Theory AbelCoset

(*  Title:      HOL/Algebra/AbelCoset.thy
Author:     Stephan Hohe, TU Muenchen
*)

theory AbelCoset
imports Coset Ring
begin

subsection ‹More Lifting from Groups to Abelian Groups›

subsubsection ‹Definitions›

text ‹Hiding <+>› from theoryHOL.Sum_Type until I come
up with better syntax here›

no_notation Sum_Type.Plus (infixr "<+>" 65)

definition
a_r_coset    :: "[_, 'a set, 'a]  'a set"    (infixl "+>ı" 60)
where "a_r_coset G = r_coset (add_monoid G)"

definition
a_l_coset    :: "[_, 'a, 'a set]  'a set"    (infixl "<+ı" 60)
where "a_l_coset G = l_coset (add_monoid G)"

definition
A_RCOSETS  :: "[_, 'a set]  ('a set)set"   ("a'_rcosetsı _" [81] 80)
where "A_RCOSETS G H = RCOSETS (add_monoid G) H"

definition
set_add  :: "[_, 'a set ,'a set]  'a set" (infixl "<+>ı" 60)

definition
A_SET_INV :: "[_,'a set]  'a set"  ("a'_set'_invı _" [81] 80)
where "A_SET_INV G H = SET_INV (add_monoid G) H"

definition
a_r_congruent :: "[('a,'b)ring_scheme, 'a set]  ('a*'a)set"  ("racongı")
where "a_r_congruent G = r_congruent (add_monoid G)"

definition
A_FactGroup :: "[('a,'b) ring_scheme, 'a set]  ('a set) monoid" (infixl "A'_Mod" 65)
― ‹Actually defined for groups rather than monoids›
where "A_FactGroup G H = FactGroup (add_monoid G) H"

definition
a_kernel :: "('a, 'm) ring_scheme  ('b, 'n) ring_scheme   ('a  'b)  'a set"
― ‹the kernel of a homomorphism (additive)›
where "a_kernel G H h = kernel (add_monoid G) (add_monoid H) h"

locale abelian_group_hom = G?: abelian_group G + H?: abelian_group H
for G (structure) and H (structure) +
fixes h

lemmas a_r_coset_defs =
a_r_coset_def r_coset_def

lemma a_r_coset_def':
fixes G (structure)
shows "H +> a  hH. {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':
fixes G (structure)
shows "a <+ H  hH. {a  h}"
unfolding a_l_coset_defs by simp

lemmas A_RCOSETS_defs =
A_RCOSETS_def RCOSETS_def

lemma A_RCOSETS_def':
fixes G (structure)
shows "a_rcosets H  acarrier G. {H +> a}"
unfolding A_RCOSETS_defs by (fold a_r_coset_def, simp)

fixes G (structure)
shows "H <+> K  hH. kK. {h  k}"

lemmas A_SET_INV_defs =
A_SET_INV_def SET_INV_def

lemma A_SET_INV_def':
fixes G (structure)
shows "a_set_inv H  hH. { h}"
unfolding A_SET_INV_defs by (fold a_inv_def)

subsubsection ‹Cosets›

sublocale abelian_group <
rewrites
and "    one (add_monoid G) =      zero G"
and "  m_inv (add_monoid G) =     a_inv G"
and "finprod (add_monoid G) =    finsum G"
and
and
and "(λa k. pow (add_monoid G) a k) = (λa k. add_pow G k a)"
by (rule a_group)
(auto simp: m_inv_def a_inv_def finsum_def a_r_coset_def a_l_coset_def add_pow_def)

context abelian_group
begin

(*
*)

end

"[| 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])

"M  carrier G ==> M +> 𝟬 = M"
by (rule group.coset_mult_one [OF a_group,
folded a_r_coset_def, simplified monoid_record_simps])

"[| M +> (x  ( 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])

