src/HOL/Algebra/Coset.thy
 author haftmann Fri Apr 20 11:21:42 2007 +0200 (2007-04-20) changeset 22744 5cbe966d67a2 parent 21404 eb85850d3eb7 child 23350 50c5b0912a0c permissions -rw-r--r--
Isar definitions are now added explicitly to code theorem table
```     1 (*  Title:      HOL/Algebra/Coset.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Florian Kammueller, with new proofs by L C Paulson, and
```
```     4                 Stephan Hohe
```
```     5 *)
```
```     6
```
```     7 theory Coset imports Group Exponent begin
```
```     8
```
```     9
```
```    10 section {*Cosets and Quotient Groups*}
```
```    11
```
```    12 constdefs (structure G)
```
```    13   r_coset    :: "[_, 'a set, 'a] \<Rightarrow> 'a set"    (infixl "#>\<index>" 60)
```
```    14   "H #> a \<equiv> \<Union>h\<in>H. {h \<otimes> a}"
```
```    15
```
```    16   l_coset    :: "[_, 'a, 'a set] \<Rightarrow> 'a set"    (infixl "<#\<index>" 60)
```
```    17   "a <# H \<equiv> \<Union>h\<in>H. {a \<otimes> h}"
```
```    18
```
```    19   RCOSETS  :: "[_, 'a set] \<Rightarrow> ('a set)set"   ("rcosets\<index> _" [81] 80)
```
```    20   "rcosets H \<equiv> \<Union>a\<in>carrier G. {H #> a}"
```
```    21
```
```    22   set_mult  :: "[_, 'a set ,'a set] \<Rightarrow> 'a set" (infixl "<#>\<index>" 60)
```
```    23   "H <#> K \<equiv> \<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes> k}"
```
```    24
```
```    25   SET_INV :: "[_,'a set] \<Rightarrow> 'a set"  ("set'_inv\<index> _" [81] 80)
```
```    26   "set_inv H \<equiv> \<Union>h\<in>H. {inv h}"
```
```    27
```
```    28
```
```    29 locale normal = subgroup + group +
```
```    30   assumes coset_eq: "(\<forall>x \<in> carrier G. H #> x = x <# H)"
```
```    31
```
```    32 abbreviation
```
```    33   normal_rel :: "['a set, ('a, 'b) monoid_scheme] \<Rightarrow> bool"  (infixl "\<lhd>" 60) where
```
```    34   "H \<lhd> G \<equiv> normal H G"
```
```    35
```
```    36
```
```    37 subsection {*Basic Properties of Cosets*}
```
```    38
```
```    39 lemma (in group) coset_mult_assoc:
```
```    40      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
```
```    41       ==> (M #> g) #> h = M #> (g \<otimes> h)"
```
```    42 by (force simp add: r_coset_def m_assoc)
```
```    43
```
```    44 lemma (in group) coset_mult_one [simp]: "M \<subseteq> carrier G ==> M #> \<one> = M"
```
```    45 by (force simp add: r_coset_def)
```
```    46
```
```    47 lemma (in group) coset_mult_inv1:
```
```    48      "[| M #> (x \<otimes> (inv y)) = M;  x \<in> carrier G ; y \<in> carrier G;
```
```    49          M \<subseteq> carrier G |] ==> M #> x = M #> y"
```
```    50 apply (erule subst [of concl: "%z. M #> x = z #> y"])
```
```    51 apply (simp add: coset_mult_assoc m_assoc)
```
```    52 done
```
```    53
```
```    54 lemma (in group) coset_mult_inv2:
```
```    55      "[| M #> x = M #> y;  x \<in> carrier G;  y \<in> carrier G;  M \<subseteq> carrier G |]
```
```    56       ==> M #> (x \<otimes> (inv y)) = M "
```
```    57 apply (simp add: coset_mult_assoc [symmetric])
```
```    58 apply (simp add: coset_mult_assoc)
```
```    59 done
```
```    60
```
```    61 lemma (in group) coset_join1:
```
```    62      "[| H #> x = H;  x \<in> carrier G;  subgroup H G |] ==> x \<in> H"
```
```    63 apply (erule subst)
```
```    64 apply (simp add: r_coset_def)
```
```    65 apply (blast intro: l_one subgroup.one_closed sym)
```
```    66 done
```
```    67
```
```    68 lemma (in group) solve_equation:
```
```    69     "\<lbrakk>subgroup H G; x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> \<exists>h\<in>H. y = h \<otimes> x"
```
```    70 apply (rule bexI [of _ "y \<otimes> (inv x)"])
```
```    71 apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc
```
```    72                       subgroup.subset [THEN subsetD])
```
```    73 done
```
```    74
```
```    75 lemma (in group) repr_independence:
```
```    76      "\<lbrakk>y \<in> H #> x;  x \<in> carrier G; subgroup H G\<rbrakk> \<Longrightarrow> H #> x = H #> y"
```
```    77 by (auto simp add: r_coset_def m_assoc [symmetric]
```
```    78                    subgroup.subset [THEN subsetD]
```
```    79                    subgroup.m_closed solve_equation)
```
```    80
```
```    81 lemma (in group) coset_join2:
```
```    82      "\<lbrakk>x \<in> carrier G;  subgroup H G;  x\<in>H\<rbrakk> \<Longrightarrow> H #> x = H"
```
```    83   --{*Alternative proof is to put @{term "x=\<one>"} in @{text repr_independence}.*}
```
```    84 by (force simp add: subgroup.m_closed r_coset_def solve_equation)
```
```    85
```
```    86 lemma (in monoid) r_coset_subset_G:
```
```    87      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> H #> x \<subseteq> carrier G"
```
```    88 by (auto simp add: r_coset_def)
```
```    89
```
```    90 lemma (in group) rcosI:
```
```    91      "[| h \<in> H; H \<subseteq> carrier G; x \<in> carrier G|] ==> h \<otimes> x \<in> H #> x"
```
```    92 by (auto simp add: r_coset_def)
```
```    93
```
```    94 lemma (in group) rcosetsI:
```
```    95      "\<lbrakk>H \<subseteq> carrier G; x \<in> carrier G\<rbrakk> \<Longrightarrow> H #> x \<in> rcosets H"
```
```    96 by (auto simp add: RCOSETS_def)
```
```    97
```
```    98 text{*Really needed?*}
```
```    99 lemma (in group) transpose_inv:
```
```   100      "[| x \<otimes> y = z;  x \<in> carrier G;  y \<in> carrier G;  z \<in> carrier G |]
```
```   101       ==> (inv x) \<otimes> z = y"
```
```   102 by (force simp add: m_assoc [symmetric])
```
```   103
```
```   104 lemma (in group) rcos_self: "[| x \<in> carrier G; subgroup H G |] ==> x \<in> H #> x"
```
```   105 apply (simp add: r_coset_def)
```
```   106 apply (blast intro: sym l_one subgroup.subset [THEN subsetD]
```
```   107                     subgroup.one_closed)
```
```   108 done
```
```   109
```
```   110 text {* Opposite of @{thm [locale=group,source] "repr_independence"} *}
```
```   111 lemma (in group) repr_independenceD:
```
```   112   includes subgroup H G
```
```   113   assumes ycarr: "y \<in> carrier G"
```
```   114       and repr:  "H #> x = H #> y"
```
