src/HOL/Algebra/Coset.thy
 author wenzelm Sun Mar 21 15:57:40 2010 +0100 (2010-03-21) changeset 35847 19f1f7066917 parent 35416 d8d7d1b785af child 35848 5443079512ea permissions -rw-r--r--
eliminated old constdefs;
```     1 (*  Title:      HOL/Algebra/Coset.thy
```
```     2     Author:     Florian Kammueller, with new proofs by L C Paulson, and
```
```     3                 Stephan Hohe
```
```     4 *)
```
```     5
```
```     6 theory Coset imports Group begin
```
```     7
```
```     8
```
```     9 section {*Cosets and Quotient Groups*}
```
```    10
```
```    11 definition
```
```    12   r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "#>\<index>" 60)
```
```    13   where "H #>\<^bsub>G\<^esub> a \<equiv> \<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a}"
```
```    14
```
```    15 definition
```
```    16   l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<#\<index>" 60)
```
```    17   where "a <#\<^bsub>G\<^esub> H \<equiv> \<Union>h\<in>H. {a \<otimes>\<^bsub>G\<^esub> h}"
```
```    18
```
```    19 definition
```
```    20   RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("rcosets\<index> _"  80)
```
```    21   where "rcosets\<^bsub>G\<^esub> H \<equiv> \<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a}"
```
```    22
```
```    23 definition
```
```    24   set_mult  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<#>\<index>" 60)
```
```    25   where "H <#>\<^bsub>G\<^esub> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes>\<^bsub>G\<^esub> k}"
```
```    26
```
```    27 definition
```
```    28   SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("set'_inv\<index> _"  80)
```
```    29   where "set_inv\<^bsub>G\<^esub> H \<equiv> \<Union>h\<in>H. {inv\<^bsub>G\<^esub> h}"
```
```    30
```
```    31
```
```    32 locale normal = subgroup + group +
```
```    33   assumes coset_eq: "(\<forall>x \<in> carrier G. H #> x = x <# H)"
```
```    34
```
```    35 abbreviation
```
```    36   normal_rel :: "['a set, ('a, 'b) monoid_scheme] \<Rightarrow> bool"  (infixl "\<lhd>" 60) where
```
```    37   "H \<lhd> G \<equiv> normal H G"
```
```    38
```
```    39
```
```    40 subsection {*Basic Properties of Cosets*}
```
```    41
```
```    42 lemma (in group) coset_mult_assoc:
```
```    43      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
```
```    44       ==> (M #> g) #> h = M #> (g \<otimes> h)"
```
```    45 by (force simp add: r_coset_def m_assoc)
```
```    46
```
```    47 lemma (in group) coset_mult_one [simp]: "M \<subseteq> carrier G ==> M #> \<one> = M"
```
```    48 by (force simp add: r_coset_def)
```
```    49
```
```    50 lemma (in group) coset_mult_inv1:
```
```    51      "[| M #> (x \<otimes> (inv y)) = M;  x \<in> carrier G ; y \<in> carrier G;
```
```    52          M \<subseteq> carrier G |] ==> M #> x = M #> y"
```
```    53 apply (erule subst [of concl: "%z. M #> x = z #> y"])
```
```    54 apply (simp add: coset_mult_assoc m_assoc)
```
```    55 done
```
```    56
```
```    57 lemma (in group) coset_mult_inv2:
```
```    58      "[| M #> x = M #> y;  x \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
```
```    59       ==> M #> (x \<otimes> (inv y)) = M "
```
```    60 apply (simp add: coset_mult_assoc [symmetric])
```
```    61 apply (simp add: coset_mult_assoc)
```
```    62 done
```
```    63
```
```    64 lemma (in group) coset_join1:
```
```    65      "[| H #> x = H;  x \<in> carrier G;  subgroup H G |] ==> x \<in> H"
```
```    66 apply (erule subst)
```
```    67 apply (simp add: r_coset_def)
```
```    68 apply (blast intro: l_one subgroup.one_closed sym)
```
```    69 done
```
```    70
```
```    71 lemma (in group) solve_equation:
```
```    72     "\<lbrakk>subgroup H G; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<otimes> x"
```
```    73 apply (rule bexI [of _ "y \<otimes> (inv x)"])
```
```    74 apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc
```
```    75                       subgroup.subset [THEN subsetD])
```
```    76 done
```
```    77
```
```    78 lemma (in group) repr_independence:
```
```    79      "\<lbrakk>y \<in> H #> x;  x \<in> carrier G; subgroup H G\<rbrakk> \<Longrightarrow> H #> x = H #> y"
```
```    80 by (auto simp add: r_coset_def m_assoc [symmetric]
```
```    81                    subgroup.subset [THEN subsetD]
```
```    82                    subgroup.m_closed solve_equation)
```
```    83
```
```    84 lemma (in group) coset_join2:
```
```    85      "\<lbrakk>x \<in> carrier G;  subgroup H G;  x\<in>H\<rbrakk> \<Longrightarrow> H #> x = H"
```
```    86   --{*Alternative proof is to put @{term "x=\<one>"} in @{text repr_independence}.*}
```
```    87 by (force simp add: subgroup.m_closed r_coset_def solve_equation)
```
```    88
```
```    89 lemma (in monoid) r_coset_subset_G:
```
```    90      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<subseteq> carrier G"
```
```    91 by (auto simp add: r_coset_def)
```
```    92
```
```    93 lemma (in group) rcosI:
```
```    94      "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<otimes> x \<in> H #> x"
```
```    95 by (auto simp add: r_coset_def)
```
```    96
```
```    97 lemma (in group) rcosetsI:
```
```    98      "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H #> x \<in> rcosets H"
```
```    99 by (auto simp add: RCOSETS_def)
```
```   100
```
```   101 text{*Really needed?*}
```
```   102 lemma (in group) transpose_inv:
```
```   103      "[| x \<otimes> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
```
```   104       ==> (inv x) \<otimes> z = y"
```
```   105 by (force simp add: m_assoc [symmetric])
```
```   106
```
```   107 lemma (in group) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x"
```
```   108 apply (simp add: r_coset_def)
```
```   109 apply (blast intro: sym l_one subgroup.subset [THEN subsetD]
```
```   110                     subgroup.one_closed)
```
```   111 done
```
```   112
```
```   113 text (in group) {* Opposite of @{thm [source] "repr_independence"} *}
```
```   114 lemma (in group) repr_independenceD:
```
```   115   assumes "subgroup H G"
```
```   116   assumes ycarr: "y \<in> carrier G"
```
```   117       and repr:  "H #> x = H #> y"
```
```   118   shows "y \<in> H #> x"
```
```   119 proof -
```
