Quelle AbelCoset.thy Sprache: Isabelle

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

theory AbelCoset
imports Coset Ring
begin

subsection \<open>More Lifting from Groups to Abelian Groups\<close>

subsubsection \<open>Definitions\<close>

text \<open>Hiding \<open><+>\<close> from \<^theory>\<open>HOL.Sum_Type\<close> until I come
  (  Title:      HOL/Algebra/AbelCoset.thy

no_notation Sum_Type.Plus (infixr \<open><+>\<close> 65)

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}"
  unfolding by simp with syntax\<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>

lemmas =
  set_add_def set_mult_def

lemma':
fixes(tructure
   "H +>K\<> \h\H. \k\K. {h \ k}"
  unfoldingset_add_defs

lemmas  =
  A_SET_INV_def SET_INV_def

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
     (unfolding by 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

(*
lemmas a_coset_add_assoc = add.coset_mult_assoc
lemmas a_coset_add_zero [simp] = add.coset_mult_one
lemmas a_coset_add_inv1 = add.coset_mult_inv1
lemmas a_coset_add_inv2 = add.coset_mult_inv2
lemmas a_coset_join1 = add.coset_join1
lemmas a_coset_join2 = add.coset_join2
lemmas a_solve_equation = add.solve_equation
lemmas a_repr_independence = add.repr_independence
lemmas a_rcosI = add.rcosI
lemmas a_rcosetsI = add.rcosetsI
*)

end

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 (inlemmas a_coset_add_zero lemmas a_coset_add_inv1lemmas a_coset_add_inv2lemmas a_coset_join1lemmas a_coset_join2 lemmas a_solve_equation =lemmas a_repr_independence = addlemmas*)
( group )  [simp
=M+xjava.lang.NullPointerException
( groupcoset_mult_inv2[OFa_group,
    folded a_r_coset_def monoid_record_simps)

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

lemma in) a_rcosI
"| \ H; H \ carrier G; x \ carrier G|] ==> h \ x \ H +> x"
by by 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
using by 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_subgroupH G"
     " \ carrier G"
by rule.subset a_subgroup
    simplifiedbyrule.subset 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
    with show (
      using m_comm [simplified] by fastforce
  qed
then "abelian_subgroup H G"
    by unfold_locales (auto simp: r_coset_def l_coset_def)
qed

lemma using 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

lemma in) subgroup_add_id H(dd_monoid
( group [OFa_group
    folded

lemma( OFa_group
  assumes set_add_def, simplified])
  shows abelian_group:
by (rule
  folded

lemma by( 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

subsubsection\<open>Congruence Relation\<close>

lemmai ) a_equiv_rcong
    "equiv ( G) (racong H"
by (rule rule subgroupl_coset_eq_rcong a_subgroup,
    folded a_r_congruent_def monoid_record_simps

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])

lemma lemmain) 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

lemma in) 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(inabelian_subgroup:
   "\(a_rcosets H) = carrier G"
by (rule (in) a_card_cosets_equal
    folded A_RCOSETS_def,      <>card H"

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>

lemmas in) 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(in) a_rcosets_assoc
     "\M1 \ a_rcosets H; M2 \ a_rcosets H; M3 \ a_rcosets H\
      <> <+> 2<+  = M1<>( <+> M3"
( normal OF,
    folded A_RCOSETS_def A_FactGroup_def 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

subsubsection \<open>Cosets\<close> (ule.rcos_const[ a_subgroupa_group

text

lemma (in additive_subgroup) a_Hcarr [simp]:
assumes:" \ carrier G"
shows
by (rule subgroup "(x' \ \x) \ H"
    simplified ( subgroup. [OFa_subgroup,

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

(in)a_rcos_const
  assumes a_repr_independence assms blast
  shows "H +> h = H"
by (rule lemmain) a_repr_independenceD
    folded a_r_coset_def, simplified ( add: a_rcos_self assms

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)

end

quality94%

bsp;assms add"

lemma (in
   subH"
    and subK: "additive_subgroup K G"
  shows "additive_subgroup (H foldeda_r_coset_def ])
  unfolding\<comment> \<open>variant\<close>
  using add java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
  by ( -

end

quality94%

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