```   115   shows "y \<in> H #> x"
```
```   116   by (subst repr, intro rcos_self)
```
```   117
```
```   118 text {* Elements of a right coset are in the carrier *}
```
```   119 lemma (in subgroup) elemrcos_carrier:
```
```   120   includes group
```
```   121   assumes acarr: "a \<in> carrier G"
```
```   122     and a': "a' \<in> H #> a"
```
```   123   shows "a' \<in> carrier G"
```
```   124 proof -
```
```   125   from subset and acarr
```
```   126   have "H #> a \<subseteq> carrier G" by (rule r_coset_subset_G)
```
```   127   from this and a'
```
```   128   show "a' \<in> carrier G"
```
```   129     by fast
```
```   130 qed
```
```   131
```
```   132 lemma (in subgroup) rcos_const:
```
```   133   includes group
```
```   134   assumes hH: "h \<in> H"
```
```   135   shows "H #> h = H"
```
```   136   apply (unfold r_coset_def)
```
```   137   apply rule apply rule
```
```   138   apply clarsimp
```
```   139   apply (intro subgroup.m_closed)
```
```   140   apply assumption+
```
```   141   apply rule
```
```   142   apply simp
```
```   143 proof -
```
```   144   fix h'
```
```   145   assume h'H: "h' \<in> H"
```
```   146   note carr = hH[THEN mem_carrier] h'H[THEN mem_carrier]
```
```   147   from carr
```
```   148   have a: "h' = (h' \<otimes> inv h) \<otimes> h" by (simp add: m_assoc)
```
```   149   from h'H hH
```
```   150   have "h' \<otimes> inv h \<in> H" by simp
```
```   151   from this and a
```
```   152   show "\<exists>x\<in>H. h' = x \<otimes> h" by fast
```
```   153 qed
```
```   154
```
```   155 text {* Step one for lemma @{text "rcos_module"} *}
```
```   156 lemma (in subgroup) rcos_module_imp:
```
```   157   includes group
```
```   158   assumes xcarr: "x \<in> carrier G"
```
```   159       and x'cos: "x' \<in> H #> x"
```
```   160   shows "(x' \<otimes> inv x) \<in> H"
```
```   161 proof -
```
```   162   from xcarr x'cos
```
```   163       have x'carr: "x' \<in> carrier G"
```
```   164       by (rule elemrcos_carrier[OF is_group])
```
```   165   from xcarr
```
```   166       have ixcarr: "inv x \<in> carrier G"
```
```   167       by simp
```
```   168   from x'cos
```
```   169       have "\<exists>h\<in>H. x' = h \<otimes> x"
```
```   170       unfolding r_coset_def
```
```   171       by fast
```
```   172   from this
```
```   173       obtain h
```
```   174         where hH: "h \<in> H"
```
```   175         and x': "x' = h \<otimes> x"
```
```   176       by auto
```
```   177   from hH and subset
```
```   178       have hcarr: "h \<in> carrier G" by fast
```
```   179   note carr = xcarr x'carr hcarr
```
```   180   from x' and carr
```
```   181       have "x' \<otimes> (inv x) = (h \<otimes> x) \<otimes> (inv x)" by fast
```
```   182   also from carr
```
```   183       have "\<dots> = h \<otimes> (x \<otimes> inv x)" by (simp add: m_assoc)
```
```   184   also from carr
```
```   185       have "\<dots> = h \<otimes> \<one>" by simp
```
```   186   also from carr
```
```   187       have "\<dots> = h" by simp
```
```   188   finally
```
```   189       have "x' \<otimes> (inv x) = h" by simp
```
```   190   from hH this
```
```   191       show "x' \<otimes> (inv x) \<in> H" by simp
```
```   192 qed
```
```   193
```
```   194 text {* Step two for lemma @{text "rcos_module"} *}
```
```   195 lemma (in subgroup) rcos_module_rev:
```
```   196   includes group
```
```   197   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
```
```   198       and xixH: "(x' \<otimes> inv x) \<in> H"
```
```   199   shows "x' \<in> H #> x"
```
```   200 proof -
```
```   201   from xixH
```
```   202       have "\<exists>h\<in>H. x' \<otimes> (inv x) = h" by fast
```
```   203   from this
```
```   204       obtain h
```
```   205         where hH: "h \<in> H"
```
```   206         and hsym: "x' \<otimes> (inv x) = h"
```
```   207       by fast
```
```   208   from hH subset have hcarr: "h \<in> carrier G" by simp
```
```   209   note carr = carr hcarr
```
```   210   from hsym[symmetric] have "h \<otimes> x = x' \<otimes> (inv x) \<otimes> x" by fast
```
```   211   also from carr
```
```   212       have "\<dots> = x' \<otimes> ((inv x) \<otimes> x)" by (simp add: m_assoc)
```
```   213   also from carr
```
```   214       have "\<dots> = x' \<otimes> \<one>" by (simp add: l_inv)
```
```   215   also from carr
```
```   216       have "\<dots> = x'" by simp
```
```   217   finally
```
```   218       have "h \<otimes> x = x'" by simp
```
```   219   from this[symmetric] and hH
```
```   220       show "x' \<in> H #> x"
```
```   221       unfolding r_coset_def
```
```   222       by fast
```
```   223 qed
```
```   224
```
```   225 text {* Module property of right cosets *}
```
```   226 lemma (in subgroup) rcos_module:
```
```   227   includes group
```
```   228   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
```
```   229   shows "(x' \<in> H #> x) = (x' \<otimes> inv x \<in> H)"
```
```   230 proof
```
```   231   assume "x' \<in> H #> x"
```
```   232   from this and carr
```
```   233       show "x' \<otimes> inv x \<in> H"
```
```   234       by (intro rcos_module_imp[OF is_group])
```
```   235 next
```
```   236   assume "x' \<otimes> inv x \<in> H"
```
```   237   from this and carr
```
```   238       show "x' \<in> H #> x"
```
```   239       by (intro rcos_module_rev[OF is_group])
```
```   240 qed
```
```   241
```
```   242 text {* Right cosets are subsets of the carrier. *}
```
```   243 lemma (in subgroup) rcosets_carrier:
```
```   244   includes group
```
```   245   assumes XH: "X \<in> rcosets H"
```
```   246   shows "X \<subseteq> carrier G"
```
```   247 proof -
```
```   248   from XH have "\<exists>x\<in> carrier G. X = H #> x"
```
```   249       unfolding RCOSETS_def
```
```   250       by fast
```
```   251   from this
```
```   252       obtain x
```
```   253         where xcarr: "x\<in> carrier G"
```
```   254         and X: "X = H #> x"
```
```   255       by fast
```
```   256   from subset and xcarr
```
```   257       show "X \<subseteq> carrier G"
```
```   258       unfolding X
```
```   259       by (rule r_coset_subset_G)
```
```   260 qed
```
