src/HOL/Algebra/AbelCoset.thy
 author wenzelm Sun Mar 21 17:12:31 2010 +0100 (2010-03-21) changeset 35849 b5522b51cb1e parent 35848 5443079512ea child 39910 10097e0a9dbd permissions -rw-r--r--
1 (*  Title:      HOL/Algebra/AbelCoset.thy
2     Author:     Stephan Hohe, TU Muenchen
3 *)
5 theory AbelCoset
6 imports Coset Ring
7 begin
9 subsection {* More Lifting from Groups to Abelian Groups *}
11 subsubsection {* Definitions *}
13 text {* Hiding @{text "<+>"} from @{theory Sum_Type} until I come
14   up with better syntax here *}
16 no_notation Plus (infixr "<+>" 65)
18 definition
19   a_r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "+>\<index>" 60)
20   where "a_r_coset G = r_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
22 definition
23   a_l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<+\<index>" 60)
24   where "a_l_coset G = l_coset \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
26 definition
27   A_RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("a'_rcosets\<index> _"  80)
28   where "A_RCOSETS G H = RCOSETS \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
30 definition
31   set_add  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<+>\<index>" 60)
32   where "set_add G = set_mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
34 definition
35   A_SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("a'_set'_inv\<index> _"  80)
36   where "A_SET_INV G H = SET_INV \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
38 definition
39   a_r_congruent :: "[('a,'b)ring_scheme, 'a set] \<Rightarrow> ('a*'a)set"  ("racong\<index> _")
40   where "a_r_congruent G = r_congruent \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
42 definition
43   A_FactGroup :: "[('a,'b) ring_scheme, 'a set] \<Rightarrow> ('a set) monoid" (infixl "A'_Mod" 65)
44     --{*Actually defined for groups rather than monoids*}
45   where "A_FactGroup G H = FactGroup \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> H"
47 definition
48   a_kernel :: "('a, 'm) ring_scheme \<Rightarrow> ('b, 'n) ring_scheme \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> 'a set"
49     --{*the kernel of a homomorphism (additive)*}
50   where "a_kernel G H h =
51     kernel \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>
52       \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr> h"
54 locale abelian_group_hom = G: abelian_group G + H: abelian_group H
55     for G (structure) and H (structure) +
56   fixes h
57   assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
58                                   (| carrier = carrier H, mult = add H, one = zero H |) h"
60 lemmas a_r_coset_defs =
61   a_r_coset_def r_coset_def
63 lemma a_r_coset_def':
64   fixes G (structure)
65   shows "H +> a \<equiv> \<Union>h\<in>H. {h \<oplus> a}"
66 unfolding a_r_coset_defs
67 by simp
69 lemmas a_l_coset_defs =
70   a_l_coset_def l_coset_def
72 lemma a_l_coset_def':
73   fixes G (structure)
74   shows "a <+ H \<equiv> \<Union>h\<in>H. {a \<oplus> h}"
75 unfolding a_l_coset_defs
76 by simp
78 lemmas A_RCOSETS_defs =
79   A_RCOSETS_def RCOSETS_def
81 lemma A_RCOSETS_def':
82   fixes G (structure)
83   shows "a_rcosets H \<equiv> \<Union>a\<in>carrier G. {H +> a}"
84 unfolding A_RCOSETS_defs
85 by (fold a_r_coset_def, simp)
87 lemmas set_add_defs =
91   fixes G (structure)
92   shows "H <+> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<oplus> k}"
94 by simp
96 lemmas A_SET_INV_defs =
97   A_SET_INV_def SET_INV_def
99 lemma A_SET_INV_def':
100   fixes G (structure)
101   shows "a_set_inv H \<equiv> \<Union>h\<in>H. {\<ominus> h}"
102 unfolding A_SET_INV_defs
103 by (fold a_inv_def)
106 subsubsection {* Cosets *}
108 lemma (in abelian_group) a_coset_add_assoc:
109      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
110       ==> (M +> g) +> h = M +> (g \<oplus> h)"
111 by (rule group.coset_mult_assoc [OF a_group,
