src/HOL/Algebra/AbelCoset.thy
author wenzelm
Thu Nov 23 20:34:21 2006 +0100 (2006-11-23)
changeset 21502 7f3ea2b3bab6
parent 20318 0e0ea63fe768
child 23350 50c5b0912a0c
permissions -rw-r--r--
prefer antiquotations over LaTeX macros;
     1 (*
     2   Title:     HOL/Algebra/AbelCoset.thy
     3   Id:        $Id$
     4   Author:    Stephan Hohe, TU Muenchen
     5 *)
     6 
     7 theory AbelCoset
     8 imports Coset Ring
     9 begin
    10 
    11 
    12 section {* More Lifting from Groups to Abelian Groups *}
    13 
    14 subsection {* Definitions *}
    15 
    16 text {* Hiding @{text "<+>"} from @{theory Sum_Type} until I come
    17   up with better syntax here *}
    18 
    19 hide const Plus
    20 
    21 constdefs (structure G)
    22   a_r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "+>\<index>" 60)
    23   "a_r_coset G \<equiv> r_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
    24 
    25   a_l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<+\<index>" 60)
    26   "a_l_coset G \<equiv> l_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
    27 
    28   A_RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("a'_rcosets\<index> _" [81] 80)
    29   "A_RCOSETS G H \<equiv> RCOSETS \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
    30 
    31   set_add  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<+>\<index>" 60)
    32   "set_add G \<equiv> set_mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
    33 
    34   A_SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("a'_set'_inv\<index> _" [81] 80)
    35   "A_SET_INV G H \<equiv> SET_INV \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
    36 
    37 constdefs (structure G)
    38   a_r_congruent :: "[('a,'b)ring_scheme, 'a set] \<Rightarrow> ('a*'a)set"
    39                   ("racong\<index> _")
    40    "a_r_congruent G \<equiv> r_congruent \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
    41 
    42 constdefs
    43   A_FactGroup :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) monoid"
    44      (infixl "A'_Mod" 65)
    45     --{*Actually defined for groups rather than monoids*}
    46   "A_FactGroup G H \<equiv> FactGroup \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
    47 
    48 constdefs
    49   a_kernel :: "('a, 'm) ring_scheme \<Rightarrow> ('b, 'n) ring_scheme \<Rightarrow> 
    50              ('a \<Rightarrow> 'b) \<Rightarrow> 'a set" 
    51     --{*the kernel of a homomorphism (additive)*}
    52   "a_kernel G H h \<equiv> kernel \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
    53                               \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
    54 
    55 locale abelian_group_hom = abelian_group G + abelian_group H + var h +
    56   assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
    57                                   (| carrier = carrier H, mult = add H, one = zero H |) h"
    58 
    59 lemmas a_r_coset_defs =
    60   a_r_coset_def r_coset_def
    61 
    62 lemma a_r_coset_def':
    63   includes struct G
    64   shows "H +> a \<equiv> \<Union>h\<in>H. {h \<oplus> a}"
    65 unfolding a_r_coset_defs
    66 by simp
    67 
    68 lemmas a_l_coset_defs =
    69   a_l_coset_def l_coset_def
    70 
    71 lemma a_l_coset_def':
    72   includes struct G
    73   shows "a <+ H \<equiv> \<Union>h\<in>H. {a \<oplus> h}"
    74 unfolding a_l_coset_defs
    75 by simp
    76 
    77 lemmas A_RCOSETS_defs =
    78   A_RCOSETS_def RCOSETS_def
    79 
    80 lemma A_RCOSETS_def':
    81   includes struct G
    82   shows "a_rcosets H \<equiv> \<Union>a\<in>carrier G. {H +> a}"
    83 unfolding A_RCOSETS_defs
    84 by (fold a_r_coset_def, simp)
    85 
    86 lemmas set_add_defs =
    87   set_add_def set_mult_def
    88 
    89 lemma set_add_def':
    90   includes struct G
    91   shows "H <+> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus> k}"
    92 unfolding set_add_defs
    93 by simp
    94 
    95 lemmas A_SET_INV_defs =
    96   A_SET_INV_def SET_INV_def
    97 
    98 lemma A_SET_INV_def':
    99   includes struct G
