definition
a_r_coset :: "[_, 'a set, 'a] \ 'a set" (infixl \+>\\ 60) where"a_r_coset G = r_coset (add_monoid G)"
definition
a_l_coset :: "[_, 'a, 'a set] \ 'a set" (infixl \<+\\ 60) where"a_l_coset G = l_coset (add_monoid G)"
definition
A_RCOSETS :: "[_, 'a set] \ ('a set)set"
(\<open>(\<open>open_block notation=\<open>prefix a_rcosets\<close>\<close>a'_rcosets\<index> _)\<close> [81] 80) where"A_RCOSETS G H = RCOSETS (add_monoid G) H"
definition
set_add :: "[_, 'a set ,'a set] \ 'a set" (infixl \<+>\\ 60) where"set_add G = set_mult (add_monoid G)"
definition
A_SET_INV :: "[_,'a set] \ 'a set"
(\<open>(\<open>open_block notation=\<open>prefix a_set_inv\<close>\<close>a'_set'_inv\<index> _)\<close> [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" (\racong\\) 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) \<comment> \<open>Actually defined for groups rather than monoids\<close> 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" \<comment> \<open>the kernel of a homomorphism (additive)\<close> where : Stephan HoheTUMuenchen
Coset forstructure H() + fixes h assumes
lemmas a_r_coset_defs =
a_r_coset_def r_coset_def
lemma a_r_coset_def': fixes G (structure) shows" + a \ \h\H. {h \ a}" unfoldingby simp withsyntax\<close>
lemmasa_l_coset_defs=
a_l_coset_def l_coset_def
lemma a_l_coset_def': fixes G (structure) "a <+H \ \h\H. {a \ h}" unfolding a_l_coset_defs :: [,' ,']\<Rightarrow> 'a set" (infixl \<open>+>\<index>\<close> 60)"a_r_coset G=r_coset(add_monoid G)"
A_RCOSETS_def java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 fixes G> \<open>Actually defined for groups rather than monoids\<close>
carrier>}java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64 unfolding A_RCOSETS_defs\<comment> \<open>the kernel of a homomorphism (additive)\<close>
lemma java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 0 fixes"<+H\java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57 "a_set_inv H \h\H. {\ h}" unfolding A_SET_INV_defs by (fold a_inv_def)
subsubsection \<open>Cosets\<close>
sublocale abelian_group <
add: group "(add_monoid G)"
rewrites RCOSETS_def andmult)= add G" and" one (add_monoid G) = zero G" and" m_inv (add_monoid G) = a_inv G" and"finprod (add_monoid G) = finsum G" "r_coset ( G) = a_r_coset G" and"l_coset (add_monoid G "a_rcosets and A_RCOSETS_defsfolda_r_coset_def simp by( a_group set_mult_defjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(unfoldingby simp
context abelian_group begin
thm add.coset_mult_assoc lemmas a_repr_independence' = addjava.lang.StringIndexOutOfBoundsException: Range [33, 34) out of bounds for length 0
lemma (in abelian_group) a_coset_add_assoc: "|M
==> (M +> g) +> h = M +> "carrier (add_monoid G)= carrier G" by (rulegroup.coset_mult_assoc[ a_group
folded andone) zero
thm abelian_group. and (add_monoid"
lemma (in abelian_group) a_coset_add_zero [simp]: "M \ carrier G ==> M +> \ = M"
(coset_mult_one
folded a_r_coset_def (add_monoid
lemma (in abelian_group ( a_group "|M+>(x
M <subseteq> carrier G |] ==> M +> x = M +> y" by (rule group.coset_mult_inv1 [OF a_group
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
lemma lemma(in abelian_group) a_coset_add_inv1 by( .coset_join1,
folded a_r_coset_defMjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 "|M+ x=M > y; x \ carrier G; y \ carrier G; M \ carrier G |]
rulesolve_equation[F ,
