src/HOL/Algebra/Group.thy
 author ballarin Tue Jul 15 16:50:09 2008 +0200 (2008-07-15) changeset 27611 2c01c0bdb385 parent 26805 27941d7d9a11 child 27698 197f0517f0bd permissions -rw-r--r--
Removed uses of context element includes.
     1 (*

     2   Title:  HOL/Algebra/Group.thy

     3   Id:     $Id$

     4   Author: Clemens Ballarin, started 4 February 2003

     5

     6 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.

     7 *)

     8

     9 theory Group imports FuncSet Lattice begin

    10

    11

    12 section {* Monoids and Groups *}

    13

    14 subsection {* Definitions *}

    15

    16 text {*

    17   Definitions follow \cite{Jacobson:1985}.

    18 *}

    19

    20 record 'a monoid =  "'a partial_object" +

    21   mult    :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)

    22   one     :: 'a ("\<one>\<index>")

    23

    24 constdefs (structure G)

    25   m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _"  80)

    26   "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"

    27

    28   Units :: "_ => 'a set"

    29   --{*The set of invertible elements*}

    30   "Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"

    31

    32 consts

    33   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)

    34

    35 defs (overloaded)

    36   nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"

    37   int_pow_def: "pow G a z ==

    38     let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)

    39     in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"

    40

    41 locale monoid =

    42   fixes G (structure)

    43   assumes m_closed [intro, simp]:

    44          "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"

    45       and m_assoc:

    46          "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk>

    47           \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

    48       and one_closed [intro, simp]: "\<one> \<in> carrier G"

    49       and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x"

    50       and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x"

    51

    52 lemma monoidI:

    53   fixes G (structure)

    54   assumes m_closed:

    55       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

    56     and one_closed: "\<one> \<in> carrier G"

    57     and m_assoc:

    58       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

    59       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

    60     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

    61     and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"

    62   shows "monoid G"

    63   by (fast intro!: monoid.intro intro: prems)

    64

    65 lemma (in monoid) Units_closed [dest]:

    66   "x \<in> Units G ==> x \<in> carrier G"

    67   by (unfold Units_def) fast

    68

    69 lemma (in monoid) inv_unique:

    70   assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"

    71     and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"

    72   shows "y = y'"

    73 proof -

    74   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp

    75   also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)

    76   also from G eq have "... = y'" by simp

    77   finally show ?thesis .

    78 qed

    79

    80 lemma (in monoid) Units_one_closed [intro, simp]:

    81   "\<one> \<in> Units G"

    82   by (unfold Units_def) auto

    83

    84 lemma (in monoid) Units_inv_closed [intro, simp]:

    85   "x \<in> Units G ==> inv x \<in> carrier G"

    86   apply (unfold Units_def m_inv_def, auto)

    87   apply (rule theI2, fast)

    88    apply (fast intro: inv_unique, fast)

    89   done

    90

    91 lemma (in monoid) Units_l_inv_ex:

    92   "x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

    93   by (unfold Units_def) auto

    94

    95 lemma (in monoid) Units_r_inv_ex:

    96   "x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

    97   by (unfold Units_def) auto

    98

    99 lemma (in monoid) Units_l_inv:

   100   "x \<in> Units G ==> inv x \<otimes> x = \<one>"

   101   apply (unfold Units_def m_inv_def, auto)

   102   apply (rule theI2, fast)

   103    apply (fast intro: inv_unique, fast)

   104   done

   105

   106 lemma (in monoid) Units_r_inv:

   107   "x \<in> Units G ==> x \<otimes> inv x = \<one>"

   108   apply (unfold Units_def m_inv_def, auto)

   109   apply (rule theI2, fast)

   110    apply (fast intro: inv_unique, fast)

   111   done

   112

   113 lemma (in monoid) Units_inv_Units [intro, simp]:

   114   "x \<in> Units G ==> inv x \<in> Units G"

   115 proof -

   116   assume x: "x \<in> Units G"

