src/HOL/Algebra/Group.thy
 author paulson Tue Jun 01 11:25:01 2004 +0200 (2004-06-01) changeset 14852 fffab59e0050 parent 14803 f7557773cc87 child 14963 d584e32f7d46 permissions -rw-r--r--
tidied
     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 header {* Groups *}

    10

    11 theory Group = FuncSet + Lattice:

    12

    13

    14 section {* From Magmas to Groups *}

    15

    16 text {*

    17   Definitions follow \cite{Jacobson:1985}; with the exception of

    18   \emph{magma} which, following Bourbaki, is a set together with a

    19   binary, closed operation.

    20 *}

    21

    22 subsection {* Definitions *}

    23

    24 record 'a semigroup = "'a partial_object" +

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

    26

    27 record 'a monoid = "'a semigroup" +

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

    29

    30 constdefs (structure G)

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

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

    33

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

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

    36   "Units G == {y. y \<in> carrier G & (EX x : carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"

    37

    38 consts

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

    40

    41 defs (overloaded)

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

    43   int_pow_def: "pow G a z ==

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

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

    46

    47 locale magma = struct G +

    48   assumes m_closed [intro, simp]:

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

    50

    51 locale semigroup = magma +

    52   assumes m_assoc:

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

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

    55

    56 locale monoid = semigroup +

    57   assumes one_closed [intro, simp]: "\<one> \<in> carrier G"

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

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

    60

    61 lemma monoidI:

    62   includes struct G

    63   assumes m_closed:

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

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

    66     and m_assoc:

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

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

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

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

    71   shows "monoid G"

    72   by (fast intro!: monoid.intro magma.intro semigroup_axioms.intro

    73     semigroup.intro monoid_axioms.intro

    74     intro: prems)

    75

    76 lemma (in monoid) Units_closed [dest]:

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

    78   by (unfold Units_def) fast

    79

    80 lemma (in monoid) inv_unique:

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

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

    83   shows "y = y'"

    84 proof -

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

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

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

    88   finally show ?thesis .

    89 qed

    90

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

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

    93   by (unfold Units_def) auto

    94

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

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

    97   apply (unfold Units_def m_inv_def, auto)

    98   apply (rule theI2, fast)

    99    apply (fast intro: inv_unique, fast)

   100   done

   101

   102 lemma (in monoid) Units_l_inv:

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

   104   apply (unfold Units_def m_inv_def, auto)

   105   apply (rule theI2, fast)

   106    apply (fast intro: inv_unique, fast)

   107   done

   108

   109 lemma (in monoid) Units_r_inv:

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

   111   apply (unfold Units_def m_inv_def, auto)

   112   apply (rule theI2, fast)

   113    apply (fast intro: inv_unique, fast)

   114   done

   115

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

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

   118 proof -

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

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

   121     by (auto simp add: Units_def

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

   123 qed

   124

   125 lemma (in monoid) Units_l_cancel [simp]:

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

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

   128 proof

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

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

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

   132     by (simp add: m_assoc Units_closed)

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

   134 next

   135   assume eq: "y = z"

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

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

   138 qed

   139

   140 lemma (in monoid) Units_inv_inv [simp]:

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

   142 proof -

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

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

   145     by (simp add: Units_l_inv Units_r_inv)

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

   147 qed

   148

   149 lemma (in monoid) inv_inj_on_Units:

   150   "inj_on (m_inv G) (Units G)"

   151 proof (rule inj_onI)

   152   fix x y

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

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

   155   with G show "x = y" by simp

   156 qed

   157

   158 lemma (in monoid) Units_inv_comm:

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

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

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

   162 proof -

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

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

   165 qed

   166

   167 text {* Power *}

   168

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

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

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

   172

   173 lemma (in monoid) nat_pow_0 [simp]:

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

   175   by (simp add: nat_pow_def)

   176

   177 lemma (in monoid) nat_pow_Suc [simp]:

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

   179   by (simp add: nat_pow_def)

   180

   181 lemma (in monoid) nat_pow_one [simp]:

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

   183   by (induct n) simp_all

   184

   185 lemma (in monoid) nat_pow_mult:

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

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

   188

   189 lemma (in monoid) nat_pow_pow:

