src/HOL/Algebra/Group.thy
 author ballarin Fri May 02 20:02:50 2003 +0200 (2003-05-02) changeset 13949 0ce528cd6f19 parent 13944 9b34607cd83e child 13975 c8e9a89883ce permissions -rw-r--r--
HOL-Algebra complete for release Isabelle2003 (modulo section headers).
     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 (* axclass number < type

    14

    15 instance nat :: number ..

    16 instance int :: number .. *)

    17

    18 section {* From Magmas to Groups *}

    19

    20 text {*

    21   Definitions follow Jacobson, Basic Algebra I, Freeman, 1985; with

    22   the exception of \emph{magma} which, following Bourbaki, is a set

    23   together with a binary, closed operation.

    24 *}

    25

    26 subsection {* Definitions *}

    27

    28 record 'a semigroup =

    29   carrier :: "'a set"

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

    31

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

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

    34

    35 constdefs

    36   m_inv :: "[('a, 'm) monoid_scheme, 'a] => 'a" ("inv\<index> _"  80)

    37   "m_inv G x == (THE y. y \<in> carrier G &

    38                   mult G x y = one G & mult G y x = one G)"

    39

    40   Units :: "('a, 'm) monoid_scheme => 'a set"

    41   "Units G == {y. y \<in> carrier G &

    42                   (EX x : carrier G. mult G x y = one G & mult G y x = one G)}"

    43

    44 consts

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

    46

    47 defs (overloaded)

    48   nat_pow_def: "pow G a n == nat_rec (one G) (%u b. mult G b a) n"

    49   int_pow_def: "pow G a z ==

    50     let p = nat_rec (one G) (%u b. mult G b a)

    51     in if neg z then m_inv G (p (nat (-z))) else p (nat z)"

    52

    53 locale magma = struct G +

    54   assumes m_closed [intro, simp]:

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

    56

    57 locale semigroup = magma +

    58   assumes m_assoc:

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

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

    61

    62 locale monoid = semigroup +

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

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

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

    66

    67 lemma monoidI:

    68   assumes m_closed:

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

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

    71     and m_assoc:

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

    73       mult G (mult G x y) z = mult G x (mult G y z)"

    74     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"

    75     and r_one: "!!x. x \<in> carrier G ==> mult G x (one G) = x"

    76   shows "monoid G"

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

    78     semigroup.intro monoid_axioms.intro

    79     intro: prems)

    80

    81 lemma (in monoid) Units_closed [dest]:

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

    83   by (unfold Units_def) fast

    84

    85 lemma (in monoid) inv_unique:

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

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

    88   shows "y = y'"

    89 proof -

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

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

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

    93   finally show ?thesis .

    94 qed

    95

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

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

    98   by (unfold Units_def) auto

    99

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

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

   102   apply (unfold Units_def m_inv_def, auto)

   103   apply (rule theI2, fast)

   104    apply (fast intro: inv_unique, fast)

   105   done

   106

   107 lemma (in monoid) Units_l_inv:

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

   109   apply (unfold Units_def m_inv_def, auto)

   110   apply (rule theI2, fast)

   111    apply (fast intro: inv_unique, fast)

   112   done

   113

   114 lemma (in monoid) Units_r_inv:

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

   116   apply (unfold Units_def m_inv_def, auto)

   117   apply (rule theI2, fast)

   118    apply (fast intro: inv_unique, fast)

   119   done

   120

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

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

   123 proof -

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

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

   126     by (auto simp add: Units_def

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

   128 qed

   129

   130 lemma (in monoid) Units_l_cancel [simp]:

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

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

   133 proof

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

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

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

   137     by (simp add: m_assoc Units_closed)

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

   139 next

   140   assume eq: "y = z"

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

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

   143 qed

   144

   145 lemma (in monoid) Units_inv_inv [simp]:

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

   147 proof -

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

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

   150     by (simp add: Units_l_inv Units_r_inv)

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

   152 qed

   153

   154 lemma (in monoid) inv_inj_on_Units:

   155   "inj_on (m_inv G) (Units G)"

   156 proof (rule inj_onI)

   157   fix x y

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

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

   160   with G show "x = y" by simp

   161 qed

   162

   163 lemma (in monoid) Units_inv_comm:

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

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

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

   167 proof -

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

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

   170 qed

   171

   172 text {* Power *}

   173

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

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

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

   177

   178 lemma (in monoid) nat_pow_0 [simp]:

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

   180   by (simp add: nat_pow_def)