   100   shows "a_set_inv H \<equiv> \<Union>h\<in>H. {\<ominus> h}"
   101 unfolding A_SET_INV_defs
   102 by (fold a_inv_def)
   103 
   104 
   105 subsection {* Cosets *}
   106 
   107 lemma (in abelian_group) a_coset_add_assoc:
   108      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
   109       ==> (M +> g) +> h = M +> (g \<oplus> h)"
   110 by (rule group.coset_mult_assoc [OF a_group,
   111     folded a_r_coset_def, simplified monoid_record_simps])
   112 
   113 lemma (in abelian_group) a_coset_add_zero [simp]:
   114   "M \<subseteq> carrier G ==> M +> \<zero> = M"
   115 by (rule group.coset_mult_one [OF a_group,
   116     folded a_r_coset_def, simplified monoid_record_simps])
   117 
   118 lemma (in abelian_group) a_coset_add_inv1:
   119      "[| M +> (x \<oplus> (\<ominus> y)) = M;  x \<in> carrier G ; y \<in> carrier G;
   120          M \<subseteq> carrier G |] ==> M +> x = M +> y"
   121 by (rule group.coset_mult_inv1 [OF a_group,
   122     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
   123 
   124 lemma (in abelian_group) a_coset_add_inv2:
   125      "[| M +> x = M +> y;  x \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
   126       ==> M +> (x \<oplus> (\<ominus> y)) = M"
   127 by (rule group.coset_mult_inv2 [OF a_group,
   128     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
   129 
   130 lemma (in abelian_group) a_coset_join1:
   131      "[| H +> x = H;  x \<in> carrier G;  subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x \<in> H"
   132 by (rule group.coset_join1 [OF a_group,
   133     folded a_r_coset_def, simplified monoid_record_simps])
   134 
   135 lemma (in abelian_group) a_solve_equation:
   136     "\<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"
   137 by (rule group.solve_equation [OF a_group,
   138     folded a_r_coset_def, simplified monoid_record_simps])
   139 
   140 lemma (in abelian_group) a_repr_independence:
   141      "\<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"
   142 by (rule group.repr_independence [OF a_group,
   143     folded a_r_coset_def, simplified monoid_record_simps])
   144 
   145 lemma (in abelian_group) a_coset_join2:
   146      "\<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"
   147 by (rule group.coset_join2 [OF a_group,
   148     folded a_r_coset_def, simplified monoid_record_simps])
   149 
   150 lemma (in abelian_monoid) a_r_coset_subset_G:
   151      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H +> x \<subseteq> carrier G"
   152 by (rule monoid.r_coset_subset_G [OF a_monoid,
   153     folded a_r_coset_def, simplified monoid_record_simps])
   154 
   155 lemma (in abelian_group) a_rcosI:
   156      "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<oplus> x \<in> H +> x"
   157 by (rule group.rcosI [OF a_group,
   158     folded a_r_coset_def, simplified monoid_record_simps])
   159 
   160 lemma (in abelian_group) a_rcosetsI:
   161      "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H +> x \<in> a_rcosets H"
   162 by (rule group.rcosetsI [OF a_group,
   163     folded a_r_coset_def A_RCOSETS_def, simplified monoid_record_simps])
   164 
   165 text{*Really needed?*}
   166 lemma (in abelian_group) a_transpose_inv:
   167      "[| x \<oplus> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
   168       ==> (\<ominus> x) \<oplus> z = y"
   169 by (rule group.transpose_inv [OF a_group,
   170     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
   171 
   172 (*
   173 --"duplicate"
   174 lemma (in abelian_group) a_rcos_self:
   175      "[| x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> x \<in> H +> x"
   176 by (rule group.rcos_self [OF a_group,
   177     folded a_r_coset_def, simplified monoid_record_simps])
   178 *)
   179 
   180 
   181 subsection {* Subgroups *}
   182 
   183 locale additive_subgroup = var H + struct G +
   184   assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   185 
   186 lemma (in additive_subgroup) is_additive_subgroup:
   187   shows "additive_subgroup H G"
   188 by -
   189 
   190 lemma additive_subgroupI:
   191   includes struct G