112     folded a_r_coset_def, simplified monoid_record_simps])
114 lemma (in abelian_group) a_coset_add_zero [simp]:
115   "M \<subseteq> carrier G ==> M +> \<zero> = M"
116 by (rule group.coset_mult_one [OF a_group,
117     folded a_r_coset_def, simplified monoid_record_simps])
119 lemma (in abelian_group) a_coset_add_inv1:
120      "[| M +> (x \<oplus> (\<ominus> y)) = M;  x \<in> carrier G ; y \<in> carrier G;
121          M \<subseteq> carrier G |] ==> M +> x = M +> y"
122 by (rule group.coset_mult_inv1 [OF a_group,
123     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
125 lemma (in abelian_group) a_coset_add_inv2:
126      "[| M +> x = M +> y;  x \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
127       ==> M +> (x \<oplus> (\<ominus> y)) = M"
128 by (rule group.coset_mult_inv2 [OF a_group,
129     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
131 lemma (in abelian_group) a_coset_join1:
132      "[| H +> x = H;  x \<in> carrier G;  subgroup H (|carrier = carrier G, mult = add G, one = zero G|) |] ==> x \<in> H"
133 by (rule group.coset_join1 [OF a_group,
134     folded a_r_coset_def, simplified monoid_record_simps])
136 lemma (in abelian_group) a_solve_equation:
137     "\<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"
138 by (rule group.solve_equation [OF a_group,
139     folded a_r_coset_def, simplified monoid_record_simps])
141 lemma (in abelian_group) a_repr_independence:
142      "\<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"
143 by (rule group.repr_independence [OF a_group,
144     folded a_r_coset_def, simplified monoid_record_simps])
146 lemma (in abelian_group) a_coset_join2:
147      "\<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"
148 by (rule group.coset_join2 [OF a_group,
149     folded a_r_coset_def, simplified monoid_record_simps])
151 lemma (in abelian_monoid) a_r_coset_subset_G:
152      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H +> x \<subseteq> carrier G"
153 by (rule monoid.r_coset_subset_G [OF a_monoid,
154     folded a_r_coset_def, simplified monoid_record_simps])
156 lemma (in abelian_group) a_rcosI:
157      "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<oplus> x \<in> H +> x"
158 by (rule group.rcosI [OF a_group,
159     folded a_r_coset_def, simplified monoid_record_simps])
161 lemma (in abelian_group) a_rcosetsI:
162      "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H +> x \<in> a_rcosets H"
163 by (rule group.rcosetsI [OF a_group,
164     folded a_r_coset_def A_RCOSETS_def, simplified monoid_record_simps])
166 text{*Really needed?*}
167 lemma (in abelian_group) a_transpose_inv:
168      "[| x \<oplus> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
169       ==> (\<ominus> x) \<oplus> z = y"
170 by (rule group.transpose_inv [OF a_group,
171     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
173 (*
174 --"duplicate"
175 lemma (in abelian_group) a_rcos_self:
176      "[| x \<in> carrier G; subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> |] ==> x \<in> H +> x"
177 by (rule group.rcos_self [OF a_group,
178     folded a_r_coset_def, simplified monoid_record_simps])
179 *)
182 subsubsection {* Subgroups *}
184 locale additive_subgroup =
185   fixes H and G (structure)
186   assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
189   shows "additive_subgroup H G"
190 by (rule additive_subgroup_axioms)
193   fixes G (structure)
194   assumes a_subgroup: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
195   shows "additive_subgroup H G"
196 by (rule additive_subgroup.intro) (rule a_subgroup)
198 lemma (in additive_subgroup) a_subset:
199      "H \<subseteq> carrier G"
200 by (rule subgroup.subset[OF a_subgroup,
201     simplified monoid_record_simps])
203 lemma (in additive_subgroup) a_closed [intro, simp]:
204      "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<oplus> y \<in> H"