```   261
```
```   262 text {* Multiplication of general subsets *}
```
```   263 lemma (in monoid) set_mult_closed:
```
```   264   assumes Acarr: "A \<subseteq> carrier G"
```
```   265       and Bcarr: "B \<subseteq> carrier G"
```
```   266   shows "A <#> B \<subseteq> carrier G"
```
```   267 apply rule apply (simp add: set_mult_def, clarsimp)
```
```   268 proof -
```
```   269   fix a b
```
```   270   assume "a \<in> A"
```
```   271   from this and Acarr
```
```   272       have acarr: "a \<in> carrier G" by fast
```
```   273
```
```   274   assume "b \<in> B"
```
```   275   from this and Bcarr
```
```   276       have bcarr: "b \<in> carrier G" by fast
```
```   277
```
```   278   from acarr bcarr
```
```   279       show "a \<otimes> b \<in> carrier G" by (rule m_closed)
```
```   280 qed
```
```   281
```
```   282 lemma (in comm_group) mult_subgroups:
```
```   283   assumes subH: "subgroup H G"
```
```   284       and subK: "subgroup K G"
```
```   285   shows "subgroup (H <#> K) G"
```
```   286 apply (rule subgroup.intro)
```
```   287    apply (intro set_mult_closed subgroup.subset[OF subH] subgroup.subset[OF subK])
```
```   288   apply (simp add: set_mult_def) apply clarsimp defer 1
```
```   289   apply (simp add: set_mult_def) defer 1
```
```   290   apply (simp add: set_mult_def, clarsimp) defer 1
```
```   291 proof -
```
```   292   fix ha hb ka kb
```
```   293   assume haH: "ha \<in> H" and hbH: "hb \<in> H" and kaK: "ka \<in> K" and kbK: "kb \<in> K"
```
```   294   note carr = haH[THEN subgroup.mem_carrier[OF subH]] hbH[THEN subgroup.mem_carrier[OF subH]]
```
```   295               kaK[THEN subgroup.mem_carrier[OF subK]] kbK[THEN subgroup.mem_carrier[OF subK]]
```
```   296   from carr
```
```   297       have "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = ha \<otimes> (ka \<otimes> hb) \<otimes> kb" by (simp add: m_assoc)
```
```   298   also from carr
```
```   299       have "\<dots> = ha \<otimes> (hb \<otimes> ka) \<otimes> kb" by (simp add: m_comm)
```
```   300   also from carr
```
```   301       have "\<dots> = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" by (simp add: m_assoc)
```
```   302   finally
```
```   303       have eq: "(ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = (ha \<otimes> hb) \<otimes> (ka \<otimes> kb)" .
```
```   304
```
```   305   from haH hbH have hH: "ha \<otimes> hb \<in> H" by (simp add: subgroup.m_closed[OF subH])
```
```   306   from kaK kbK have kK: "ka \<otimes> kb \<in> K" by (simp add: subgroup.m_closed[OF subK])
```
```   307
```
```   308   from hH and kK and eq
```
```   309       show "\<exists>h'\<in>H. \<exists>k'\<in>K. (ha \<otimes> ka) \<otimes> (hb \<otimes> kb) = h' \<otimes> k'" by fast
```
```   310 next
```
```   311   have "\<one> = \<one> \<otimes> \<one>" by simp
```
```   312   from subgroup.one_closed[OF subH] subgroup.one_closed[OF subK] this
```
```   313       show "\<exists>h\<in>H. \<exists>k\<in>K. \<one> = h \<otimes> k" by fast
```
```   314 next
```
```   315   fix h k
```
```   316   assume hH: "h \<in> H"
```
```   317      and kK: "k \<in> K"
```
```   318
```
```   319   from hH[THEN subgroup.mem_carrier[OF subH]] kK[THEN subgroup.mem_carrier[OF subK]]
```
```   320       have "inv (h \<otimes> k) = inv h \<otimes> inv k" by (simp add: inv_mult_group m_comm)
```
```   321
```
```   322   from subgroup.m_inv_closed[OF subH hH] and subgroup.m_inv_closed[OF subK kK] and this
```
```   323       show "\<exists>ha\<in>H. \<exists>ka\<in>K. inv (h \<otimes> k) = ha \<otimes> ka" by fast
```
```   324 qed
```
```   325
```
```   326 lemma (in subgroup) lcos_module_rev:
```
```   327   includes group
```
```   328   assumes carr: "x \<in> carrier G" "x' \<in> carrier G"
```
```   329       and xixH: "(inv x \<otimes> x') \<in> H"
```
```   330   shows "x' \<in> x <# H"
```
```   331 proof -
```
```   332   from xixH
```
```   333       have "\<exists>h\<in>H. (inv x) \<otimes> x' = h" by fast
```
```   334   from this
```
```   335       obtain h
```
```   336         where hH: "h \<in> H"
```
```   337         and hsym: "(inv x) \<otimes> x' = h"
```
```   338       by fast
```
```   339
```
```   340   from hH subset have hcarr: "h \<in> carrier G" by simp
```
```   341   note carr = carr hcarr
```
```   342   from hsym[symmetric] have "x \<otimes> h = x \<otimes> ((inv x) \<otimes> x')" by fast
```
```   343   also from carr
```
```   344       have "\<dots> = (x \<otimes> (inv x)) \<otimes> x'" by (simp add: m_assoc[symmetric])
```
```   345   also from carr
```
```   346       have "\<dots> = \<one> \<otimes> x'" by simp
```
```   347   also from carr
```
```   348       have "\<dots> = x'" by simp
```
```   349   finally
```
```   350       have "x \<otimes> h = x'" by simp
```
```   351
```
```   352   from this[symmetric] and hH
```
```   353       show "x' \<in> x <# H"
```
```   354       unfolding l_coset_def
```
```   355       by fast
```
```   356 qed
```
```   357
```
```   358
```
```   359 subsection {* Normal subgroups *}
```
```   360
```
```   361 lemma normal_imp_subgroup: "H \<lhd> G \<Longrightarrow> subgroup H G"
```
```   362   by (simp add: normal_def subgroup_def)
```
```   363
```
```   364 lemma (in group) normalI:
```
```   365   "subgroup H G \<Longrightarrow> (\<forall>x \<in> carrier G. H #> x = x <# H) \<Longrightarrow> H \<lhd> G";
```
```   366   by (simp add: normal_def normal_axioms_def prems)
```
```   367
```
```   368 lemma (in normal) inv_op_closed1:
```
```   369      "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> (inv x) \<otimes> h \<otimes> x \<in> H"
```
```   370 apply (insert coset_eq)
```
```   371 apply (auto simp add: l_coset_def r_coset_def)
```
```   372 apply (drule bspec, assumption)
```
```   373 apply (drule equalityD1 [THEN subsetD], blast, clarify)
```
```   374 apply (simp add: m_assoc)
```
```   375 apply (simp add: m_assoc [symmetric])
```
```   376 done
```
```   377
```
```   378 lemma (in normal) inv_op_closed2:
```
```   379      "\<lbrakk>x \<in> carrier G; h \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> h \<otimes> (inv x) \<in> H"
```
