src/HOL/Algebra/Group.thy
 author paulson Wed May 19 11:30:18 2004 +0200 (2004-05-19) changeset 14761 28b5eb4a867f parent 14751 0d7850e27fed child 14803 f7557773cc87 permissions -rw-r--r--
     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 => '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   "Units G == {y. y \<in> carrier G & (EX x : carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"

    36

    37 consts

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

    39

    40 defs (overloaded)

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

    42   int_pow_def: "pow G a z ==

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

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

    45

    46 locale magma = struct G +

    47   assumes m_closed [intro, simp]:

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

    49

    50 locale semigroup = magma +

    51   assumes m_assoc:

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

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

    54

    55 locale monoid = semigroup +

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

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

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

    59

    60 lemma monoidI:

    61   includes struct G

    62   assumes m_closed:

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

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

    65     and m_assoc:

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

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

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

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

    70   shows "monoid G"

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

    72     semigroup.intro monoid_axioms.intro

    73     intro: prems)

    74

    75 lemma (in monoid) Units_closed [dest]:

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

    77   by (unfold Units_def) fast

    78

    79 lemma (in monoid) inv_unique:

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

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

    82   shows "y = y'"

    83 proof -

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

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

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

    87   finally show ?thesis .

    88 qed

    89

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

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

    92   by (unfold Units_def) auto

    93

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

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

    96   apply (unfold Units_def m_inv_def, auto)

    97   apply (rule theI2, fast)

    98    apply (fast intro: inv_unique, fast)

    99   done

   100

   101 lemma (in monoid) Units_l_inv:

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

   103   apply (unfold Units_def m_inv_def, auto)

   104   apply (rule theI2, fast)

   105    apply (fast intro: inv_unique, fast)

   106   done

   107

   108 lemma (in monoid) Units_r_inv:

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

   110   apply (unfold Units_def m_inv_def, auto)

   111   apply (rule theI2, fast)

   112    apply (fast intro: inv_unique, fast)

   113   done

   114

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

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

   117 proof -

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

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

   120     by (auto simp add: Units_def

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

   122 qed

   123

   124 lemma (in monoid) Units_l_cancel [simp]:

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

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

   127 proof

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

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

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

   131     by (simp add: m_assoc Units_closed)

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

   133 next

   134   assume eq: "y = z"

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

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

   137 qed

   138

   139 lemma (in monoid) Units_inv_inv [simp]:

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

   141 proof -

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

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

   144     by (simp add: Units_l_inv Units_r_inv)

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

   146 qed

   147

   148 lemma (in monoid) inv_inj_on_Units:

   149   "inj_on (m_inv G) (Units G)"

   150 proof (rule inj_onI)

   151   fix x y

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

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

   154   with G show "x = y" by simp

   155 qed

   156

   157 lemma (in monoid) Units_inv_comm:

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

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

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

   161 proof -

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

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

   164 qed

   165

   166 text {* Power *}

   167

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

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

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

   171

   172 lemma (in monoid) nat_pow_0 [simp]:

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

   174   by (simp add: nat_pow_def)

   175

   176 lemma (in monoid) nat_pow_Suc [simp]:

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

   178   by (simp add: nat_pow_def)

   179

   180 lemma (in monoid) nat_pow_one [simp]:

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

   182   by (induct n) simp_all

   183

   184 lemma (in monoid) nat_pow_mult:

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

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

   187

   188 lemma (in monoid) nat_pow_pow:

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

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

   191

   192 text {*

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

   194 *}

   195

   196 locale group = monoid +

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

   198

   199

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

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

   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 lemma submagma_imp_subset:

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

   381   by (rule submagma.subset)

   382

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

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

   385   using subset by blast

   386

   387 lemma (in submagma) magmaI [intro]:

   388   includes magma G

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

   390   by rule simp

   391

   392 lemma (in submagma) semigroup_axiomsI [intro]:

   393   includes semigroup G

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

   395     by rule (simp add: m_assoc)

   396

   397 lemma (in submagma) semigroupI [intro]:

   398   includes semigroup G

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

   400   using prems by fast

   401

   402

   403 locale subgroup = submagma H G +

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

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

   406

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

   408

   409 lemma (in subgroup) group_axiomsI [intro]:

   410   includes group G

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

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

   413

   414 lemma (in subgroup) groupI [intro]:

   415   includes group G

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

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

   418

   419 text {*

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

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

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

   423 *}

   424

   425 lemma (in group) one_in_subset:

   426   "[| 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 |]

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

   428 by (force simp add: l_inv)

   429

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

   431

   432 lemma (in group) subgroupI:

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

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

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

   436   shows "subgroup H G"

   437 proof (rule subgroup.intro)

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

   439 next

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

   441   with inv show "subgroup_axioms H G"

   442     by (intro subgroup_axioms.intro) simp_all

   443 qed

   444

   445 text {*

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

   447 *}

   448

   449 lemma (in subgroup) subset:

   450   "H \<subseteq> carrier G"

   451   ..

   452

   453 lemma (in subgroup) m_closed:

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

   455   ..

   456

   457 declare magma.m_closed [simp]

   458

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

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

   461

   462 lemma subgroup_nonempty:

   463   "~ subgroup {} G"

   464   by (blast dest: subgroup.one_closed)

   465

   466 lemma (in subgroup) finite_imp_card_positive:

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

   468 proof (rule classical)

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

   470   assume fin: "finite (carrier G)"

   471     and zero: "~ 0 < card H"

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

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

   474   with subgroup_nonempty show ?thesis by contradiction

   475 qed

   476

   477 (*

   478 lemma (in monoid) Units_subgroup:

   479   "subgroup (Units G) G"

   480 *)

   481

   482 subsection {* Direct Products *}

   483

   484 constdefs (structure G and H)

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

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

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

   488

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

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

   491

   492 lemma DirProdSemigroup_magma:

   493   includes magma G + magma H

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

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

   496

   497 lemma DirProdSemigroup_semigroup_axioms:

   498   includes semigroup G + semigroup H

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

   500   by (rule semigroup_axioms.intro)

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

   502

   503 lemma DirProdSemigroup_semigroup:

   504   includes semigroup G + semigroup H

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

   506   using prems

   507   by (fast intro: semigroup.intro

   508     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)

   509

   510 lemma DirProdGroup_magma:

   511   includes magma G + magma H

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

   513   by (rule magma.intro)

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

   515

   516 lemma DirProdGroup_semigroup_axioms:

   517   includes semigroup G + semigroup H

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

   519   by (rule semigroup_axioms.intro)

   520     (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs

   521       G.m_assoc H.m_assoc)

   522

   523 lemma DirProdGroup_semigroup:

   524   includes semigroup G + semigroup H

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

   526   using prems

   527   by (fast intro: semigroup.intro

   528     DirProdGroup_magma DirProdGroup_semigroup_axioms)

   529

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

   531

   532 lemma DirProdGroup_group:

   533   includes group G + group H

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

   535   by (rule groupI)

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

   537       simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)

   538

   539 lemma carrier_DirProdGroup [simp]:

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

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

   542

   543 lemma one_DirProdGroup [simp]:

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

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

   546

   547 lemma mult_DirProdGroup [simp]:

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

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

   550

   551 lemma inv_DirProdGroup [simp]:

   552   includes group G + group H

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

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

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

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

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

   558   done

   559

   560 subsection {* Isomorphisms *}

   561

   562 constdefs (structure G and H)

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

   564   "hom G H ==

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

   566       (\<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)}"

   567

   568 lemma (in semigroup) hom:

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

   570 proof (rule semigroup.intro)

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

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

   573 next

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

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

   576 qed

   577

   578 lemma hom_mult:

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

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

   581   by (simp add: hom_def)

   582

   583 lemma hom_closed:

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

   585   by (auto simp add: hom_def funcset_mem)

   586

   587 lemma (in group) hom_compose:

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

   589 apply (auto simp add: hom_def funcset_compose)

   590 apply (simp add: compose_def funcset_mem)

   591 done

   592

   593

   594 subsection {* Isomorphisms *}

   595

   596 constdefs (structure G and H)

   597   iso :: "_ => _ => ('a => 'b) set"

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

   599

   600 lemma iso_refl: "(%x. x) \<in> iso G G"

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

   602

   603 lemma (in group) iso_sym:

   604      "h \<in> iso G H \<Longrightarrow> Inv (carrier G) h \<in> iso H G"