205 by (rule subgroup.m_closed[OF a_subgroup,
206     simplified monoid_record_simps])
208 lemma (in additive_subgroup) zero_closed [simp]:
209      "\<zero> \<in> H"
210 by (rule subgroup.one_closed[OF a_subgroup,
211     simplified monoid_record_simps])
213 lemma (in additive_subgroup) a_inv_closed [intro,simp]:
214      "x \<in> H \<Longrightarrow> \<ominus> x \<in> H"
215 by (rule subgroup.m_inv_closed[OF a_subgroup,
216     folded a_inv_def, simplified monoid_record_simps])
219 subsubsection {* Additive subgroups are normal *}
221 text {* Every subgroup of an @{text "abelian_group"} is normal *}
223 locale abelian_subgroup = additive_subgroup + abelian_group G +
224   assumes a_normal: "normal H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
226 lemma (in abelian_subgroup) is_abelian_subgroup:
227   shows "abelian_subgroup H G"
228 by (rule abelian_subgroup_axioms)
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)
238   show "abelian_subgroup H G"
239   proof qed (simp add: a_comm)
240 qed
242 lemma abelian_subgroupI2:
243   fixes G (structure)
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)
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
267 lemma abelian_subgroupI3:
268   fixes G (structure)
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
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])
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])
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])
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])
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])
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])
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])
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])
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])
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 apply (rule group.l_repr_imp_subset [OF a_group,
332     folded a_l_coset_def, simplified monoid_record_simps])
333 apply (rule y)
334 apply (rule x)
335 apply (rule sb)
336 done
338 lemma (in abelian_group) a_l_repr_independence:
339   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>"
340   shows "x <+ H = y <+ H"
341 apply (rule group.l_repr_independence [OF a_group,
342     folded a_l_coset_def, simplified monoid_record_simps])
343 apply (rule y)
344 apply (rule x)
345 apply (rule sb)
346 done
348 lemma (in abelian_group) setadd_subset_G:
349      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <+> K \<subseteq> carrier G"
350 by (rule group.setmult_subset_G [OF a_group,
351     folded set_add_def, simplified monoid_record_simps])
353 lemma (in abelian_group) subgroup_add_id: "subgroup H \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> \<Longrightarrow> H <+> H = H"
354 by (rule group.subgroup_mult_id [OF a_group,
355     folded set_add_def, simplified monoid_record_simps])
357 lemma (in abelian_subgroup) a_rcos_inv:
358   assumes x:     "x \<in> carrier G"
359   shows "a_set_inv (H +> x) = H +> (\<ominus> x)"
360 by (rule normal.rcos_inv [OF a_normal,
361   folded a_r_coset_def a_inv_def A_SET_INV_def, simplified monoid_record_simps]) (rule x)
363 lemma (in abelian_group) a_setmult_rcos_assoc:
364      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
365       \<Longrightarrow> H <+> (K +> x) = (H <+> K) +> x"
366 by (rule group.setmult_rcos_assoc [OF a_group,
367     folded set_add_def a_r_coset_def, simplified monoid_record_simps])
369 lemma (in abelian_group) a_rcos_assoc_lcos:
370      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
371       \<Longrightarrow> (H +> x) <+> K = H <+> (x <+ K)"
372 by (rule group.rcos_assoc_lcos [OF a_group,
373      folded set_add_def a_r_coset_def a_l_coset_def, simplified monoid_record_simps])
375 lemma (in abelian_subgroup) a_rcos_sum:
376      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
