src/HOL/Algebra/Group.thy
 author ballarin Wed Apr 30 10:01:35 2003 +0200 (2003-04-30) changeset 13936 d3671b878828 parent 13854 91c9ab25fece child 13940 c67798653056 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 {* Algebraic Structures up to Commutative 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_inv_closed [intro, simp]:

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

    98   apply (unfold Units_def m_inv_def)

    99   apply auto

   100   apply (rule theI2, fast)

   101    apply (fast intro: inv_unique)

   102   apply 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)

   108   apply auto

   109   apply (rule theI2, fast)

   110    apply (fast intro: inv_unique)

   111   apply 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)

   117   apply auto

   118   apply (rule theI2, fast)

   119    apply (fast intro: inv_unique)

   120   apply fast

   121   done

   122

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

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

   125 proof -

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

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

   128     by (auto simp add: Units_def

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

   130 qed

   131

   132 lemma (in monoid) Units_l_cancel [simp]:

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

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

   135 proof

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

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

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

   139     by (simp add: m_assoc Units_closed)

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

   141 next

   142   assume eq: "y = z"

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

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

   145 qed

   146

   147 lemma (in monoid) Units_inv_inv [simp]:

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

   149 proof -

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

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

   152     by (simp add: Units_l_inv Units_r_inv)

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

   154 qed

   155

   156 lemma (in monoid) inv_inj_on_Units:

   157   "inj_on (m_inv G) (Units G)"

   158 proof (rule inj_onI)

   159   fix x y

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

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

   162   with G show "x = y" by simp

   163 qed

   164

   165 text {* Power *}

   166

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

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

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

   170

   171 lemma (in monoid) nat_pow_0 [simp]:

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

   173   by (simp add: nat_pow_def)

   174

   175 lemma (in monoid) nat_pow_Suc [simp]:

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

   177   by (simp add: nat_pow_def)

   178

   179 lemma (in monoid) nat_pow_one [simp]:

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

   181   by (induct n) simp_all

   182

   183 lemma (in monoid) nat_pow_mult:

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

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

   186

   187 lemma (in monoid) nat_pow_pow:

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

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

   190

   191 text {*

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

   193 *}

   194

   195 locale group = monoid +

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

   197

   198 theorem groupI:

   199   assumes m_closed [simp]:

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

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

   202     and m_assoc:

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

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

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

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

   207   shows "group G"

   208 proof -

   209   have l_cancel [simp]:

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

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

   212   proof

   213     fix x y z

   214     assume eq: "mult G x y = mult G x z"

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

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

   217       and l_inv: "mult G x_inv x = one G" by fast

   218     from G eq xG have "mult G (mult G x_inv x) y = mult G (mult G x_inv x) z"

   219       by (simp add: m_assoc)

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

   221   next

   222     fix x y z

   223     assume eq: "y = z"

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

   225     then show "mult G x y = mult G x z" by simp

   226   qed

   227   have r_one:

   228     "!!x. x \<in> carrier G ==> mult G x (one G) = x"

   229   proof -

   230     fix x

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

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

   233       and l_inv: "mult G x_inv x = one G" by fast

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

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

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

   237   qed

   238   have inv_ex:

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

   240       mult G x y = one G"

   241   proof -

   242     fix x

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

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

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

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

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

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

   249       by simp

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

   251       mult G x y = one G"

   252       by (fast intro: l_inv r_inv)

   253   qed

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

   255     by (unfold Units_def) fast

   256   show ?thesis

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

   258       semigroup.intro monoid_axioms.intro group_axioms.intro

   259       carrier_subset_Units intro: prems r_one)

   260 qed

   261

   262 lemma (in monoid) monoid_groupI:

   263   assumes l_inv_ex:

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

   265   shows "group G"

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

   267

   268 lemma (in group) Units_eq [simp]:

   269   "Units G = carrier G"

   270 proof

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

   272 next

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

   274 qed

   275

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

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

   278   using Units_inv_closed by simp

   279

   280 lemma (in group) l_inv:

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

   282   using Units_l_inv by simp

   283

   284 subsection {* Cancellation Laws and Basic Properties *}

   285

   286 lemma (in group) l_cancel [simp]:

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

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

   289   using Units_l_inv by simp

   290 (*

   291 lemma (in group) r_one [simp]:

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

   293 proof -

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

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

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

   297   with x show ?thesis by simp

   298 qed

   299 *)

   300 lemma (in group) r_inv:

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

   302 proof -

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

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

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

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

   307 qed

   308

   309 lemma (in group) r_cancel [simp]:

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

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

   312 proof

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

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

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

   316     by (simp add: m_assoc [symmetric])

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

   318 next

   319   assume eq: "y = z"

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

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

   322 qed

   323

   324 lemma (in group) inv_one [simp]:

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

   326 proof -

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

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

   329   finally show ?thesis .