   117   show "inv x \<in> Units G"

   118     by (auto simp add: Units_def

   119       intro: Units_l_inv Units_r_inv x Units_closed [OF x])

   120 qed

   121

   122 lemma (in monoid) Units_l_cancel [simp]:

   123   "[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==>

   124    (x \<otimes> y = x \<otimes> z) = (y = z)"

   125 proof

   126   assume eq: "x \<otimes> y = x \<otimes> z"

   127     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"

   128   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"

   129     by (simp add: m_assoc Units_closed)

   130   with G show "y = z" by (simp add: Units_l_inv)

   131 next

   132   assume eq: "y = z"

   133     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"

   134   then show "x \<otimes> y = x \<otimes> z" by simp

   135 qed

   136

   137 lemma (in monoid) Units_inv_inv [simp]:

   138   "x \<in> Units G ==> inv (inv x) = x"

   139 proof -

   140   assume x: "x \<in> Units G"

   141   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x"

   142     by (simp add: Units_l_inv Units_r_inv)

   143   with x show ?thesis by (simp add: Units_closed)

   144 qed

   145

   146 lemma (in monoid) inv_inj_on_Units:

   147   "inj_on (m_inv G) (Units G)"

   148 proof (rule inj_onI)

   149   fix x y

   150   assume G: "x \<in> Units G"  "y \<in> Units G" and eq: "inv x = inv y"

   151   then have "inv (inv x) = inv (inv y)" by simp

   152   with G show "x = y" by simp

   153 qed

   154

   155 lemma (in monoid) Units_inv_comm:

   156   assumes inv: "x \<otimes> y = \<one>"

   157     and G: "x \<in> Units G"  "y \<in> Units G"

   158   shows "y \<otimes> x = \<one>"

   159 proof -

   160   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)

   161   with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)

   162 qed

   163

   164 text {* Power *}

   165

   166 lemma (in monoid) nat_pow_closed [intro, simp]:

   167   "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"

   168   by (induct n) (simp_all add: nat_pow_def)

   169

   170 lemma (in monoid) nat_pow_0 [simp]:

   171   "x (^) (0::nat) = \<one>"

   172   by (simp add: nat_pow_def)

   173

   174 lemma (in monoid) nat_pow_Suc [simp]:

   175   "x (^) (Suc n) = x (^) n \<otimes> x"

   176   by (simp add: nat_pow_def)

   177

   178 lemma (in monoid) nat_pow_one [simp]:

   179   "\<one> (^) (n::nat) = \<one>"

   180   by (induct n) simp_all

   181

   182 lemma (in monoid) nat_pow_mult:

   183   "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"

   184   by (induct m) (simp_all add: m_assoc [THEN sym])

   185

   186 lemma (in monoid) nat_pow_pow:

   187   "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"

   188   by (induct m) (simp, simp add: nat_pow_mult add_commute)

   189

   190 text {*

   191   A group is a monoid all of whose elements are invertible.

   192 *}

   193

   194 locale group = monoid +

   195   assumes Units: "carrier G <= Units G"

   196

   197

   198 lemma (in group) is_group: "group G" by (rule group_axioms)

   199

   200 theorem groupI:

   201   fixes G (structure)

   202   assumes m_closed [simp]:

   203       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

   204     and one_closed [simp]: "\<one> \<in> carrier G"

   205     and m_assoc:

   206       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

   207       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

   208     and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

   209     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   210   shows "group G"

   211 proof -

   212   have l_cancel [simp]:

   213     "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

   214     (x \<otimes> y = x \<otimes> z) = (y = z)"

   215   proof

   216     fix x y z

   217     assume eq: "x \<otimes> y = x \<otimes> z"

   218       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   219     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"

   220       and l_inv: "x_inv \<otimes> x = \<one>" by fast

   221     from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"

   222       by (simp add: m_assoc)

   223     with G show "y = z" by (simp add: l_inv)