"[| M +> x = M +> y;  x  carrier G;  y  carrier G;  M  carrier G |]
==> M +> (x  ( 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 (add_monoid 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 (add_monoid G); x  H; y  H  hH. 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 (add_monoid G)
H +> x = H +> y"
using a_repr_independence' by (simp add: a_r_coset_def)

lemma (in abelian_group) a_coset_join2:
"x  carrier G;  subgroup H (add_monoid G); xH  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 |]
==> ( x)  z = y"
using r_neg1 by blast

subsubsection ‹Subgroups›

fixes H and G (structure)
assumes a_subgroup: "subgroup H (add_monoid G)"

shows

fixes G (structure)
assumes a_subgroup: "subgroup H (add_monoid G)"
shows

"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])

"𝟬  H"
by (rule subgroup.one_closed[OF a_subgroup,
simplified monoid_record_simps])

"x  H   x  H"
by (rule subgroup.m_inv_closed[OF a_subgroup,
folded a_inv_def, simplified monoid_record_simps])

text ‹Every subgroup of an abelian_group› is normal›

locale abelian_subgroup = additive_subgroup + abelian_group G +
assumes a_normal: "normal H (add_monoid 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 (add_monoid G)"
and a_comm: "!!x y. [| x  carrier G; y  carrier G |] ==> x Gy = y Gx"
shows "abelian_subgroup H G"
proof -
by (rule a_normal)

show "abelian_subgroup H G"
qed

lemma abelian_subgroupI2:
fixes G (structure)
and a_subgroup: "subgroup H (add_monoid G)"
shows "abelian_subgroup H G"
proof -
by (rule a_comm_group)
by (rule a_subgroup)
have "(xaH. {xa  x}) = (xaH. {x  xa})" if "x  carrier G" for x
proof -
have "H  carrier G"
using a_subgroup that unfolding subgroup_def by simp
with that show "(hH. {h Gx}) = (hH. {x Gh})"
using m_comm [simplified] by fastforce
qed
then show "abelian_subgroup H G"
by unfold_locales (auto simp: r_coset_def l_coset_def)
qed

lemma abelian_subgroupI3:
fixes G (structure)
assumes
and
shows "abelian_subgroup H G"
using assms abelian_subgroupI2 abelian_group.a_comm_group additive_subgroup_def by blast

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  ( 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  ( x)  H"
by (rule normal.inv_op_closed2 [OF a_normal,
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 ==> 𝟬 <+ 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 (add_monoid 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 (add_monoid 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  x <+ H" "x  carrier G" "subgroup H (add_monoid G)"
shows "y <+ H  x <+ H"
by (metis (full_types) a_l_coset_defs(1) add.l_repr_independence assms set_eq_subset)

lemma (in abelian_group) a_l_repr_independence:
assumes y: "y  x <+ H" and x: "x  carrier G" and sb: "subgroup H (add_monoid 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

"H  carrier G; K  carrier G  H <+> K  carrier G"
by (rule group.setmult_subset_G [OF a_group,

lemma (in abelian_group) subgroup_add_id: "subgroup H (add_monoid G)  H <+> H = H"
by (rule group.subgroup_mult_id [OF a_group,

lemma (in abelian_subgroup) a_rcos_inv:
assumes x:     "x  carrier G"
shows "a_set_inv (H +> x) = H +> ( 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,

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,

"M  a_rcosets H  H <+> M = M"
― ‹generalizes subgroup_mult_id›
by (rule normal.rcosets_mult_eq [OF a_normal,

subsubsection ‹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  (hH. {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:
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 "(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"

lemma (in abelian_group) rcosets_subset_PowG:

by (rule group.rcosets_subset_PowG [OF a_group,
folded A_RCOSETS_def, simplified monoid_record_simps],

theorem (in abelian_group) a_lagrange:
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])

subsubsection ‹Factorization›

lemmas A_FactGroup_defs = A_FactGroup_def FactGroup_def

lemma A_FactGroup_def':
fixes G (structure)
shows "G A_Mod H  carrier = a_rcosetsGH, mult = set_add G, one = H"
unfolding A_FactGroup_defs

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,

lemma (in abelian_subgroup) a_setinv_closed:

