src/HOL/Algebra/AbelCoset.thy
 author wenzelm Thu, 26 Mar 2009 20:08:55 +0100 changeset 30729 461ee3e49ad3 parent 29237 e90d9d51106b child 35416 d8d7d1b785af permissions -rw-r--r--
interpretation/interpret: prefixes are mandatory by default;

(*
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 @{text "<+>"} from @{theory Sum_Type} until I come
up with better syntax here *}

no_notation Plus (infixr "<+>" 65)

constdefs (structure G)
a_r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "+>\<index>" 60)
"a_r_coset G \<equiv> r_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

a_l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<+\<index>" 60)
"a_l_coset G \<equiv> l_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

A_RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("a'_rcosets\<index> _" [81] 80)
"A_RCOSETS G H \<equiv> RCOSETS \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"

set_add  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<+>\<index>" 60)
"set_add G \<equiv> set_mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

A_SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("a'_set'_inv\<index> _" [81] 80)
"A_SET_INV G H \<equiv> SET_INV \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"

constdefs (structure G)
a_r_congruent :: "[('a,'b)ring_scheme, 'a set] \<Rightarrow> ('a*'a)set"
("racong\<index> _")
"a_r_congruent G \<equiv> r_congruent \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

constdefs
A_FactGroup :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) monoid"
(infixl "A'_Mod" 65)
--{*Actually defined for groups rather than monoids*}
"A_FactGroup G H \<equiv> FactGroup \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"

constdefs
a_kernel :: "('a, 'm) ring_scheme \<Rightarrow> ('b, 'n) ring_scheme \<Rightarrow>
('a \<Rightarrow> 'b) \<Rightarrow> 'a set"
--{*the kernel of a homomorphism (additive)*}
"a_kernel G H h \<equiv> kernel \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
\<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"

locale abelian_group_hom = G: abelian_group G + H: abelian_group H
for G (structure) and H (structure) +
fixes 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':
fixes G (structure)
shows "H +> a \<equiv> \<Union>h\<in>H. {h \<oplus> 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 \<equiv> \<Union>h\<in>H. {a \<oplus> 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 \<equiv> \<Union>a\<in>carrier G. {H +> a}"
unfolding A_RCOSETS_defs
by (fold a_r_coset_def, simp)

fixes G (structure)
shows "H <+> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus> k}"
by simp

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 \<equiv> \<Union>h\<in>H. {\<ominus> h}"
unfolding A_SET_INV_defs
by (fold a_inv_def)

subsubsection {* Cosets *}

"[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
==> (M +> g) +> h = M +> (g \<oplus> h)"
by (rule group.coset_mult_assoc [OF a_group,
folded a_r_coset_def, simplified monoid_record_simps])

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

"[| M +> (x \<oplus> (\<ominus> y)) = M;  x \<in> carrier G ; y \<in> carrier G;
M \<subseteq> 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 \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
==> M +> (x \<oplus> (\<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 \<in> carrier G;  subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x \<in> 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:
"\<lbrakk>subgroup H (|carrier = carrier G, mult = add G, one = zero G|); x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> 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:
"\<lbrakk>y \<in> H +> x;  x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<rbrakk> \<Longrightarrow> 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:
"\<lbrakk>x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>; x\<in>H\<rbrakk> \<Longrightarrow> 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 \<subseteq> carrier G; x \<in> carrier G |] ==> H +> x \<subseteq> 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 \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<oplus> x \<in> H +> x"
by (rule group.rcosI [OF a_group,
folded a_r_coset_def, simplified monoid_record_simps])

lemma (in abelian_group) a_rcosetsI:
"\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H +> x \<in> 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 \<oplus> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
==> (\<ominus> x) \<oplus> 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 \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> x \<in> H +> x"
by (rule group.rcos_self [OF a_group,
folded a_r_coset_def, simplified monoid_record_simps])
*)

subsubsection {* Subgroups *}

fixes H and G (structure)
assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

fixes G (structure)
assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

"H \<subseteq> carrier G"
by (rule subgroup.subset[OF a_subgroup,
simplified monoid_record_simps])

lemma (in additive_subgroup) a_closed [intro, simp]:
"\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H"
by (rule subgroup.m_closed[OF a_subgroup,
simplified monoid_record_simps])