   224   next

   225     fix x y z

   226     assume eq: "y = z"

   227       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   228     then show "x \<otimes> y = x \<otimes> z" by simp

   229   qed

   230   have r_one:

   231     "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"

   232   proof -

   233     fix x

   234     assume x: "x \<in> carrier G"

   235     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"

   236       and l_inv: "x_inv \<otimes> x = \<one>" by fast

   237     from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"

   238       by (simp add: m_assoc [symmetric] l_inv)

   239     with x xG show "x \<otimes> \<one> = x" by simp

   240   qed

   241   have inv_ex:

   242     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   243   proof -

   244     fix x

   245     assume x: "x \<in> carrier G"

   246     with l_inv_ex obtain y where y: "y \<in> carrier G"

   247       and l_inv: "y \<otimes> x = \<one>" by fast

   248     from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"

   249       by (simp add: m_assoc [symmetric] l_inv r_one)

   250     with x y have r_inv: "x \<otimes> y = \<one>"

   251       by simp

   252     from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   253       by (fast intro: l_inv r_inv)

   254   qed

   255   then have carrier_subset_Units: "carrier G <= Units G"

   256     by (unfold Units_def) fast

   257   show ?thesis

   258     by (fast intro!: group.intro monoid.intro group_axioms.intro

   259       carrier_subset_Units intro: prems r_one)

   260 qed

   261

   262 lemma (in monoid) monoid_groupI:

   263   assumes l_inv_ex:

   264     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   265   shows "group G"

   266   by (rule groupI) (auto intro: m_assoc l_inv_ex)

   267

   268 lemma (in group) Units_eq [simp]:

   269   "Units G = carrier G"

   270 proof

   271   show "Units G <= carrier G" by fast

   272 next

   273   show "carrier G <= Units G" by (rule Units)

   274 qed

   275

   276 lemma (in group) inv_closed [intro, simp]:

   277   "x \<in> carrier G ==> inv x \<in> carrier G"

   278   using Units_inv_closed by simp

   279

   280 lemma (in group) l_inv_ex [simp]:

   281   "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   282   using Units_l_inv_ex by simp

   283

   284 lemma (in group) r_inv_ex [simp]:

   285   "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

   286   using Units_r_inv_ex by simp

   287

   288 lemma (in group) l_inv [simp]:

   289   "x \<in> carrier G ==> inv x \<otimes> x = \<one>"

   290   using Units_l_inv by simp

   291

   292

   293 subsection {* Cancellation Laws and Basic Properties *}

   294

   295 lemma (in group) l_cancel [simp]:

   296   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

   297    (x \<otimes> y = x \<otimes> z) = (y = z)"

   298   using Units_l_inv by simp

   299

   300 lemma (in group) r_inv [simp]:

   301   "x \<in> carrier G ==> x \<otimes> inv x = \<one>"

   302 proof -

   303   assume x: "x \<in> carrier G"

   304   then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"

   305     by (simp add: m_assoc [symmetric] l_inv)

   306   with x show ?thesis by (simp del: r_one)

   307 qed

   308

   309 lemma (in group) r_cancel [simp]:

   310   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

   311    (y \<otimes> x = z \<otimes> x) = (y = z)"

   312 proof

   313   assume eq: "y \<otimes> x = z \<otimes> x"

   314     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   315   then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"

   316     by (simp add: m_assoc [symmetric] del: r_inv)

   317   with G show "y = z" by simp

   318 next

   319   assume eq: "y = z"

   320     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   321   then show "y \<otimes> x = z \<otimes> x" by simp

   322 qed

   323

   324 lemma (in group) inv_one [simp]:

   325   "inv \<one> = \<one>"

   326 proof -

   327   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv)

   328   moreover have "... = \<one>" by simp

   329   finally show ?thesis .

