src/HOL/Algebra/Group.thy
author paulson
Tue Jun 01 11:25:01 2004 +0200 (2004-06-01)
changeset 14852 fffab59e0050
parent 14803 f7557773cc87
child 14963 d584e32f7d46
permissions -rw-r--r--
tidied
     1 (*
     2   Title:  HOL/Algebra/Group.thy
     3   Id:     $Id$
     4   Author: Clemens Ballarin, started 4 February 2003
     5 
     6 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
     7 *)
     8 
     9 header {* Groups *}
    10 
    11 theory Group = FuncSet + Lattice:
    12 
    13 
    14 section {* From Magmas to Groups *}
    15 
    16 text {*
    17   Definitions follow \cite{Jacobson:1985}; with the exception of
    18   \emph{magma} which, following Bourbaki, is a set together with a
    19   binary, closed operation.
    20 *}
    21 
    22 subsection {* Definitions *}
    23 
    24 record 'a semigroup = "'a partial_object" +
    25   mult :: "['a, 'a] => 'a" (infixl "\<otimes>\<index>" 70)
    26 
    27 record 'a monoid = "'a semigroup" +
    28   one :: 'a ("\<one>\<index>")
    29 
    30 constdefs (structure G)
    31   m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80)
    32   "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"
    33 
    34   Units :: "_ => 'a set"
    35   --{*The set of invertible elements*}
    36   "Units G == {y. y \<in> carrier G & (EX x : carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"
    37 
    38 consts
    39   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)
    40 
    41 defs (overloaded)
    42   nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
    43   int_pow_def: "pow G a z ==
    44     let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
    45     in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"
    46 
    47 locale magma = struct G +
    48   assumes m_closed [intro, simp]:
    49     "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
    50 
    51 locale semigroup = magma +
    52   assumes m_assoc:
    53     "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
    54     (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
    55 
    56 locale monoid = semigroup +
    57   assumes one_closed [intro, simp]: "\<one> \<in> carrier G"
    58     and l_one [simp]: "x \<in> carrier G ==> \<one> \<otimes> x = x"
    59     and r_one [simp]: "x \<in> carrier G ==> x \<otimes> \<one> = x"
    60 
    61 lemma monoidI:
    62   includes struct G
    63   assumes m_closed:
    64       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
    65     and one_closed: "\<one> \<in> carrier G"
    66     and m_assoc:
    67       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
    68       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
    69     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
    70     and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
    71   shows "monoid G"
    72   by (fast intro!: monoid.intro magma.intro semigroup_axioms.intro
    73     semigroup.intro monoid_axioms.intro
    74     intro: prems)
    75 
    76 lemma (in monoid) Units_closed [dest]:
    77   "x \<in> Units G ==> x \<in> carrier G"
    78   by (unfold Units_def) fast
    79 
    80 lemma (in monoid) inv_unique:
    81   assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"
    82     and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
    83   shows "y = y'"
    84 proof -
    85   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp
    86   also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)
    87   also from G eq have "... = y'" by simp
    88   finally show ?thesis .
    89 qed
    90 
    91 lemma (in monoid) Units_one_closed [intro, simp]:
    92   "\<one> \<in> Units G"
    93   by (unfold Units_def) auto
    94 
    95 lemma (in monoid) Units_inv_closed [intro, simp]:
    96   "x \<in> Units G ==> inv x \<in> carrier G"
    97   apply (unfold Units_def m_inv_def, auto)
    98   apply (rule theI2, fast)
    99    apply (fast intro: inv_unique, fast)
   100   done
   101 
   102 lemma (in monoid) Units_l_inv:
   103   "x \<in> Units G ==> inv x \<otimes> x = \<one>"
   104   apply (unfold Units_def m_inv_def, auto)
   105   apply (rule theI2, fast)
   106    apply (fast intro: inv_unique, fast)