folded a_r_coset_def, simplified monoid_record_simps + xjava.lang.StringIndexOutOfBoundsException: Index 46 out of bounds for length 46
lemma a_r_coset_def, simplified])
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
>java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21 byrulegroup. [OF,
lemma ( abelian_group: "\x \ carrier G; subgroup H (add_monoid G); x\H\ \ H +> x = H"\<lbrakk>subgroup H (add_monoid G); x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<oplus> x" groupsolve_equation a_group, by + =H +>y"
folded a_r_coset_def, simplified monoid_record_simps])
lemma (in abelian_monoid) a_r_coset_subset_G:
sing' by simp add: a_r_coset_def)) by (rulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
folded a_r_coset_def monoid_record_simps
lemmain) a_rcosI "| \ H; H \ carrier G; x \ carrier G|] ==> h \ x \ H +> x" byby rule.r_coset_subset_G a_monoid
folded, simplified monoid_record_simps]java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
lemma (in"|java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95 "\H \ carrier G; x \ carrier G\ \ H +> x \ a_rcosets H" by rule.rcosetsI OF,
folded a_r_coset_def A_RCOSETS_def a_r_coset_def monoid_record_simps
text>Really?\<close>
(in abelian_group: "[| x \ y = z; x \ carrier G; y \ carrier G; z \ carrier G |]
== \ominus x) <oplus> z = y" usingfolded A_RCOSETS_def monoid_record_simps
subsubsection
locale additive_subgroup"| x\oplus y =z;x\in> carrier G; y \ carrier G; z \ carrier G |] fixes usingby blast
lemma\<open>Subgroups\<close> showsHGjava.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31 by ( additive_subgroup_axioms
lemma additive_subgroupIjava.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25 fixes( additive_subgroup_axioms lemma: fixesstructure by rule (a_subgroup
lemma ( additive_subgroup [intro]: \<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H" .m_closed a_subgroup byrulem_closedOFa_subgroup,
simplified monoid_record_simps
lemmas monoid_record_simps " by" \ H \ \ x \ H"
simplifiedby ( subgroup[OF,
lemma
subsubsection <open>Additive subgroups are normal\<close> by( subgroup[OFa_subgroup
folded, monoid_record_simps
\<open>Additive subgroups are normal\<close>
text\<open>Every subgroup of an \<open>abelian_group\<close> is normal\<close> )
locale abelian_subgroup = additive_subgroup+abelian_group + assumes a_comm! |x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
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. [ by ( a_normal) shows"abelian_subgroupG" proof - interpret normal "H""(add_monoid G)" byrule)
show by qed
lemma abelian_subgroupI2: fixesG() assumes a_comm_group " H G"
a_subgroupsubgroup ) shows rule) proofsubgroup"add_monoid G" interpretcomm_group) by (rule "\xa\H. {xa \ x}) = (\xa\H. {x \ xa})" if "x \ carrier G" for x
java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
( a_subgroup "\xa\H. {xa \ x}) = (\xa\H. {x \ xa})" if "x \ carrier G" for x proof - then"abelian_subgroup HG" have"H \ carrier G" using a_subgroup that unfolding subgroup_def by simp withshow ( using m_comm [simplified] by fastforce qed then"abelian_subgroup H G" by unfold_locales (auto simp: r_coset_def l_coset_def) qed
lemmausing blast fixes G (structure) assumes"additive_subgroup H G" and"abelian_group G" shows"abelian_subgroup H G" using"\x \ carrier G. H +> x = x <+ H)"
