src/HOL/Algebra/Group.thy
 author wenzelm Thu May 06 14:14:18 2004 +0200 (2004-05-06) changeset 14706 71590b7733b7 parent 14693 4deda204e1d8 child 14751 0d7850e27fed permissions -rw-r--r--
tuned document;
     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:

    12

    13 section {* From Magmas to Groups *}

    14

    15 text {*

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

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

    18   binary, closed operation.

    19 *}

    20

    21 subsection {* Definitions *}

    22

    23 (* Object with a carrier set. *)

    24

    25 record 'a partial_object =

    26   carrier :: "'a set"

    27

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

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

    30

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

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

    33

    34 constdefs (structure G)

    35   m_inv :: "_ => 'a => 'a" ("inv\<index> _"  80)

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

    37

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

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

    40

    41 consts

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

    43

    44 defs (overloaded)

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

    46   int_pow_def: "pow G a z ==

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

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

    49

    50 locale magma = struct G +

    51   assumes m_closed [intro, simp]:

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

    53

    54 locale semigroup = magma +

    55   assumes m_assoc:

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

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

    58

    59 locale monoid = semigroup +

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

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

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

    63

    64 lemma monoidI:

    65   includes struct G

    66   assumes m_closed:

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

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

    69     and m_assoc:

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

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

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

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

    74   shows "monoid G"

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

    76     semigroup.intro monoid_axioms.intro

    77     intro: prems)

    78

    79 lemma (in monoid) Units_closed [dest]:

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

    81   by (unfold Units_def) fast

    82

    83 lemma (in monoid) inv_unique:

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

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

    86   shows "y = y'"

    87 proof -

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

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

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

    91   finally show ?thesis .

    92 qed

    93

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

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

    96   by (unfold Units_def) auto

    97

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

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

   100   apply (unfold Units_def m_inv_def, auto)

   101   apply (rule theI2, fast)

   102    apply (fast intro: inv_unique, fast)

   103   done

   104

   105 lemma (in monoid) Units_l_inv:

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

   107   apply (unfold Units_def m_inv_def, auto)

   108   apply (rule theI2, fast)

   109    apply (fast intro: inv_unique, fast)

   110   done

   111

   112 lemma (in monoid) Units_r_inv:

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

   114   apply (unfold Units_def m_inv_def, auto)

   115   apply (rule theI2, fast)

   116    apply (fast intro: inv_unique, fast)

   117   done

   118

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

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

   121 proof -

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

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

   124     by (auto simp add: Units_def

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

   126 qed

   127

   128 lemma (in monoid) Units_l_cancel [simp]:

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

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

   131 proof

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

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

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

   135     by (simp add: m_assoc Units_closed)

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

   137 next

   138   assume eq: "y = z"

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

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

   141 qed

   142

   143 lemma (in monoid) Units_inv_inv [simp]:

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

   145 proof -

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

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

   148     by (simp add: Units_l_inv Units_r_inv)

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

   150 qed

   151

   152 lemma (in monoid) inv_inj_on_Units:

   153   "inj_on (m_inv G) (Units G)"

   154 proof (rule inj_onI)

   155   fix x y

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

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

   158   with G show "x = y" by simp

   159 qed

   160

   161 lemma (in monoid) Units_inv_comm:

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

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

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

   165 proof -

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

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

   168 qed

   169

   170 text {* Power *}

   171

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

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

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

   175

   176 lemma (in monoid) nat_pow_0 [simp]:

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

   178   by (simp add: nat_pow_def)

   179

   180 lemma (in monoid) nat_pow_Suc [simp]:

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

   182   by (simp add: nat_pow_def)

   183

   184 lemma (in monoid) nat_pow_one [simp]:

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

   186   by (induct n) simp_all

   187

   188 lemma (in monoid) nat_pow_mult:

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

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

   191

   192 lemma (in monoid) nat_pow_pow:

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

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

   195

   196 text {*

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

   198 *}

   199

   200 locale group = monoid +

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

   202

   203 theorem groupI:

   204   includes struct G

   205   assumes m_closed [simp]:

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

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

   208     and m_assoc:

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

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

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

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

   213   shows "group G"

   214 proof -

   215   have l_cancel [simp]:

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

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

   218   proof

   219     fix x y z

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

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

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

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

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

   225       by (simp add: m_assoc)

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

   227   next

   228     fix x y z

   229     assume eq: "y = z"

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

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

   232   qed

   233   have r_one:

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

   235   proof -

   236     fix x

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

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

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

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

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

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

   243   qed

   244   have inv_ex:

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

   246   proof -

   247     fix x

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

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

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

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

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

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

   254       by simp

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

   256       by (fast intro: l_inv r_inv)

   257   qed

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

   259     by (unfold Units_def) fast

   260   show ?thesis

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

   262       semigroup.intro monoid_axioms.intro group_axioms.intro

   263       carrier_subset_Units intro: prems r_one)