   192   assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   193   shows "additive_subgroup H G"
   194 by (rule additive_subgroup.intro)
   195 
   196 lemma (in additive_subgroup) a_subset:
   197      "H \<subseteq> carrier G"
   198 by (rule subgroup.subset[OF a_subgroup,
   199     simplified monoid_record_simps])
   200 
   201 lemma (in additive_subgroup) a_closed [intro, simp]:
   202      "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H"
   203 by (rule subgroup.m_closed[OF a_subgroup,
   204     simplified monoid_record_simps])
   205 
   206 lemma (in additive_subgroup) zero_closed [simp]:
   207      "\<zero> \<in> H"
   208 by (rule subgroup.one_closed[OF a_subgroup,
   209     simplified monoid_record_simps])
   210 
   211 lemma (in additive_subgroup) a_inv_closed [intro,simp]:
   212      "x \<in> H \<Longrightarrow> \<ominus> x \<in> H"
   213 by (rule subgroup.m_inv_closed[OF a_subgroup,
   214     folded a_inv_def, simplified monoid_record_simps])
   215 
   216 
   217 subsection {* Normal additive subgroups *}
   218 
   219 subsubsection {* Definition of @{text "abelian_subgroup"} *}
   220 
   221 text {* Every subgroup of an @{text "abelian_group"} is normal *}
   222 
   223 locale abelian_subgroup = additive_subgroup H G + abelian_group G +
   224   assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   225 
   226 lemma (in abelian_subgroup) is_abelian_subgroup:
   227   shows "abelian_subgroup H G"
   228 by -
   229 
   230 lemma abelian_subgroupI:
   231   assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   232       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"
   233   shows "abelian_subgroup H G"
   234 proof -
   235   interpret normal ["H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"]
   236   by (rule a_normal)
   237 
   238   show "abelian_subgroup H G"
   239   by (unfold_locales, simp add: a_comm)
   240 qed
   241 
   242 lemma abelian_subgroupI2:
   243   includes struct G
   244   assumes a_comm_group: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   245       and a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
   246   shows "abelian_subgroup H G"
   247 proof -
   248   interpret comm_group ["\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"]
   249   by (rule a_comm_group)
   250   interpret subgroup ["H" "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"]
   251   by (rule a_subgroup)
   252 
   253   show "abelian_subgroup H G"
   254   apply unfold_locales
   255   proof (simp add: r_coset_def l_coset_def, clarsimp)
   256     fix x
   257     assume xcarr: "x \<in> carrier G"
   258     from a_subgroup
   259         have Hcarr: "H \<subseteq> carrier G" by (unfold subgroup_def, simp)
   260     from xcarr Hcarr
   261         show "(\<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> x}) = (\<Union>h\<in>H. {x \<oplus>\<^bsub>G\<^esub> h})"
   262         using m_comm[simplified]
   263         by fast
   264   qed
   265 qed
   266 
   267 lemma abelian_subgroupI3:
   268   includes struct G
   269   assumes asg: "additive_subgroup H G"
   270       and ag: "abelian_group G"
   271   shows "abelian_subgroup H G"
   272 apply (rule abelian_subgroupI2)
   273  apply (rule abelian_group.a_comm_group[OF ag])
   274 apply (rule additive_subgroup.a_subgroup[OF asg])
   275 done
   276 
   277 lemma (in abelian_subgroup) a_coset_eq:
   278      "(\<forall>x \<in> carrier G. H +> x = x <+ H)"
   279 by (rule normal.coset_eq[OF a_normal,
   280     folded a_r_coset_def a_l_coset_def, simplified monoid_record_simps])
   281 
   282 lemma (in abelian_subgroup) a_inv_op_closed1:
   283   shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (\<ominus> x) \<oplus> h \<oplus> x \<in> H"
   284 by (rule normal.inv_op_closed1 [OF a_normal,
   285     folded a_inv_def, simplified monoid_record_simps])
   286 
   287 lemma (in abelian_subgroup) a_inv_op_closed2:
   288   shows "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> h \<oplus> (\<ominus> x) \<in> H"
   289 by (rule normal.inv_op_closed2 [OF a_normal,
   290     folded a_inv_def, simplified monoid_record_simps])
   291 
