x ∈ H
lemma a_solve_equation:
[| subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |); x ∈ H;
y ∈ H |]
==> ∃h∈H. y = h ⊕ x
lemma a_repr_independence:
[| y ∈ H +> x; x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) |]
==> H +> x = H +> y
lemma a_coset_join2:
[| x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |); x ∈ H |]
==> H +> x = H
lemma a_r_coset_subset_G:
[| H ⊆ carrier G; x ∈ carrier G |] ==> H +> x ⊆ carrier G
lemma a_rcosI:
[| h ∈ H; H ⊆ carrier G; x ∈ carrier G |] ==> h ⊕ x ∈ H +> x
lemma a_rcosetsI:
[| H ⊆ carrier G; x ∈ carrier G |] ==> H +> x ∈ a_rcosets H
lemma a_transpose_inv:
[| x ⊕ y = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |]
==> \<ominus> x ⊕ z = y
lemma is_additive_subgroup:
additive_subgroup H G
lemma additive_subgroupI:
subgroup H (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
==> additive_subgroup H G
lemma a_subset:
H ⊆ carrier G
lemma a_closed:
[| x ∈ H; y ∈ H |] ==> x ⊕ y ∈ H
lemma zero_closed:
\<zero> ∈ H
lemma a_inv_closed:
x ∈ H ==> \<ominus> x ∈ H
lemma is_abelian_subgroup:
abelian_subgroup H G
lemma abelian_subgroupI:
[| H \<lhd> (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |);
!!x y. [| x ∈ carrier G; y ∈ carrier G |] ==> x ⊕G y = y ⊕G x |]
==> abelian_subgroup H G
lemma abelian_subgroupI2:
[| comm_group (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |);
subgroup H (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |) |]
==> abelian_subgroup H G
lemma abelian_subgroupI3:
[| additive_subgroup H G; abelian_group G |] ==> abelian_subgroup H G
lemma a_coset_eq:
∀x∈carrier G. H +> x = x <+ H
lemma a_inv_op_closed1:
[| x ∈ carrier G; h ∈ H |] ==> \<ominus> x ⊕ h ⊕ x ∈ H
lemma a_inv_op_closed2:
[| x ∈ carrier G; h ∈ H |] ==> x ⊕ h ⊕ \<ominus> x ∈ H
lemma a_normal_inv_iff:
N \<lhd> (| carrier = carrier G, mult = op ⊕, one = \<zero> |) =
(subgroup N (| carrier = carrier G, mult = op ⊕, one = \<zero> |) ∧
(∀x∈carrier G. ∀h∈N. x ⊕ h ⊕ \<ominus> x ∈ N))
lemma a_lcos_m_assoc:
[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <+ (h <+ M) = g ⊕ h <+ M
lemma a_lcos_mult_one:
M ⊆ carrier G ==> \<zero> <+ M = M
lemma a_l_coset_subset_G:
[| H ⊆ carrier G; x ∈ carrier G |] ==> x <+ H ⊆ carrier G
lemma a_l_coset_swap:
[| y ∈ x <+ H; x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) |]
==> x ∈ y <+ H
lemma a_l_coset_carrier:
[| y ∈ x <+ H; x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) |]
==> y ∈ carrier G
lemma a_l_repr_imp_subset:
[| y ∈ x <+ H; x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) |]
==> y <+ H ⊆ x <+ H
lemma a_l_repr_independence:
[| y ∈ x <+ H; x ∈ carrier G;
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) |]
==> x <+ H = y <+ H
lemma setadd_subset_G:
[| H ⊆ carrier G; K ⊆ carrier G |] ==> H <+> K ⊆ carrier G
lemma subgroup_add_id:
subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero> |) ==> H <+> H = H
lemma a_rcos_inv:
x ∈ carrier G ==> a_set_inv (H +> x) = H +> \<ominus> x
lemma a_setmult_rcos_assoc:
[| H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G |]
==> H <+> (K +> x) = H <+> K +> x
lemma a_rcos_assoc_lcos:
[| H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G |]
==> H +> x <+> K = H <+> (x <+ K)
lemma a_rcos_sum:
[| x ∈ carrier G; y ∈ carrier G |] ==> H +> x <+> (H +> y) = H +> x ⊕ y
lemma rcosets_add_eq:
M ∈ a_rcosets H ==> H <+> M = M
lemma a_equiv_rcong:
equiv (carrier G) racong H
lemma a_l_coset_eq_rcong:
a ∈ carrier G ==> a <+ H = racong H `` {a}
lemma a_rcos_equation:
[| ha ⊕ a = h ⊕ b; a ∈ carrier G; b ∈ carrier G; h ∈ H; ha ∈ H; hb ∈ H |]
==> hb ⊕ a ∈ (UN h:H. {h ⊕ b})
lemma a_rcos_disjoint:
[| a ∈ a_rcosets H; b ∈ a_rcosets H; a ≠ b |] ==> a ∩ b = {}
lemma a_rcos_self:
x ∈ carrier G ==> x ∈ H +> x
lemma a_rcosets_part_G:
Union (a_rcosets H) = carrier G
lemma a_cosets_finite:
[| c ∈ a_rcosets H; H ⊆ carrier G; finite (carrier G) |] ==> finite c
lemma a_card_cosets_equal:
[| c ∈ a_rcosets H; H ⊆ carrier G; finite (carrier G) |] ==> card c = card H