   330 qed

   331

   332 lemma (in group) inv_inv [simp]:

   333   "x \<in> carrier G ==> inv (inv x) = x"

   334   using Units_inv_inv by simp

   335

   336 lemma (in group) inv_inj:

   337   "inj_on (m_inv G) (carrier G)"

   338   using inv_inj_on_Units by simp

   339

   340 lemma (in group) inv_mult_group:

   341   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"

   342 proof -

   343   assume G: "x \<in> carrier G"  "y \<in> carrier G"

   344   then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"

   345     by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric])

   346   with G show ?thesis by (simp del: l_inv)

   347 qed

   348

   349 lemma (in group) inv_comm:

   350   "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"

   351   by (rule Units_inv_comm) auto

   352

   353 lemma (in group) inv_equality:

   354      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"

   355 apply (simp add: m_inv_def)

   356 apply (rule the_equality)

   357  apply (simp add: inv_comm [of y x])

   358 apply (rule r_cancel [THEN iffD1], auto)

   359 done

   360

   361 text {* Power *}

   362

   363 lemma (in group) int_pow_def2:

   364   "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"

   365   by (simp add: int_pow_def nat_pow_def Let_def)

   366

   367 lemma (in group) int_pow_0 [simp]:

   368   "x (^) (0::int) = \<one>"

   369   by (simp add: int_pow_def2)

   370

   371 lemma (in group) int_pow_one [simp]:

   372   "\<one> (^) (z::int) = \<one>"

   373   by (simp add: int_pow_def2)

   374

   375

   376 subsection {* Subgroups *}

   377

   378 locale subgroup =

   379   fixes H and G (structure)

   380   assumes subset: "H \<subseteq> carrier G"

   381     and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"

   382     and one_closed [simp]: "\<one> \<in> H"

   383     and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H"

   384

   385 lemma (in subgroup) is_subgroup:

   386   "subgroup H G" by (rule subgroup_axioms)

   387

   388 declare (in subgroup) group.intro [intro]

   389

   390 lemma (in subgroup) mem_carrier [simp]:

   391   "x \<in> H \<Longrightarrow> x \<in> carrier G"

   392   using subset by blast

   393

   394 lemma subgroup_imp_subset:

   395   "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"

   396   by (rule subgroup.subset)

   397

   398 lemma (in subgroup) subgroup_is_group [intro]:

   399   assumes "group G"

   400   shows "group (G\<lparr>carrier := H\<rparr>)"

   401 proof -

   402   interpret group [G] by fact

   403   show ?thesis

   404     by (rule groupI) (auto intro: m_assoc l_inv mem_carrier)

   405 qed

   406

   407 text {*

   408   Since @{term H} is nonempty, it contains some element @{term x}.  Since

   409   it is closed under inverse, it contains @{text "inv x"}.  Since

   410   it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.

   411 *}

   412

   413 lemma (in group) one_in_subset:

   414   "[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |]

   415    ==> \<one> \<in> H"

   416 by (force simp add: l_inv)

   417

   418 text {* A characterization of subgroups: closed, non-empty subset. *}

   419

   420 lemma (in group) subgroupI:

   421   assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"

   422     and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"

   423     and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"

   424   shows "subgroup H G"

   425 proof (simp add: subgroup_def prems)

   426   show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)

   427 qed

   428

   429 declare monoid.one_closed [iff] group.inv_closed [simp]

   430   monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]

   431

   432 lemma subgroup_nonempty:

   433   "~ subgroup {} G"

   434   by (blast dest: subgroup.one_closed)

   435

   436 lemma (in subgroup) finite_imp_card_positive:

   437   "finite (carrier G) ==> 0 < card H"

   438 proof (rule classical)

   439   assume "finite (carrier G)" "~ 0 < card H"

   440   then have "finite H" by (blast intro: finite_subset [OF subset])

   441   with prems have "subgroup {} G" by simp

   442   with subgroup_nonempty show ?thesis by contradiction

   443 qed

   444

   445 (*

   446 lemma (in monoid) Units_subgroup:

   447   "subgroup (Units G) G"