   292 text{*Alternative characterization of normal subgroups*}
   293 lemma (in abelian_group) a_normal_inv_iff:
   294      "(N \<lhd> \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>) = 
   295       (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))"
   296       (is "_ = ?rhs")
   297 by (rule group.normal_inv_iff [OF a_group,
   298     folded a_inv_def, simplified monoid_record_simps])
   299 
   300 lemma (in abelian_group) a_lcos_m_assoc:
   301      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
   302       ==> g <+ (h <+ M) = (g \<oplus> h) <+ M"
   303 by (rule group.lcos_m_assoc [OF a_group,
   304     folded a_l_coset_def, simplified monoid_record_simps])
   305 
   306 lemma (in abelian_group) a_lcos_mult_one:
   307      "M \<subseteq> carrier G ==> \<zero> <+ M = M"
   308 by (rule group.lcos_mult_one [OF a_group,
   309     folded a_l_coset_def, simplified monoid_record_simps])
   310 
   311 
   312 lemma (in abelian_group) a_l_coset_subset_G:
   313      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <+ H \<subseteq> carrier G"
   314 by (rule group.l_coset_subset_G [OF a_group,
   315     folded a_l_coset_def, simplified monoid_record_simps])
   316 
   317 
   318 lemma (in abelian_group) a_l_coset_swap:
   319      "\<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"
   320 by (rule group.l_coset_swap [OF a_group,
   321     folded a_l_coset_def, simplified monoid_record_simps])
   322 
   323 lemma (in abelian_group) a_l_coset_carrier:
   324      "[| 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"
   325 by (rule group.l_coset_carrier [OF a_group,
   326     folded a_l_coset_def, simplified monoid_record_simps])
   327 
   328 lemma (in abelian_group) a_l_repr_imp_subset:
   329   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>"
   330   shows "y <+ H \<subseteq> x <+ H"
   331 by (rule group.l_repr_imp_subset [OF a_group,
   332     folded a_l_coset_def, simplified monoid_record_simps])
   333 
   334 lemma (in abelian_group) a_l_repr_independence:
   335   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>"
   336   shows "x <+ H = y <+ H"
   337 by (rule group.l_repr_independence [OF a_group,
   338     folded a_l_coset_def, simplified monoid_record_simps])
   339 
   340 lemma (in abelian_group) setadd_subset_G:
   341      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <+> K \<subseteq> carrier G"
   342 by (rule group.setmult_subset_G [OF a_group,
   343     folded set_add_def, simplified monoid_record_simps])
   344 
   345 lemma (in abelian_group) subgroup_add_id: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<Longrightarrow> H <+> H = H"
   346 by (rule group.subgroup_mult_id [OF a_group,
   347     folded set_add_def, simplified monoid_record_simps])
   348 
   349 lemma (in abelian_subgroup) a_rcos_inv:
   350   assumes x:     "x \<in> carrier G"
   351   shows "a_set_inv (H +> x) = H +> (\<ominus> x)" 
   352 by (rule normal.rcos_inv [OF a_normal,
   353     folded a_r_coset_def a_inv_def A_SET_INV_def, simplified monoid_record_simps])
   354 
   355 lemma (in abelian_group) a_setmult_rcos_assoc:
   356      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
   357       \<Longrightarrow> H <+> (K +> x) = (H <+> K) +> x"
   358 by (rule group.setmult_rcos_assoc [OF a_group,
   359     folded set_add_def a_r_coset_def, simplified monoid_record_simps])
   360 
   361 lemma (in abelian_group) a_rcos_assoc_lcos:
   362      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
   363       \<Longrightarrow> (H +> x) <+> K = H <+> (x <+ K)"
   364 by (rule group.rcos_assoc_lcos [OF a_group,
   365      folded set_add_def a_r_coset_def a_l_coset_def, simplified monoid_record_simps])
   366 
   367 lemma (in abelian_subgroup) a_rcos_sum:
   368      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
   369       \<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)"
   370 by (rule normal.rcos_sum [OF a_normal,
   371     folded set_add_def a_r_coset_def, simplified monoid_record_simps])
   372 
   373 lemma (in abelian_subgroup) rcosets_add_eq:
   374   "M \<in> a_rcosets H \<Longrightarrow> H <+> M = M"