```   380 apply (subgoal_tac "inv (inv x) \<otimes> h \<otimes> (inv x) \<in> H")
```
```   381 apply (simp add: );
```
```   382 apply (blast intro: inv_op_closed1)
```
```   383 done
```
```   384
```
```   385 text{*Alternative characterization of normal subgroups*}
```
```   386 lemma (in group) normal_inv_iff:
```
```   387      "(N \<lhd> G) =
```
```   388       (subgroup N G & (\<forall>x \<in> carrier G. \<forall>h \<in> N. x \<otimes> h \<otimes> (inv x) \<in> N))"
```
```   389       (is "_ = ?rhs")
```
```   390 proof
```
```   391   assume N: "N \<lhd> G"
```
```   392   show ?rhs
```
```   393     by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup)
```
```   394 next
```
```   395   assume ?rhs
```
```   396   hence sg: "subgroup N G"
```
```   397     and closed: "\<And>x. x\<in>carrier G \<Longrightarrow> \<forall>h\<in>N. x \<otimes> h \<otimes> inv x \<in> N" by auto
```
```   398   hence sb: "N \<subseteq> carrier G" by (simp add: subgroup.subset)
```
```   399   show "N \<lhd> G"
```
```   400   proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify)
```
```   401     fix x
```
```   402     assume x: "x \<in> carrier G"
```
```   403     show "(\<Union>h\<in>N. {h \<otimes> x}) = (\<Union>h\<in>N. {x \<otimes> h})"
```
```   404     proof
```
```   405       show "(\<Union>h\<in>N. {h \<otimes> x}) \<subseteq> (\<Union>h\<in>N. {x \<otimes> h})"
```
```   406       proof clarify
```
```   407         fix n
```
```   408         assume n: "n \<in> N"
```
```   409         show "n \<otimes> x \<in> (\<Union>h\<in>N. {x \<otimes> h})"
```
```   410         proof
```
```   411           from closed [of "inv x"]
```
```   412           show "inv x \<otimes> n \<otimes> x \<in> N" by (simp add: x n)
```
```   413           show "n \<otimes> x \<in> {x \<otimes> (inv x \<otimes> n \<otimes> x)}"
```
```   414             by (simp add: x n m_assoc [symmetric] sb [THEN subsetD])
```
```   415         qed
```
```   416       qed
```
```   417     next
```
```   418       show "(\<Union>h\<in>N. {x \<otimes> h}) \<subseteq> (\<Union>h\<in>N. {h \<otimes> x})"
```
```   419       proof clarify
```
```   420         fix n
```
```   421         assume n: "n \<in> N"
```
```   422         show "x \<otimes> n \<in> (\<Union>h\<in>N. {h \<otimes> x})"
```
```   423         proof
```
```   424           show "x \<otimes> n \<otimes> inv x \<in> N" by (simp add: x n closed)
```
```   425           show "x \<otimes> n \<in> {x \<otimes> n \<otimes> inv x \<otimes> x}"
```
```   426             by (simp add: x n m_assoc sb [THEN subsetD])
```
```   427         qed
```
```   428       qed
```
```   429     qed
```
```   430   qed
```
```   431 qed
```
```   432
```
```   433
```
```   434 subsection{*More Properties of Cosets*}
```
```   435
```
```   436 lemma (in group) lcos_m_assoc:
```
```   437      "[| M \<subseteq> carrier G; g \<in> carrier G; h \<in> carrier G |]
```
```   438       ==> g <# (h <# M) = (g \<otimes> h) <# M"
```
```   439 by (force simp add: l_coset_def m_assoc)
```
```   440
```
```   441 lemma (in group) lcos_mult_one: "M \<subseteq> carrier G ==> \<one> <# M = M"
```
```   442 by (force simp add: l_coset_def)
```
```   443
```
```   444 lemma (in group) l_coset_subset_G:
```
```   445      "[| H \<subseteq> carrier G; x \<in> carrier G |] ==> x <# H \<subseteq> carrier G"
```
```   446 by (auto simp add: l_coset_def subsetD)
```
```   447
```
```   448 lemma (in group) l_coset_swap:
```
```   449      "\<lbrakk>y \<in> x <# H;  x \<in> carrier G;  subgroup H G\<rbrakk> \<Longrightarrow> x \<in> y <# H"
```
```   450 proof (simp add: l_coset_def)
```
```   451   assume "\<exists>h\<in>H. y = x \<otimes> h"
```
```   452     and x: "x \<in> carrier G"
```
```   453     and sb: "subgroup H G"
```
```   454   then obtain h' where h': "h' \<in> H & x \<otimes> h' = y" by blast
```
```   455   show "\<exists>h\<in>H. x = y \<otimes> h"
```
```   456   proof
```
```   457     show "x = y \<otimes> inv h'" using h' x sb
```
```   458       by (auto simp add: m_assoc subgroup.subset [THEN subsetD])
```
```   459     show "inv h' \<in> H" using h' sb
```
```   460       by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed)
```
```   461   qed
```
```   462 qed
```
```   463
```
```   464 lemma (in group) l_coset_carrier:
```
```   465      "[| y \<in> x <# H;  x \<in> carrier G;  subgroup H G |] ==> y \<in> carrier G"
```
```   466 by (auto simp add: l_coset_def m_assoc
```
```   467                    subgroup.subset [THEN subsetD] subgroup.m_closed)
```
```   468
```
```   469 lemma (in group) l_repr_imp_subset:
```
```   470   assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
```
```   471   shows "y <# H \<subseteq> x <# H"
```
```   472 proof -
```
```   473   from y
```
```   474   obtain h' where "h' \<in> H" "x \<otimes> h' = y" by (auto simp add: l_coset_def)
```
```   475   thus ?thesis using x sb
```
```   476     by (auto simp add: l_coset_def m_assoc
```
```   477                        subgroup.subset [THEN subsetD] subgroup.m_closed)
```
```   478 qed
```
```   479
```
```   480 lemma (in group) l_repr_independence:
```
```   481   assumes y: "y \<in> x <# H" and x: "x \<in> carrier G" and sb: "subgroup H G"
```
```   482   shows "x <# H = y <# H"
```
```   483 proof
```
```   484   show "x <# H \<subseteq> y <# H"
```
```   485     by (rule l_repr_imp_subset,
```
```   486         (blast intro: l_coset_swap l_coset_carrier y x sb)+)
```
```   487   show "y <# H \<subseteq> x <# H" by (rule l_repr_imp_subset [OF y x sb])
```
```   488 qed
```
```   489
```
```   490 lemma (in group) setmult_subset_G:
```
```   491      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G\<rbrakk> \<Longrightarrow> H <#> K \<subseteq> carrier G"
```
```   492 by (auto simp add: set_mult_def subsetD)
```
```   493
```
```   494 lemma (in group) subgroup_mult_id: "subgroup H G \<Longrightarrow> H <#> H = H"
```
```   495 apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def)
```
```   496 apply (rule_tac x = x in bexI)
```
```   497 apply (rule bexI [of _ "\<one>"])
```
```   498 apply (auto simp add: subgroup.m_closed subgroup.one_closed