```   120   interpret subgroup H G by fact
```
```   121   show ?thesis  apply (subst repr)
```
```   122   apply (intro rcos_self)
```
```   123    apply (rule ycarr)
```
```   124    apply (rule is_subgroup)
```
```   125   done
```
```   126 qed
```
```   127
```
```   128 text {* Elements of a right coset are in the carrier *}
```
```   129 lemma (in subgroup) elemrcos_carrier:
```
```   130   assumes "group G"
```
```   131   assumes acarr: "a \<in> carrier G"
```
```   132     and a': "a' \<in> H #> a"
```
```   133   shows "a' \<in> carrier G"
```
```   134 proof -
```
```   135   interpret group G by fact
```
```   136   from subset and acarr
```
```   137   have "H #> a \<subseteq> carrier G" by (rule r_coset_subset_G)
```
```   138   from this and a'
```
```   139   show "a' \<in> carrier G"
```
```   140     by fast
```
```   141 qed
```
```   142
```
```   143 lemma (in subgroup) rcos_const:
```
```   144   assumes "group G"
```
```   145   assumes hH: "h \<in> H"
```
```   146   shows "H #> h = H"
```
```   147 proof -
```
```   148   interpret group G by fact
```
```   149   show ?thesis apply (unfold r_coset_def)
```
```   150     apply rule
```
```   151     apply rule
```
```   152     apply clarsimp
```
```   153     apply (intro subgroup.m_closed)
```
```   154     apply (rule is_subgroup)
```
```   155     apply assumption
```
```   156     apply (rule hH)
```
```   157     apply rule
```
```   158     apply simp
```
```   159   proof -
```
```   160     fix h'
```
```   161     assume h'H: "h' \<in> H"
```
```   162     note carr = hH[THEN mem_carrier] h'H[THEN mem_carrier]
```
```   163     from carr
```
```   164     have a: "h' = (h' \<otimes> inv h) \<otimes> h" by (simp add: m_assoc)
```
```   165     from h'H hH
```
```   166     have "h' \<otimes> inv h \<in> H" by simp
```
```   167     from this and a
```
```   168     show "\<exists>x\<in>H. h' = x \<otimes> h" by fast
```
```   169   qed
```
```   170 qed
```
```   171
```
```   172 text {* Step one for lemma @{text "rcos_module"} *}
```
```   173 lemma (in subgroup) rcos_module_imp:
```
```   174   assumes "group G"
```
```   175   assumes xcarr: "x \<in> carrier G"
```
```   176       and x'cos: "x' \<in> H #> x"
```
```   177   shows "(x' \<otimes> inv x) \<in> H"
```
```   178 proof -
```
```   179   interpret group G by fact
```
```   180   from xcarr x'cos
```
```   181       have x'carr: "x' \<in> carrier G"
```
```   182       by (rule elemrcos_carrier[OF is_group])
```
```   183   from xcarr
```
```   184       have ixcarr: "inv x \<in> carrier G"
```
```   185       by simp
```
```   186   from x'cos
```
```   187       have "\<exists>h\<in>H. x' = h \<otimes> x"
```
```   188       unfolding r_coset_def
```
```   189       by fast
```
```   190   from this
```
```   191       obtain h
```
```   192         where hH: "h \<in> H"
```
```   193         and x': "x' = h \<otimes> x"
```
```   194       by auto
```
```   195   from hH and subset
```
```   196       have hcarr: "h \<in> carrier G" by fast
```
```   197   note carr = xcarr x'carr hcarr
```
```   198   from x' and carr
```
```   199       have "x' \<otimes> (inv x) = (h \<otimes> x) \<otimes> (inv x)" by fast
```
```   200   also from carr
```
```   201       have "\<dots> = h \<otimes> (x \<otimes> inv x)" by (simp add: m_assoc)
```
```   202   also from carr
```
```   203       have "\<dots> = h \<otimes> \<one>" by simp
```
```   204   also from carr
```
```   205       have "\<dots> = h" by simp
```
```   206   finally
```
```   207       have "x' \<otimes> (inv x) = h" by simp
```
```   208   from hH this
```
```   209       show "x' \<otimes> (inv x) \<in> H" by simp
```
```   210 qed
```
```   211
```
```   212 text {* Step two for lemma @{text "rcos_module"} *}
```
```   213 lemma (in subgroup) rcos_module_rev:
```
```   214   assumes "group G"
```
```   215   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
```
```   216       and xixH: "(x' \<otimes> inv x) \<in> H"
```
```   217   shows "x' \<in> H #> x"
```
```   218 proof -
```
```   219   interpret group G by fact
```
```   220   from xixH
```
```   221       have "\<exists>h\<in>H. x' \<otimes> (inv x) = h" by fast
```
```   222   from this
```
```   223       obtain h
```
```   224         where hH: "h \<in> H"
```
```   225         and hsym: "x' \<otimes> (inv x) = h"
```
```   226       by fast
```
```   227   from hH subset have hcarr: "h \<in> carrier G" by simp
```
```   228   note carr = carr hcarr
```
```   229   from hsym[symmetric] have "h \<otimes> x = x' \<otimes> (inv x) \<otimes> x" by fast
```
```   230   also from carr
```
```   231       have "\<dots> = x' \<otimes> ((inv x) \<otimes> x)" by (simp add: m_assoc)
```
```   232   also from carr
```
```   233       have "\<dots> = x' \<otimes> \<one>" by (simp add: l_inv)
```
```   234   also from carr
```
```   235       have "\<dots> = x'" by simp
```
```   236   finally
```
```   237       have "h \<otimes> x = x'" by simp
```
```   238   from this[symmetric] and hH
```
```   239       show "x' \<in> H #> x"
```
```   240       unfolding r_coset_def
```
```   241       by fast
```
```   242 qed
```
```   243
```
```   244 text {* Module property of right cosets *}
```
```   245 lemma (in subgroup) rcos_module:
```
```   246   assumes "group G"
```
```   247   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
```
```   248   shows "(x' \<in> H #> x) = (x' \<otimes> inv x \<in> H)"
```
```   249 proof -
```
```   250   interpret group G by fact
```
```   251   show ?thesis proof  assume "x' \<in> H #> x"
```
```   252     from this and carr
```
```   253     show "x' \<otimes> inv x \<in> H"
```
```   254       by (intro rcos_module_imp[OF is_group])
```
```   255   next
```
```   256     assume "x' \<otimes> inv x \<in> H"
```
```   257     from this and carr
```
```   258     show "x' \<in> H #> x"
```
```   259       by (intro rcos_module_rev[OF is_group])
```
```   260   qed
```
```   261 qed
```
```   262
```
```   263 text {* Right cosets are subsets of the carrier. *}
```
```   264 lemma (in subgroup) rcosets_carrier:
```
```   265   assumes "group G"
```
```   266   assumes XH: "X \<in> rcosets H"
```
```   267   shows "X \<subseteq> carrier G"
```
```   268 proof -
```