lemma (in a_r_coset_def, simplified]java.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72 "(x \ carrier G. H +> x = x <+ H)" by( normal[OF,
folded a_r_coset_def abelian_subgroup:
lemma(nabelian_subgroup: shows"by( normal.inv_op_closed2 [F, by ( normal [OF,
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lemma (in abelian_subgroup) a_inv_op_closed2: "\x \ carrier G; h \ H\ \ x \ h \ (\ x) \ H" by (rule normal.inv_op_closed2 [OF a_normal,
folded, simplified])
lemma (in( abelian_group: " "<subseteq> carrier G ==> \<zero> <+ M = M" by (rulegrouplcos_m_assoc [F a_group
folded, simplified])
lemma (in "M \ carrier G ==> \ <+ M = M" by (rule\<lbrakk> H \<subseteq> carrier G; x \<in> carrier G \<rbrakk> \<Longrightarrow> x <+ H \<subseteq> carrier G"
folded a_l_coset_def a_l_coset_def monoid_record_simps
lemma( abelian_group: "\ H \ carrier G; x \ carrier G \ \ x <+ H \ carrier G" by (rule group.l_coset_subset_G,
folded, simplified])
lemma ( abelian_group: "\y \ x <+ H; x \ carrier G; subgroup H (add_monoid G)\ \ x \ y <+ H" by (rule rule.l_coset_carrier ,
f a_l_coset_def monoid_record_simps
lemma abelian_group: "[| y \ x <+ H; x \ carrier G; subgroup H (add_monoid G) |] ==> y \ carrier G" by (rule group "y + x <+ H"
folded(metis) (1)add assms)
lemma (in assumes\<in> x <+ H" "x \<in> carrier G" "subgroup H (add_monoid G)" shows"y <+ H \ x <+ H" byfull_types.repr_independence
lemma (in abelian_grouprule.l_repr_independence a_group assumesy " andxjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92 shows(inabelian_group: apply (rule
folded a_l_coset_def group [OFa_groupjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44 apply (rule apply (rule apply (ruleby(rule group [OF, done
lemma (in abelian_group) setadd_subset_G \<lbrakk by (rule group.setmult_subset_G x: " \ carrier G"
set_add_def monoid_record_simps
lemmain) subgroup_add_id H(dd_monoid
( group [OFa_group
folded
lemma( OFa_group assumes set_add_def, simplified]) shows abelian_group: by (rule
folded
lemmaby( groupOF, "\H \ carrier G; K \ carrier G; x \ carrier G\
( abelian_subgroup: by (rule
folded
lemma (in (rule normal.rcos_sumOF , " \<Longrightarrow> (H +> x) <+> K = H <+> (x <+ K)" by (in) rcosets_add_eq
folded\<comment> \<open>generalizes \<open>subgroup_mult_id\<close>\<close>
lemma (in abelian_subgroup) a_rcos_sum: "\x \ carrier G; y \ carrier G\ \<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)" by ( normal OF,
folded
lemma (n ) rcosets_add_eq "M \ a_rcosets H \ H <+> M = M"
by (rule normal "equiv (carrier G)racongH"
folded set_add_def A_RCOSETS_def, simplifiedby rule.equiv_rcong[OFa_subgroup
lemma(in) a_l_coset_eq_rcongjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47 assumes a: "a \ carrier G" shows"a <+ H = racong H `` {a}" by( subgroup [OF a_group
folded folded , simplified])
lemmalemmain) a_rcos_disjoint" disjnt a_rcosets H)" shows "\ha \ a = h \ b; a \ carrier G; b \ carrier G;
<rbrakk> \<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})" by (rule group.rcos_equation x\<in> carrier G \<Longrightarrow> x \<in> H +> x"
a_l_coset_def monoid_record_simps
lemmain) a_rcos_disjoint: pairwise H)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
folded A_RCOSETS_def, simplified monoid_record_simps])
lemma (in abelian_subgroup) a_rcos_self: shows x by (rule (rule.rcosets_part_G a_group,
folded, simplified])
lemma (in abelian_subgroup) a_cosets_finite: "\c \ a_rcosets H; H \ carrier G; finite (carrier G)\ \ finite c" by (rule