lemma rcosets_subset_PowG:
additive_subgroup H G ==> a_rcosets H ⊆ Pow (carrier G)
theorem a_lagrange:
[| finite (carrier G); additive_subgroup H G |]
==> card (a_rcosets H) * card H = Coset.order G
lemma A_FactGroup_defs:
G A_Mod H == (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |) Mod H
G Mod H == (| carrier = rcosetsG H, mult = op <#>G, one = H |)
lemma A_FactGroup_def':
G A_Mod H == (| carrier = a_rcosetsG H, mult = op <+>G, one = H |)
lemma a_setmult_closed:
[| K1.0 ∈ a_rcosets H; K2.0 ∈ a_rcosets H |] ==> K1.0 <+> K2.0 ∈ a_rcosets H
lemma a_setinv_closed:
K ∈ a_rcosets H ==> a_set_inv K ∈ a_rcosets H
lemma a_rcosets_assoc:
[| M1.0 ∈ a_rcosets H; M2.0 ∈ a_rcosets H; M3.0 ∈ a_rcosets H |]
==> M1.0 <+> M2.0 <+> M3.0 = M1.0 <+> (M2.0 <+> M3.0)
lemma a_subgroup_in_rcosets:
H ∈ a_rcosets H
lemma a_rcosets_inv_mult_group_eq:
M ∈ a_rcosets H ==> a_set_inv M <+> M = H
theorem a_factorgroup_is_group:
group (G A_Mod H)
theorem a_factorgroup_is_comm_group:
comm_group (G A_Mod H)
lemma add_A_FactGroup:
X ⊗G A_Mod H X' = X <+>G X'
lemma a_inv_FactGroup:
X ∈ carrier (G A_Mod H) ==> invG A_Mod H X = a_set_inv X
lemma a_r_coset_hom_A_Mod:
op +> H ∈ hom (| carrier = carrier G, mult = op ⊕, one = \<zero> |) (G A_Mod H)
lemma a_kernel_defs:
a_kernel G H h ==
kernel (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
(| carrier = carrier H, mult = op ⊕H, one = \<zero>H |) h
kernel G H h == {x : carrier G. h x = \<one>H}
lemma a_kernel_def':
a_kernel R S h == {x : carrier R. h x = \<zero>S}
lemma abelian_group_homI:
[| abelian_group G; abelian_group H;
group_hom (| carrier = carrier G, mult = op ⊕G, one = \<zero>G |)
(| carrier = carrier H, mult = op ⊕H, one = \<zero>H |) h |]
==> abelian_group_hom G H h
lemma is_abelian_group_hom:
abelian_group_hom G H h
lemma hom_add:
[| x ∈ carrier G; y ∈ carrier G |] ==> h (x ⊕ y) = h x ⊕H h y
lemma hom_closed:
x ∈ carrier G ==> h x ∈ carrier H
lemma zero_closed:
h \<zero> ∈ carrier H
lemma hom_zero:
h \<zero> = \<zero>H
lemma a_inv_closed:
x ∈ carrier G ==> h (\<ominus> x) ∈ carrier H
lemma hom_a_inv:
x ∈ carrier G ==> h (\<ominus> x) = \<ominus>H h x
lemma additive_subgroup_a_kernel:
additive_subgroup (a_kernel G H h) G
lemma abelian_subgroup_a_kernel:
abelian_subgroup (a_kernel G H h) G
lemma A_FactGroup_nonempty:
X ∈ carrier (G A_Mod a_kernel G H h) ==> X ≠ {}
lemma FactGroup_contents_mem:
X ∈ carrier (G A_Mod a_kernel G H h) ==> contents (h ` X) ∈ carrier H
lemma A_FactGroup_hom:
(λX. contents (h ` X))
∈ hom (G A_Mod a_kernel G H h)
(| carrier = carrier H, mult = op ⊕H, one = \<zero>H |)
lemma A_FactGroup_inj_on:
inj_on (λX. contents (h ` X)) (carrier (G A_Mod a_kernel G H h))
lemma A_FactGroup_onto:
h ` carrier G = carrier H
==> (λX. contents (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H
theorem A_FactGroup_iso:
h ` carrier G = carrier H
==> (λX. contents (h ` X))
∈ G A_Mod a_kernel G H h ≅
(| carrier = carrier H, mult = op ⊕H, one = \<zero>H |)
lemma a_Hcarr:
h ∈ H ==> h ∈ carrier G
lemma a_elemrcos_carrier:
[| a ∈ carrier G; a' ∈ H +> a |] ==> a' ∈ carrier G
lemma a_rcos_const:
h ∈ H ==> H +> h = H
lemma a_rcos_module_imp:
[| x ∈ carrier G; x' ∈ H +> x |] ==> x' ⊕ \<ominus> x ∈ H
lemma a_rcos_module_rev:
[| x ∈ carrier G; x' ∈ carrier G; x' ⊕ \<ominus> x ∈ H |] ==> x' ∈ H +> x
lemma a_rcos_module:
[| x ∈ carrier G; x' ∈ carrier G |] ==> (x' ∈ H +> x) = (x' ⊕ \<ominus> x ∈ H)
lemma a_rcos_module_minus:
[| Ring.ring G; x ∈ carrier G; x' ∈ carrier G |]
==> (x' ∈ H +> x) = (x' \<ominus> x ∈ H)
lemma a_repr_independence':
[| y ∈ H +> x; x ∈ carrier G |] ==> H +> x = H +> y
lemma a_repr_independenceD:
[| y ∈ carrier G; H +> x = H +> y |] ==> y ∈ H +> x
lemma a_rcosets_carrier:
X ∈ a_rcosets H ==> X ⊆ carrier G
lemma set_add_closed:
[| A ⊆ carrier G; B ⊆ carrier G |] ==> A <+> B ⊆ carrier G
lemma add_additive_subgroups:
[| additive_subgroup H G; additive_subgroup K G |]
==> additive_subgroup (H <+> K) G