```   269   interpret group G by fact
```
```   270   from XH have "\<exists>x\<in> carrier G. X = H #> x"
```
```   271       unfolding RCOSETS_def
```
```   272       by fast
```
```   273   from this
```
```   274       obtain x
```
```   275         where xcarr: "x\<in> carrier G"
```
```   276         and X: "X = H #> x"
```
```   277       by fast
```
```   278   from subset and xcarr
```
```   279       show "X \<subseteq> carrier G"
```
```   280       unfolding X
```
```   281       by (rule r_coset_subset_G)
```
```   282 qed
```
```   283
```
```   284 text {* Multiplication of general subsets *}
```
```   285 lemma (in monoid) set_mult_closed:
```
```   286   assumes Acarr: "A \<subseteq> carrier G"
```
```   287       and Bcarr: "B \<subseteq> carrier G"
```
```   288   shows "A <#> B \<subseteq> carrier G"
```
```   289 apply rule apply (simp add: set_mult_def, clarsimp)
```
```   290 proof -
```
```   291   fix a b
```
```   292   assume "a \<in> A"
```
```   293   from this and Acarr
```
```   294       have acarr: "a \<in> carrier G" by fast
```
```   295
```
```   296   assume "b \<in> B"
```
```   297   from this and Bcarr
```
```   298       have bcarr: "b \<in> carrier G" by fast
```
```   299
```
```   300   from acarr bcarr
```
```   301       show "a \<otimes> b \<in> carrier G" by (rule m_closed)
```
```   302 qed
```
```   303
```
```   304 lemma (in comm_group) mult_subgroups:
```
```   305   assumes subH: "subgroup H G"
```
```   306       and subK: "subgroup K G"
```
```   307   shows "subgroup (H <#> K) G"
```
```   308 apply (rule subgroup.intro)
```
```   309    apply (intro set_mult_closed subgroup.subset[OF subH] subgroup.subset[OF subK])
```
```   310   apply (simp add: set_mult_def) apply clarsimp defer 1
```
```   311   apply (simp add: set_mult_def) defer 1
```
```   312   apply (simp add: set_mult_def, clarsimp) defer 1
```
```   313 proof -
```
```   314   fix ha hb ka kb
```
```   315   assume haH: "ha \<in> H" and hbH: "hb \<in> H" and kaK: "ka \<in> K" and kbK: "kb \<in> K"
```
```   316   note carr = haH[THEN subgroup.mem_carrier[OF subH]] hbH[THEN subgroup.mem_carrier[OF subH]]
```
```   317               kaK[THEN subgroup.mem_carrier[OF subK]] kbK[THEN subgroup.mem_carrier[OF subK]]
```
```   318   from carr
```
```   319       have "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = ha \<otimes> (ka \<otimes> hb) \<otimes> kb" by (simp add: m_assoc)
```
```   320   also from carr
```
```   321       have "\<dots> = ha \<otimes> (hb \<otimes> ka) \<otimes> kb" by (simp add: m_comm)
```
```   322   also from carr
```
```   323       have "\<dots> = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" by (simp add: m_assoc)
```
```   324   finally
```
```   325       have eq: "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" .
```
```   326
```
```   327   from haH hbH have hH: "ha \<otimes> hb \<in> H" by (simp add: subgroup.m_closed[OF subH])
```
```   328   from kaK kbK have kK: "ka \<otimes> kb \<in> K" by (simp add: subgroup.m_closed[OF subK])
```
```   329
```
```   330   from hH and kK and eq
```
```   331       show "\<exists>h'\<in>H. \<exists>k'\<in>K. (ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = h' \<otimes> k'" by fast
```
```   332 next
```
```   333   have "\<one> = \<one> \<otimes> \<one>" by simp
```
```   334   from subgroup.one_closed[OF subH] subgroup.one_closed[OF subK] this
```
```   335       show "\<exists>h\<in>H. \<exists>k\<in>K. \<one> = h \<otimes> k" by fast
```
```   336 next
```
```   337   fix h k
```
```   338   assume hH: "h \<in> H"
```
```   339      and kK: "k \<in> K"
```
```   340
```
```   341   from hH[THEN subgroup.mem_carrier[OF subH]] kK[THEN subgroup.mem_carrier[OF subK]]
```
```   342       have "inv (h \<otimes> k) = inv h \<otimes> inv k" by (simp add: inv_mult_group m_comm)
```
```   343
```
```   344   from subgroup.m_inv_closed[OF subH hH] and subgroup.m_inv_closed[OF subK kK] and this
```
```   345       show "\<exists>ha\<in>H. \<exists>ka\<in>K. inv (h \<otimes> k) = ha \<otimes> ka" by fast
```
```   346 qed
```
```   347
```
```   348 lemma (in subgroup) lcos_module_rev:
```
```   349   assumes "group G"
```
```   350   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
```
```   351       and xixH: "(inv x \<otimes> x') \<in> H"
```
```   352   shows "x' \<in> x <# H"
```
```   353 proof -
```
```   354   interpret group G by fact
```
```   355   from xixH
```
```   356       have "\<exists>h\<in>H. (inv x) \<otimes> x' = h" by fast
```
```   357   from this
```
```   358       obtain h
```
```   359         where hH: "h \<in> H"
```
```   360         and hsym: "(inv x) \<otimes> x' = h"
```
```   361       by fast
```
```   362
```
```   363   from hH subset have hcarr: "h \<in> carrier G" by simp
```
```   364   note carr = carr hcarr
```
```   365   from hsym[symmetric] have "x \<otimes> h = x \<otimes> ((inv x) \<otimes> x')" by fast
```
```   366   also from carr
```
```   367       have "\<dots> = (x \<otimes> (inv x)) \<otimes> x'" by (simp add: m_assoc[symmetric])
```
```   368   also from carr
```
```   369       have "\<dots> = \<one> \<otimes> x'" by simp
```
```   370   also from carr
```
```   371       have "\<dots> = x'" by simp
```
```   372   finally
```
```   373       have "x \<otimes> h = x'" by simp
```
```   374
```
```   375   from this[symmetric] and hH
```
```   376       show "x' \<in> x <# H"
```
```   377       unfolding l_coset_def
```
```   378       by fast
```
```   379 qed
```
```   380
```
```   381
```
```   382 subsection {* Normal subgroups *}
```
```   383
```
```   384 lemma normal_imp_subgroup: "H \<lhd> G \<Longrightarrow> subgroup H G"
```
```   385   by (simp add: normal_def subgroup_def)
```
```   386
```
```   387 lemma (in group) normalI:
```
```   388   "subgroup H G \<Longrightarrow> (\<forall>x \<in> carrier G. H #> x = x <# H) \<Longrightarrow> H \<lhd> G"
```
```   389   by (simp add: normal_def normal_axioms_def prems)
```
```   390
```
```   391 lemma (in normal) inv_op_closed1:
```
```   392      "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (inv x) \<otimes> h \<otimes> x \<in> H"
```
```   393 apply (insert coset_eq)
```
```   394 apply (auto simp add: l_coset_def r_coset_def)
```
```   395 apply (drule bspec, assumption)