   448 *)

   449

   450

   451 subsection {* Direct Products *}

   452

   453 constdefs

   454   DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid"  (infixr "\<times>\<times>" 80)

   455   "G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H,

   456                 mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),

   457                 one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"

   458

   459 lemma DirProd_monoid:

   460   assumes "monoid G" and "monoid H"

   461   shows "monoid (G \<times>\<times> H)"

   462 proof -

   463   interpret G: monoid [G] by fact

   464   interpret H: monoid [H] by fact

   465   from prems

   466   show ?thesis by (unfold monoid_def DirProd_def, auto)

   467 qed

   468

   469

   470 text{*Does not use the previous result because it's easier just to use auto.*}

   471 lemma DirProd_group:

   472   assumes "group G" and "group H"

   473   shows "group (G \<times>\<times> H)"

   474 proof -

   475   interpret G: group [G] by fact

   476   interpret H: group [H] by fact

   477   show ?thesis by (rule groupI)

   478      (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv

   479            simp add: DirProd_def)

   480 qed

   481

   482 lemma carrier_DirProd [simp]:

   483      "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"

   484   by (simp add: DirProd_def)

   485

   486 lemma one_DirProd [simp]:

   487      "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"

   488   by (simp add: DirProd_def)

   489

   490 lemma mult_DirProd [simp]:

   491      "(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"

   492   by (simp add: DirProd_def)

   493

   494 lemma inv_DirProd [simp]:

   495   assumes "group G" and "group H"

   496   assumes g: "g \<in> carrier G"

   497       and h: "h \<in> carrier H"

   498   shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"

   499 proof -

   500   interpret G: group [G] by fact

   501   interpret H: group [H] by fact

   502   show ?thesis apply (rule group.inv_equality [OF DirProd_group])

   503   apply (simp_all add: prems group.l_inv)

   504   done

   505 qed

   506

   507 text{*This alternative proof of the previous result demonstrates interpret.

   508    It uses @{text Prod.inv_equality} (available after @{text interpret})

   509    instead of @{text "group.inv_equality [OF DirProd_group]"}. *}

   510 lemma

   511   assumes "group G" and "group H"

   512   assumes g: "g \<in> carrier G"

   513       and h: "h \<in> carrier H"

   514   shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"

   515 proof -

   516   interpret G: group [G] by fact

   517   interpret H: group [H] by fact

   518   interpret Prod: group ["G \<times>\<times> H"]

   519     by (auto intro: DirProd_group group.intro group.axioms prems)

   520   show ?thesis by (simp add: Prod.inv_equality g h)

   521 qed

   522

   523

   524 subsection {* Homomorphisms and Isomorphisms *}

   525

   526 constdefs (structure G and H)

   527   hom :: "_ => _ => ('a => 'b) set"

   528   "hom G H ==

   529     {h. h \<in> carrier G -> carrier H &

   530       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}"

   531

   532 lemma hom_mult:

   533   "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]

   534    ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"

   535   by (simp add: hom_def)

   536

   537 lemma hom_closed:

   538   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"

   539   by (auto simp add: hom_def funcset_mem)

   540

   541 lemma (in group) hom_compose:

   542      "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"

   543 apply (auto simp add: hom_def funcset_compose)

   544 apply (simp add: compose_def funcset_mem)

   545 done

   546

   547 constdefs

   548   iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)

   549   "G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"

   550

   551 lemma iso_refl: "(%x. x) \<in> G \<cong> G"

   552 by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

   553

   554 lemma (in group) iso_sym:

   555      "h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G"

   556 apply (simp add: iso_def bij_betw_Inv)

   557 apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G")

   558  prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv])

   559 apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f)

   560 done

   561

   562 lemma (in group) iso_trans:

   563      "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"

   564 by (auto simp add: iso_def hom_compose bij_betw_compose)

   565

   566 lemma DirProd_commute_iso:

   567   shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"