   375   -- {* generalizes @{text subgroup_mult_id} *}
   376 by (rule normal.rcosets_mult_eq [OF a_normal,
   377     folded set_add_def A_RCOSETS_def, simplified monoid_record_simps])
   378 
   379 
   380 subsection {* Congruence Relation *}
   381 
   382 lemma (in abelian_subgroup) a_equiv_rcong:
   383    shows "equiv (carrier G) (racong H)"
   384 by (rule subgroup.equiv_rcong [OF a_subgroup a_group,
   385     folded a_r_congruent_def, simplified monoid_record_simps])
   386 
   387 lemma (in abelian_subgroup) a_l_coset_eq_rcong:
   388   assumes a: "a \<in> carrier G"
   389   shows "a <+ H = racong H `` {a}"
   390 by (rule subgroup.l_coset_eq_rcong [OF a_subgroup a_group,
   391     folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps])
   392 
   393 lemma (in abelian_subgroup) a_rcos_equation:
   394   shows
   395      "\<lbrakk>ha \<oplus> a = h \<oplus> b; a \<in> carrier G;  b \<in> carrier G;  
   396         h \<in> H;  ha \<in> H;  hb \<in> H\<rbrakk>
   397       \<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})"
   398 by (rule group.rcos_equation [OF a_group a_subgroup,
   399     folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps])
   400 
   401 lemma (in abelian_subgroup) a_rcos_disjoint:
   402   shows "\<lbrakk>a \<in> a_rcosets H; b \<in> a_rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
   403 by (rule group.rcos_disjoint [OF a_group a_subgroup,
   404     folded A_RCOSETS_def, simplified monoid_record_simps])
   405 
   406 lemma (in abelian_subgroup) a_rcos_self:
   407   shows "x \<in> carrier G \<Longrightarrow> x \<in> H +> x"
   408 by (rule group.rcos_self [OF a_group a_subgroup,
   409     folded a_r_coset_def, simplified monoid_record_simps])
   410 
   411 lemma (in abelian_subgroup) a_rcosets_part_G:
   412   shows "\<Union>(a_rcosets H) = carrier G"
   413 by (rule group.rcosets_part_G [OF a_group a_subgroup,
   414     folded A_RCOSETS_def, simplified monoid_record_simps])
   415 
   416 lemma (in abelian_subgroup) a_cosets_finite:
   417      "\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
   418 by (rule group.cosets_finite [OF a_group,
   419     folded A_RCOSETS_def, simplified monoid_record_simps])
   420 
   421 lemma (in abelian_group) a_card_cosets_equal:
   422      "\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
   423       \<Longrightarrow> card c = card H"
   424 by (rule group.card_cosets_equal [OF a_group,
   425     folded A_RCOSETS_def, simplified monoid_record_simps])
   426 
   427 lemma (in abelian_group) rcosets_subset_PowG:
   428      "additive_subgroup H G  \<Longrightarrow> a_rcosets H \<subseteq> Pow(carrier G)"
   429 by (rule group.rcosets_subset_PowG [OF a_group,
   430     folded A_RCOSETS_def, simplified monoid_record_simps],
   431     rule additive_subgroup.a_subgroup)
   432 
   433 theorem (in abelian_group) a_lagrange:
   434      "\<lbrakk>finite(carrier G); additive_subgroup H G\<rbrakk>
   435       \<Longrightarrow> card(a_rcosets H) * card(H) = order(G)"
   436 by (rule group.lagrange [OF a_group,
   437     folded A_RCOSETS_def, simplified monoid_record_simps order_def, folded order_def])
   438     (fast intro!: additive_subgroup.a_subgroup)+
   439 
   440 
   441 subsection {* Factorization *}
   442 
   443 lemmas A_FactGroup_defs = A_FactGroup_def FactGroup_def
   444 
   445 lemma A_FactGroup_def':
   446   includes struct G
   447   shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = set_add G, one = H\<rparr>"
   448 unfolding A_FactGroup_defs
   449 by (fold A_RCOSETS_def set_add_def)
   450 
   451 
   452 lemma (in abelian_subgroup) a_setmult_closed:
   453      "\<lbrakk>K1 \<in> a_rcosets H; K2 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> K1 <+> K2 \<in> a_rcosets H"
   454 by (rule normal.setmult_closed [OF a_normal,
   455     folded A_RCOSETS_def set_add_def, simplified monoid_record_simps])
   456 
   457 lemma (in abelian_subgroup) a_setinv_closed:
   458      "K \<in> a_rcosets H \<Longrightarrow> a_set_inv K \<in> a_rcosets H"
   459 by (rule normal.setinv_closed [OF a_normal,