```
```   396 apply (drule equalityD1 [THEN subsetD], blast, clarify)
```
```   397 apply (simp add: m_assoc)
```
```   398 apply (simp add: m_assoc [symmetric])
```
```   399 done
```
```   400
```
```   401 lemma (in normal) inv_op_closed2:
```
```   402      "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> (inv x) \<in> H"
```
```   403 apply (subgoal_tac "inv (inv x) \<otimes> h \<otimes> (inv x) \<in> H")
```
```   404 apply (simp add: )
```
```   405 apply (blast intro: inv_op_closed1)
```
```   406 done
```
```   407
```
```   408 text{*Alternative characterization of normal subgroups*}
```
```   409 lemma (in group) normal_inv_iff:
```
```   410      "(N \<lhd> G) =
```
```   411       (subgroup N G & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<otimes> h \<otimes> (inv x) \<in> N))"
```
```   412       (is "_ = ?rhs")
```
```   413 proof
```
```   414   assume N: "N \<lhd> G"
```
```   415   show ?rhs
```
```   416     by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup)
```
```   417 next
```
```   418   assume ?rhs
```
```   419   hence sg: "subgroup N G"
```
```   420     and closed: "\<And>x. x\<in>carrier G \<Longrightarrow> \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N" by auto
```
```   421   hence sb: "N \<subseteq> carrier G" by (simp add: subgroup.subset)
```
```   422   show "N \<lhd> G"
```
```   423   proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify)
```
```   424     fix x
```
```   425     assume x: "x \<in> carrier G"
```
```   426     show "(\<Union>h\<in>N. {h \<otimes> x}) = (\<Union>h\<in>N. {x \<otimes> h})"
```
```   427     proof
```
```   428       show "(\<Union>h\<in>N. {h \<otimes> x}) \<subseteq> (\<Union>h\<in>N. {x \<otimes> h})"
```
```   429       proof clarify
```
```   430         fix n
```
```   431         assume n: "n \<in> N"
```
```   432         show "n \<otimes> x \<in> (\<Union>h\<in>N. {x \<otimes> h})"
```
```   433         proof
```
```   434           from closed [of "inv x"]
```
```   435           show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n)
```
```   436           show "n \<otimes> x \<in> {x \<otimes> (inv x \<otimes> n \<otimes> x)}"
```
```   437             by (simp add: x n m_assoc [symmetric] sb [THEN subsetD])
```
```   438         qed
```
```   439       qed
```
```   440     next
```
```   441       show "(\<Union>h\<in>N. {x \<otimes> h}) \<subseteq> (\<Union>h\<in>N. {h \<otimes> x})"
```
```   442       proof clarify
```
```   443         fix n
```
```   444         assume n: "n \<in> N"
```
```   445         show "x \<otimes> n \<in> (\<Union>h\<in>N. {h \<otimes> x})"
```
```   446         proof
```
```   447           show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed)
```
```   448           show "x \<otimes> n \<in> {x \<otimes> n \<otimes> inv x \<otimes> x}"
```
```   449             by (simp add: x n m_assoc sb [THEN subsetD])
```
```   450         qed
```
```   451       qed
```
```   452     qed
```
```   453   qed
```
```   454 qed
```
```   455
```
```   456
```
```   457 subsection{*More Properties of Cosets*}
```
```   458
```
```   459 lemma (in group) lcos_m_assoc:
```
```   460      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
```
```   461       ==> g <# (h <# M) = (g \<otimes> h) <# M"
```
```   462 by (force simp add: l_coset_def m_assoc)
```
```   463
```
```   464 lemma (in group) lcos_mult_one: "M \<subseteq> carrier G ==> \<one> <# M = M"
```
```   465 by (force simp add: l_coset_def)
```
```   466
```
```   467 lemma (in group) l_coset_subset_G:
```
```   468      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <# H \<subseteq> carrier G"
```
```   469 by (auto simp add: l_coset_def subsetD)
```
```   470
```
```   471 lemma (in group) l_coset_swap:
```
```   472      "\<lbrakk>y \<in> x <# H;  x \<in> carrier G;  subgroup H G\<rbrakk> \<Longrightarrow> x \<in> y <# H"
```
```   473 proof (simp add: l_coset_def)
```
```   474   assume "\<exists>h\<in>H. y = x \<otimes> h"
```
```   475     and x: "x \<in> carrier G"
```
```   476     and sb: "subgroup H G"
```
```   477   then obtain h' where h': "h' \<in> H & x \<otimes> h' = y" by blast
```
```   478   show "\<exists>h\<in>H. x = y \<otimes> h"
```
```   479   proof
```
```   480     show "x = y \<otimes> inv h'" using h' x sb
```
```   481       by (auto simp add: m_assoc subgroup.subset [THEN subsetD])
```
```   482     show "inv h' \<in> H" using h' sb
```
```   483       by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed)
```
```   484   qed
```
```   485 qed
```
```   486
```
```   487 lemma (in group) l_coset_carrier:
```
```   488      "[| y \<in> x <# H;  x \<in> carrier G;  subgroup H G |] ==> y \<in> carrier G"
```
```   489 by (auto simp add: l_coset_def m_assoc
```
```   490                    subgroup.subset [THEN subsetD] subgroup.m_closed)
```
```   491
```
```   492 lemma (in group) l_repr_imp_subset:
```
```   493   assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
```
```   494   shows "y <# H \<subseteq> x <# H"
```
```   495 proof -
```
```   496   from y
```
```   497   obtain h' where "h' \<in> H" "x \<otimes> h' = y" by (auto simp add: l_coset_def)
```
```   498   thus ?thesis using x sb
```
```   499     by (auto simp add: l_coset_def m_assoc
```
```   500                        subgroup.subset [THEN subsetD] subgroup.m_closed)
```
```   501 qed
```
```   502
```
```   503 lemma (in group) l_repr_independence:
```
```   504   assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
```
```   505   shows "x <# H = y <# H"
```
```   506 proof
```
```   507   show "x <# H \<subseteq> y <# H"
```
```   508     by (rule l_repr_imp_subset,
```
```   509         (blast intro: l_coset_swap l_coset_carrier y x sb)+)
```
```   510   show "y <# H \<subseteq> x <# H" by (rule l_repr_imp_subset [OF y x sb])
```
```   511 qed
```
```   512
```
```   513 lemma (in group) setmult_subset_G:
```
```   514      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <#> K \<subseteq> carrier G"
```
```   515 by (auto simp add: set_mult_def subsetD)
```
```   516
```
```   517 lemma (in group) subgroup_mult_id: "subgroup H G \<Longrightarrow> H <#> H = H"
```
```   518 apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def)
```