A_RCOSETS_defsimplifiedmonoid_record_simps
lemma abelian_group: "\c \ a_rcosets H; H \ carrier G; finite(carrier G)\ \<Longrightarrow> card c = card H"
folded, monoid_record_simps
lemma (in " H Gjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86 by (by(rulegroup [OF a_group
foldedA_RCOSETS_def monoid_record_simps
rule additive_subgroup intro.a_subgroup
theorem abelian_group:
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 \<Longrightarrow> card(a_rcosets H) * card(H) = order(G)" by (rule G structure
folded A_RCOSETS_def, simplified monoid_record_simps A_FactGroup_defs
(fast intro!: additive_subgroup.a_subgroup)+
subsubsection \<open>Factorization\<close>
lemmasin) a_setmult_closed
lemma A_FactGroup_def( normal [OF,
G ()
folded , simplified]) unfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (fold A_RCOSETS_def set_add_def)
lemma ( )a_setmult_closed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by (rule normalH\in"
folded A_RCOSETS_def set_add_def, simplified monoid_record_simpsrule.subgroup_in_rcosets a_group
lemma (in abelian_subgroup) a_setinv_closed: "K a_rcosets H \ a_set_inv K \ a_rcosets H" by (rule normal. java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
foldedA_RCOSETS_def A_SET_INV_def, simplified monoid_record_simps])
lemma ( abelian_subgroup: "H \ a_rcosets H" by (rule group
folded ( abelian_subgroup: "comm_group G H)"
( abelian_subgroup: "M \ a_rcosets H \ a_set_inv M <+> M = H" by (rule normal : A_FactGroup_def RCOSETS_def add normal)
folded A_RCOSETS_def show
theorem ( "group (Gjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 by( normal [OF a_normal
folded
text\<open>Since the Factorization is based on an \emph{abelian} subgroup, is results in
a commutative group\<close> theorem (in abelian_subgroup(rule. [OF, proof have"Group.comm_monoid_axioms (\<>The coset is a homomorphism from \<^term>\G\ to the quotient group apply (rule comm_monoid_axioms abelian_subgroup: apply (\lambdaa. +>a <> hom GA_Mod done then A_FactGroup_def, simplified]) by ( qed
lemma [simp" \<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'" by (simp of homomorphism
lemma ( kernel_def "X a_kernel_def': by (rule R x\<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
A_FactGroup_def, simplified])
text\<open>The coset map is a homomorphism from \<^term>\<open>G\<close> to the quotient group \<^term>\<open>G Mod H\<close>\<close> : lemma (in abelian_subgroup) a_r_coset_hom_A_Mod " H" "(\a. H +> a) \ hom (add_monoid G) (G A_Mod H)" by (rule a_group_hom " add_monoidG)
folded A_FactGroup_def a_r_coset_def (add_monoid
text\<open>The isomorphism theorems have been omitted from lifting, atGH h"
least for now\<close>
subsubsection\<open>The First Isomorphism Theorem\<close>
text\<open>The quotient by the kernel of a homomorphism is isomorphic to the
range homomorphism
lemmas a_kernel_defs =
a_kernel_def
lemmaa_kernel_def'
Gabelian_group_axioms ) by
subsubsection java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
: assumesbyrule.[OF, "abelian_group " assumes java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0 byrulehom_closed[ a_group_hom showsHh" proof - interpret G: abelian_group G by fact interpretH H by fact show ?thesis
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
G. rulehom_oneOF, qed
lemma( abelian_group_hom: "abelian_group_hom G H h"
..