377       \<Longrightarrow> (H +> x) <+> (H +> y) = H +> (x \<oplus> y)"
378 by (rule normal.rcos_sum [OF a_normal,
379     folded set_add_def a_r_coset_def, simplified monoid_record_simps])
381 lemma (in abelian_subgroup) rcosets_add_eq:
382   "M \<in> a_rcosets H \<Longrightarrow> H <+> M = M"
383   -- {* generalizes @{text subgroup_mult_id} *}
384 by (rule normal.rcosets_mult_eq [OF a_normal,
385     folded set_add_def A_RCOSETS_def, simplified monoid_record_simps])
388 subsubsection {* Congruence Relation *}
390 lemma (in abelian_subgroup) a_equiv_rcong:
391    shows "equiv (carrier G) (racong H)"
392 by (rule subgroup.equiv_rcong [OF a_subgroup a_group,
393     folded a_r_congruent_def, simplified monoid_record_simps])
395 lemma (in abelian_subgroup) a_l_coset_eq_rcong:
396   assumes a: "a \<in> carrier G"
397   shows "a <+ H = racong H  {a}"
398 by (rule subgroup.l_coset_eq_rcong [OF a_subgroup a_group,
399     folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps]) (rule a)
401 lemma (in abelian_subgroup) a_rcos_equation:
402   shows
403      "\<lbrakk>ha \<oplus> a = h \<oplus> b; a \<in> carrier G;  b \<in> carrier G;
404         h \<in> H;  ha \<in> H;  hb \<in> H\<rbrakk>
405       \<Longrightarrow> hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})"
406 by (rule group.rcos_equation [OF a_group a_subgroup,
407     folded a_r_congruent_def a_l_coset_def, simplified monoid_record_simps])
409 lemma (in abelian_subgroup) a_rcos_disjoint:
410   shows "\<lbrakk>a \<in> a_rcosets H; b \<in> a_rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
411 by (rule group.rcos_disjoint [OF a_group a_subgroup,
412     folded A_RCOSETS_def, simplified monoid_record_simps])
414 lemma (in abelian_subgroup) a_rcos_self:
415   shows "x \<in> carrier G \<Longrightarrow> x \<in> H +> x"
416 by (rule group.rcos_self [OF a_group _ a_subgroup,
417     folded a_r_coset_def, simplified monoid_record_simps])
419 lemma (in abelian_subgroup) a_rcosets_part_G:
420   shows "\<Union>(a_rcosets H) = carrier G"
421 by (rule group.rcosets_part_G [OF a_group a_subgroup,
422     folded A_RCOSETS_def, simplified monoid_record_simps])
424 lemma (in abelian_subgroup) a_cosets_finite:
425      "\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
426 by (rule group.cosets_finite [OF a_group,
427     folded A_RCOSETS_def, simplified monoid_record_simps])
429 lemma (in abelian_group) a_card_cosets_equal:
430      "\<lbrakk>c \<in> a_rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
431       \<Longrightarrow> card c = card H"
432 by (rule group.card_cosets_equal [OF a_group,
433     folded A_RCOSETS_def, simplified monoid_record_simps])
435 lemma (in abelian_group) rcosets_subset_PowG:
436      "additive_subgroup H G  \<Longrightarrow> a_rcosets H \<subseteq> Pow(carrier G)"
437 by (rule group.rcosets_subset_PowG [OF a_group,
438     folded A_RCOSETS_def, simplified monoid_record_simps],
441 theorem (in abelian_group) a_lagrange:
442      "\<lbrakk>finite(carrier G); additive_subgroup H G\<rbrakk>
443       \<Longrightarrow> card(a_rcosets H) * card(H) = order(G)"
444 by (rule group.lagrange [OF a_group,
445     folded A_RCOSETS_def, simplified monoid_record_simps order_def, folded order_def])
446     (fast intro!: additive_subgroup.a_subgroup)+
449 subsubsection {* Factorization *}
451 lemmas A_FactGroup_defs = A_FactGroup_def FactGroup_def
453 lemma A_FactGroup_def':
454   fixes G (structure)
455   shows "G A_Mod H \<equiv> \<lparr>carrier = a_rcosets\<^bsub>G\<^esub> H, mult = set_add G, one = H\<rparr>"
456 unfolding A_FactGroup_defs
457 by (fold A_RCOSETS_def set_add_def)
460 lemma (in abelian_subgroup) a_setmult_closed:
461      "\<lbrakk>K1 \<in> a_rcosets H; K2 \<in> a_rcosets H\<rbrakk> \<Longrightarrow> K1 <+> K2 \<in> a_rcosets H"
462 by (rule normal.setmult_closed [OF a_normal,