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

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

   192

   193 text {*

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

   195 *}

   196

   197 locale group = monoid +

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

   199

   200

   201 lemma (in group) is_group: "group G"

   202   by (rule group.intro [OF prems])

   203

   204 theorem groupI:

   205   includes struct G

   206   assumes m_closed [simp]:

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

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

   209     and m_assoc:

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

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

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

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

   214   shows "group G"

   215 proof -

   216   have l_cancel [simp]:

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

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

   219   proof

   220     fix x y z

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

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

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

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

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

   226       by (simp add: m_assoc)

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

   228   next

   229     fix x y z

   230     assume eq: "y = z"

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

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

   233   qed

   234   have r_one:

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

   236   proof -

   237     fix x

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

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

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

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

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

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

   244   qed

   245   have inv_ex:

   246     "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   247   proof -

   248     fix x

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

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

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

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

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

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

   255       by simp

   256     from x y show "EX y : carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   257       by (fast intro: l_inv r_inv)

   258   qed

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

   260     by (unfold Units_def) fast

   261   show ?thesis

   262     by (fast intro!: group.intro magma.intro semigroup_axioms.intro

   263       semigroup.intro monoid_axioms.intro group_axioms.intro

   264       carrier_subset_Units intro: prems r_one)

   265 qed

   266

   267 lemma (in monoid) monoid_groupI:

   268   assumes l_inv_ex:

   269     "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"

   270   shows "group G"

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

   272

   273 lemma (in group) Units_eq [simp]:

   274   "Units G = carrier G"

   275 proof

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

   277 next

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

   279 qed

   280

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

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

   283   using Units_inv_closed by simp

   284

   285 lemma (in group) l_inv:

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

   287   using Units_l_inv by simp

   288

   289 subsection {* Cancellation Laws and Basic Properties *}

   290

   291 lemma (in group) l_cancel [simp]:

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

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

   294   using Units_l_inv by simp

   295

   296 lemma (in group) r_inv:

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

   298 proof -

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

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

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

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

   303 qed

   304

   305 lemma (in group) r_cancel [simp]:

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

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

   308 proof

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

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

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

   312     by (simp add: m_assoc [symmetric])

   313   with G show "y = z" by (simp add: r_inv)

   314 next

   315   assume eq: "y = z"

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

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

   318 qed

   319

   320 lemma (in group) inv_one [simp]:

   321   "inv \<one> = \<one>"

   322 proof -

   323   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by simp

   324   moreover have "... = \<one>" by (simp add: r_inv)

   325   finally show ?thesis .

   326 qed

   327

   328 lemma (in group) inv_inv [simp]:

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

   330   using Units_inv_inv by simp

   331

   332 lemma (in group) inv_inj:

   333   "inj_on (m_inv G) (carrier G)"

   334   using inv_inj_on_Units by simp

   335

   336 lemma (in group) inv_mult_group:

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

   338 proof -

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

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

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

   342   with G show ?thesis by simp

   343 qed

   344

   345 lemma (in group) inv_comm:

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

   347   by (rule Units_inv_comm) auto

   348

   349 lemma (in group) inv_equality:

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

   351 apply (simp add: m_inv_def)

   352 apply (rule the_equality)

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

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

   355 done

   356

   357 text {* Power *}

   358

   359 lemma (in group) int_pow_def2:

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

   361   by (simp add: int_pow_def nat_pow_def Let_def)

   362

   363 lemma (in group) int_pow_0 [simp]:

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

   365   by (simp add: int_pow_def2)

   366

   367 lemma (in group) int_pow_one [simp]:

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

   369   by (simp add: int_pow_def2)

   370

   371 subsection {* Substructures *}

   372

   373 locale submagma = var H + struct G +

   374   assumes subset [intro, simp]: "H \<subseteq> carrier G"

   375     and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"

   376

   377 declare (in submagma) magma.intro [intro] semigroup.intro [intro]

   378   semigroup_axioms.intro [intro]

   379

   380 lemma submagma_imp_subset:

   381   "submagma H G ==> H \<subseteq> carrier G"

   382   by (rule submagma.subset)

   383

   384 lemma (in submagma) subsetD [dest, simp]:

   385   "x \<in> H ==> x \<in> carrier G"

   386   using subset by blast

   387

   388 lemma (in submagma) magmaI [intro]:

   389   includes magma G

   390   shows "magma (G(| carrier := H |))"

   391   by rule simp

   392

   393 lemma (in submagma) semigroup_axiomsI [intro]:

   394   includes semigroup G

   395   shows "semigroup_axioms (G(| carrier := H |))"

   396     by rule (simp add: m_assoc)

   397

   398 lemma (in submagma) semigroupI [intro]:

   399   includes semigroup G

   400   shows "semigroup (G(| carrier := H |))"

   401   using prems by fast

   402

   403

   404 locale subgroup = submagma H G +

   405   assumes one_closed [intro, simp]: "\<one> \<in> H"

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

   407

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

   409

   410 lemma (in subgroup) group_axiomsI [intro]:

   411   includes group G

   412   shows "group_axioms (G(| carrier := H |))"

   413   by (rule group_axioms.intro) (auto intro: l_inv r_inv simp add: Units_def)

   414

   415 lemma (in subgroup) groupI [intro]:

   416   includes group G

   417   shows "group (G(| carrier := H |))"

   418   by (rule groupI) (auto intro: m_assoc l_inv)

   419

   420 text {*

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

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

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

   424 *}

   425

   426 lemma (in group) one_in_subset:

   427   "[| 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 |]

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

   429 by (force simp add: l_inv)

   430

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

   432

   433 lemma (in group) subgroupI:

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

   435     and inv: "!!a. a \<in> H ==> inv a \<in> H"

   436     and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"

   437   shows "subgroup H G"

   438 proof (rule subgroup.intro)

   439   from subset and mult show "submagma H G" by (rule submagma.intro)

   440 next

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

   442   with inv show "subgroup_axioms H G"

   443     by (intro subgroup_axioms.intro) simp_all

   444 qed

   445

   446 text {*

   447   Repeat facts of submagmas for subgroups.  Necessary???

   448 *}

   449

   450 lemma (in subgroup) subset:

   451   "H \<subseteq> carrier G"

   452   ..

   453

   454 lemma (in subgroup) m_closed:

   455   "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"

   456   ..

   457

   458 declare magma.m_closed [simp]

   459

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

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

   462

   463 lemma subgroup_nonempty:

   464   "~ subgroup {} G"

   465   by (blast dest: subgroup.one_closed)

   466

   467 lemma (in subgroup) finite_imp_card_positive:

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

   469 proof (rule classical)

   470   have sub: "subgroup H G" using prems by (rule subgroup.intro)

   471   assume fin: "finite (carrier G)"

   472     and zero: "~ 0 < card H"

   473   then have "finite H" by (blast intro: finite_subset dest: subset)

   474   with zero sub have "subgroup {} G" by simp

   475   with subgroup_nonempty show ?thesis by contradiction

   476 qed

   477

   478 (*

   479 lemma (in monoid) Units_subgroup:

   480   "subgroup (Units G) G"

   481 *)

   482

   483 subsection {* Direct Products *}

   484

   485 constdefs (structure G and H)

   486   DirProdSemigroup :: "_ => _ => ('a \<times> 'b) semigroup"  (infixr "\<times>\<^sub>s" 80)

   487   "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,

   488     mult = (%(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')) |)"

   489

   490   DirProdGroup :: "_ => _ => ('a \<times> 'b) monoid"  (infixr "\<times>\<^sub>g" 80)

   491   "G \<times>\<^sub>g H == semigroup.extend (G \<times>\<^sub>s H) (| one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>) |)"

   492

   493 lemma DirProdSemigroup_magma:

   494   includes magma G + magma H

   495   shows "magma (G \<times>\<^sub>s H)"

   496   by (rule magma.intro) (auto simp add: DirProdSemigroup_def)

   497

   498 lemma DirProdSemigroup_semigroup_axioms:

   499   includes semigroup G + semigroup H

   500   shows "semigroup_axioms (G \<times>\<^sub>s H)"

   501   by (rule semigroup_axioms.intro)

   502     (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)

   503

   504 lemma DirProdSemigroup_semigroup:

   505   includes semigroup G + semigroup H

   506   shows "semigroup (G \<times>\<^sub>s H)"

   507   using prems

   508   by (fast intro: semigroup.intro

   509     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)

   510

   511 lemma DirProdGroup_magma:

   512   includes magma G + magma H

   513   shows "magma (G \<times>\<^sub>g H)"

   514   by (rule magma.intro)

   515     (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   516

   517 lemma DirProdGroup_semigroup_axioms:

   518   includes semigroup G + semigroup H

   519   shows "semigroup_axioms (G \<times>\<^sub>g H)"

   520   by (rule semigroup_axioms.intro)

   521     (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs

   522       G.m_assoc H.m_assoc)

   523

   524 lemma DirProdGroup_semigroup:

   525   includes semigroup G + semigroup H

   526   shows "semigroup (G \<times>\<^sub>g H)"

   527   using prems

   528   by (fast intro: semigroup.intro

   529     DirProdGroup_magma DirProdGroup_semigroup_axioms)

   530

   531 text {* \dots\ and further lemmas for group \dots *}

   532

   533 lemma DirProdGroup_group:

   534   includes group G + group H

   535   shows "group (G \<times>\<^sub>g H)"

   536   by (rule groupI)

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

   538       simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   539

   540 lemma carrier_DirProdGroup [simp]:

   541      "carrier (G \<times>\<^sub>g H) = carrier G \<times> carrier H"

   542   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   543

   544 lemma one_DirProdGroup [simp]:

   545      "\<one>\<^bsub>(G \<times>\<^sub>g H)\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"

   546   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   547

   548 lemma mult_DirProdGroup [simp]:

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

   550   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   551

   552 lemma inv_DirProdGroup [simp]:

   553   includes group G + group H

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

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

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

   557   apply (rule group.inv_equality [OF DirProdGroup_group])

   558   apply (simp_all add: prems group_def group.l_inv)

   559   done

   560

   561 subsection {* Isomorphisms *}

   562

   563 constdefs (structure G and H)

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

   565   "hom G H ==

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

   567       (\<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)}"

   568

   569 lemma (in semigroup) hom:

   570      "semigroup (| carrier = hom G G, mult = op o |)"

   571 proof (rule semigroup.intro)

   572   show "magma (| carrier = hom G G, mult = op o |)"

   573     by (rule magma.intro) (simp add: Pi_def hom_def)

   574 next

   575   show "semigroup_axioms (| carrier = hom G G, mult = op o |)"

   576     by (rule semigroup_axioms.intro) (simp add: o_assoc)

   577 qed

   578

   579 lemma hom_mult:

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

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

   582   by (simp add: hom_def)

   583

   584 lemma hom_closed:

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

   586   by (auto simp add: hom_def funcset_mem)

   587

   588 lemma (in group) hom_compose:

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

   590 apply (auto simp add: hom_def funcset_compose)

   591 apply (simp add: compose_def funcset_mem)

   592 done

   593

   594

   595 subsection {* Isomorphisms *}

   596

   597 constdefs

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

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

   600

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

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

   603

   604 lemma (in group) iso_sym:

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

   606 apply (simp add: iso_def bij_betw_Inv)

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

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

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

   610 done

   611

   612 lemma (in group) iso_trans:

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

   614 by (auto simp add: iso_def hom_compose bij_betw_compose)

   615

   616 lemma DirProdGroup_commute_iso:

   617   shows "(%(x,y). (y,x)) \<in> (G \<times>\<^sub>g H) \<cong> (H \<times>\<^sub>g G)"

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

   619

   620 lemma DirProdGroup_assoc_iso:

   621   shows "(%(x,y,z). (x,(y,z))) \<in> (G \<times>\<^sub>g H \<times>\<^sub>g I) \<cong> (G \<times>\<^sub>g (H \<times>\<^sub>g I))"

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

   623

   624

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

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

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

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

   629

   630 lemma (in group_hom) one_closed [simp]:

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

   632   by simp

   633

   634 lemma (in group_hom) hom_one [simp]:

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

   636 proof -

   637   have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^sub>2 h \<one>"

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

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

   640 qed

   641

   642 lemma (in group_hom) inv_closed [simp]:

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

   644   by simp

   645

   646 lemma (in group_hom) hom_inv [simp]:

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

   648 proof -

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

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

   651     by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)

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

   653     by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)

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

   655   with x show ?thesis by simp

   656 qed

   657

   658 subsection {* Commutative Structures *}

   659

   660 text {*

   661   Naming convention: multiplicative structures that are commutative

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

   663   \emph{Abelian}.