   460     folded A_RCOSETS_def A_SET_INV_def, simplified monoid_record_simps])
   461 
   462 lemma (in abelian_subgroup) a_rcosets_assoc:
   463      "\<lbrakk>M1 \<in> a_rcosets H; M2 \<in> a_rcosets H; M3 \<in> a_rcosets H\<rbrakk>
   464       \<Longrightarrow> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)"
   465 by (rule normal.rcosets_assoc [OF a_normal,
   466     folded A_RCOSETS_def set_add_def, simplified monoid_record_simps])
   467 
   468 lemma (in abelian_subgroup) a_subgroup_in_rcosets:
   469      "H \<in> a_rcosets H"
   470 by (rule subgroup.subgroup_in_rcosets [OF a_subgroup a_group,
   471     folded A_RCOSETS_def, simplified monoid_record_simps])
   472 
   473 lemma (in abelian_subgroup) a_rcosets_inv_mult_group_eq:
   474      "M \<in> a_rcosets H \<Longrightarrow> a_set_inv M <+> M = H"
   475 by (rule normal.rcosets_inv_mult_group_eq [OF a_normal,
   476     folded A_RCOSETS_def A_SET_INV_def set_add_def, simplified monoid_record_simps])
   477 
   478 theorem (in abelian_subgroup) a_factorgroup_is_group:
   479   "group (G A_Mod H)"
   480 by (rule normal.factorgroup_is_group [OF a_normal,
   481     folded A_FactGroup_def, simplified monoid_record_simps])
   482 
   483 text {* Since the Factorization is based on an \emph{abelian} subgroup, is results in 
   484         a commutative group *}
   485 theorem (in abelian_subgroup) a_factorgroup_is_comm_group:
   486   "comm_group (G A_Mod H)"
   487 apply (intro comm_group.intro comm_monoid.intro) prefer 3
   488   apply (rule a_factorgroup_is_group)
   489  apply (rule group.axioms[OF a_factorgroup_is_group])
   490 apply (rule comm_monoid_axioms.intro)
   491 apply (unfold A_FactGroup_def FactGroup_def RCOSETS_def, fold set_add_def a_r_coset_def, clarsimp)
   492 apply (simp add: a_rcos_sum a_comm)
   493 done
   494 
   495 lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"
   496 by (simp add: A_FactGroup_def set_add_def)
   497 
   498 lemma (in abelian_subgroup) a_inv_FactGroup:
   499      "X \<in> carrier (G A_Mod H) \<Longrightarrow> inv\<^bsub>G A_Mod H\<^esub> X = a_set_inv X"
   500 by (rule normal.inv_FactGroup [OF a_normal,
   501     folded A_FactGroup_def A_SET_INV_def, simplified monoid_record_simps])
   502 
   503 text{*The coset map is a homomorphism from @{term G} to the quotient group
   504   @{term "G Mod H"}*}
   505 lemma (in abelian_subgroup) a_r_coset_hom_A_Mod:
   506   "(\<lambda>a. H +> a) \<in> hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> (G A_Mod H)"
   507 by (rule normal.r_coset_hom_Mod [OF a_normal,
   508     folded A_FactGroup_def a_r_coset_def, simplified monoid_record_simps])
   509 
   510 text {* The isomorphism theorems have been omitted from lifting, at
   511   least for now *}
   512 
   513 subsection{*The First Isomorphism Theorem*}
   514 
   515 text{*The quotient by the kernel of a homomorphism is isomorphic to the 
   516   range of that homomorphism.*}
   517 
   518 lemmas a_kernel_defs =
   519   a_kernel_def kernel_def
   520 
   521 lemma a_kernel_def':
   522   "a_kernel R S h \<equiv> {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
   523 by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps])
   524 
   525 
   526 subsection {* Homomorphisms *}
   527 
   528 lemma abelian_group_homI:
   529   includes abelian_group G
   530   includes abelian_group H
   531   assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
   532                                   (| carrier = carrier H, mult = add H, one = zero H |) h"
   533   shows "abelian_group_hom G H h"
   534 by (intro abelian_group_hom.intro abelian_group_hom_axioms.intro)
   535 
   536 lemma (in abelian_group_hom) is_abelian_group_hom:
   537   "abelian_group_hom G H h"
   538 by (unfold_locales)
   539 
   540 lemma (in abelian_group_hom) hom_add [simp]:
   541   "[| x : carrier G; y : carrier G |]
   542         ==> h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> h y"
   543 by (rule group_hom.hom_mult[OF a_group_hom,
   544     simplified ring_record_simps])
   545 
   546 lemma (in abelian_group_hom) hom_closed [simp]:
