Theory AbelCoset

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theory AbelCoset
imports Coset Ring
begin

(*
  Title:     HOL/Algebra/AbelCoset.thy
  Id:        $Id: AbelCoset.thy,v 1.5 2007/06/21 15:28:53 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 fact

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 fact

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

More Lifting from Groups to Abelian Groups

Definitions

lemma a_r_coset_defs:

  op +>G == op #>(| carrier = carrier G, mult = op ⊕G, one = \<zero>G, ... = () |)
  H #>G a == UN h:H. {hG a}

lemma a_r_coset_def':

  H +>G a == UN h:H. {hG a}

lemma a_l_coset_defs:

  op <+G == op <#(| carrier = carrier G, mult = op ⊕G, one = \<zero>G, ... = () |)
  a <#G H == UN h:H. {aG h}

lemma a_l_coset_def':

  a <+G H == UN h:H. {aG 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. {hG k}

lemma set_add_def':

  H <+>G K == UN h:H. UN k:K. {hG 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}

Cosets

lemma a_coset_add_assoc:

  [| M  carrier G; g ∈ carrier G; h ∈ carrier G |] ==> M +> g +> h = M +> gh

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>, ... = () |) |]
  ==> xH

lemma a_solve_equation:

  [| subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero>, ... = () |);
     xH; yH |]
  ==> ∃hH. y = hx

lemma a_repr_independence:

  [| yH +> 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>, ... = () |);
     xH |]
  ==> H +> x = H

lemma a_r_coset_subset_G:

  [| H  carrier G; x ∈ carrier G |] ==> H +> x  carrier G

lemma a_rcosI:

  [| hH; H  carrier G; x ∈ carrier G |] ==> hxH +> x

lemma a_rcosetsI:

  [| H  carrier G; x ∈ carrier G |] ==> H +> x ∈ a_rcosets H

lemma a_transpose_inv:

  [| xy = z; x ∈ carrier G; y ∈ carrier G; z ∈ carrier G |]
  ==> \<ominus> xz = y

Subgroups

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:

  [| xH; yH |] ==> xyH

lemma zero_closed:

  \<zero> ∈ H

lemma a_inv_closed:

  xH ==> \<ominus> xH

Normal additive subgroups

Definition of @{text "abelian_subgroup"}

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 |] ==> xG y = yG 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; hH |] ==> \<ominus> xhxH

lemma a_inv_op_closed2:

  [| x ∈ carrier G; hH |] ==> xh ⊕ \<ominus> xH

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. ∀hN. xh ⊕ \<ominus> xN))

lemma a_lcos_m_assoc:

  [| M  carrier G; g ∈ carrier G; h ∈ carrier G |] ==> g <+ (h <+ M) = gh <+ 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:

  [| yx <+ H; x ∈ carrier G;
     subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero>, ... = () |) |]
  ==> xy <+ H

lemma a_l_coset_carrier:

  [| yx <+ H; x ∈ carrier G;
     subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero>, ... = () |) |]
  ==> y ∈ carrier G

lemma a_l_repr_imp_subset:

  [| yx <+ H; x ∈ carrier G;
     subgroup H (| carrier = carrier G, mult = op ⊕, one = \<zero>, ... = () |) |]
  ==> y <+ H  x <+ H

lemma a_l_repr_independence:

  [| yx <+ 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 +> xy

lemma rcosets_add_eq:

  M ∈ a_rcosets H ==> H <+> M = M

Congruence Relation

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:

  [| haa = hb; a ∈ carrier G; b ∈ carrier G; hH; haH; hbH |]
  ==> hba ∈ (UN h:H. {hb})

lemma a_rcos_disjoint:

  [| a ∈ a_rcosets H; b ∈ a_rcosets H; a  b |] ==> ab = {}

lemma a_rcos_self:

  x ∈ carrier G ==> xH +> 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

Factorization

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:

  XG 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)

The First Isomorphism Theorem

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}

Homomorphisms

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 (xy) = h xH 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:

  X. 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, ... = () |)

Lemmas Lifted from CosetExt.thy

General Lemmas from \texttt{AlgebraExt.thy}

lemma a_Hcarr:

  hH ==> h ∈ carrier G

Lemmas for Right Cosets

lemma a_elemrcos_carrier:

  [| a ∈ carrier G; a'H +> a |] ==> a' ∈ carrier G

lemma a_rcos_const:

  hH ==> H +> h = H

lemma a_rcos_module_imp:

  [| x ∈ carrier G; x'H +> x |] ==> x' ⊕ \<ominus> xH

lemma a_rcos_module_rev:

  [| x ∈ carrier G; x' ∈ carrier G; x' ⊕ \<ominus> xH |] ==> x'H +> x

lemma a_rcos_module:

  [| x ∈ carrier G; x' ∈ carrier G |] ==> (x'H +> x) = (x' ⊕ \<ominus> xH)

lemma a_rcos_module_minus:

  [| Ring.ring G; x ∈ carrier G; x' ∈ carrier G |]
  ==> (x'H +> x) = (x' \<ominus> xH)

lemma a_repr_independence':

  [| yH +> x; x ∈ carrier G |] ==> H +> x = H +> y

lemma a_repr_independenceD:

  [| y ∈ carrier G; H +> x = H +> y |] ==> yH +> x

Lemmas for the Set of Right Cosets

lemma a_rcosets_carrier:

  X ∈ a_rcosets H ==> X  carrier G

Addition of Subgroups

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