   605 apply (simp add: iso_def bij_betw_Inv)

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

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

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

   609 done

   610

   611 lemma (in group) iso_trans:

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

   613 by (auto simp add: iso_def hom_compose bij_betw_compose)

   614

   615 lemma DirProdGroup_commute_iso:

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

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

   618

   619 lemma DirProdGroup_assoc_iso:

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

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

   622

   623

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

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

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

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

   628

   629 lemma (in group_hom) one_closed [simp]:

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

   631   by simp

   632

   633 lemma (in group_hom) hom_one [simp]:

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

   635 proof -

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

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

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

   639 qed

   640

   641 lemma (in group_hom) inv_closed [simp]:

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

   643   by simp

   644

   645 lemma (in group_hom) hom_inv [simp]:

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

   647 proof -

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

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

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

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

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

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

   654   with x show ?thesis by simp

   655 qed

   656

   657 subsection {* Commutative Structures *}

   658

   659 text {*

   660   Naming convention: multiplicative structures that are commutative

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

   662   \emph{Abelian}.

   663 *}

   664

   665 subsection {* Definition *}

   666

   667 locale comm_semigroup = semigroup +

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

   669

   670 lemma (in comm_semigroup) m_lcomm:

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

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

   673 proof -

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

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

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

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

   678   finally show ?thesis .

   679 qed

   680

   681 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm

   682

   683 locale comm_monoid = comm_semigroup + monoid

   684

   685 lemma comm_monoidI:

   686   includes struct G

   687   assumes m_closed:

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

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

   690     and m_assoc:

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

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

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

   694     and m_comm:

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

   696   shows "comm_monoid G"

   697   using l_one

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

   699     comm_semigroup_axioms.intro monoid_axioms.intro

   700     intro: prems simp: m_closed one_closed m_comm)

   701

   702 lemma (in monoid) monoid_comm_monoidI:

   703   assumes m_comm:

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

   705   shows "comm_monoid G"

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

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

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

   709 proof -

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

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

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

   713   finally show ?thesis .

   714 qed*)

   715 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   719

   720 locale comm_group = comm_monoid + group

   721

   722 lemma (in group) group_comm_groupI:

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

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

   725   shows "comm_group G"

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

   727                   is_group prems)

   728

   729 lemma comm_groupI:

   730   includes struct G

   731   assumes m_closed:

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

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

   734     and m_assoc:

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

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

   737     and m_comm:

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

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

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

   741   shows "comm_group G"

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

   743

   744 lemma (in comm_group) inv_mult:

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

   746   by (simp add: m_ac inv_mult_group)

   747

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

   749

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

   751

   752 theorem (in group) subgroups_partial_order:

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

   754   by (rule partial_order.intro) simp_all

   755

   756 lemma (in group) subgroup_self:

   757   "subgroup (carrier G) G"

   758   by (rule subgroupI) auto

   759

   760 lemma (in group) subgroup_imp_group:

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

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

   763

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

   765   "monoid G"

   766   by (rule monoid.intro)

   767

   768 lemma (in group) subgroup_inv_equality:

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

   770 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   775 done

   776

   777 theorem (in group) subgroups_Inter:

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

   779     and not_empty: "A ~= {}"

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

   781 proof (rule subgroupI)

   782   from subgr [THEN subgroup.subset] and not_empty

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

   784 next

   785   from subgr [THEN subgroup.one_closed]

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

   787 next

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

   789   with subgr [THEN subgroup.m_inv_closed]

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

   791 next

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

   793   with subgr [THEN subgroup.m_closed]

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

   795 qed

   796

   797 theorem (in group) subgroups_complete_lattice:

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

   799     (is "complete_lattice ?L")

   800 proof (rule partial_order.complete_lattice_criterion1)

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

   802 next

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

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

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

   806 next

   807   fix A

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

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

   810     by (fastsimp intro: subgroups_Inter)

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

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

   813   proof (rule greatest_LowerI)

   814     fix H

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

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

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

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

   819       by (rule subgroup_imp_group)

   820     from groupH have monoidH: "monoid ?H"

   821       by (rule group.is_monoid)

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

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

   824   next

   825     fix H

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

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

   828   next

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

   830   next

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

   832   qed

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

   834 qed

   835

   836 end