   547   "x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"
   548 by (rule group_hom.hom_closed[OF a_group_hom,
   549     simplified ring_record_simps])
   550 
   551 lemma (in abelian_group_hom) zero_closed [simp]:
   552   "h \<zero> \<in> carrier H"
   553 by (rule group_hom.one_closed[OF a_group_hom,
   554     simplified ring_record_simps])
   555 
   556 lemma (in abelian_group_hom) hom_zero [simp]:
   557   "h \<zero> = \<zero>\<^bsub>H\<^esub>"
   558 by (rule group_hom.hom_one[OF a_group_hom,
   559     simplified ring_record_simps])
   560 
   561 lemma (in abelian_group_hom) a_inv_closed [simp]:
   562   "x \<in> carrier G ==> h (\<ominus>x) \<in> carrier H"
   563 by (rule group_hom.inv_closed[OF a_group_hom,
   564     folded a_inv_def, simplified ring_record_simps])
   565 
   566 lemma (in abelian_group_hom) hom_a_inv [simp]:
   567   "x \<in> carrier G ==> h (\<ominus>x) = \<ominus>\<^bsub>H\<^esub> (h x)"
   568 by (rule group_hom.hom_inv[OF a_group_hom,
   569     folded a_inv_def, simplified ring_record_simps])
   570 
   571 lemma (in abelian_group_hom) additive_subgroup_a_kernel:
   572   "additive_subgroup (a_kernel G H h) G"
   573 apply (rule additive_subgroup.intro)
   574 apply (rule group_hom.subgroup_kernel[OF a_group_hom,
   575        folded a_kernel_def, simplified ring_record_simps])
   576 done
   577 
   578 text{*The kernel of a homomorphism is an abelian subgroup*}
   579 lemma (in abelian_group_hom) abelian_subgroup_a_kernel:
   580   "abelian_subgroup (a_kernel G H h) G"
   581 apply (rule abelian_subgroupI)
   582 apply (rule group_hom.normal_kernel[OF a_group_hom,
   583        folded a_kernel_def, simplified ring_record_simps])
   584 apply (simp add: G.a_comm)
   585 done
   586 
   587 lemma (in abelian_group_hom) A_FactGroup_nonempty:
   588   assumes X: "X \<in> carrier (G A_Mod a_kernel G H h)"
   589   shows "X \<noteq> {}"
   590 by (rule group_hom.FactGroup_nonempty[OF a_group_hom,
   591     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
   592 
   593 lemma (in abelian_group_hom) FactGroup_contents_mem:
   594   assumes X: "X \<in> carrier (G A_Mod (a_kernel G H h))"
   595   shows "contents (h`X) \<in> carrier H"
   596 by (rule group_hom.FactGroup_contents_mem[OF a_group_hom,
   597     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
   598 
   599 lemma (in abelian_group_hom) A_FactGroup_hom:
   600      "(\<lambda>X. contents (h`X)) \<in> hom (G A_Mod (a_kernel G H h))
   601           \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>"
   602 by (rule group_hom.FactGroup_hom[OF a_group_hom,
   603     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
   604 
   605 lemma (in abelian_group_hom) A_FactGroup_inj_on:
   606      "inj_on (\<lambda>X. contents (h ` X)) (carrier (G A_Mod a_kernel G H h))"
   607 by (rule group_hom.FactGroup_inj_on[OF a_group_hom,
   608     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
   609 
   610 text{*If the homomorphism @{term h} is onto @{term H}, then so is the
   611 homomorphism from the quotient group*}
   612 lemma (in abelian_group_hom) A_FactGroup_onto:
   613   assumes h: "h ` carrier G = carrier H"
   614   shows "(\<lambda>X. contents (h ` X)) ` carrier (G A_Mod a_kernel G H h) = carrier H"
   615 by (rule group_hom.FactGroup_onto[OF a_group_hom,
   616     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
   617 
   618 text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
   619  quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
   620 theorem (in abelian_group_hom) A_FactGroup_iso:
   621   "h ` carrier G = carrier H
   622    \<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G A_Mod (a_kernel G H h)) \<cong>
   623           (| carrier = carrier H, mult = add H, one = zero H |)"
   624 by (rule group_hom.FactGroup_iso[OF a_group_hom,
   625     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
   626 
   627 section {* Lemmas Lifted from CosetExt.thy *}
   628 
   629 text {* Not eveything from \texttt{CosetExt.thy} is lifted here. *}
   630 
   631 subsection {* General Lemmas from \texttt{AlgebraExt.thy} *}
   632 