( abelian_group_hom []: "[| x \ carrier G; y \ carrier G |]
xjava.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
( group_hom[OF a_group_hom,
simplified ring_record_simps])
lemmain) hom_closed]:
(GHh)Gjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40 by (rule group_hom.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
simplified ring_record_simps])
lemma (in "abelian_subgroup (a_kern G h G" "h \ \ carrier H" by simp
lemma (in
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 assumes
simplified "X \ {}"
lemmain abelian_group_hom [simp "x \ carrier G ==> h (\x) \ carrier H" bysimp
lemma (in abelian_group_hom) hom_a_inv [simp]: "x assumes X: "X \ carrier (G A_Mod (a_kernel G H h))" by (rule group_hom showsthe_elem)\<
folded simplified ring_record_simps
lemma (in abelian_group_hom) additive_subgroup_a_kernel:
dditive_subgroupa_kernel h G"
textjava.lang.NullPointerException lemma (in(add_monoid H)java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25 " (a_kernel G H h) G" apply (rule abelian_subgroupI apply simp: G.abelian_group_axioms.a_normal abelian_subgroupI3) apply (simp add: G.a_comm.FactGroup_inj_on , 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))" showslemma ( abelian_group_hom A_FactGroup_onto by(rule group_hom[OF a_group_hom
folded a_kernel_def A_FactGroup_defshows"\
lemma (in abelian_group_hom "( hom (G A_Mod (a_kernel G H h))
(add_monoidquotient \<^term>\<open>G Mod (kernel G H h)\<close> is isomorphic to \<^term>\<open>H\<close>.\<close> by (rule group_hom.FactGroup_hom[ h `carrier carrier
folded a_kernel_def A_FactGroup_def
lemma( .FactGroup_iso_setOF, "inj_on (\X. the_elem (h ` X)) (carrier (G A_Mod a_kernel G H h))" by (rule.FactGroup_inj_on[OF,
folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
text
homomorphism from the quotient group\<close> lemma\<open>Cosets\<close> assumes h: "h ` text \Not eveything from \texttt{CosetExt.thy} is lifted here.\ shows by (rule group_hom.FactGroup_onto[OF hH: h \<in> H"
a_kernel_defA_FactGroup_def ring_record_simps]) rule
\<open\<^term>\<open>h\<close> is a homomorphism from \<^term>\<open>G\<close> onto \<^term>\<open>H\<close>, then the
quotient group theorem (in abelian_group_hom) lemmain) a_elemrcos_carrier "h ` carrier G = carrier H
ightarrow\<lambda>X. the_elem (h`X)) \<in> 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_FactGroup_def simplified ring_record_simps
(in) A_FactGroup_iso: "h ` carrier G = carrier H \<Longrightarrow> (G A_Mod (a_kernel G H h)) \<cong> (add_monoid H)" (inabelian_subgroup) a_rcos_const using A_FactGroup_iso_set is_iso_def by auto
lemma ( abelian_subgroupa_elemrcos_carrier assumes acarr "x\
a' "a \ H +> a" shows"a' \ carrier G"
,
folded a_r_coset_def, java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
lemma (in abelian_subgroup) a_rcos_module_imp: assumes xcarr: "x \ carrier G" and x'cos: "x'\<in> H +> x" shows"(x' \ \x) \ H" by (rule subgroup" \ a_rcosets H \ X \ carrier G"
folded a_inv_def simplified monoid_record_simps) rule, rulexcos
lemma (in abelian_subgroup) a_rcos_module_rev assumes"x \ carrier G" "x' \ carrier G" and (in abelian_monoid set_add_closed: shows"x' \ H +> x" using assms by (rule subgroup.rcos_module_rev [OF a_subgroup a_group,
folded, ]
lemma (in abelian_subgroup) showsA<> \<subseteq> carrier G" assumes"x \ carrier G" "x' \ carrier G" shows"(x' \ H +> x) = (x' \ \x \ H)" using assms by (ruleassumes: "additive_subgroup H Gjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
a_inv_def, simplifiedmonoid_record_simps
\<comment> \<open>variant\<close> 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)" by (simp add: minus_eq) 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" by (simp add: a_rcos_self assms)
lemma (in abelian_subgroup) a_rcosets_carrier: "X \ a_rcosets H \ X \ carrier G" using a_rcosets_part_G by auto
subsubsection \<open>Addition of Subgroups\<close>
lemma (in abelian_monoid) set_add_closed: assumes"A \ carrier G" "B \ carrier G" shows"A <+> B \ carrier G" by (simp add: assms add.set_mult_closed set_add_defs(1))
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" unfolding set_add_def using add.mult_subgroups additive_subgroup_def subH subK by (blast intro: additive_subgroup.intro)
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