```
```   499                       r_one subgroup.subset [THEN subsetD])
```
```   500 done
```
```   501
```
```   502
```
```   503 subsubsection {* Set of Inverses of an @{text r_coset}. *}
```
```   504
```
```   505 lemma (in normal) rcos_inv:
```
```   506   assumes x:     "x \<in> carrier G"
```
```   507   shows "set_inv (H #> x) = H #> (inv x)"
```
```   508 proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe)
```
```   509   fix h
```
```   510   assume "h \<in> H"
```
```   511   show "inv x \<otimes> inv h \<in> (\<Union>j\<in>H. {j \<otimes> inv x})"
```
```   512   proof
```
```   513     show "inv x \<otimes> inv h \<otimes> x \<in> H"
```
```   514       by (simp add: inv_op_closed1 prems)
```
```   515     show "inv x \<otimes> inv h \<in> {inv x \<otimes> inv h \<otimes> x \<otimes> inv x}"
```
```   516       by (simp add: prems m_assoc)
```
```   517   qed
```
```   518 next
```
```   519   fix h
```
```   520   assume "h \<in> H"
```
```   521   show "h \<otimes> inv x \<in> (\<Union>j\<in>H. {inv x \<otimes> inv j})"
```
```   522   proof
```
```   523     show "x \<otimes> inv h \<otimes> inv x \<in> H"
```
```   524       by (simp add: inv_op_closed2 prems)
```
```   525     show "h \<otimes> inv x \<in> {inv x \<otimes> inv (x \<otimes> inv h \<otimes> inv x)}"
```
```   526       by (simp add: prems m_assoc [symmetric] inv_mult_group)
```
```   527   qed
```
```   528 qed
```
```   529
```
```   530
```
```   531 subsubsection {*Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.*}
```
```   532
```
```   533 lemma (in group) setmult_rcos_assoc:
```
```   534      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
```
```   535       \<Longrightarrow> H <#> (K #> x) = (H <#> K) #> x"
```
```   536 by (force simp add: r_coset_def set_mult_def m_assoc)
```
```   537
```
```   538 lemma (in group) rcos_assoc_lcos:
```
```   539      "\<lbrakk>H \<subseteq> carrier G; K \<subseteq> carrier G; x \<in> carrier G\<rbrakk>
```
```   540       \<Longrightarrow> (H #> x) <#> K = H <#> (x <# K)"
```
```   541 by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc)
```
```   542
```
```   543 lemma (in normal) rcos_mult_step1:
```
```   544      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   545       \<Longrightarrow> (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y"
```
```   546 by (simp add: setmult_rcos_assoc subset
```
```   547               r_coset_subset_G l_coset_subset_G rcos_assoc_lcos)
```
```   548
```
```   549 lemma (in normal) rcos_mult_step2:
```
```   550      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   551       \<Longrightarrow> (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y"
```
```   552 by (insert coset_eq, simp add: normal_def)
```
```   553
```
```   554 lemma (in normal) rcos_mult_step3:
```
```   555      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   556       \<Longrightarrow> (H <#> (H #> x)) #> y = H #> (x \<otimes> y)"
```
```   557 by (simp add: setmult_rcos_assoc coset_mult_assoc
```
```   558               subgroup_mult_id normal.axioms subset prems)
```
```   559
```
```   560 lemma (in normal) rcos_sum:
```
```   561      "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk>
```
```   562       \<Longrightarrow> (H #> x) <#> (H #> y) = H #> (x \<otimes> y)"
```
```   563 by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3)
```
```   564
```
```   565 lemma (in normal) rcosets_mult_eq: "M \<in> rcosets H \<Longrightarrow> H <#> M = M"
```
```   566   -- {* generalizes @{text subgroup_mult_id} *}
```
```   567   by (auto simp add: RCOSETS_def subset
```
```   568         setmult_rcos_assoc subgroup_mult_id normal.axioms prems)
```
```   569
```
```   570
```
```   571 subsubsection{*An Equivalence Relation*}
```
```   572
```
```   573 constdefs (structure G)
```
```   574   r_congruent :: "[('a,'b)monoid_scheme, 'a set] \<Rightarrow> ('a*'a)set"
```
```   575                   ("rcong\<index> _")
```
```   576    "rcong H \<equiv> {(x,y). x \<in> carrier G & y \<in> carrier G & inv x \<otimes> y \<in> H}"
```
```   577
```
```   578
```
```   579 lemma (in subgroup) equiv_rcong:
```
```   580    includes group G
```
```   581    shows "equiv (carrier G) (rcong H)"
```
```   582 proof (intro equiv.intro)
```
```   583   show "refl (carrier G) (rcong H)"
```
```   584     by (auto simp add: r_congruent_def refl_def)
```
```   585 next
```
```   586   show "sym (rcong H)"
```
```   587   proof (simp add: r_congruent_def sym_def, clarify)
```
```   588     fix x y
```
```   589     assume [simp]: "x \<in> carrier G" "y \<in> carrier G"
```
```   590        and "inv x \<otimes> y \<in> H"
```
```   591     hence "inv (inv x \<otimes> y) \<in> H" by (simp add: m_inv_closed)
```
```   592     thus "inv y \<otimes> x \<in> H" by (simp add: inv_mult_group)
```
```   593   qed
```
```   594 next
```
```   595   show "trans (rcong H)"
```
```   596   proof (simp add: r_congruent_def trans_def, clarify)
```
```   597     fix x y z
```
```   598     assume [simp]: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
```
```   599        and "inv x \<otimes> y \<in> H" and "inv y \<otimes> z \<in> H"
```
```   600     hence "(inv x \<otimes> y) \<otimes> (inv y \<otimes> z) \<in> H" by simp
```
```   601     hence "inv x \<otimes> (y \<otimes> inv y) \<otimes> z \<in> H" by (simp add: m_assoc del: r_inv)
```
```   602     thus "inv x \<otimes> z \<in> H" by simp
```
```   603   qed
```
```   604 qed
```
```   605
```
```   606 text{*Equivalence classes of @{text rcong} correspond to left cosets.
```
```   607   Was there a mistake in the definitions? I'd have expected them to
```
```   608   correspond to right cosets.*}
```
```   609
```
```   610 (* CB: This is correct, but subtle.
```
```   611    We call H #> a the right coset of a relative to H.  According to
```
```   612    Jacobson, this is what the majority of group theory literature does.
```
```   613    He then defines the notion of congruence relation ~ over monoids as
```
```   614    equivalence relation with a ~ a' & b ~ b' \<Longrightarrow> a*b ~ a'*b'.