   107   done
   108 
   109 lemma (in monoid) Units_r_inv:
   110   "x \<in> Units G ==> x \<otimes> inv x = \<one>"
   111   apply (unfold Units_def m_inv_def, auto)
   112   apply (rule theI2, fast)
   113    apply (fast intro: inv_unique, fast)
   114   done
   115 
   116 lemma (in monoid) Units_inv_Units [intro, simp]:
   117   "x \<in> Units G ==> inv x \<in> Units G"
   118 proof -
   119   assume x: "x \<in> Units G"
   120   show "inv x \<in> Units G"
   121     by (auto simp add: Units_def
   122       intro: Units_l_inv Units_r_inv x Units_closed [OF x])
   123 qed
   124 
   125 lemma (in monoid) Units_l_cancel [simp]:
   126   "[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==>
   127    (x \<otimes> y = x \<otimes> z) = (y = z)"
   128 proof
   129   assume eq: "x \<otimes> y = x \<otimes> z"
   130     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
   131   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"
   132     by (simp add: m_assoc Units_closed)
   133   with G show "y = z" by (simp add: Units_l_inv)
   134 next
   135   assume eq: "y = z"
   136     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
   137   then show "x \<otimes> y = x \<otimes> z" by simp
   138 qed
   139 
   140 lemma (in monoid) Units_inv_inv [simp]:
   141   "x \<in> Units G ==> inv (inv x) = x"
   142 proof -
   143   assume x: "x \<in> Units G"
   144   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x"
   145     by (simp add: Units_l_inv Units_r_inv)
   146   with x show ?thesis by (simp add: Units_closed)
   147 qed
   148 
   149 lemma (in monoid) inv_inj_on_Units:
   150   "inj_on (m_inv G) (Units G)"
   151 proof (rule inj_onI)
   152   fix x y
   153   assume G: "x \<in> Units G"  "y \<in> Units G" and eq: "inv x = inv y"
   154   then have "inv (inv x) = inv (inv y)" by simp
   155   with G show "x = y" by simp
   156 qed
   157 
   158 lemma (in monoid) Units_inv_comm:
   159   assumes inv: "x \<otimes> y = \<one>"
   160     and G: "x \<in> Units G"  "y \<in> Units G"
   161   shows "y \<otimes> x = \<one>"
   162 proof -
   163   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
   164   with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
   165 qed
   166 
   167 text {* Power *}
   168 
   169 lemma (in monoid) nat_pow_closed [intro, simp]:
   170   "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"
   171   by (induct n) (simp_all add: nat_pow_def)
   172 
   173 lemma (in monoid) nat_pow_0 [simp]:
   174   "x (^) (0::nat) = \<one>"
   175   by (simp add: nat_pow_def)
   176 
   177 lemma (in monoid) nat_pow_Suc [simp]:
   178   "x (^) (Suc n) = x (^) n \<otimes> x"
   179   by (simp add: nat_pow_def)
   180 
   181 lemma (in monoid) nat_pow_one [simp]:
   182   "\<one> (^) (n::nat) = \<one>"
   183   by (induct n) simp_all
   184 
   185 lemma (in monoid) nat_pow_mult:
   186   "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"
   187   by (induct m) (simp_all add: m_assoc [THEN sym])
   188 
   189 lemma (in monoid) nat_pow_pow:
   190   "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
   191   by (induct m) (simp, simp add: nat_pow_mult add_commute)
   192 
   193 text {*
   194   A group is a monoid all of whose elements are invertible.
   195 *}
   196 
   197 locale group = monoid +
   198   assumes Units: "carrier G <= Units G"
   199 
   200 
   201 lemma (in group) is_group: "group G"
   202   by (rule group.intro [OF prems]) 
   203 
   204 theorem groupI:
   205   includes struct G
   206   assumes m_closed [simp]:
   207       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
   208     and one_closed [simp]: "\<one> \<in> carrier G"
   209     and m_assoc:
   210       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   211       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
   212     and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
   213     and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
   214   shows "group G"
   215 proof -
   216   have l_cancel [simp]:
   217     "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   218     (x \<otimes> y = x \<otimes> z) = (y = z)"
   219   proof
   220     fix x y z
   221     assume eq: "x \<otimes> y = x \<otimes> z"