   264 qed

   265

   266 lemma (in monoid) monoid_groupI:

   267   assumes l_inv_ex:

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

   269   shows "group G"

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

   271

   272 lemma (in group) Units_eq [simp]:

   273   "Units G = carrier G"

   274 proof

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

   276 next

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

   278 qed

   279

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

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

   282   using Units_inv_closed by simp

   283

   284 lemma (in group) l_inv:

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

   286   using Units_l_inv by simp

   287

   288 subsection {* Cancellation Laws and Basic Properties *}

   289

   290 lemma (in group) l_cancel [simp]:

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

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

   293   using Units_l_inv by simp

   294

   295 lemma (in group) r_inv:

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

   297 proof -

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

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

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

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

   302 qed

   303

   304 lemma (in group) r_cancel [simp]:

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

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

   307 proof

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

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

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

   311     by (simp add: m_assoc [symmetric])

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

   313 next

   314   assume eq: "y = z"

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

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

   317 qed

   318

   319 lemma (in group) inv_one [simp]:

   320   "inv \<one> = \<one>"

   321 proof -

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

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

   324   finally show ?thesis .

   325 qed

   326

   327 lemma (in group) inv_inv [simp]:

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

   329   using Units_inv_inv by simp

   330

   331 lemma (in group) inv_inj:

   332   "inj_on (m_inv G) (carrier G)"

   333   using inv_inj_on_Units by simp

   334

   335 lemma (in group) inv_mult_group:

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

   337 proof -

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

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

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

   341   with G show ?thesis by simp

   342 qed

   343

   344 lemma (in group) inv_comm:

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

   346   by (rule Units_inv_comm) auto

   347

   348 lemma (in group) inv_equality:

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

   350 apply (simp add: m_inv_def)

   351 apply (rule the_equality)

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

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

   354 done

   355

   356 text {* Power *}

   357

   358 lemma (in group) int_pow_def2:

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

   360   by (simp add: int_pow_def nat_pow_def Let_def)

   361

   362 lemma (in group) int_pow_0 [simp]:

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

   364   by (simp add: int_pow_def2)

   365

   366 lemma (in group) int_pow_one [simp]:

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

   368   by (simp add: int_pow_def2)

   369

   370 subsection {* Substructures *}

   371

   372 locale submagma = var H + struct G +

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

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

   375

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

   377   semigroup_axioms.intro [intro]

   378 (*

   379 alternative definition of submagma

   380

   381 locale submagma = var H + struct G +

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

   383     and m_equal [simp]: "mult H = mult G"

   384     and m_closed [intro, simp]:

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

   386 *)

   387

   388 lemma submagma_imp_subset:

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

   390   by (rule submagma.subset)

   391

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

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

   394   using subset by blast

   395

   396 lemma (in submagma) magmaI [intro]:

   397   includes magma G

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

   399   by rule simp

   400

   401 lemma (in submagma) semigroup_axiomsI [intro]:

   402   includes semigroup G

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

   404     by rule (simp add: m_assoc)

   405

   406 lemma (in submagma) semigroupI [intro]:

   407   includes semigroup G

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

   409   using prems by fast

   410

   411

   412 locale subgroup = submagma H G +

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

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

   415

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

   417

   418 lemma (in subgroup) group_axiomsI [intro]:

   419   includes group G

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

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

   422

   423 lemma (in subgroup) groupI [intro]:

   424   includes group G

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

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

   427

   428 text {*

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

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

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

   432 *}

   433

   434 lemma (in group) one_in_subset:

   435   "[| 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 |]

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

   437 by (force simp add: l_inv)

   438

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

   440

   441 lemma (in group) subgroupI:

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

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

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

   445   shows "subgroup H G"

   446 proof (rule subgroup.intro)

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

   448 next

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

   450   with inv show "subgroup_axioms H G"

   451     by (intro subgroup_axioms.intro) simp_all

   452 qed

   453

   454 text {*

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

   456 *}

   457

   458 lemma (in subgroup) subset:

   459   "H \<subseteq> carrier G"

   460   ..

   461

   462 lemma (in subgroup) m_closed:

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

   464   ..