   330 qed

   331

   332 lemma (in group) inv_inv [simp]:

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

   334   using Units_inv_inv by simp

   335

   336 lemma (in group) inv_inj:

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

   338   using inv_inj_on_Units by simp

   339

   340 lemma (in group) inv_mult_group:

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

   342 proof -

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

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

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

   346   with G show ?thesis by simp

   347 qed

   348

   349 text {* Power *}

   350

   351 lemma (in group) int_pow_def2:

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

   353   by (simp add: int_pow_def nat_pow_def Let_def)

   354

   355 lemma (in group) int_pow_0 [simp]:

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

   357   by (simp add: int_pow_def2)

   358

   359 lemma (in group) int_pow_one [simp]:

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

   361   by (simp add: int_pow_def2)

   362

   363 subsection {* Substructures *}

   364

   365 locale submagma = var H + struct G +

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

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

   368

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

   370   semigroup_axioms.intro [intro]

   371 (*

   372 alternative definition of submagma

   373

   374 locale submagma = var H + struct G +

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

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

   377     and m_closed [intro, simp]:

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

   379 *)

   380

   381 lemma submagma_imp_subset:

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

   383   by (rule submagma.subset)

   384

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

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

   387   using subset by blast

   388

   389 lemma (in submagma) magmaI [intro]:

   390   includes magma G

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

   392   by rule simp

   393

   394 lemma (in submagma) semigroup_axiomsI [intro]:

   395   includes semigroup G

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

   397     by rule (simp add: m_assoc)

   398

   399 lemma (in submagma) semigroupI [intro]:

   400   includes semigroup G

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

   402   using prems by fast

   403

   404 locale subgroup = submagma H G +

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

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

   407

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

   409 (*

   410 lemma (in subgroup) l_oneI [intro]:

   411   includes l_one G

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

   413   by rule simp_all

   414 *)

   415 lemma (in subgroup) group_axiomsI [intro]:

   416   includes group G

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

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

   419

   420 lemma (in subgroup) groupI [intro]:

   421   includes group G

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

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

   424

   425 text {*

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

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

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

   429 *}

   430

   431 lemma (in group) one_in_subset:

   432   "[| 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 |]

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

   434 by (force simp add: l_inv)

   435

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

   437

   438 lemma (in group) subgroupI:

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

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

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

   442   shows "subgroup H G"

   443 proof

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

   445 next

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

   447   with inv show "subgroup_axioms H G"

   448     by (intro subgroup_axioms.intro) simp_all

   449 qed

   450

   451 text {*

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

   453 *}

   454

   455 lemma (in subgroup) subset:

   456   "H \<subseteq> carrier G"

   457   ..

   458

   459 lemma (in subgroup) m_closed:

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

   461   ..

   462

   463 declare magma.m_closed [simp]

   464

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

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

   467

   468 lemma subgroup_nonempty:

   469   "~ subgroup {} G"

   470   by (blast dest: subgroup.one_closed)

   471

   472 lemma (in subgroup) finite_imp_card_positive:

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

   474 proof (rule classical)

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

   476   assume fin: "finite (carrier G)"

   477     and zero: "~ 0 < card H"

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

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

   480   with subgroup_nonempty show ?thesis by contradiction

   481 qed

   482

   483 (*

   484 lemma (in monoid) Units_subgroup:

   485   "subgroup (Units G) G"

   486 *)

   487

   488 subsection {* Direct Products *}

   489

   490 constdefs

   491   DirProdSemigroup ::

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

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

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

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

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

   497

   498   DirProdGroup ::

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

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

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

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

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

   504 (*

   505   DirProdGroup ::

   506     "[('a, 'm) group_scheme, ('b, 'n) group_scheme] => ('a \<times> 'b) group"

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

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

   509     mult = mult (G \<times>\<^sub>m H),

   510     one = one (G \<times>\<^sub>m H),

   511     m_inv = (%(g, h). (m_inv G g, m_inv H h)) |)"