   222       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   223     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
   224       and l_inv: "x_inv \<otimes> x = \<one>" by fast
   225     from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"
   226       by (simp add: m_assoc)
   227     with G show "y = z" by (simp add: l_inv)
   228   next
   229     fix x y z
   230     assume eq: "y = z"
   231       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   232     then show "x \<otimes> y = x \<otimes> z" by simp
   233   qed
   234   have r_one:
   235     "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
   236   proof -
   237     fix x
   238     assume x: "x \<in> carrier G"
   239     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
   240       and l_inv: "x_inv \<otimes> x = \<one>" by fast
   241     from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"
   242       by (simp add: m_assoc [symmetric] l_inv)
   243     with x xG show "x \<otimes> \<one> = x" by simp
   244   qed
   245   have inv_ex:
   246     "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
   247   proof -
   248     fix x
   249     assume x: "x \<in> carrier G"
   250     with l_inv_ex obtain y where y: "y \<in> carrier G"
   251       and l_inv: "y \<otimes> x = \<one>" by fast
   252     from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"
   253       by (simp add: m_assoc [symmetric] l_inv r_one)
   254     with x y have r_inv: "x \<otimes> y = \<one>"
   255       by simp
   256     from x y show "EX y : carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
   257       by (fast intro: l_inv r_inv)
   258   qed
   259   then have carrier_subset_Units: "carrier G <= Units G"
   260     by (unfold Units_def) fast
   261   show ?thesis
   262     by (fast intro!: group.intro magma.intro semigroup_axioms.intro
   263       semigroup.intro monoid_axioms.intro group_axioms.intro
   264       carrier_subset_Units intro: prems r_one)
   265 qed
   266 
   267 lemma (in monoid) monoid_groupI:
   268   assumes l_inv_ex:
   269     "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
   270   shows "group G"
   271   by (rule groupI) (auto intro: m_assoc l_inv_ex)
   272 
   273 lemma (in group) Units_eq [simp]:
   274   "Units G = carrier G"
   275 proof
   276   show "Units G <= carrier G" by fast
   277 next
   278   show "carrier G <= Units G" by (rule Units)
   279 qed
   280 
   281 lemma (in group) inv_closed [intro, simp]:
   282   "x \<in> carrier G ==> inv x \<in> carrier G"
   283   using Units_inv_closed by simp
   284 
   285 lemma (in group) l_inv:
   286   "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
   287   using Units_l_inv by simp
   288 
   289 subsection {* Cancellation Laws and Basic Properties *}
   290 
   291 lemma (in group) l_cancel [simp]:
   292   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   293    (x \<otimes> y = x \<otimes> z) = (y = z)"
   294   using Units_l_inv by simp
   295 
   296 lemma (in group) r_inv:
   297   "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
   298 proof -
   299   assume x: "x \<in> carrier G"
   300   then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
   301     by (simp add: m_assoc [symmetric] l_inv)
   302   with x show ?thesis by (simp del: r_one)
   303 qed
   304 
   305 lemma (in group) r_cancel [simp]:
   306   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   307    (y \<otimes> x = z \<otimes> x) = (y = z)"
   308 proof
   309   assume eq: "y \<otimes> x = z \<otimes> x"
   310     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   311   then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
   312     by (simp add: m_assoc [symmetric])
   313   with G show "y = z" by (simp add: r_inv)
   314 next
   315   assume eq: "y = z"
   316     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   317   then show "y \<otimes> x = z \<otimes> x" by simp
   318 qed
   319 
   320 lemma (in group) inv_one [simp]:
   321   "inv \<one> = \<one>"
   322 proof -
   323   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by simp
   324   moreover have "... = \<one>" by (simp add: r_inv)
   325   finally show ?thesis .