   181

   182 lemma (in monoid) nat_pow_Suc [simp]:

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

   184   by (simp add: nat_pow_def)

   185

   186 lemma (in monoid) nat_pow_one [simp]:

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

   188   by (induct n) simp_all

   189

   190 lemma (in monoid) nat_pow_mult:

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

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

   193

   194 lemma (in monoid) nat_pow_pow:

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

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

   197

   198 text {*

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

   200 *}

   201

   202 locale group = monoid +

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

   204

   205 theorem groupI:

   206   assumes m_closed [simp]:

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

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

   209     and m_assoc:

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

   211       mult G (mult G x y) z = mult G x (mult G y z)"

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

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

   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     (mult G x y = mult G x z) = (y = z)"

   219   proof

   220     fix x y z

   221     assume eq: "mult G x y = mult G x 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: "mult G x_inv x = one G" by fast

   225     from G eq xG have "mult G (mult G x_inv x) y = mult G (mult G x_inv x) 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 "mult G x y = mult G x z" by simp

   233   qed

   234   have r_one:

   235     "!!x. x \<in> carrier G ==> mult G x (one G) = 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: "mult G x_inv x = one G" by fast

   241     from x xG have "mult G x_inv (mult G x (one G)) = mult G x_inv x"

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

   243     with x xG show "mult G x (one G) = x" by simp

   244   qed

   245   have inv_ex:

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

   247       mult G x y = one G"

   248   proof -

   249     fix x

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

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

   252       and l_inv: "mult G y x = one G" by fast

   253     from x y have "mult G y (mult G x y) = mult G y (one G)"

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

   255     with x y have r_inv: "mult G x y = one G"

   256       by simp

   257     from x y show "EX y : carrier G. mult G y x = one G &

   258       mult G x y = one G"

   259       by (fast intro: l_inv r_inv)

   260   qed

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

   262     by (unfold Units_def) fast

   263   show ?thesis

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

   265       semigroup.intro monoid_axioms.intro group_axioms.intro

   266       carrier_subset_Units intro: prems r_one)

   267 qed

   268

   269 lemma (in monoid) monoid_groupI:

   270   assumes l_inv_ex:

   271     "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"

   272   shows "group G"

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

   274

   275 lemma (in group) Units_eq [simp]:

   276   "Units G = carrier G"

   277 proof

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

   279 next

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

   281 qed

   282

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

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

   285   using Units_inv_closed by simp

   286

   287 lemma (in group) l_inv:

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

   289   using Units_l_inv by simp

   290

   291 subsection {* Cancellation Laws and Basic Properties *}

   292

   293 lemma (in group) l_cancel [simp]:

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

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

   296   using Units_l_inv by simp

   297

   298 lemma (in group) r_inv:

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

   300 proof -

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

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

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

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

   305 qed

   306

   307 lemma (in group) r_cancel [simp]:

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

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

   310 proof

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

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

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

   314     by (simp add: m_assoc [symmetric])

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

   316 next

   317   assume eq: "y = z"

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

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

   320 qed

   321

   322 lemma (in group) inv_one [simp]:

   323   "inv \<one> = \<one>"

   324 proof -

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

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

   327   finally show ?thesis .

   328 qed

   329

   330 lemma (in group) inv_inv [simp]:

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

   332   using Units_inv_inv by simp

   333

   334 lemma (in group) inv_inj:

   335   "inj_on (m_inv G) (carrier G)"

   336   using inv_inj_on_Units by simp

   337

   338 lemma (in group) inv_mult_group:

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

   340 proof -

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

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

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

   344   with G show ?thesis by simp

   345 qed

   346

   347 lemma (in group) inv_comm:

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

   349   by (rule Units_inv_comm) auto

   350

   351 lemma (in group) inv_equality:

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

   353 apply (simp add: m_inv_def)

   354 apply (rule the_equality)

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

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

   357 done

   358

   359 text {* Power *}

   360

   361 lemma (in group) int_pow_def2:

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

   363   by (simp add: int_pow_def nat_pow_def Let_def)

   364

   365 lemma (in group) int_pow_0 [simp]:

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

   367   by (simp add: int_pow_def2)

   368

   369 lemma (in group) int_pow_one [simp]:

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

   371   by (simp add: int_pow_def2)

   372

   373 subsection {* Substructures *}

   374

   375 locale submagma = var H + struct G +

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

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

   378

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

   380   semigroup_axioms.intro [intro]

   381 (*

   382 alternative definition of submagma

   383

   384 locale submagma = var H + struct G +

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

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

   387     and m_closed [intro, simp]:

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

   389 *)

   390

   391 lemma submagma_imp_subset:

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

   393   by (rule submagma.subset)

   394

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

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

   397   using subset by blast

   398

   399 lemma (in submagma) magmaI [intro]:

   400   includes magma G

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

   402   by rule simp

   403

   404 lemma (in submagma) semigroup_axiomsI [intro]:

   405   includes semigroup G

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

   407     by rule (simp add: m_assoc)

   408

   409 lemma (in submagma) semigroupI [intro]:

   410   includes semigroup G

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

   412   using prems by fast

   413

   414 locale subgroup = submagma H G +

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

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

   417

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

   419

   420 lemma (in subgroup) group_axiomsI [intro]:

   421   includes group G

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

   423   by rule (auto intro: l_inv r_inv simp add: Units_def)

   424

   425 lemma (in subgroup) groupI [intro]:

   426   includes group G

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

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

   429

   430 text {*

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

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

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

   434 *}

   435

   436 lemma (in group) one_in_subset:

   437   "[| 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 |]

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

   439 by (force simp add: l_inv)

   440

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

   442

   443 lemma (in group) subgroupI:

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

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

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

   447   shows "subgroup H G"

   448 proof

   449   from subset and mult show "submagma H G" ..

   450 next

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

   452   with inv show "subgroup_axioms H G"

   453     by (intro subgroup_axioms.intro) simp_all

   454 qed

   455

   456 text {*

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

   458 *}

   459

   460 lemma (in subgroup) subset:

   461   "H \<subseteq> carrier G"

   462   ..

   463

   464 lemma (in subgroup) m_closed:

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

   466   ..

   467

   468 declare magma.m_closed [simp]

   469

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

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

   472

   473 lemma subgroup_nonempty:

   474   "~ subgroup {} G"

   475   by (blast dest: subgroup.one_closed)

   476

   477 lemma (in subgroup) finite_imp_card_positive:

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

   479 proof (rule classical)

   480   have sub: "subgroup H G" using prems ..

   481   assume fin: "finite (carrier G)"

   482     and zero: "~ 0 < card H"

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

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

   485   with subgroup_nonempty show ?thesis by contradiction

   486 qed

   487

   488 (*

   489 lemma (in monoid) Units_subgroup:

   490   "subgroup (Units G) G"

   491 *)

   492

   493 subsection {* Direct Products *}

   494

   495 constdefs

   496   DirProdSemigroup ::

   497     "[('a, 'm) semigroup_scheme, ('b, 'n) semigroup_scheme]

   498     => ('a \<times> 'b) semigroup"

   499     (infixr "\<times>\<^sub>s" 80)

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

   501     mult = (%(xg, xh) (yg, yh). (mult G xg yg, mult H xh yh)) |)"

   502

   503   DirProdGroup ::

   504     "[('a, 'm) monoid_scheme, ('b, 'n) monoid_scheme] => ('a \<times> 'b) monoid"

   505     (infixr "\<times>\<^sub>g" 80)

   506   "G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>s H),

   507     mult = mult (G \<times>\<^sub>s H),

   508     one = (one G, one H) |)"

   509

   510 lemma DirProdSemigroup_magma:

   511   includes magma G + magma H

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

   513   by rule (auto simp add: DirProdSemigroup_def)

   514

   515 lemma DirProdSemigroup_semigroup_axioms:

   516   includes semigroup G + semigroup H

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

   518   by rule (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)

   519

   520 lemma DirProdSemigroup_semigroup:

   521   includes semigroup G + semigroup H

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

   523   using prems

   524   by (fast intro: semigroup.intro

   525     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)

   526

   527 lemma DirProdGroup_magma:

   528   includes magma G + magma H

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

   530   by rule

   531     (auto simp add: DirProdGroup_def DirProdSemigroup_def)

   532

   533 lemma DirProdGroup_semigroup_axioms:

   534   includes semigroup G + semigroup H

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

   536   by rule

   537     (auto simp add: DirProdGroup_def DirProdSemigroup_def

   538       G.m_assoc H.m_assoc)

   539

   540 lemma DirProdGroup_semigroup:

   541   includes semigroup G + semigroup H

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

   543   using prems

   544   by (fast intro: semigroup.intro

   545     DirProdGroup_magma DirProdGroup_semigroup_axioms)

   546

   547 (* ... and further lemmas for group ... *)

   548

   549 lemma DirProdGroup_group:

   550   includes group G + group H

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

   552   by (rule groupI)

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

   554       simp add: DirProdGroup_def DirProdSemigroup_def)

   555

   556 lemma carrier_DirProdGroup [simp]:

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

   558   by (simp add: DirProdGroup_def DirProdSemigroup_def)

   559

   560 lemma one_DirProdGroup [simp]:

   561      "one (G \<times>\<^sub>g H) = (one G, one H)"

   562   by (simp add: DirProdGroup_def DirProdSemigroup_def);

   563

   564 lemma mult_DirProdGroup [simp]:

   565      "mult (G \<times>\<^sub>g H) (g, h) (g', h') = (mult G g g', mult H h h')"

   566   by (simp add: DirProdGroup_def DirProdSemigroup_def)

   567

   568 lemma inv_DirProdGroup [simp]:

   569   includes group G + group H

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

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

   572   shows "m_inv (G \<times>\<^sub>g H) (g, h) = (m_inv G g, m_inv H h)"

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

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

   575   done

   576

   577 subsection {* Homomorphisms *}

   578

   579 constdefs

   580   hom :: "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme]

   581     => ('a => 'b)set"

   582   "hom G H ==

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

   584       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}"

   585

   586 lemma (in semigroup) hom:

   587   includes semigroup G

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

   589 proof

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

   591     by rule (simp add: Pi_def hom_def)

   592 next

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

   594     by rule (simp add: o_assoc)

   595 qed

   596

   597 lemma hom_mult:

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

   599    ==> h (mult G x y) = mult H (h x) (h y)"

   600   by (simp add: hom_def)

   601

   602 lemma hom_closed:

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

   604   by (auto simp add: hom_def funcset_mem)

   605

   606 lemma compose_hom:

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

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

   609 apply (simp (no_asm_simp) add: hom_def)

   610 apply (intro conjI)

   611  apply (force simp add: funcset_compose hom_def)

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

   613 done

   614

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

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

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

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

   619

   620 lemma (in group_hom) one_closed [simp]:

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

   622   by simp

   623

   624 lemma (in group_hom) hom_one [simp]:

   625   "h \<one> = \<one>\<^sub>2"

   626 proof -

   627   have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"

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

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

   630 qed

   631

   632 lemma (in group_hom) inv_closed [simp]:

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

   634   by simp

   635

   636 lemma (in group_hom) hom_inv [simp]:

   637   "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"

   638 proof -

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

   640   then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"

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

   642   also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"

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

   644   finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .

   645   with x show ?thesis by simp

   646 qed

   647

   648 subsection {* Commutative Structures *}

   649

   650 text {*

   651   Naming convention: multiplicative structures that are commutative

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

   653   \emph{Abelian}.

   654 *}

   655

   656 subsection {* Definition *}

   657

   658 locale comm_semigroup = semigroup +

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

   660

   661 lemma (in comm_semigroup) m_lcomm:

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

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

   664 proof -

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

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

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

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

   669   finally show ?thesis .

   670 qed

   671

   672 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm

   673

   674 locale comm_monoid = comm_semigroup + monoid

   675

   676 lemma comm_monoidI:

   677   assumes m_closed:

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

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

   680     and m_assoc:

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

   682       mult G (mult G x y) z = mult G x (mult G y z)"

   683     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"

   684     and m_comm:

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

   686   shows "comm_monoid G"

   687   using l_one

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

   689     comm_semigroup_axioms.intro monoid_axioms.intro

   690     intro: prems simp: m_closed one_closed m_comm)

   691

   692 lemma (in monoid) monoid_comm_monoidI:

   693   assumes m_comm:

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

   695   shows "comm_monoid G"

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

   697 (*

   698 lemma (in comm_monoid) r_one [simp]:

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

   700 proof -

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

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

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

   704   finally show ?thesis .

   705 qed

   706 *)

   707

   708 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   712

   713 locale comm_group = comm_monoid + group

   714

   715 lemma (in group) group_comm_groupI:

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

   717       mult G x y = mult G y x"

   718   shows "comm_group G"

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

   720     group.axioms prems)

   721

   722 lemma comm_groupI:

   723   assumes m_closed:

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

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

   726     and m_assoc:

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

   728       mult G (mult G x y) z = mult G x (mult G y z)"

   729     and m_comm:

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

   731     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"

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

   733   shows "comm_group G"

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

   735

   736 lemma (in comm_group) inv_mult:

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

   738   by (simp add: m_ac inv_mult_group)

   739

   740 end