```   519 apply (rule_tac x = x in bexI)
```
```   520 apply (rule bexI [of _ "\<one>"])
```
```   521 apply (auto simp add: subgroup.m_closed subgroup.one_closed
```
```   522                       r_one subgroup.subset [THEN subsetD])
```
```   523 done
```
```   524
```
```   525
```
```   526 subsubsection {* Set of Inverses of an @{text r_coset}. *}
```
```   527
```
```   528 lemma (in normal) rcos_inv:
```
```   529   assumes x:     "x \<in> carrier G"
```
```   530   shows "set_inv (H #> x) = H #> (inv x)"
```
```   531 proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe)
```
```   532   fix h
```
```   533   assume "h \<in> H"
```
```   534   show "inv x \<otimes> inv h \<in> (\<Union>j\<in>H. {j \<otimes> inv x})"
```
```   535   proof
```
```   536     show "inv x \<otimes> inv h \<otimes> x \<in> H"
```
```   537       by (simp add: inv_op_closed1 prems)
```
```   538     show "inv x \<otimes> inv h \<in> {inv x \<otimes> inv h \<otimes> x \<otimes> inv x}"
```
```   539       by (simp add: prems m_assoc)
```
```   540   qed
```
```   541 next
```
```   542   fix h
```
```   543   assume "h \<in> H"
```
```   544   show "h \<otimes> inv x \<in> (\<Union>j\<in>H. {inv x \<otimes> inv j})"
```
```   545   proof
```
```   546     show "x \<otimes> inv h \<otimes> inv x \<in> H"
```
```   547       by (simp add: inv_op_closed2 prems)
```
```   548     show "h \<otimes> inv x \<in> {inv x \<otimes> inv (x \<otimes> inv h \<otimes> inv x)}"
```
```   549       by (simp add: prems m_assoc [symmetric] inv_mult_group)
```
```   550   qed
```
```   551 qed
```
```   552
```
```   553
```
```   554 subsubsection {*Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.*}
```
```   555
```
```   556 lemma (in group) setmult_rcos_assoc:
```
```   557      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
```
```   558       \<Longrightarrow> H <#> (K #> x) = (H <#> K) #> x"
```
```   559 by (force simp add: r_coset_def set_mult_def m_assoc)
```
```   560
```
```   561 lemma (in group) rcos_assoc_lcos:
```
```   562      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
```
```   563       \<Longrightarrow> (H #> x) <#> K = H <#> (x <# K)"
```
```   564 by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc)
```
```   565
```
```   566 lemma (in normal) rcos_mult_step1:
```
```   567      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   568       \<Longrightarrow> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y"
```
```   569 by (simp add: setmult_rcos_assoc subset
```
```   570               r_coset_subset_G l_coset_subset_G rcos_assoc_lcos)
```
```   571
```
```   572 lemma (in normal) rcos_mult_step2:
```
```   573      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   574       \<Longrightarrow> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y"
```
```   575 by (insert coset_eq, simp add: normal_def)
```
```   576
```
```   577 lemma (in normal) rcos_mult_step3:
```
```   578      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   579       \<Longrightarrow> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)"
```
```   580 by (simp add: setmult_rcos_assoc coset_mult_assoc
```
```   581               subgroup_mult_id normal.axioms subset prems)
```
```   582
```
```   583 lemma (in normal) rcos_sum:
```
```   584      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   585       \<Longrightarrow> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)"
```
```   586 by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3)
```
```   587
```
```   588 lemma (in normal) rcosets_mult_eq: "M \<in> rcosets H \<Longrightarrow> H <#> M = M"
```
```   589   -- {* generalizes @{text subgroup_mult_id} *}
```
```   590   by (auto simp add: RCOSETS_def subset
```
```   591         setmult_rcos_assoc subgroup_mult_id normal.axioms prems)
```
```   592
```
```   593
```
```   594 subsubsection{*An Equivalence Relation*}
```
```   595
```
```   596 definition
```
```   597   r_congruent :: "[('a,'b)monoid_scheme, 'a set] \<Rightarrow> ('a*'a)set"  ("rcong\<index> _")
```
```   598   where "rcong\<^bsub>G\<^esub> H \<equiv> {(x,y). x \<in> carrier G & y \<in> carrier G & inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> y \<in> H}"
```
```   599
```
```   600
```
```   601 lemma (in subgroup) equiv_rcong:
```
```   602    assumes "group G"
```
```   603    shows "equiv (carrier G) (rcong H)"
```
```   604 proof -
```
```   605   interpret group G by fact
```
```   606   show ?thesis
```
```   607   proof (intro equiv.intro)
```
```   608     show "refl_on (carrier G) (rcong H)"
```
```   609       by (auto simp add: r_congruent_def refl_on_def)
```
```   610   next
```
```   611     show "sym (rcong H)"
```
```   612     proof (simp add: r_congruent_def sym_def, clarify)
```
```   613       fix x y
```
```   614       assume [simp]: "x \<in> carrier G" "y \<in> carrier G"
```
```   615          and "inv x \<otimes> y \<in> H"
```
```   616       hence "inv (inv x \<otimes> y) \<in> H" by (simp add: m_inv_closed)
```
```   617       thus "inv y \<otimes> x \<in> H" by (simp add: inv_mult_group)
```
```   618     qed
```
```   619   next
```
```   620     show "trans (rcong H)"
```
```   621     proof (simp add: r_congruent_def trans_def, clarify)
```
```   622       fix x y z
```
```   623       assume [simp]: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
```
```   624          and "inv x \<otimes> y \<in> H" and "inv y \<otimes> z \<in> H"
```
```   625       hence "(inv x \<otimes> y) \<otimes> (inv y \<otimes> z) \<in> H" by simp
```
```   626       hence "inv x \<otimes> (y \<otimes> inv y) \<otimes> z \<in> H"
```
```   627         by (simp add: m_assoc del: r_inv Units_r_inv)
```
```   628       thus "inv x \<otimes> z \<in> H" by simp
```
```   629     qed
```
```   630   qed
```
```   631 qed
```
```   632
```
```   633 text{*Equivalence classes of @{text rcong} correspond to left cosets.
```
```   634   Was there a mistake in the definitions? I'd have expected them to
```
```   635   correspond to right cosets.*}
```
```   636
```
```   637 (* CB: This is correct, but subtle.
```
```   638    We call H #> a the right coset of a relative to H.  According to
```
```   639    Jacobson, this is what the majority of group theory literature does.