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,

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)"
proof -
have
apply (rule comm_monoid_axioms.intro)
apply (auto simp: A_FactGroup_def FactGroup_def RCOSETS_def a_normal add.m_comm normal.rcos_sum)
done
then show ?thesis
by (intro comm_group.intro comm_monoid.intro) (simp_all add: a_factorgroup_is_group group.is_monoid)
qed

lemma add_A_FactGroup [simp]: "X (G A_Mod H)X' = X <+>GX'"

lemma (in abelian_subgroup) a_inv_FactGroup:
"X  carrier (G A_Mod H)  invG A_Mod HX = 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 termG to the quotient group
termG Mod H
lemma (in abelian_subgroup) a_r_coset_hom_A_Mod:
"(λa. H +> a)  hom (add_monoid 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›

subsubsection‹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 = 𝟬S}"
by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps])

subsubsection ‹Homomorphisms›

lemma abelian_group_homI:
assumes
assumes
shows "abelian_group_hom G H h"
proof -
interpret G: abelian_group G by fact
interpret H: abelian_group H by fact
show ?thesis
by (intro abelian_group_hom.intro abelian_group_hom_axioms.intro
G.abelian_group_axioms H.abelian_group_axioms a_group_hom)
qed

lemma (in abelian_group_hom) is_abelian_group_hom:
"abelian_group_hom G H h"
..

"[| x  carrier G; y  carrier G |]
==> h (x Gy) = h x Hh 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 𝟬  carrier H"
by simp

lemma (in abelian_group_hom) hom_zero [simp]:
"h 𝟬 = 𝟬H⇙"
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 (x)  carrier H"
by simp

lemma (in abelian_group_hom) hom_a_inv [simp]:
"x  carrier G ==> h (x) = H(h x)"
by (rule group_hom.hom_inv[OF a_group_hom,
folded a_inv_def, simplified ring_record_simps])

"additive_subgroup (a_kernel G H h) G"

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)
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_the_elem_mem:
assumes X: "X  carrier (G A_Mod (a_kernel G H h))"
shows "the_elem (hX)  carrier H"
by (rule group_hom.FactGroup_the_elem_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. the_elem (hX))  hom (G A_Mod (a_kernel G H 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. the_elem (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 termh is onto termH, 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. the_elem (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 termh is a homomorphism from termG onto termH, then the
quotient group termG Mod (kernel G H h) is isomorphic to termH.›
theorem (in abelian_group_hom) A_FactGroup_iso_set:
"h  carrier G = carrier H
(λX. the_elem (hX))  iso (G A_Mod (a_kernel G H h)) (add_monoid H)"
by (rule group_hom.FactGroup_iso_set[OF a_group_hom,
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])

corollary (in abelian_group_hom) A_FactGroup_iso :
"h ` carrier G = carrier H
(G A_Mod (a_kernel G H h))   (add_monoid H)"
using A_FactGroup_iso_set unfolding is_iso_def by auto

subsubsection ‹Cosets›

text ‹Not eveything from \texttt{CosetExt.thy} is lifted here.›

assumes hH: "h  H"
shows "h  carrier G"
by (rule subgroup.mem_carrier [OF a_subgroup,
simplified monoid_record_simps]) (rule hH)

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'  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'  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'  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:
assumes "ring G"
assumes carr: "x  carrier G" "x'  carrier G"
shows "(x'  H +> x) = (x'  x  H)"
proof -
interpret G: ring G by fact
from carr
have "(x'  H +> x) = (x'  x  H)" by (rule a_rcos_module)
with carr
show "(x'  H +> x) = (x'  x  H)"
qed

lemma (in abelian_subgroup) a_repr_independence':
assumes "y  H +> x" "x  carrier G"
shows "H +> x = H +> y"
using a_repr_independence a_subgroup assms by blast

lemma (in abelian_subgroup) a_repr_independenceD:
assumes "y  carrier G" "H +> x = H +> y"
shows "y  H +> x"

lemma (in abelian_subgroup) a_rcosets_carrier:
"X  a_rcosets H  X  carrier G"
using a_rcosets_part_G by auto