   512 *)

   513

   514 lemma DirProdSemigroup_magma:

   515   includes magma G + magma H

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

   517   by rule (auto simp add: DirProdSemigroup_def)

   518

   519 lemma DirProdSemigroup_semigroup_axioms:

   520   includes semigroup G + semigroup H

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

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

   523

   524 lemma DirProdSemigroup_semigroup:

   525   includes semigroup G + semigroup H

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

   527   using prems

   528   by (fast intro: semigroup.intro

   529     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)

   530

   531 lemma DirProdGroup_magma:

   532   includes magma G + magma H

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

   534   by rule

   535     (auto simp add: DirProdGroup_def DirProdSemigroup_def)

   536

   537 lemma DirProdGroup_semigroup_axioms:

   538   includes semigroup G + semigroup H

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

   540   by rule

   541     (auto simp add: DirProdGroup_def DirProdSemigroup_def

   542       G.m_assoc H.m_assoc)

   543

   544 lemma DirProdGroup_semigroup:

   545   includes semigroup G + semigroup H

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

   547   using prems

   548   by (fast intro: semigroup.intro

   549     DirProdGroup_magma DirProdGroup_semigroup_axioms)

   550

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

   552

   553 lemma DirProdGroup_group:

   554   includes group G + group H

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

   556   by (rule groupI)

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

   558       simp add: DirProdGroup_def DirProdSemigroup_def)

   559

   560 subsection {* Homomorphisms *}

   561

   562 constdefs

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

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

   565   "hom G H ==

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

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

   568

   569 lemma (in semigroup) hom:

   570   includes semigroup G

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

   572 proof

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

   574     by rule (simp add: Pi_def hom_def)

   575 next

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

   577     by rule (simp add: o_assoc)

   578 qed

   579

   580 lemma hom_mult:

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

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

   583   by (simp add: hom_def)

   584

   585 lemma hom_closed:

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

   587   by (auto simp add: hom_def funcset_mem)

   588

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

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

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

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

   593

   594 lemma (in group_hom) one_closed [simp]:

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

   596   by simp

   597

   598 lemma (in group_hom) hom_one [simp]:

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

   600 proof -

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

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

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

   604 qed

   605

   606 lemma (in group_hom) inv_closed [simp]:

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

   608   by simp

   609

   610 lemma (in group_hom) hom_inv [simp]:

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

   612 proof -

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

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

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

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

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

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

   619   with x show ?thesis by simp

   620 qed

   621

   622 section {* Commutative Structures *}

   623

   624 text {*

   625   Naming convention: multiplicative structures that are commutative

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

   627   \emph{Abelian}.

   628 *}

   629

   630 subsection {* Definition *}

   631

   632 locale comm_semigroup = semigroup +

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

   634

   635 lemma (in comm_semigroup) m_lcomm:

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

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

   638 proof -

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

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

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

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

   643   finally show ?thesis .

   644 qed

   645

   646 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm

   647

   648 locale comm_monoid = comm_semigroup + monoid

   649

   650 lemma comm_monoidI:

   651   assumes m_closed:

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

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

   654     and m_assoc:

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

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

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

   658     and m_comm:

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

   660   shows "comm_monoid G"

   661   using l_one

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

   663     comm_semigroup_axioms.intro monoid_axioms.intro

   664     intro: prems simp: m_closed one_closed m_comm)

   665

   666 lemma (in monoid) monoid_comm_monoidI:

   667   assumes m_comm:

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

   669   shows "comm_monoid G"

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

   671 (*

   672 lemma (in comm_monoid) r_one [simp]:

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

   674 proof -

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

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

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

   678   finally show ?thesis .

   679 qed

   680 *)

   681

   682 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   686

   687 locale comm_group = comm_monoid + group

   688

   689 lemma (in group) group_comm_groupI:

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

   691       mult G x y = mult G y x"

   692   shows "comm_group G"

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

   694     group.axioms prems)

   695

   696 lemma comm_groupI:

   697   assumes m_closed:

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

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

   700     and m_assoc:

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

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

   703     and m_comm:

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

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

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

   707   shows "comm_group G"

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

   709

   710 lemma (in comm_group) inv_mult:

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

   712   by (simp add: m_ac inv_mult_group)

   713

   714 end