```
```   640    He then defines the notion of congruence relation ~ over monoids as
```
```   641    equivalence relation with a ~ a' & b ~ b' \<Longrightarrow> a*b ~ a'*b'.
```
```   642    Our notion of right congruence induced by K: rcong K appears only in
```
```   643    the context where K is a normal subgroup.  Jacobson doesn't name it.
```
```   644    But in this context left and right cosets are identical.
```
```   645 *)
```
```   646
```
```   647 lemma (in subgroup) l_coset_eq_rcong:
```
```   648   assumes "group G"
```
```   649   assumes a: "a \<in> carrier G"
```
```   650   shows "a <# H = rcong H `` {a}"
```
```   651 proof -
```
```   652   interpret group G by fact
```
```   653   show ?thesis by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a )
```
```   654 qed
```
```   655
```
```   656 subsubsection{*Two Distinct Right Cosets are Disjoint*}
```
```   657
```
```   658 lemma (in group) rcos_equation:
```
```   659   assumes "subgroup H G"
```
```   660   assumes p: "ha \<otimes> a = h \<otimes> b" "a \<in> carrier G" "b \<in> carrier G" "h \<in> H" "ha \<in> H" "hb \<in> H"
```
```   661   shows "hb \<otimes> a \<in> (\<Union>h\<in>H. {h \<otimes> b})"
```
```   662 proof -
```
```   663   interpret subgroup H G by fact
```
```   664   from p show ?thesis apply (rule_tac UN_I [of "hb \<otimes> ((inv ha) \<otimes> h)"])
```
```   665     apply (simp add: )
```
```   666     apply (simp add: m_assoc transpose_inv)
```
```   667     done
```
```   668 qed
```
```   669
```
```   670 lemma (in group) rcos_disjoint:
```
```   671   assumes "subgroup H G"
```
```   672   assumes p: "a \<in> rcosets H" "b \<in> rcosets H" "a\<noteq>b"
```
```   673   shows "a \<inter> b = {}"
```
```   674 proof -
```
```   675   interpret subgroup H G by fact
```
```   676   from p show ?thesis apply (simp add: RCOSETS_def r_coset_def)
```
```   677     apply (blast intro: rcos_equation prems sym)
```
```   678     done
```
```   679 qed
```
```   680
```
```   681 subsection {* Further lemmas for @{text "r_congruent"} *}
```
```   682
```
```   683 text {* The relation is a congruence *}
```
```   684
```
```   685 lemma (in normal) congruent_rcong:
```
```   686   shows "congruent2 (rcong H) (rcong H) (\<lambda>a b. a \<otimes> b <# H)"
```
```   687 proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group)
```
```   688   fix a b c
```
```   689   assume abrcong: "(a, b) \<in> rcong H"
```
```   690     and ccarr: "c \<in> carrier G"
```
```   691
```
```   692   from abrcong
```
```   693       have acarr: "a \<in> carrier G"
```
```   694         and bcarr: "b \<in> carrier G"
```
```   695         and abH: "inv a \<otimes> b \<in> H"
```
```   696       unfolding r_congruent_def
```
```   697       by fast+
```
```   698
```
```   699   note carr = acarr bcarr ccarr
```
```   700
```
```   701   from ccarr and abH
```
```   702       have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c \<in> H" by (rule inv_op_closed1)
```
```   703   moreover
```
```   704       from carr and inv_closed
```
```   705       have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c = (inv c \<otimes> inv a) \<otimes> (b \<otimes> c)"
```
```   706       by (force cong: m_assoc)
```
```   707   moreover
```
```   708       from carr and inv_closed
```
```   709       have "\<dots> = (inv (a \<otimes> c)) \<otimes> (b \<otimes> c)"
```
```   710       by (simp add: inv_mult_group)
```
```   711   ultimately
```
```   712       have "(inv (a \<otimes> c)) \<otimes> (b \<otimes> c) \<in> H" by simp
```
```   713   from carr and this
```
```   714      have "(b \<otimes> c) \<in> (a \<otimes> c) <# H"
```
```   715      by (simp add: lcos_module_rev[OF is_group])
```
```   716   from carr and this and is_subgroup
```
```   717      show "(a \<otimes> c) <# H = (b \<otimes> c) <# H" by (intro l_repr_independence, simp+)
```
```   718 next
```
```   719   fix a b c
```
```   720   assume abrcong: "(a, b) \<in> rcong H"
```
```   721     and ccarr: "c \<in> carrier G"
```
```   722
```
```   723   from ccarr have "c \<in> Units G" by (simp add: Units_eq)
```
```   724   hence cinvc_one: "inv c \<otimes> c = \<one>" by (rule Units_l_inv)
```
```   725
```
```   726   from abrcong
```
```   727       have acarr: "a \<in> carrier G"
```
```   728        and bcarr: "b \<in> carrier G"
```
```   729        and abH: "inv a \<otimes> b \<in> H"
```
```   730       by (unfold r_congruent_def, fast+)
```
```   731
```
```   732   note carr = acarr bcarr ccarr
```
```   733
```
```   734   from carr and inv_closed
```
```   735      have "inv a \<otimes> b = inv a \<otimes> (\<one> \<otimes> b)" by simp
```
```   736   also from carr and inv_closed
```
```   737       have "\<dots> = inv a \<otimes> (inv c \<otimes> c) \<otimes> b" by simp
```
```   738   also from carr and inv_closed
```
```   739       have "\<dots> = (inv a \<otimes> inv c) \<otimes> (c \<otimes> b)" by (force cong: m_assoc)
```
```   740   also from carr and inv_closed
```
```   741       have "\<dots> = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" by (simp add: inv_mult_group)
```
```   742   finally
```
```   743       have "inv a \<otimes> b = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" .