   465

   466 declare magma.m_closed [simp]

   467

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

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

   470

   471 lemma subgroup_nonempty:

   472   "~ subgroup {} G"

   473   by (blast dest: subgroup.one_closed)

   474

   475 lemma (in subgroup) finite_imp_card_positive:

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

   477 proof (rule classical)

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

   479   assume fin: "finite (carrier G)"

   480     and zero: "~ 0 < card H"

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

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

   483   with subgroup_nonempty show ?thesis by contradiction

   484 qed

   485

   486 (*

   487 lemma (in monoid) Units_subgroup:

   488   "subgroup (Units G) G"

   489 *)

   490

   491 subsection {* Direct Products *}

   492

   493 constdefs (structure G and H)

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

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

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

   497

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

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

   500

   501 lemma DirProdSemigroup_magma:

   502   includes magma G + magma H

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

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

   505

   506 lemma DirProdSemigroup_semigroup_axioms:

   507   includes semigroup G + semigroup H

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

   509   by (rule semigroup_axioms.intro)

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

   511

   512 lemma DirProdSemigroup_semigroup:

   513   includes semigroup G + semigroup H

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

   515   using prems

   516   by (fast intro: semigroup.intro

   517     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)

   518

   519 lemma DirProdGroup_magma:

   520   includes magma G + magma H

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

   522   by (rule magma.intro)

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

   524

   525 lemma DirProdGroup_semigroup_axioms:

   526   includes semigroup G + semigroup H

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

   528   by (rule semigroup_axioms.intro)

   529     (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs

   530       G.m_assoc H.m_assoc)

   531

   532 lemma DirProdGroup_semigroup:

   533   includes semigroup G + semigroup H

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

   535   using prems

   536   by (fast intro: semigroup.intro

   537     DirProdGroup_magma DirProdGroup_semigroup_axioms)

   538

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

   540

   541 lemma DirProdGroup_group:

   542   includes group G + group H

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

   544   by (rule groupI)

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

   546       simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   547

   548 lemma carrier_DirProdGroup [simp]:

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

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

   551

   552 lemma one_DirProdGroup [simp]:

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

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

   555

   556 lemma mult_DirProdGroup [simp]:

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

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

   559

   560 lemma inv_DirProdGroup [simp]:

   561   includes group G + group H

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

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

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

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

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

   567   done

   568

   569 subsection {* Homomorphisms *}

   570

   571 constdefs (structure G and H)

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

   573   "hom G H ==

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

   575       (\<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)}"

   576

   577 lemma (in semigroup) hom:

   578   includes semigroup G

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

   580 proof (rule semigroup.intro)

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

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

   583 next

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

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

   586 qed

   587

   588 lemma hom_mult:

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

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

   591   by (simp add: hom_def)

   592

   593 lemma hom_closed:

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

   595   by (auto simp add: hom_def funcset_mem)

   596

   597 lemma compose_hom:

   598      "[|group G; h \<in> hom G G; h' \<in> hom G G; h' \<in> carrier G -> carrier G|]

   599       ==> compose (carrier G) h h' \<in> hom G G"

   600 apply (simp (no_asm_simp) add: hom_def)

   601 apply (intro conjI)

   602  apply (force simp add: funcset_compose hom_def)

   603 apply (simp add: compose_def group.axioms hom_mult funcset_mem)

   604 done

   605

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

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

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

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

   610

   611 lemma (in group_hom) one_closed [simp]:

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

   613   by simp

   614

   615 lemma (in group_hom) hom_one [simp]:

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

   617 proof -

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

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

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

   621 qed

   622

   623 lemma (in group_hom) inv_closed [simp]:

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

   625   by simp

   626

   627 lemma (in group_hom) hom_inv [simp]:

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

   629 proof -

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

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

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

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

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

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

   636   with x show ?thesis by simp

   637 qed

   638

   639 subsection {* Commutative Structures *}

   640

   641 text {*

   642   Naming convention: multiplicative structures that are commutative

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

   644   \emph{Abelian}.

   645 *}

   646

   647 subsection {* Definition *}

   648

   649 locale comm_semigroup = semigroup +

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

   651

   652 lemma (in comm_semigroup) m_lcomm:

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

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

   655 proof -

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

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

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

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

   660   finally show ?thesis .

   661 qed

   662

   663 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm

   664

   665 locale comm_monoid = comm_semigroup + monoid

   666

   667 lemma comm_monoidI:

   668   includes struct G

   669   assumes m_closed:

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

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

   672     and m_assoc:

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

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

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

   676     and m_comm:

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

   678   shows "comm_monoid G"

   679   using l_one

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

   681     comm_semigroup_axioms.intro monoid_axioms.intro

   682     intro: prems simp: m_closed one_closed m_comm)

   683

   684 lemma (in monoid) monoid_comm_monoidI:

   685   assumes m_comm:

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

   687   shows "comm_monoid G"

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

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

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

   691 proof -

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

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

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

   695   finally show ?thesis .

   696 qed*)

   697 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   701

   702 locale comm_group = comm_monoid + group

   703

   704 lemma (in group) group_comm_groupI:

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

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

   707   shows "comm_group G"

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

   709     group.axioms prems)

   710

   711 lemma comm_groupI:

   712   includes struct G

   713   assumes m_closed:

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

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

   716     and m_assoc:

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

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

   719     and m_comm:

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

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

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

   723   shows "comm_group G"

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

   725

   726 lemma (in comm_group) inv_mult:

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

   728   by (simp add: m_ac inv_mult_group)

   729

   730 end