```
```   615    Our notion of right congruence induced by K: rcong K appears only in
```
```   616    the context where K is a normal subgroup.  Jacobson doesn't name it.
```
```   617    But in this context left and right cosets are identical.
```
```   618 *)
```
```   619
```
```   620 lemma (in subgroup) l_coset_eq_rcong:
```
```   621   includes group G
```
```   622   assumes a: "a \<in> carrier G"
```
```   623   shows "a <# H = rcong H `` {a}"
```
```   624 by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a )
```
```   625
```
```   626
```
```   627 subsubsection{*Two Distinct Right Cosets are Disjoint*}
```
```   628
```
```   629 lemma (in group) rcos_equation:
```
```   630   includes subgroup H G
```
```   631   shows
```
```   632      "\<lbrakk>ha \<otimes> a = h \<otimes> b; a \<in> carrier G;  b \<in> carrier G;
```
```   633         h \<in> H;  ha \<in> H;  hb \<in> H\<rbrakk>
```
```   634       \<Longrightarrow> hb \<otimes> a \<in> (\<Union>h\<in>H. {h \<otimes> b})"
```
```   635 apply (rule UN_I [of "hb \<otimes> ((inv ha) \<otimes> h)"])
```
```   636 apply (simp add: );
```
```   637 apply (simp add: m_assoc transpose_inv)
```
```   638 done
```
```   639
```
```   640 lemma (in group) rcos_disjoint:
```
```   641   includes subgroup H G
```
```   642   shows "\<lbrakk>a \<in> rcosets H; b \<in> rcosets H; a\<noteq>b\<rbrakk> \<Longrightarrow> a \<inter> b = {}"
```
```   643 apply (simp add: RCOSETS_def r_coset_def)
```
```   644 apply (blast intro: rcos_equation prems sym)
```
```   645 done
```
```   646
```
```   647 subsection {* Further lemmas for @{text "r_congruent"} *}
```
```   648
```
```   649 text {* The relation is a congruence *}
```
```   650
```
```   651 lemma (in normal) congruent_rcong:
```
```   652   shows "congruent2 (rcong H) (rcong H) (\<lambda>a b. a \<otimes> b <# H)"
```
```   653 proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group)
```
```   654   fix a b c
```
```   655   assume abrcong: "(a, b) \<in> rcong H"
```
```   656     and ccarr: "c \<in> carrier G"
```
```   657
```
```   658   from abrcong
```
```   659       have acarr: "a \<in> carrier G"
```
```   660         and bcarr: "b \<in> carrier G"
```
```   661         and abH: "inv a \<otimes> b \<in> H"
```
```   662       unfolding r_congruent_def
```
```   663       by fast+
```
```   664
```
```   665   note carr = acarr bcarr ccarr
```
```   666
```
```   667   from ccarr and abH
```
```   668       have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c \<in> H" by (rule inv_op_closed1)
```
```   669   moreover
```
```   670       from carr and inv_closed
```
```   671       have "inv c \<otimes> (inv a \<otimes> b) \<otimes> c = (inv c \<otimes> inv a) \<otimes> (b \<otimes> c)"
```
```   672       by (force cong: m_assoc)
```
```   673   moreover
```
```   674       from carr and inv_closed
```
```   675       have "\<dots> = (inv (a \<otimes> c)) \<otimes> (b \<otimes> c)"
```
```   676       by (simp add: inv_mult_group)
```
```   677   ultimately
```
```   678       have "(inv (a \<otimes> c)) \<otimes> (b \<otimes> c) \<in> H" by simp
```
```   679   from carr and this
```
```   680      have "(b \<otimes> c) \<in> (a \<otimes> c) <# H"
```
```   681      by (simp add: lcos_module_rev[OF is_group])
```
```   682   from carr and this and is_subgroup
```
```   683      show "(a \<otimes> c) <# H = (b \<otimes> c) <# H" by (intro l_repr_independence, simp+)
```
```   684 next
```
```   685   fix a b c
```
```   686   assume abrcong: "(a, b) \<in> rcong H"
```
```   687     and ccarr: "c \<in> carrier G"
```
```   688
```
```   689   from ccarr have "c \<in> Units G" by (simp add: Units_eq)
```
```   690   hence cinvc_one: "inv c \<otimes> c = \<one>" by (rule Units_l_inv)
```
```   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       by (unfold r_congruent_def, fast+)
```
```   697
```
```   698   note carr = acarr bcarr ccarr
```
```   699
```
```   700   from carr and inv_closed
```
```   701      have "inv a \<otimes> b = inv a \<otimes> (\<one> \<otimes> b)" by simp
```
```   702   also from carr and inv_closed
```
```   703       have "\<dots> = inv a \<otimes> (inv c \<otimes> c) \<otimes> b" by simp
```
```   704   also from carr and inv_closed
```
```   705       have "\<dots> = (inv a \<otimes> inv c) \<otimes> (c \<otimes> b)" by (force cong: m_assoc)
```
```   706   also from carr and inv_closed
```
```   707       have "\<dots> = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" by (simp add: inv_mult_group)
```
```   708   finally
```
```   709       have "inv a \<otimes> b = inv (c \<otimes> a) \<otimes> (c \<otimes> b)" .