```
```   744   from abH and this
```
```   745       have "inv (c \<otimes> a) \<otimes> (c \<otimes> b) \<in> H" by simp
```
```   746
```
```   747   from carr and this
```
```   748      have "(c \<otimes> b) \<in> (c \<otimes> a) <# H"
```
```   749      by (simp add: lcos_module_rev[OF is_group])
```
```   750   from carr and this and is_subgroup
```
```   751      show "(c \<otimes> a) <# H = (c \<otimes> b) <# H" by (intro l_repr_independence, simp+)
```
```   752 qed
```
```   753
```
```   754
```
```   755 subsection {*Order of a Group and Lagrange's Theorem*}
```
```   756
```
```   757 definition order :: "('a, 'b) monoid_scheme \<Rightarrow> nat" where
```
```   758   "order S \<equiv> card (carrier S)"
```
```   759
```
```   760 lemma (in group) rcosets_part_G:
```
```   761   assumes "subgroup H G"
```
```   762   shows "\<Union>(rcosets H) = carrier G"
```
```   763 proof -
```
```   764   interpret subgroup H G by fact
```
```   765   show ?thesis
```
```   766     apply (rule equalityI)
```
```   767     apply (force simp add: RCOSETS_def r_coset_def)
```
```   768     apply (auto simp add: RCOSETS_def intro: rcos_self prems)
```
```   769     done
```
```   770 qed
```
```   771
```
```   772 lemma (in group) cosets_finite:
```
```   773      "\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
```
```   774 apply (auto simp add: RCOSETS_def)
```
```   775 apply (simp add: r_coset_subset_G [THEN finite_subset])
```
```   776 done
```
```   777
```
```   778 text{*The next two lemmas support the proof of @{text card_cosets_equal}.*}
```
```   779 lemma (in group) inj_on_f:
```
```   780     "\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> inv a) (H #> a)"
```
```   781 apply (rule inj_onI)
```
```   782 apply (subgoal_tac "x \<in> carrier G & y \<in> carrier G")
```
```   783  prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD])
```
```   784 apply (simp add: subsetD)
```
```   785 done
```
```   786
```
```   787 lemma (in group) inj_on_g:
```
```   788     "\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> a) H"
```
```   789 by (force simp add: inj_on_def subsetD)
```
```   790
```
```   791 lemma (in group) card_cosets_equal:
```
```   792      "\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
```
```   793       \<Longrightarrow> card c = card H"
```
```   794 apply (auto simp add: RCOSETS_def)
```
```   795 apply (rule card_bij_eq)
```
```   796      apply (rule inj_on_f, assumption+)
```
```   797     apply (force simp add: m_assoc subsetD r_coset_def)
```
```   798    apply (rule inj_on_g, assumption+)
```
```   799   apply (force simp add: m_assoc subsetD r_coset_def)
```
```   800  txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*}
```
```   801  apply (simp add: r_coset_subset_G [THEN finite_subset])
```
```   802 apply (blast intro: finite_subset)
```
```   803 done
```
```   804
```
```   805 lemma (in group) rcosets_subset_PowG:
```
```   806      "subgroup H G  \<Longrightarrow> rcosets H \<subseteq> Pow(carrier G)"
```
```   807 apply (simp add: RCOSETS_def)
```
```   808 apply (blast dest: r_coset_subset_G subgroup.subset)
```
```   809 done
```
```   810
```
```   811
```
```   812 theorem (in group) lagrange:
```
```   813      "\<lbrakk>finite(carrier G); subgroup H G\<rbrakk>
```
```   814       \<Longrightarrow> card(rcosets H) * card(H) = order(G)"
```
```   815 apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric])
```
```   816 apply (subst mult_commute)
```
```   817 apply (rule card_partition)
```
```   818    apply (simp add: rcosets_subset_PowG [THEN finite_subset])
```
```   819   apply (simp add: rcosets_part_G)
```
```   820  apply (simp add: card_cosets_equal subgroup.subset)
```
```   821 apply (simp add: rcos_disjoint)
```
```   822 done
```
```   823
```
```   824
```
```   825 subsection {*Quotient Groups: Factorization of a Group*}
```
```   826
```
```   827 definition FactGroup :: "[('a,'b) monoid_scheme, 'a set] \<Rightarrow> ('a set) monoid" (infixl "Mod" 65) where
```
```   828     --{*Actually defined for groups rather than monoids*}
```
```   829   "FactGroup G H \<equiv>
```
```   830     \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = set_mult G, one = H\<rparr>"
```
```   831
```
```   832 lemma (in normal) setmult_closed:
```
```   833      "\<lbrakk>K1 \<in> rcosets H; K2 \<in> rcosets H\<rbrakk> \<Longrightarrow> K1 <#> K2 \<in> rcosets H"
```
```   834 by (auto simp add: rcos_sum RCOSETS_def)
```
```   835
```
```   836 lemma (in normal) setinv_closed:
```
```   837      "K \<in> rcosets H \<Longrightarrow> set_inv K \<in> rcosets H"
```
```   838 by (auto simp add: rcos_inv RCOSETS_def)
```
```   839
```
```   840 lemma (in normal) rcosets_assoc:
```
```   841      "\<lbrakk>M1 \<in> rcosets H; M2 \<in> rcosets H; M3 \<in> rcosets H\<rbrakk>
```
```   842       \<Longrightarrow> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
```
```   843 by (auto simp add: RCOSETS_def rcos_sum m_assoc)
```
```   844
```
```   845 lemma (in subgroup) subgroup_in_rcosets:
```
```   846   assumes "group G"
```
```   847   shows "H \<in> rcosets H"
```
```   848 proof -
```
```   849   interpret group G by fact
```
```   850   from _ subgroup_axioms have "H #> \<one> = H"
```
```   851     by (rule coset_join2) auto
```
```   852   then show ?thesis
```
```   853     by (auto simp add: RCOSETS_def)
```
```   854 qed
```
```   855
```
```   856 lemma (in normal) rcosets_inv_mult_group_eq:
```
```   857      "M \<in> rcosets H \<Longrightarrow> set_inv M <#> M = H"
```
```   858 by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms prems)
```
```   859
```
```   860 theorem (in normal) factorgroup_is_group:
```
```   861   "group (G Mod H)"
```
```   862 apply (simp add: FactGroup_def)
```
```   863 apply (rule groupI)
```
```   864     apply (simp add: setmult_closed)
```
```   865    apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group])
```
```   866   apply (simp add: restrictI setmult_closed rcosets_assoc)
```
```   867  apply (simp add: normal_imp_subgroup
```
```   868                   subgroup_in_rcosets rcosets_mult_eq)
```
```   869 apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed)
```
```   870 done
```
```   871
```
```   872 lemma mult_FactGroup [simp]: "X \<otimes>\<^bsub>(G Mod H)\<^esub> X' = X <#>\<^bsub>G\<^esub> X'"
```
```   873   by (simp add: FactGroup_def)
```
```   874
```
```   875 lemma (in normal) inv_FactGroup:
```
```   876      "X \<in> carrier (G Mod H) \<Longrightarrow> inv\<^bsub>G Mod H\<^esub> X = set_inv X"
```
```   877 apply (rule group.inv_equality [OF factorgroup_is_group])
```
```   878 apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq)
```
```   879 done
```
```   880
```
```   881 text{*The coset map is a homomorphism from @{term G} to the quotient group
```
```   882   @{term "G Mod H"}*}
```
```   883 lemma (in normal) r_coset_hom_Mod:
```
```   884   "(\<lambda>a. H #> a) \<in> hom G (G Mod H)"
```