463     folded A_RCOSETS_def set_add_def, simplified monoid_record_simps])
465 lemma (in abelian_subgroup) a_setinv_closed:
466      "K \<in> a_rcosets H \<Longrightarrow> a_set_inv K \<in> a_rcosets H"
467 by (rule normal.setinv_closed [OF a_normal,
468     folded A_RCOSETS_def A_SET_INV_def, simplified monoid_record_simps])
470 lemma (in abelian_subgroup) a_rcosets_assoc:
471      "\<lbrakk>M1 \<in> a_rcosets H; M2 \<in> a_rcosets H; M3 \<in> a_rcosets H\<rbrakk>
472       \<Longrightarrow> M1 <+> M2 <+> M3 = M1 <+> (M2 <+> M3)"
473 by (rule normal.rcosets_assoc [OF a_normal,
474     folded A_RCOSETS_def set_add_def, simplified monoid_record_simps])
476 lemma (in abelian_subgroup) a_subgroup_in_rcosets:
477      "H \<in> a_rcosets H"
478 by (rule subgroup.subgroup_in_rcosets [OF a_subgroup a_group,
479     folded A_RCOSETS_def, simplified monoid_record_simps])
481 lemma (in abelian_subgroup) a_rcosets_inv_mult_group_eq:
482      "M \<in> a_rcosets H \<Longrightarrow> a_set_inv M <+> M = H"
483 by (rule normal.rcosets_inv_mult_group_eq [OF a_normal,
484     folded A_RCOSETS_def A_SET_INV_def set_add_def, simplified monoid_record_simps])
486 theorem (in abelian_subgroup) a_factorgroup_is_group:
487   "group (G A_Mod H)"
488 by (rule normal.factorgroup_is_group [OF a_normal,
489     folded A_FactGroup_def, simplified monoid_record_simps])
491 text {* Since the Factorization is based on an \emph{abelian} subgroup, is results in
492         a commutative group *}
493 theorem (in abelian_subgroup) a_factorgroup_is_comm_group:
494   "comm_group (G A_Mod H)"
495 apply (intro comm_group.intro comm_monoid.intro) prefer 3
496   apply (rule a_factorgroup_is_group)
497  apply (rule group.axioms[OF a_factorgroup_is_group])
498 apply (rule comm_monoid_axioms.intro)
499 apply (unfold A_FactGroup_def FactGroup_def RCOSETS_def, fold set_add_def a_r_coset_def, clarsimp)
500 apply (simp add: a_rcos_sum a_comm)
501 done
503 lemma add_A_FactGroup [simp]: "X \<otimes>\<^bsub>(G A_Mod H)\<^esub> X' = X <+>\<^bsub>G\<^esub> X'"
506 lemma (in abelian_subgroup) a_inv_FactGroup:
507      "X \<in> carrier (G A_Mod H) \<Longrightarrow> inv\<^bsub>G A_Mod H\<^esub> X = a_set_inv X"
508 by (rule normal.inv_FactGroup [OF a_normal,
509     folded A_FactGroup_def A_SET_INV_def, simplified monoid_record_simps])
511 text{*The coset map is a homomorphism from @{term G} to the quotient group
512   @{term "G Mod H"}*}
513 lemma (in abelian_subgroup) a_r_coset_hom_A_Mod:
514   "(\<lambda>a. H +> a) \<in> hom \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> (G A_Mod H)"
515 by (rule normal.r_coset_hom_Mod [OF a_normal,
516     folded A_FactGroup_def a_r_coset_def, simplified monoid_record_simps])
518 text {* The isomorphism theorems have been omitted from lifting, at
519   least for now *}
522 subsubsection{*The First Isomorphism Theorem*}
524 text{*The quotient by the kernel of a homomorphism is isomorphic to the
525   range of that homomorphism.*}
527 lemmas a_kernel_defs =
528   a_kernel_def kernel_def
530 lemma a_kernel_def':
531   "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
532 by (rule a_kernel_def[unfolded kernel_def, simplified ring_record_simps])
535 subsubsection {* Homomorphisms *}
537 lemma abelian_group_homI:
538   assumes "abelian_group G"
539   assumes "abelian_group H"
540   assumes a_group_hom: "group_hom (| carrier = carrier G, mult = add G, one = zero G |)
541                                   (| carrier = carrier H, mult = add H, one = zero H |) h"
542   shows "abelian_group_hom G H h"
543 proof -
544   interpret G: abelian_group G by fact
545   interpret H: abelian_group H by fact
546   show ?thesis apply (intro abelian_group_hom.intro abelian_group_hom_axioms.intro)
547     apply fact
548     apply fact
549     apply (rule a_group_hom)
550     done
551 qed
553 lemma (in abelian_group_hom) is_abelian_group_hom:
554   "abelian_group_hom G H h"
555   ..