"\<zero> \<in> H"
by (rule subgroup.one_closed[OF a_subgroup,
simplified monoid_record_simps])

"x \<in> H \<Longrightarrow> \<ominus> x \<in> H"
by (rule subgroup.m_inv_closed[OF a_subgroup,
folded a_inv_def, simplified monoid_record_simps])

subsubsection {* Additive subgroups are normal *}

text {* Every subgroup of an @{text "abelian_group"} is normal *}

locale abelian_subgroup = additive_subgroup + abelian_group G +
assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"

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 \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
and a_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
shows "abelian_subgroup H G"
proof -
interpret normal "H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
by (rule a_normal)

show "abelian_subgroup H G"
qed

lemma abelian_subgroupI2:
fixes G (structure)
assumes a_comm_group: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
and a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
shows "abelian_subgroup H G"
proof -
interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
by (rule a_comm_group)
interpret subgroup "H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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 \<in> carrier G"
from a_subgroup
have Hcarr: "H \<subseteq> carrier G" by (unfold subgroup_def, simp)
from xcarr Hcarr
show "(\<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> x}) = (\<Union>h\<in>H. {x \<oplus>\<^bsub>G\<^esub> h})"
using m_comm[simplified]
by fast
qed
qed

lemma abelian_subgroupI3:
fixes G (structure)
and ag: "abelian_group G"
shows "abelian_subgroup H G"
apply (rule abelian_subgroupI2)
apply (rule abelian_group.a_comm_group[OF ag])
done

lemma (in abelian_subgroup) a_coset_eq:
"(\<forall>x \<in> 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 "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (\<ominus> x) \<oplus> h \<oplus> x \<in> 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 "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> h \<oplus> (\<ominus> x) \<in> 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> \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>) =
(subgroup N \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<oplus> h \<oplus> (\<ominus> x) \<in> 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 \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
==> g <+ (h <+ M) = (g \<oplus> 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 \<subseteq> 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 \<subseteq> carrier G; x \<in> carrier G |] ==> x <+ H \<subseteq> 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:
"\<lbrakk>y \<in> x <+ H;  x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>\<rbrakk> \<Longrightarrow> x \<in> 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 \<in> x <+ H;  x \<in> carrier G;  subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> y \<in> 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 \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
shows "y <+ H \<subseteq> 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 \<in> x <+ H" and x: "x \<in> carrier G" and sb: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
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

"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <+> K \<subseteq> carrier G"
by (rule group.setmult_subset_G [OF a_group,

lemma (in abelian_group) subgroup_add_id: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<Longrightarrow> H <+> H = H"
by (rule group.subgroup_mult_id [OF a_group,

lemma (in abelian_subgroup) a_rcos_inv:
assumes x:     "x \<in> 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:
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
\<Longrightarrow> H <+> (K +> x) = (H <+> K) +> x"
by (rule group.setmult_rcos_assoc [OF a_group,

lemma (in abelian_group) a_rcos_assoc_lcos:
"\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
\<Longrightarrow> (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:
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
\<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)"
by (rule normal.rcos_sum [OF a_normal,

"M \<in> a_rcosets H \<Longrightarrow> H <+> M = M"
-- {* generalizes @{text 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 \<in> 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
"\<lbrakk>ha \<oplus> a = h \<oplus> b; a \<in> carrier G;  b \<in> carrier G;
h \<in> H;  ha \<in> H;  hb \<in> H\<rbrakk>
\<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> 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 "\<lbrakk>a \<in> a_rcosets H; b \<in> a_rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> 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 \<in> carrier G \<Longrightarrow> x \<in> 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:
"\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> 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:
"\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
\<Longrightarrow> 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  \<Longrightarrow> a_rcosets H \<subseteq> Pow(carrier G)"
by (rule group.rcosets_subset_PowG [OF a_group,
folded A_RCOSETS_def, simplified monoid_record_simps],

theorem (in abelian_group) a_lagrange:
\<Longrightarrow> 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 \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = set_add G, one = H\<rparr>"
unfolding A_FactGroup_defs