```   885   by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum)
```
```   886
```
```   887
```
```   888 subsection{*The First Isomorphism Theorem*}
```
```   889
```
```   890 text{*The quotient by the kernel of a homomorphism is isomorphic to the
```
```   891   range of that homomorphism.*}
```
```   892
```
```   893 definition kernel :: "('a, 'm) monoid_scheme \<Rightarrow> ('b, 'n) monoid_scheme \<Rightarrow>
```
```   894              ('a \<Rightarrow> 'b) \<Rightarrow> 'a set" where
```
```   895     --{*the kernel of a homomorphism*}
```
```   896   "kernel G H h \<equiv> {x. x \<in> carrier G & h x = \<one>\<^bsub>H\<^esub>}"
```
```   897
```
```   898 lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G"
```
```   899 apply (rule subgroup.intro)
```
```   900 apply (auto simp add: kernel_def group.intro prems)
```
```   901 done
```
```   902
```
```   903 text{*The kernel of a homomorphism is a normal subgroup*}
```
```   904 lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G"
```
```   905 apply (simp add: G.normal_inv_iff subgroup_kernel)
```
```   906 apply (simp add: kernel_def)
```
```   907 done
```
```   908
```
```   909 lemma (in group_hom) FactGroup_nonempty:
```
```   910   assumes X: "X \<in> carrier (G Mod kernel G H h)"
```
```   911   shows "X \<noteq> {}"
```
```   912 proof -
```
```   913   from X
```
```   914   obtain g where "g \<in> carrier G"
```
```   915              and "X = kernel G H h #> g"
```
```   916     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   917   thus ?thesis
```
```   918    by (auto simp add: kernel_def r_coset_def image_def intro: hom_one)
```
```   919 qed
```
```   920
```
```   921
```
```   922 lemma (in group_hom) FactGroup_contents_mem:
```
```   923   assumes X: "X \<in> carrier (G Mod (kernel G H h))"
```
```   924   shows "contents (h`X) \<in> carrier H"
```
```   925 proof -
```
```   926   from X
```
```   927   obtain g where g: "g \<in> carrier G"
```
```   928              and "X = kernel G H h #> g"
```
```   929     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   930   hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g)
```
```   931   thus ?thesis by (auto simp add: g)
```
```   932 qed
```
```   933
```
```   934 lemma (in group_hom) FactGroup_hom:
```
```   935      "(\<lambda>X. contents (h`X)) \<in> hom (G Mod (kernel G H h)) H"
```
```   936 apply (simp add: hom_def FactGroup_contents_mem  normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed)
```
```   937 proof (intro ballI)
```
```   938   fix X and X'
```
```   939   assume X:  "X  \<in> carrier (G Mod kernel G H h)"
```
```   940      and X': "X' \<in> carrier (G Mod kernel G H h)"
```
```   941   then
```
```   942   obtain g and g'
```
```   943            where "g \<in> carrier G" and "g' \<in> carrier G"
```
```   944              and "X = kernel G H h #> g" and "X' = kernel G H h #> g'"
```
```   945     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   946   hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'"
```
```   947     and Xsub: "X \<subseteq> carrier G" and X'sub: "X' \<subseteq> carrier G"
```
```   948     by (force simp add: kernel_def r_coset_def image_def)+
```
```   949   hence "h ` (X <#> X') = {h g \<otimes>\<^bsub>H\<^esub> h g'}" using X X'
```
```   950     by (auto dest!: FactGroup_nonempty
```
```   951              simp add: set_mult_def image_eq_UN
```
```   952                        subsetD [OF Xsub] subsetD [OF X'sub])
```
```   953   thus "contents (h ` (X <#> X')) = contents (h ` X) \<otimes>\<^bsub>H\<^esub> contents (h ` X')"
```
```   954     by (simp add: all image_eq_UN FactGroup_nonempty X X')
```
```   955 qed
```
```   956
```
```   957
```
```   958 text{*Lemma for the following injectivity result*}
```
```   959 lemma (in group_hom) FactGroup_subset:
```
```   960      "\<lbrakk>g \<in> carrier G; g' \<in> carrier G; h g = h g'\<rbrakk>
```
```   961       \<Longrightarrow>  kernel G H h #> g \<subseteq> kernel G H h #> g'"
```
```   962 apply (clarsimp simp add: kernel_def r_coset_def image_def)
```
```   963 apply (rename_tac y)
```
```   964 apply (rule_tac x="y \<otimes> g \<otimes> inv g'" in exI)
```
```   965 apply (simp add: G.m_assoc)
```
```   966 done
```
```   967
```
```   968 lemma (in group_hom) FactGroup_inj_on:
```
```   969      "inj_on (\<lambda>X. contents (h ` X)) (carrier (G Mod kernel G H h))"
```
```   970 proof (simp add: inj_on_def, clarify)
```
```   971   fix X and X'
```
```   972   assume X:  "X  \<in> carrier (G Mod kernel G H h)"
```
```   973      and X': "X' \<in> carrier (G Mod kernel G H h)"
```
```   974   then
```
```   975   obtain g and g'
```
```   976            where gX: "g \<in> carrier G"  "g' \<in> carrier G"
```
```   977               "X = kernel G H h #> g" "X' = kernel G H h #> g'"
```
```   978     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   979   hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'"
```
```   980     by (force simp add: kernel_def r_coset_def image_def)+
```
```   981   assume "contents (h ` X) = contents (h ` X')"
```
```   982   hence h: "h g = h g'"
```
```   983     by (simp add: image_eq_UN all FactGroup_nonempty X X')
```
```   984   show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX)
```
```   985 qed
```
```   986
```
```   987 text{*If the homomorphism @{term h} is onto @{term H}, then so is the
```
```   988 homomorphism from the quotient group*}
```
```   989 lemma (in group_hom) FactGroup_onto:
```
```   990   assumes h: "h ` carrier G = carrier H"
```
```   991   shows "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
```
```   992 proof
```
```   993   show "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) \<subseteq> carrier H"
```
```   994     by (auto simp add: FactGroup_contents_mem)
```
```   995   show "carrier H \<subseteq> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
```
```   996   proof
```
```   997     fix y
```
```   998     assume y: "y \<in> carrier H"
```
```   999     with h obtain g where g: "g \<in> carrier G" "h g = y"
```
```  1000       by (blast elim: equalityE)
```
```  1001     hence "(\<Union>x\<in>kernel G H h #> g. {h x}) = {y}"
```
```  1002       by (auto simp add: y kernel_def r_coset_def)
```
```  1003     with g show "y \<in> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
```
```  1004       by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN)
```
```  1005   qed
```
```  1006 qed
```
```  1007
```
```  1008
```
```  1009 text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
```
```  1010  quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
```
```  1011 theorem (in group_hom) FactGroup_iso:
```
```  1012   "h ` carrier G = carrier H
```
```  1013    \<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G Mod (kernel G H h)) \<cong> H"
```
```  1014 by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def
```
```  1015               FactGroup_onto)
```
```  1016
```
```  1017
```
```  1018 end
```