```
```   710   from abH and this
```
```   711       have "inv (c \<otimes> a) \<otimes> (c \<otimes> b) \<in> H" by simp
```
```   712
```
```   713   from carr and this
```
```   714      have "(c \<otimes> b) \<in> (c \<otimes> a) <# H"
```
```   715      by (simp add: lcos_module_rev[OF is_group])
```
```   716   from carr and this and is_subgroup
```
```   717      show "(c \<otimes> a) <# H = (c \<otimes> b) <# H" by (intro l_repr_independence, simp+)
```
```   718 qed
```
```   719
```
```   720
```
```   721 subsection {*Order of a Group and Lagrange's Theorem*}
```
```   722
```
```   723 constdefs
```
```   724   order :: "('a, 'b) monoid_scheme \<Rightarrow> nat"
```
```   725   "order S \<equiv> card (carrier S)"
```
```   726
```
```   727 lemma (in group) rcos_self:
```
```   728   includes subgroup
```
```   729   shows "x \<in> carrier G \<Longrightarrow> x \<in> H #> x"
```
```   730 apply (simp add: r_coset_def)
```
```   731 apply (rule_tac x="\<one>" in bexI)
```
```   732 apply (auto simp add: );
```
```   733 done
```
```   734
```
```   735 lemma (in group) rcosets_part_G:
```
```   736   includes subgroup
```
```   737   shows "\<Union>(rcosets H) = carrier G"
```
```   738 apply (rule equalityI)
```
```   739  apply (force simp add: RCOSETS_def r_coset_def)
```
```   740 apply (auto simp add: RCOSETS_def intro: rcos_self prems)
```
```   741 done
```
```   742
```
```   743 lemma (in group) cosets_finite:
```
```   744      "\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G;  finite (carrier G)\<rbrakk> \<Longrightarrow> finite c"
```
```   745 apply (auto simp add: RCOSETS_def)
```
```   746 apply (simp add: r_coset_subset_G [THEN finite_subset])
```
```   747 done
```
```   748
```
```   749 text{*The next two lemmas support the proof of @{text card_cosets_equal}.*}
```
```   750 lemma (in group) inj_on_f:
```
```   751     "\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> inv a) (H #> a)"
```
```   752 apply (rule inj_onI)
```
```   753 apply (subgoal_tac "x \<in> carrier G & y \<in> carrier G")
```
```   754  prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD])
```
```   755 apply (simp add: subsetD)
```
```   756 done
```
```   757
```
```   758 lemma (in group) inj_on_g:
```
```   759     "\<lbrakk>H \<subseteq> carrier G;  a \<in> carrier G\<rbrakk> \<Longrightarrow> inj_on (\<lambda>y. y \<otimes> a) H"
```
```   760 by (force simp add: inj_on_def subsetD)
```
```   761
```
```   762 lemma (in group) card_cosets_equal:
```
```   763      "\<lbrakk>c \<in> rcosets H;  H \<subseteq> carrier G; finite(carrier G)\<rbrakk>
```
```   764       \<Longrightarrow> card c = card H"
```
```   765 apply (auto simp add: RCOSETS_def)
```
```   766 apply (rule card_bij_eq)
```
```   767      apply (rule inj_on_f, assumption+)
```
```   768     apply (force simp add: m_assoc subsetD r_coset_def)
```
```   769    apply (rule inj_on_g, assumption+)
```
```   770   apply (force simp add: m_assoc subsetD r_coset_def)
```
```   771  txt{*The sets @{term "H #> a"} and @{term "H"} are finite.*}
```
```   772  apply (simp add: r_coset_subset_G [THEN finite_subset])
```
```   773 apply (blast intro: finite_subset)
```
```   774 done
```
```   775
```
```   776 lemma (in group) rcosets_subset_PowG:
```
```   777      "subgroup H G  \<Longrightarrow> rcosets H \<subseteq> Pow(carrier G)"
```
```   778 apply (simp add: RCOSETS_def)
```
```   779 apply (blast dest: r_coset_subset_G subgroup.subset)
```
```   780 done
```
```   781
```
```   782
```
```   783 theorem (in group) lagrange:
```
```   784      "\<lbrakk>finite(carrier G); subgroup H G\<rbrakk>
```
```   785       \<Longrightarrow> card(rcosets H) * card(H) = order(G)"
```
```   786 apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric])
```
```   787 apply (subst mult_commute)
```
```   788 apply (rule card_partition)
```
```   789    apply (simp add: rcosets_subset_PowG [THEN finite_subset])
```
```   790   apply (simp add: rcosets_part_G)
```
```   791  apply (simp add: card_cosets_equal subgroup.subset)
```
```   792 apply (simp add: rcos_disjoint)
```
```   793 done
```
```   794
```
```   795
```
```   796 subsection {*Quotient Groups: Factorization of a Group*}
```
```   797
```
```   798 constdefs
```
```   799   FactGroup :: "[('a,'b) monoid_scheme, 'a set] \<Rightarrow> ('a set) monoid"
```
```   800      (infixl "Mod" 65)
```
```   801     --{*Actually defined for groups rather than monoids*}
```
```   802   "FactGroup G H \<equiv>
```
```   803     \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, mult = set_mult G, one = H\<rparr>"
```
```   804
```
```   805 lemma (in normal) setmult_closed:
```
```   806      "\<lbrakk>K1 \<in> rcosets H; K2 \<in> rcosets H\<rbrakk> \<Longrightarrow> K1 <#> K2 \<in> rcosets H"
```
```   807 by (auto simp add: rcos_sum RCOSETS_def)
```
```   808
```
```   809 lemma (in normal) setinv_closed:
```
```   810      "K \<in> rcosets H \<Longrightarrow> set_inv K \<in> rcosets H"
```
```   811 by (auto simp add: rcos_inv RCOSETS_def)
```
```   812
```
```   813 lemma (in normal) rcosets_assoc:
```
```   814      "\<lbrakk>M1 \<in> rcosets H; M2 \<in> rcosets H; M3 \<in> rcosets H\<rbrakk>
```
```   815       \<Longrightarrow> M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
```
```   816 by (auto simp add: RCOSETS_def rcos_sum m_assoc)
```
```   817
```
```   818 lemma (in subgroup) subgroup_in_rcosets:
```
```   819   includes group G
```
```   820   shows "H \<in> rcosets H"
```
```   821 proof -
```
```   822   have "H #> \<one> = H"
```
```   823     by (rule coset_join2, auto)
```
```   824   then show ?thesis
```
```   825     by (auto simp add: RCOSETS_def)
```
```   826 qed
```
```   827
```
```   828 lemma (in normal) rcosets_inv_mult_group_eq:
```
```   829      "M \<in> rcosets H \<Longrightarrow> set_inv M <#> M = H"
```
```   830 by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms prems)
```
```   831
```
```   832 theorem (in normal) factorgroup_is_group:
```
```   833   "group (G Mod H)"
```
```   834 apply (simp add: FactGroup_def)
```
```   835 apply (rule groupI)
```
```   836     apply (simp add: setmult_closed)
```
```   837    apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group])
```
```   838   apply (simp add: restrictI setmult_closed rcosets_assoc)
```
```   839  apply (simp add: normal_imp_subgroup
```
```   840                   subgroup_in_rcosets rcosets_mult_eq)
```
```   841 apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed)
```
```   842 done
```
```   843
```
```   844 lemma mult_FactGroup [simp]: "X \<otimes>\<^bsub>(G Mod H)\<^esub> X' = X <#>\<^bsub>G\<^esub> X'"
```
```   845   by (simp add: FactGroup_def)
```
```   846
```
```   847 lemma (in normal) inv_FactGroup:
```
```   848      "X \<in> carrier (G Mod H) \<Longrightarrow> inv\<^bsub>G Mod H\<^esub> X = set_inv X"
```
```   849 apply (rule group.inv_equality [OF factorgroup_is_group])
```
```   850 apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq)
```
```   851 done
```
```   852
```
```   853 text{*The coset map is a homomorphism from @{term G} to the quotient group
```
```   854   @{term "G Mod H"}*}
```
```   855 lemma (in normal) r_coset_hom_Mod:
```
```   856   "(\<lambda>a. H #> a) \<in> hom G (G Mod H)"
```
```   857   by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum)
```
```   858
```
```   859
```
```   860 subsection{*The First Isomorphism Theorem*}
```
```   861
```
```   862 text{*The quotient by the kernel of a homomorphism is isomorphic to the
```
```   863   range of that homomorphism.*}
```
```   864
```
```   865 constdefs
```
```   866   kernel :: "('a, 'm) monoid_scheme \<Rightarrow> ('b, 'n) monoid_scheme \<Rightarrow>
```
```   867              ('a \<Rightarrow> 'b) \<Rightarrow> 'a set"
```
```   868     --{*the kernel of a homomorphism*}
```
```   869   "kernel G H h \<equiv> {x. x \<in> carrier G & h x = \<one>\<^bsub>H\<^esub>}";
```
```   870
```
```   871 lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G"
```
```   872 apply (rule subgroup.intro)
```
```   873 apply (auto simp add: kernel_def group.intro prems)
```
```   874 done
```
```   875
```
```   876 text{*The kernel of a homomorphism is a normal subgroup*}
```
```   877 lemma (in group_hom) normal_kernel: "(kernel G H h) \<lhd> G"
```
```   878 apply (simp add: G.normal_inv_iff subgroup_kernel)
```
```   879 apply (simp add: kernel_def)
```
```   880 done
```
```   881
```
```   882 lemma (in group_hom) FactGroup_nonempty:
```
```   883   assumes X: "X \<in> carrier (G Mod kernel G H h)"
```
```   884   shows "X \<noteq> {}"
```
```   885 proof -
```
```   886   from X
```
```   887   obtain g where "g \<in> carrier G"
```
```   888              and "X = kernel G H h #> g"
```
```   889     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   890   thus ?thesis
```
```   891    by (auto simp add: kernel_def r_coset_def image_def intro: hom_one)
```
```   892 qed
```
```   893
```
```   894
```
```   895 lemma (in group_hom) FactGroup_contents_mem:
```
```   896   assumes X: "X \<in> carrier (G Mod (kernel G H h))"
```
```   897   shows "contents (h`X) \<in> carrier H"
```
```   898 proof -
```
```   899   from X
```
```   900   obtain g where g: "g \<in> carrier G"
```
```   901              and "X = kernel G H h #> g"
```
```   902     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   903   hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g)
```
```   904   thus ?thesis by (auto simp add: g)
```
```   905 qed
```
```   906
```
```   907 lemma (in group_hom) FactGroup_hom:
```
```   908      "(\<lambda>X. contents (h`X)) \<in> hom (G Mod (kernel G H h)) H"
```
```   909 apply (simp add: hom_def FactGroup_contents_mem  normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed)
```
```   910 proof (simp add: hom_def funcsetI FactGroup_contents_mem, intro ballI)
```
```   911   fix X and X'
```
```   912   assume X:  "X  \<in> carrier (G Mod kernel G H h)"
```
```   913      and X': "X' \<in> carrier (G Mod kernel G H h)"
```
```   914   then
```
```   915   obtain g and g'
```
```   916            where "g \<in> carrier G" and "g' \<in> carrier G"
```
```   917              and "X = kernel G H h #> g" and "X' = kernel G H h #> g'"
```
```   918     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   919   hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'"
```
```   920     and Xsub: "X \<subseteq> carrier G" and X'sub: "X' \<subseteq> carrier G"
```
```   921     by (force simp add: kernel_def r_coset_def image_def)+
```
```   922   hence "h ` (X <#> X') = {h g \<otimes>\<^bsub>H\<^esub> h g'}" using X X'
```
```   923     by (auto dest!: FactGroup_nonempty
```
```   924              simp add: set_mult_def image_eq_UN
```
```   925                        subsetD [OF Xsub] subsetD [OF X'sub])
```
```   926   thus "contents (h ` (X <#> X')) = contents (h ` X) \<otimes>\<^bsub>H\<^esub> contents (h ` X')"
```
```   927     by (simp add: all image_eq_UN FactGroup_nonempty X X')
```
```   928 qed
```
```   929
```
```   930
```
```   931 text{*Lemma for the following injectivity result*}
```
```   932 lemma (in group_hom) FactGroup_subset:
```
```   933      "\<lbrakk>g \<in> carrier G; g' \<in> carrier G; h g = h g'\<rbrakk>
```
```   934       \<Longrightarrow>  kernel G H h #> g \<subseteq> kernel G H h #> g'"
```
```   935 apply (clarsimp simp add: kernel_def r_coset_def image_def);
```
```   936 apply (rename_tac y)
```
```   937 apply (rule_tac x="y \<otimes> g \<otimes> inv g'" in exI)
```
```   938 apply (simp add: G.m_assoc);
```
```   939 done
```
```   940
```
```   941 lemma (in group_hom) FactGroup_inj_on:
```
```   942      "inj_on (\<lambda>X. contents (h ` X)) (carrier (G Mod kernel G H h))"
```
```   943 proof (simp add: inj_on_def, clarify)
```
```   944   fix X and X'
```
```   945   assume X:  "X  \<in> carrier (G Mod kernel G H h)"
```
```   946      and X': "X' \<in> carrier (G Mod kernel G H h)"
```
```   947   then
```
```   948   obtain g and g'
```
```   949            where gX: "g \<in> carrier G"  "g' \<in> carrier G"
```
```   950               "X = kernel G H h #> g" "X' = kernel G H h #> g'"
```
```   951     by (auto simp add: FactGroup_def RCOSETS_def)
```
```   952   hence all: "\<forall>x\<in>X. h x = h g" "\<forall>x\<in>X'. h x = h g'"
```
```   953     by (force simp add: kernel_def r_coset_def image_def)+
```
```   954   assume "contents (h ` X) = contents (h ` X')"
```
```   955   hence h: "h g = h g'"
```
```   956     by (simp add: image_eq_UN all FactGroup_nonempty X X')
```
```   957   show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX)
```
```   958 qed
```
```   959
```
```   960 text{*If the homomorphism @{term h} is onto @{term H}, then so is the
```
```   961 homomorphism from the quotient group*}
```
```   962 lemma (in group_hom) FactGroup_onto:
```
```   963   assumes h: "h ` carrier G = carrier H"
```
```   964   shows "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
```
```   965 proof
```
```   966   show "(\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h) \<subseteq> carrier H"
```
```   967     by (auto simp add: FactGroup_contents_mem)
```
```   968   show "carrier H \<subseteq> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
```
```   969   proof
```
```   970     fix y
```
```   971     assume y: "y \<in> carrier H"
```
```   972     with h obtain g where g: "g \<in> carrier G" "h g = y"
```
```   973       by (blast elim: equalityE);
```
```   974     hence "(\<Union>x\<in>kernel G H h #> g. {h x}) = {y}"
```
```   975       by (auto simp add: y kernel_def r_coset_def)
```
```   976     with g show "y \<in> (\<lambda>X. contents (h ` X)) ` carrier (G Mod kernel G H h)"
```
```   977       by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN)
```
```   978   qed
```
```   979 qed
```
```   980
```
```   981
```
```   982 text{*If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
```
```   983  quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.*}
```
```   984 theorem (in group_hom) FactGroup_iso:
```
```   985   "h ` carrier G = carrier H
```
```   986    \<Longrightarrow> (\<lambda>X. contents (h`X)) \<in> (G Mod (kernel G H h)) \<cong> H"
```
```   987 by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def
```
```   988               FactGroup_onto)
```
```   989
```
```   990
```
```   991 end
```