   326 qed
   327 
   328 lemma (in group) inv_inv [simp]:
   329   "x \<in> carrier G ==> inv (inv x) = x"
   330   using Units_inv_inv by simp
   331 
   332 lemma (in group) inv_inj:
   333   "inj_on (m_inv G) (carrier G)"
   334   using inv_inj_on_Units by simp
   335 
   336 lemma (in group) inv_mult_group:
   337   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
   338 proof -
   339   assume G: "x \<in> carrier G"  "y \<in> carrier G"
   340   then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
   341     by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv)
   342   with G show ?thesis by simp
   343 qed
   344 
   345 lemma (in group) inv_comm:
   346   "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
   347   by (rule Units_inv_comm) auto
   348 
   349 lemma (in group) inv_equality:
   350      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
   351 apply (simp add: m_inv_def)
   352 apply (rule the_equality)
   353  apply (simp add: inv_comm [of y x])
   354 apply (rule r_cancel [THEN iffD1], auto)
   355 done
   356 
   357 text {* Power *}
   358 
   359 lemma (in group) int_pow_def2:
   360   "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
   361   by (simp add: int_pow_def nat_pow_def Let_def)
   362 
   363 lemma (in group) int_pow_0 [simp]:
   364   "x (^) (0::int) = \<one>"
   365   by (simp add: int_pow_def2)
   366 
   367 lemma (in group) int_pow_one [simp]:
   368   "\<one> (^) (z::int) = \<one>"
   369   by (simp add: int_pow_def2)
   370 
   371 subsection {* Substructures *}
   372 
   373 locale submagma = var H + struct G +
   374   assumes subset [intro, simp]: "H \<subseteq> carrier G"
   375     and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
   376 
   377 declare (in submagma) magma.intro [intro] semigroup.intro [intro]
   378   semigroup_axioms.intro [intro]
   379 
   380 lemma submagma_imp_subset:
   381   "submagma H G ==> H \<subseteq> carrier G"
   382   by (rule submagma.subset)
   383 
   384 lemma (in submagma) subsetD [dest, simp]:
   385   "x \<in> H ==> x \<in> carrier G"
   386   using subset by blast
   387 
   388 lemma (in submagma) magmaI [intro]:
   389   includes magma G
   390   shows "magma (G(| carrier := H |))"
   391   by rule simp
   392 
   393 lemma (in submagma) semigroup_axiomsI [intro]:
   394   includes semigroup G
   395   shows "semigroup_axioms (G(| carrier := H |))"
   396     by rule (simp add: m_assoc)
   397 
   398 lemma (in submagma) semigroupI [intro]:
   399   includes semigroup G
   400   shows "semigroup (G(| carrier := H |))"
   401   using prems by fast
   402 
   403 
   404 locale subgroup = submagma H G +
   405   assumes one_closed [intro, simp]: "\<one> \<in> H"
   406     and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H"
   407 
   408 declare (in subgroup) group.intro [intro]
   409 
   410 lemma (in subgroup) group_axiomsI [intro]:
   411   includes group G
   412   shows "group_axioms (G(| carrier := H |))"
   413   by (rule group_axioms.intro) (auto intro: l_inv r_inv simp add: Units_def)
   414 
   415 lemma (in subgroup) groupI [intro]:
   416   includes group G
   417   shows "group (G(| carrier := H |))"
   418   by (rule groupI) (auto intro: m_assoc l_inv)
   419 
   420 text {*
   421   Since @{term H} is nonempty, it contains some element @{term x}.  Since
   422   it is closed under inverse, it contains @{text "inv x"}.  Since
   423   it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
   424 *}
   425 
   426 lemma (in group) one_in_subset:
   427   "[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |]
   428    ==> \<one> \<in> H"
   429 by (force simp add: l_inv)
   430 
   431 text {* A characterization of subgroups: closed, non-empty subset. *}
   432 
   433 lemma (in group) subgroupI:
   434   assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
   435     and inv: "!!a. a \<in> H ==> inv a \<in> H"
   436     and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"
   437   shows "subgroup H G"
   438 proof (rule subgroup.intro)
   439   from subset and mult show "submagma H G" by (rule submagma.intro)
   440 next
   441   have "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
   442   with inv show "subgroup_axioms H G"
   443     by (intro subgroup_axioms.intro) simp_all
   444 qed
   445 
   446 text {*
   447   Repeat facts of submagmas for subgroups.  Necessary???
   448 *}
   449 
   450 lemma (in subgroup) subset:
   451   "H \<subseteq> carrier G"
   452   ..
   453 
   454 lemma (in subgroup) m_closed:
   455   "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
   456   ..