557 lemma (in abelian_group_hom) hom_add [simp]:
558   "[| x : carrier G; y : carrier G |]
559         ==> h (x \<oplus>\<^bsub>G\<^esub> y) = h x \<oplus>\<^bsub>H\<^esub> h y"
560 by (rule group_hom.hom_mult[OF a_group_hom,
561     simplified ring_record_simps])
563 lemma (in abelian_group_hom) hom_closed [simp]:
564   "x \<in> carrier G \<Longrightarrow> h x \<in> carrier H"
565 by (rule group_hom.hom_closed[OF a_group_hom,
566     simplified ring_record_simps])
568 lemma (in abelian_group_hom) zero_closed [simp]:
569   "h \<zero> \<in> carrier H"
570 by (rule group_hom.one_closed[OF a_group_hom,
571     simplified ring_record_simps])
573 lemma (in abelian_group_hom) hom_zero [simp]:
574   "h \<zero> = \<zero>\<^bsub>H\<^esub>"
575 by (rule group_hom.hom_one[OF a_group_hom,
576     simplified ring_record_simps])
578 lemma (in abelian_group_hom) a_inv_closed [simp]:
579   "x \<in> carrier G ==> h (\<ominus>x) \<in> carrier H"
580 by (rule group_hom.inv_closed[OF a_group_hom,
581     folded a_inv_def, simplified ring_record_simps])
583 lemma (in abelian_group_hom) hom_a_inv [simp]:
584   "x \<in> carrier G ==> h (\<ominus>x) = \<ominus>\<^bsub>H\<^esub> (h x)"
585 by (rule group_hom.hom_inv[OF a_group_hom,
586     folded a_inv_def, simplified ring_record_simps])
588 lemma (in abelian_group_hom) additive_subgroup_a_kernel:
589   "additive_subgroup (a_kernel G H h) G"
590 apply (rule additive_subgroup.intro)
591 apply (rule group_hom.subgroup_kernel[OF a_group_hom,
592        folded a_kernel_def, simplified ring_record_simps])
593 done
595 text{*The kernel of a homomorphism is an abelian subgroup*}
596 lemma (in abelian_group_hom) abelian_subgroup_a_kernel:
597   "abelian_subgroup (a_kernel G H h) G"
598 apply (rule abelian_subgroupI)
599 apply (rule group_hom.normal_kernel[OF a_group_hom,
600        folded a_kernel_def, simplified ring_record_simps])
601 apply (simp add: G.a_comm)
602 done
604 lemma (in abelian_group_hom) A_FactGroup_nonempty:
605   assumes X: "X \<in> carrier (G A_Mod a_kernel G H h)"
606   shows "X \<noteq> {}"
607 by (rule group_hom.FactGroup_nonempty[OF a_group_hom,
608     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X)
610 lemma (in abelian_group_hom) FactGroup_contents_mem:
611   assumes X: "X \<in> carrier (G A_Mod (a_kernel G H h))"
612   shows "contents (hX) \<in> carrier H"
613 by (rule group_hom.FactGroup_contents_mem[OF a_group_hom,
614     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule X)
616 lemma (in abelian_group_hom) A_FactGroup_hom:
617      "(\<lambda>X. contents (hX)) \<in> hom (G A_Mod (a_kernel G H h))
618           \<lparr>carrier = carrier H, mult = add H, one = zero H\<rparr>"
619 by (rule group_hom.FactGroup_hom[OF a_group_hom,
620     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
622 lemma (in abelian_group_hom) A_FactGroup_inj_on:
623      "inj_on (\<lambda>X. contents (h  X)) (carrier (G A_Mod a_kernel G H h))"
624 by (rule group_hom.FactGroup_inj_on[OF a_group_hom,
625     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
627 text{*If the homomorphism @{term h} is onto @{term H}, then so is the
628 homomorphism from the quotient group*}
629 lemma (in abelian_group_hom) A_FactGroup_onto:
630   assumes h: "h  carrier G = carrier H"
631   shows "(\<lambda>X. contents (h  X))  carrier (G A_Mod a_kernel G H h) = carrier H"
632 by (rule group_hom.FactGroup_onto[OF a_group_hom,
633     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps]) (rule h)
635 text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
636  quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
637 theorem (in abelian_group_hom) A_FactGroup_iso:
638   "h  carrier G = carrier H
639    \<Longrightarrow> (\<lambda>X. contents (hX)) \<in> (G A_Mod (a_kernel G H h)) \<cong>