   568 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

   569

   570 lemma DirProd_assoc_iso:

   571   shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"

   572 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

   573

   574

   575 text{*Basis for homomorphism proofs: we assume two groups @{term G} and

   576   @{term H}, with a homomorphism @{term h} between them*}

   577 locale group_hom = group G + group H + var h +

   578   assumes homh: "h \<in> hom G H"

   579   notes hom_mult [simp] = hom_mult [OF homh]

   580     and hom_closed [simp] = hom_closed [OF homh]

   581

   582 lemma (in group_hom) one_closed [simp]:

   583   "h \<one> \<in> carrier H"

   584   by simp

   585

   586 lemma (in group_hom) hom_one [simp]:

   587   "h \<one> = \<one>\<^bsub>H\<^esub>"

   588 proof -

   589   have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"

   590     by (simp add: hom_mult [symmetric] del: hom_mult)

   591   then show ?thesis by (simp del: r_one)

   592 qed

   593

   594 lemma (in group_hom) inv_closed [simp]:

   595   "x \<in> carrier G ==> h (inv x) \<in> carrier H"

   596   by simp

   597

   598 lemma (in group_hom) hom_inv [simp]:

   599   "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"

   600 proof -

   601   assume x: "x \<in> carrier G"

   602   then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"

   603     by (simp add: hom_mult [symmetric] del: hom_mult)

   604   also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"

   605     by (simp add: hom_mult [symmetric] del: hom_mult)

   606   finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .

   607   with x show ?thesis by (simp del: H.r_inv)

   608 qed

   609

   610

   611 subsection {* Commutative Structures *}

   612

   613 text {*

   614   Naming convention: multiplicative structures that are commutative

   615   are called \emph{commutative}, additive structures are called

   616   \emph{Abelian}.

   617 *}

   618

   619 locale comm_monoid = monoid +

   620   assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"

   621

   622 lemma (in comm_monoid) m_lcomm:

   623   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>

   624    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"

   625 proof -

   626   assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"

   627   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)

   628   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)

   629   also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)

   630   finally show ?thesis .

   631 qed

   632

   633 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   634

   635 lemma comm_monoidI:

   636   fixes G (structure)

   637   assumes m_closed:

   638       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

   639     and one_closed: "\<one> \<in> carrier G"

   640     and m_assoc:

   641       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

   642       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

   643     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

   644     and m_comm:

   645       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

   646   shows "comm_monoid G"

   647   using l_one

   648     by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro

   649              intro: prems simp: m_closed one_closed m_comm)

   650

   651 lemma (in monoid) monoid_comm_monoidI:

   652   assumes m_comm:

   653       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

   654   shows "comm_monoid G"

   655   by (rule comm_monoidI) (auto intro: m_assoc m_comm)

   656

   657 (*lemma (in comm_monoid) r_one [simp]:

   658   "x \<in> carrier G ==> x \<otimes> \<one> = x"

   659 proof -

   660   assume G: "x \<in> carrier G"

   661   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)

   662   also from G have "... = x" by simp

   663   finally show ?thesis .

   664 qed*)

   665

   666 lemma (in comm_monoid) nat_pow_distr:

   667   "[| x \<in> carrier G; y \<in> carrier G |] ==>

   668   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"

   669   by (induct n) (simp, simp add: m_ac)

   670

   671 locale comm_group = comm_monoid + group

   672

   673 lemma (in group) group_comm_groupI:

   674   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>

   675       x \<otimes> y = y \<otimes> x"

   676   shows "comm_group G"

   677   by unfold_locales (simp_all add: m_comm)

   678

   679 lemma comm_groupI:

   680   fixes G (structure)

   681   assumes m_closed:

   682       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"

   683     and one_closed: "\<one> \<in> carrier G"

   684     and m_assoc:

   685       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>

   686       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

   687     and m_comm:

   688       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

   689     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

   690     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   691   shows "comm_group G"