   457 
   458 declare magma.m_closed [simp]
   459 
   460 declare monoid.one_closed [iff] group.inv_closed [simp]
   461   monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
   462 
   463 lemma subgroup_nonempty:
   464   "~ subgroup {} G"
   465   by (blast dest: subgroup.one_closed)
   466 
   467 lemma (in subgroup) finite_imp_card_positive:
   468   "finite (carrier G) ==> 0 < card H"
   469 proof (rule classical)
   470   have sub: "subgroup H G" using prems by (rule subgroup.intro)
   471   assume fin: "finite (carrier G)"
   472     and zero: "~ 0 < card H"
   473   then have "finite H" by (blast intro: finite_subset dest: subset)
   474   with zero sub have "subgroup {} G" by simp
   475   with subgroup_nonempty show ?thesis by contradiction
   476 qed
   477 
   478 (*
   479 lemma (in monoid) Units_subgroup:
   480   "subgroup (Units G) G"
   481 *)
   482 
   483 subsection {* Direct Products *}
   484 
   485 constdefs (structure G and H)
   486   DirProdSemigroup :: "_ => _ => ('a \<times> 'b) semigroup"  (infixr "\<times>\<^sub>s" 80)
   487   "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,
   488     mult = (%(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')) |)"
   489 
   490   DirProdGroup :: "_ => _ => ('a \<times> 'b) monoid"  (infixr "\<times>\<^sub>g" 80)
   491   "G \<times>\<^sub>g H == semigroup.extend (G \<times>\<^sub>s H) (| one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>) |)"
   492 
   493 lemma DirProdSemigroup_magma:
   494   includes magma G + magma H
   495   shows "magma (G \<times>\<^sub>s H)"
   496   by (rule magma.intro) (auto simp add: DirProdSemigroup_def)
   497 
   498 lemma DirProdSemigroup_semigroup_axioms:
   499   includes semigroup G + semigroup H
   500   shows "semigroup_axioms (G \<times>\<^sub>s H)"
   501   by (rule semigroup_axioms.intro)
   502     (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)
   503 
   504 lemma DirProdSemigroup_semigroup:
   505   includes semigroup G + semigroup H
   506   shows "semigroup (G \<times>\<^sub>s H)"
   507   using prems
   508   by (fast intro: semigroup.intro
   509     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)
   510 
   511 lemma DirProdGroup_magma:
   512   includes magma G + magma H
   513   shows "magma (G \<times>\<^sub>g H)"
   514   by (rule magma.intro)
   515     (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
   516 
   517 lemma DirProdGroup_semigroup_axioms:
   518   includes semigroup G + semigroup H
   519   shows "semigroup_axioms (G \<times>\<^sub>g H)"
   520   by (rule semigroup_axioms.intro)
   521     (auto simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs
   522       G.m_assoc H.m_assoc)
   523 
   524 lemma DirProdGroup_semigroup:
   525   includes semigroup G + semigroup H
   526   shows "semigroup (G \<times>\<^sub>g H)"
   527   using prems
   528   by (fast intro: semigroup.intro
   529     DirProdGroup_magma DirProdGroup_semigroup_axioms)
   530 
   531 text {* \dots\ and further lemmas for group \dots *}
   532 
   533 lemma DirProdGroup_group:
   534   includes group G + group H
   535   shows "group (G \<times>\<^sub>g H)"
   536   by (rule groupI)
   537     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
   538       simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
   539 
   540 lemma carrier_DirProdGroup [simp]:
   541      "carrier (G \<times>\<^sub>g H) = carrier G \<times> carrier H"
   542   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
   543 
   544 lemma one_DirProdGroup [simp]:
   545      "\<one>\<^bsub>(G \<times>\<^sub>g H)\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
   546   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
   547 
   548 lemma mult_DirProdGroup [simp]:
   549      "(g, h) \<otimes>\<^bsub>(G \<times>\<^sub>g H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
   550   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
   551 
   552 lemma inv_DirProdGroup [simp]:
   553   includes group G + group H
   554   assumes g: "g \<in> carrier G"
   555       and h: "h \<in> carrier H"
   556   shows "m_inv (G \<times>\<^sub>g H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
   557   apply (rule group.inv_equality [OF DirProdGroup_group])
   558   apply (simp_all add: prems group_def group.l_inv)
   559   done
   560 
   561 subsection {* Isomorphisms *}
   562 
   563 constdefs (structure G and H)
   564   hom :: "_ => _ => ('a => 'b) set"
   565   "hom G H ==
   566     {h. h \<in> carrier G -> carrier H &
   567       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}"
   568 
   569 lemma (in semigroup) hom:
   570      "semigroup (| carrier = hom G G, mult = op o |)"
   571 proof (rule semigroup.intro)
   572   show "magma (| carrier = hom G G, mult = op o |)"
   573     by (rule magma.intro) (simp add: Pi_def hom_def)
   574 next
   575   show "semigroup_axioms (| carrier = hom G G, mult = op o |)"
   576     by (rule semigroup_axioms.intro) (simp add: o_assoc)
   577 qed
   578 
   579 lemma hom_mult:
   580   "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
   581    ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
   582   by (simp add: hom_def)
   583 
   584 lemma hom_closed:
   585   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
   586   by (auto simp add: hom_def funcset_mem)
   587 
   588 lemma (in group) hom_compose:
   589      "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
   590 apply (auto simp add: hom_def funcset_compose) 
   591 apply (simp add: compose_def funcset_mem)
   592 done
   593 
   594 
   595 subsection {* Isomorphisms *}
   596 
   597 constdefs
   598   iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
   599   "G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"
   600 
   601 lemma iso_refl: "(%x. x) \<in> G \<cong> G"
   602 by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
   603 
   604 lemma (in group) iso_sym:
   605      "h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G"
   606 apply (simp add: iso_def bij_betw_Inv) 
   607 apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G") 
   608  prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv]) 
   609 apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f) 
   610 done
   611 
   612 lemma (in group) iso_trans: 
   613      "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"
   614 by (auto simp add: iso_def hom_compose bij_betw_compose)
   615 
   616 lemma DirProdGroup_commute_iso:
   617   shows "(%(x,y). (y,x)) \<in> (G \<times>\<^sub>g H) \<cong> (H \<times>\<^sub>g G)"
   618 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
   619 
   620 lemma DirProdGroup_assoc_iso:
   621   shows "(%(x,y,z). (x,(y,z))) \<in> (G \<times>\<^sub>g H \<times>\<^sub>g I) \<cong> (G \<times>\<^sub>g (H \<times>\<^sub>g I))"
   622 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
   623 
   624 
   625 locale group_hom = group G + group H + var h +
   626   assumes homh: "h \<in> hom G H"
   627   notes hom_mult [simp] = hom_mult [OF homh]
   628     and hom_closed [simp] = hom_closed [OF homh]
   629 
   630 lemma (in group_hom) one_closed [simp]:
   631   "h \<one> \<in> carrier H"
   632   by simp
   633 
   634 lemma (in group_hom) hom_one [simp]:
   635   "h \<one> = \<one>\<^bsub>H\<^esub>"
   636 proof -
   637   have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^sub>2 h \<one>"
   638     by (simp add: hom_mult [symmetric] del: hom_mult)
   639   then show ?thesis by (simp del: r_one)
   640 qed
   641 
   642 lemma (in group_hom) inv_closed [simp]:
   643   "x \<in> carrier G ==> h (inv x) \<in> carrier H"
   644   by simp
   645 
   646 lemma (in group_hom) hom_inv [simp]:
   647   "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
   648 proof -
   649   assume x: "x \<in> carrier G"
   650   then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
   651     by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
   652   also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
   653     by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
   654   finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .
   655   with x show ?thesis by simp
   656 qed
   657 
   658 subsection {* Commutative Structures *}
   659 
   660 text {*
   661   Naming convention: multiplicative structures that are commutative
   662   are called \emph{commutative}, additive structures are called
   663   \emph{Abelian}.
   664 *}
   665 
   666 subsection {* Definition *}
   667 
   668 locale comm_semigroup = semigroup +
   669   assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   670 
   671 lemma (in comm_semigroup) m_lcomm:
   672   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   673    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
   674 proof -
   675   assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   676   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
   677   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
   678   also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
   679   finally show ?thesis .
   680 qed
   681 
   682 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm
   683 
   684 locale comm_monoid = comm_semigroup + monoid
   685 
   686 lemma comm_monoidI:
   687   includes struct G
   688   assumes m_closed:
   689       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
   690     and one_closed: "\<one> \<in> carrier G"
   691     and m_assoc:
   692       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   693       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
   694     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
   695     and m_comm:
   696       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   697   shows "comm_monoid G"
   698   using l_one
   699   by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro
   700     comm_semigroup_axioms.intro monoid_axioms.intro
   701     intro: prems simp: m_closed one_closed m_comm)
   702 
   703 lemma (in monoid) monoid_comm_monoidI:
   704   assumes m_comm:
   705       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   706   shows "comm_monoid G"
   707   by (rule comm_monoidI) (auto intro: m_assoc m_comm)
   708 (*lemma (in comm_monoid) r_one [simp]:
   709   "x \<in> carrier G ==> x \<otimes> \<one> = x"
   710 proof -
   711   assume G: "x \<in> carrier G"
   712   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
   713   also from G have "... = x" by simp
   714   finally show ?thesis .