   664 *}

   665

   666 subsection {* Definition *}

   667

   668 locale comm_semigroup = semigroup +

   669   assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

   670

   671 lemma (in comm_semigroup) m_lcomm:

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

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

   674 proof -

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

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

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

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

   679   finally show ?thesis .

   680 qed

   681

   682 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm

   683

   684 locale comm_monoid = comm_semigroup + monoid

   685

   686 lemma comm_monoidI:

   687   includes struct G

   688   assumes m_closed:

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

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

   691     and m_assoc:

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

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

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

   695     and m_comm:

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

   697   shows "comm_monoid G"

   698   using l_one

   699   by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro

   700     comm_semigroup_axioms.intro monoid_axioms.intro

   701     intro: prems simp: m_closed one_closed m_comm)

   702

   703 lemma (in monoid) monoid_comm_monoidI:

   704   assumes m_comm:

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

   706   shows "comm_monoid G"

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

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

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

   710 proof -

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

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

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

   714   finally show ?thesis .

   715 qed*)

   716 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   720

   721 locale comm_group = comm_monoid + group

   722

   723 lemma (in group) group_comm_groupI:

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

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

   726   shows "comm_group G"

   727   by (fast intro: comm_group.intro comm_semigroup_axioms.intro

   728                   is_group prems)

   729

   730 lemma comm_groupI:

   731   includes struct G

   732   assumes m_closed:

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

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

   735     and m_assoc:

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

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

   738     and m_comm:

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

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

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

   742   shows "comm_group G"

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

   744

   745 lemma (in comm_group) inv_mult:

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

   747   by (simp add: m_ac inv_mult_group)

   748

   749 subsection {* Lattice of subgroups of a group *}

   750

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

   752

   753 theorem (in group) subgroups_partial_order:

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

   755   by (rule partial_order.intro) simp_all

   756

   757 lemma (in group) subgroup_self:

   758   "subgroup (carrier G) G"

   759   by (rule subgroupI) auto

   760

   761 lemma (in group) subgroup_imp_group:

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

   763   using subgroup.groupI [OF _ group.intro] .

   764

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

   766   "monoid G"

   767   by (rule monoid.intro)

   768

   769 lemma (in group) subgroup_inv_equality:

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

   771 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   776 done

   777

   778 theorem (in group) subgroups_Inter:

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

   780     and not_empty: "A ~= {}"

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

   782 proof (rule subgroupI)

   783   from subgr [THEN subgroup.subset] and not_empty

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

   785 next

   786   from subgr [THEN subgroup.one_closed]

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

   788 next

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

   790   with subgr [THEN subgroup.m_inv_closed]

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

   792 next

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

   794   with subgr [THEN subgroup.m_closed]

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

   796 qed

   797

   798 theorem (in group) subgroups_complete_lattice:

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

   800     (is "complete_lattice ?L")

   801 proof (rule partial_order.complete_lattice_criterion1)

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

   803 next

   804   have "greatest ?L (carrier G) (carrier ?L)"

   805     by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)

   806   then show "EX G. greatest ?L G (carrier ?L)" ..

   807 next

   808   fix A

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

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

   811     by (fastsimp intro: subgroups_Inter)

   812   have "greatest ?L (\<Inter>A) (Lower ?L A)"

   813     (is "greatest ?L ?Int _")

   814   proof (rule greatest_LowerI)

   815     fix H

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

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

   818     from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)

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

   820       by (rule subgroup_imp_group)

   821     from groupH have monoidH: "monoid ?H"

   822       by (rule group.is_monoid)

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

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

   825   next

   826     fix H

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

   828     with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)

   829   next

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

   831   next

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

   833   qed

   834   then show "EX I. greatest ?L I (Lower ?L A)" ..

   835 qed

   836

   837 end