lemma (in abelian_subgroup) a_setmult_closed:
"\<lbrakk>K1 \<in> a_rcosets H; K2 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> K1 <+> K2 \<in> a_rcosets H"
by (rule normal.setmult_closed [OF a_normal,

lemma (in abelian_subgroup) a_setinv_closed:
"K \<in> a_rcosets H \<Longrightarrow> a_set_inv K \<in> 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:
"\<lbrakk>M1 \<in> a_rcosets H; M2 \<in> a_rcosets H; M3 \<in> a_rcosets H\<rbrakk>
\<Longrightarrow> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)"
by (rule normal.rcosets_assoc [OF a_normal,

lemma (in abelian_subgroup) a_subgroup_in_rcosets:
"H \<in> 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 \<in> a_rcosets H \<Longrightarrow> 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)
done

lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"

lemma (in abelian_subgroup) a_inv_FactGroup:
"X \<in> carrier (G A_Mod H) \<Longrightarrow> inv\<^bsub>G A_Mod H\<^esub> 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:
"(\<lambda>a. H +> a) \<in> hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> (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 \<equiv> {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps])

subsubsection {* Homomorphisms *}

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

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

"[| x : carrier G; y : carrier G |]
==> h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> 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 \<in> carrier G \<Longrightarrow> h x \<in> 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> \<in> 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>\<^bsub>H\<^esub>"
by (rule group_hom.hom_one[OF a_group_hom,
simplified ring_record_simps])

lemma (in abelian_group_hom) a_inv_closed [simp]:
"x \<in> carrier G ==> h (\<ominus>x) \<in> 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 \<in> carrier G ==> h (\<ominus>x) = \<ominus>\<^bsub>H\<^esub> (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"
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])
done

lemma (in abelian_group_hom) A_FactGroup_nonempty:
assumes X: "X \<in> carrier (G A_Mod a_kernel G H h)"
shows "X \<noteq> {}"
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 \<in> carrier (G A_Mod (a_kernel G H h))"
shows "contents (hX) \<in> 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:
"(\<lambda>X. contents (hX)) \<in> hom (G A_Mod (a_kernel G H h))
\<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>"
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 (\<lambda>X. 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 "(\<lambda>X. 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
\<Longrightarrow> (\<lambda>X. contents (hX)) \<in> (G A_Mod (a_kernel G H h)) \<cong>
(| 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])

subsubsection {* Cosets *}

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

assumes hH: "h \<in> H"
shows "h \<in> 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 \<in> carrier G"
and a': "a' \<in> H +> a"
shows "a' \<in> 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 \<in> 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 \<in> carrier G"
and x'cos: "x' \<in> H +> x"
shows "(x' \<oplus> \<ominus>x) \<in> 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 \<in> carrier G" "x' \<in> carrier G"
and "(x' \<oplus> \<ominus>x) \<in> H"
shows "x' \<in> 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 \<in> carrier G" "x' \<in> carrier G"
shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> 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 \<in> carrier G" "x' \<in> carrier G"
shows "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
proof -
interpret G: ring G by fact
from carr
have "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)" by (rule a_rcos_module)
with carr
show "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
qed

lemma (in abelian_subgroup) a_repr_independence':
assumes y: "y \<in> H +> x"
and xcarr: "x \<in> 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 \<in> carrier G"
and repr:  "H +> x = H +> y"
shows "y \<in> 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)

lemma (in abelian_subgroup) a_rcosets_carrier:
"X \<in> a_rcosets H \<Longrightarrow> X \<subseteq> carrier G"
by (rule subgroup.rcosets_carrier [OF a_subgroup a_group,
folded A_RCOSETS_def, simplified monoid_record_simps])

subsubsection {* Addition of Subgroups *}

assumes Acarr: "A \<subseteq> carrier G"
and Bcarr: "B \<subseteq> carrier G"
shows "A <+> B \<subseteq> carrier G"
by (rule monoid.set_mult_closed [OF a_monoid,
folded set_add_def, simplified monoid_record_simps]) (rule Acarr, rule Bcarr)