   633 lemma (in additive_subgroup) a_Hcarr [simp]:
   634   assumes hH: "h \<in> H"
   635   shows "h \<in> carrier G"
   636 by (rule subgroup.mem_carrier [OF a_subgroup,
   637     simplified monoid_record_simps])
   638 
   639 
   640 subsection {* Lemmas for Right Cosets *}
   641 
   642 lemma (in abelian_subgroup) a_elemrcos_carrier:
   643   assumes acarr: "a \<in> carrier G"
   644       and a': "a' \<in> H +> a"
   645   shows "a' \<in> carrier G"
   646 by (rule subgroup.elemrcos_carrier [OF a_subgroup a_group,
   647     folded a_r_coset_def, simplified monoid_record_simps])
   648 
   649 lemma (in abelian_subgroup) a_rcos_const:
   650   assumes hH: "h \<in> H"
   651   shows "H +> h = H"
   652 by (rule subgroup.rcos_const [OF a_subgroup a_group,
   653     folded a_r_coset_def, simplified monoid_record_simps])
   654 
   655 lemma (in abelian_subgroup) a_rcos_module_imp:
   656   assumes xcarr: "x \<in> carrier G"
   657       and x'cos: "x' \<in> H +> x"
   658   shows "(x' \<oplus> \<ominus>x) \<in> H"
   659 by (rule subgroup.rcos_module_imp [OF a_subgroup a_group,
   660     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
   661 
   662 lemma (in abelian_subgroup) a_rcos_module_rev:
   663   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
   664       and xixH: "(x' \<oplus> \<ominus>x) \<in> H"
   665   shows "x' \<in> H +> x"
   666 by (rule subgroup.rcos_module_rev [OF a_subgroup a_group,
   667     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
   668 
   669 lemma (in abelian_subgroup) a_rcos_module:
   670   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
   671   shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)"
   672 by (rule subgroup.rcos_module [OF a_subgroup a_group,
   673     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
   674 
   675 --"variant"
   676 lemma (in abelian_subgroup) a_rcos_module_minus:
   677   includes ring G
   678   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
   679   shows "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
   680 proof -
   681   from carr
   682       have "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)" by (rule a_rcos_module)
   683   from this and carr
   684       show "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
   685       by (simp add: minus_eq)
   686 qed
   687 
   688 lemma (in abelian_subgroup) a_repr_independence':
   689   assumes "y \<in> H +> x"
   690       and "x \<in> carrier G"
   691   shows "H +> x = H +> y"
   692 apply (rule a_repr_independence, assumption+)
   693 apply (rule a_subgroup)
   694 done
   695 
   696 lemma (in abelian_subgroup) a_repr_independenceD:
   697   assumes ycarr: "y \<in> carrier G"
   698       and repr:  "H +> x = H +> y"
   699   shows "y \<in> H +> x"
   700 by (rule group.repr_independenceD [OF a_group a_subgroup,
   701     folded a_r_coset_def, simplified monoid_record_simps])
   702 
   703 
   704 subsection {* Lemmas for the Set of Right Cosets *}
   705 
   706 lemma (in abelian_subgroup) a_rcosets_carrier:
   707   "X \<in> a_rcosets H \<Longrightarrow> X \<subseteq> carrier G"
   708 by (rule subgroup.rcosets_carrier [OF a_subgroup a_group,
   709     folded A_RCOSETS_def, simplified monoid_record_simps])
   710 
   711 
   712 
   713 subsection {* Addition of Subgroups *}
   714 
   715 lemma (in abelian_monoid) set_add_closed:
   716   assumes Acarr: "A \<subseteq> carrier G"
   717       and Bcarr: "B \<subseteq> carrier G"
   718   shows "A <+> B \<subseteq> carrier G"
   719 by (rule monoid.set_mult_closed [OF a_monoid,
   720     folded set_add_def, simplified monoid_record_simps])
   721 
   722 lemma (in abelian_group) add_additive_subgroups:
   723   assumes subH: "additive_subgroup H G"
   724       and subK: "additive_subgroup K G"
   725   shows "additive_subgroup (H <+> K) G"
   726 apply (rule additive_subgroup.intro)
   727 apply (unfold set_add_def)
   728 apply (intro comm_group.mult_subgroups)
   729   apply (rule a_comm_group)
   730  apply (rule additive_subgroup.a_subgroup[OF subH])
   731 apply (rule additive_subgroup.a_subgroup[OF subK])
   732 done
   733 
   734 end