640           (| carrier = carrier H, mult = add H, one = zero H |)"
641 by (rule group_hom.FactGroup_iso[OF a_group_hom,
642     folded a_kernel_def A_FactGroup_def, simplified ring_record_simps])
645 subsubsection {* Cosets *}
647 text {* Not eveything from \texttt{CosetExt.thy} is lifted here. *}
649 lemma (in additive_subgroup) a_Hcarr [simp]:
650   assumes hH: "h \<in> H"
651   shows "h \<in> carrier G"
652 by (rule subgroup.mem_carrier [OF a_subgroup,
653     simplified monoid_record_simps]) (rule hH)
656 lemma (in abelian_subgroup) a_elemrcos_carrier:
657   assumes acarr: "a \<in> carrier G"
658       and a': "a' \<in> H +> a"
659   shows "a' \<in> carrier G"
660 by (rule subgroup.elemrcos_carrier [OF a_subgroup a_group,
661     folded a_r_coset_def, simplified monoid_record_simps]) (rule acarr, rule a')
663 lemma (in abelian_subgroup) a_rcos_const:
664   assumes hH: "h \<in> H"
665   shows "H +> h = H"
666 by (rule subgroup.rcos_const [OF a_subgroup a_group,
667     folded a_r_coset_def, simplified monoid_record_simps]) (rule hH)
669 lemma (in abelian_subgroup) a_rcos_module_imp:
670   assumes xcarr: "x \<in> carrier G"
671       and x'cos: "x' \<in> H +> x"
672   shows "(x' \<oplus> \<ominus>x) \<in> H"
673 by (rule subgroup.rcos_module_imp [OF a_subgroup a_group,
674     folded a_r_coset_def a_inv_def, simplified monoid_record_simps]) (rule xcarr, rule x'cos)
676 lemma (in abelian_subgroup) a_rcos_module_rev:
677   assumes "x \<in> carrier G" "x' \<in> carrier G"
678       and "(x' \<oplus> \<ominus>x) \<in> H"
679   shows "x' \<in> H +> x"
680 using assms
681 by (rule subgroup.rcos_module_rev [OF a_subgroup a_group,
682     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
684 lemma (in abelian_subgroup) a_rcos_module:
685   assumes "x \<in> carrier G" "x' \<in> carrier G"
686   shows "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)"
687 using assms
688 by (rule subgroup.rcos_module [OF a_subgroup a_group,
689     folded a_r_coset_def a_inv_def, simplified monoid_record_simps])
691 --"variant"
692 lemma (in abelian_subgroup) a_rcos_module_minus:
693   assumes "ring G"
694   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
695   shows "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
696 proof -
697   interpret G: ring G by fact
698   from carr
699   have "(x' \<in> H +> x) = (x' \<oplus> \<ominus>x \<in> H)" by (rule a_rcos_module)
700   with carr
701   show "(x' \<in> H +> x) = (x' \<ominus> x \<in> H)"
702     by (simp add: minus_eq)
703 qed
705 lemma (in abelian_subgroup) a_repr_independence':
706   assumes y: "y \<in> H +> x"
707       and xcarr: "x \<in> carrier G"
708   shows "H +> x = H +> y"
709   apply (rule a_repr_independence)
710     apply (rule y)
711    apply (rule xcarr)
712   apply (rule a_subgroup)
713   done
715 lemma (in abelian_subgroup) a_repr_independenceD:
716   assumes ycarr: "y \<in> carrier G"
717       and repr:  "H +> x = H +> y"
718   shows "y \<in> H +> x"
719 by (rule group.repr_independenceD [OF a_group a_subgroup,
720     folded a_r_coset_def, simplified monoid_record_simps]) (rule ycarr, rule repr)
723 lemma (in abelian_subgroup) a_rcosets_carrier:
724   "X \<in> a_rcosets H \<Longrightarrow> X \<subseteq> carrier G"
725 by (rule subgroup.rcosets_carrier [OF a_subgroup a_group,
726     folded A_RCOSETS_def, simplified monoid_record_simps])
729 subsubsection {* Addition of Subgroups *}
731 lemma (in abelian_monoid) set_add_closed:
732   assumes Acarr: "A \<subseteq> carrier G"
733       and Bcarr: "B \<subseteq> carrier G"
734   shows "A <+> B \<subseteq> carrier G"
735 by (rule monoid.set_mult_closed [OF a_monoid,
736     folded set_add_def, simplified monoid_record_simps]) (rule Acarr, rule Bcarr)