   692   by (fast intro: group.group_comm_groupI groupI prems)

   693

   694 lemma (in comm_group) inv_mult:

   695   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"

   696   by (simp add: m_ac inv_mult_group)

   697

   698

   699 subsection {* The Lattice of Subgroups of a Group *}

   700

   701 text_raw {* \label{sec:subgroup-lattice} *}

   702

   703 theorem (in group) subgroups_partial_order:

   704   "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"

   705   by (rule partial_order.intro) simp_all

   706

   707 lemma (in group) subgroup_self:

   708   "subgroup (carrier G) G"

   709   by (rule subgroupI) auto

   710

   711 lemma (in group) subgroup_imp_group:

   712   "subgroup H G ==> group (G(| carrier := H |))"

   713   by (erule subgroup.subgroup_is_group) (rule group_axioms)

   714

   715 lemma (in group) is_monoid [intro, simp]:

   716   "monoid G"

   717   by (auto intro: monoid.intro m_assoc)

   718

   719 lemma (in group) subgroup_inv_equality:

   720   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"

   721 apply (rule_tac inv_equality [THEN sym])

   722   apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)

   723  apply (rule subsetD [OF subgroup.subset], assumption+)

   724 apply (rule subsetD [OF subgroup.subset], assumption)

   725 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)

   726 done

   727

   728 theorem (in group) subgroups_Inter:

   729   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"

   730     and not_empty: "A ~= {}"

   731   shows "subgroup (\<Inter>A) G"

   732 proof (rule subgroupI)

   733   from subgr [THEN subgroup.subset] and not_empty

   734   show "\<Inter>A \<subseteq> carrier G" by blast

   735 next

   736   from subgr [THEN subgroup.one_closed]

   737   show "\<Inter>A ~= {}" by blast

   738 next

   739   fix x assume "x \<in> \<Inter>A"

   740   with subgr [THEN subgroup.m_inv_closed]

   741   show "inv x \<in> \<Inter>A" by blast

   742 next

   743   fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"

   744   with subgr [THEN subgroup.m_closed]

   745   show "x \<otimes> y \<in> \<Inter>A" by blast

   746 qed

   747

   748 theorem (in group) subgroups_complete_lattice:

   749   "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"

   750     (is "complete_lattice ?L")

   751 proof (rule partial_order.complete_lattice_criterion1)

   752   show "partial_order ?L" by (rule subgroups_partial_order)

   753 next

   754   show "\<exists>G. greatest ?L G (carrier ?L)"

   755   proof

   756     show "greatest ?L (carrier G) (carrier ?L)"

   757       by (unfold greatest_def)

   758         (simp add: subgroup.subset subgroup_self)

   759   qed

   760 next

   761   fix A

   762   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"

   763   then have Int_subgroup: "subgroup (\<Inter>A) G"

   764     by (fastsimp intro: subgroups_Inter)

   765   show "\<exists>I. greatest ?L I (Lower ?L A)"

   766   proof

   767     show "greatest ?L (\<Inter>A) (Lower ?L A)"

   768       (is "greatest _ ?Int _")

   769     proof (rule greatest_LowerI)

   770       fix H

   771       assume H: "H \<in> A"

   772       with L have subgroupH: "subgroup H G" by auto

   773       from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")

   774 	by (rule subgroup_imp_group)

   775       from groupH have monoidH: "monoid ?H"

   776 	by (rule group.is_monoid)

   777       from H have Int_subset: "?Int \<subseteq> H" by fastsimp

   778       then show "le ?L ?Int H" by simp

   779     next

   780       fix H

   781       assume H: "H \<in> Lower ?L A"

   782       with L Int_subgroup show "le ?L H ?Int"

   783 	by (fastsimp simp: Lower_def intro: Inter_greatest)

   784     next

   785       show "A \<subseteq> carrier ?L" by (rule L)

   786     next

   787       show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)

   788     qed

   789   qed

   790 qed

   791

   792 end