   715 qed*)
   716 lemma (in comm_monoid) nat_pow_distr:
   717   "[| x \<in> carrier G; y \<in> carrier G |] ==>
   718   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
   719   by (induct n) (simp, simp add: m_ac)
   720 
   721 locale comm_group = comm_monoid + group
   722 
   723 lemma (in group) group_comm_groupI:
   724   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
   725       x \<otimes> y = y \<otimes> x"
   726   shows "comm_group G"
   727   by (fast intro: comm_group.intro comm_semigroup_axioms.intro
   728                   is_group prems)
   729 
   730 lemma comm_groupI:
   731   includes struct G
   732   assumes m_closed:
   733       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
   734     and one_closed: "\<one> \<in> carrier G"
   735     and m_assoc:
   736       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   737       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
   738     and m_comm:
   739       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   740     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
   741     and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
   742   shows "comm_group G"
   743   by (fast intro: group.group_comm_groupI groupI prems)
   744 
   745 lemma (in comm_group) inv_mult:
   746   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
   747   by (simp add: m_ac inv_mult_group)
   748 
   749 subsection {* Lattice of subgroups of a group *}
   750 
   751 text_raw {* \label{sec:subgroup-lattice} *}
   752 
   753 theorem (in group) subgroups_partial_order:
   754   "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
   755   by (rule partial_order.intro) simp_all
   756 
   757 lemma (in group) subgroup_self:
   758   "subgroup (carrier G) G"
   759   by (rule subgroupI) auto
   760 
   761 lemma (in group) subgroup_imp_group:
   762   "subgroup H G ==> group (G(| carrier := H |))"
   763   using subgroup.groupI [OF _ group.intro] .
   764 
   765 lemma (in group) is_monoid [intro, simp]:
   766   "monoid G"
   767   by (rule monoid.intro)
   768 
   769 lemma (in group) subgroup_inv_equality:
   770   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
   771 apply (rule_tac inv_equality [THEN sym])
   772   apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
   773  apply (rule subsetD [OF subgroup.subset], assumption+)
   774 apply (rule subsetD [OF subgroup.subset], assumption)
   775 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)
   776 done
   777 
   778 theorem (in group) subgroups_Inter:
   779   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
   780     and not_empty: "A ~= {}"
   781   shows "subgroup (\<Inter>A) G"
   782 proof (rule subgroupI)
   783   from subgr [THEN subgroup.subset] and not_empty
   784   show "\<Inter>A \<subseteq> carrier G" by blast
   785 next
   786   from subgr [THEN subgroup.one_closed]
   787   show "\<Inter>A ~= {}" by blast
   788 next
   789   fix x assume "x \<in> \<Inter>A"
   790   with subgr [THEN subgroup.m_inv_closed]
   791   show "inv x \<in> \<Inter>A" by blast
   792 next
   793   fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
   794   with subgr [THEN subgroup.m_closed]
   795   show "x \<otimes> y \<in> \<Inter>A" by blast
   796 qed
   797 
   798 theorem (in group) subgroups_complete_lattice:
   799   "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
   800     (is "complete_lattice ?L")
   801 proof (rule partial_order.complete_lattice_criterion1)
   802   show "partial_order ?L" by (rule subgroups_partial_order)
   803 next
   804   have "greatest ?L (carrier G) (carrier ?L)"
   805     by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
   806   then show "EX G. greatest ?L G (carrier ?L)" ..
   807 next
   808   fix A
   809   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
   810   then have Int_subgroup: "subgroup (\<Inter>A) G"
   811     by (fastsimp intro: subgroups_Inter)
   812   have "greatest ?L (\<Inter>A) (Lower ?L A)"
   813     (is "greatest ?L ?Int _")
   814   proof (rule greatest_LowerI)
   815     fix H
   816     assume H: "H \<in> A"
   817     with L have subgroupH: "subgroup H G" by auto
   818     from subgroupH have submagmaH: "submagma H G" by (rule subgroup.axioms)
   819     from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
   820       by (rule subgroup_imp_group)
   821     from groupH have monoidH: "monoid ?H"
   822       by (rule group.is_monoid)
   823     from H have Int_subset: "?Int \<subseteq> H" by fastsimp
   824     then show "le ?L ?Int H" by simp
   825   next
   826     fix H
   827     assume H: "H \<in> Lower ?L A"
   828     with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
   829   next
   830     show "A \<subseteq> carrier ?L" by (rule L)
   831   next
   832     show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
   833   qed
   834   then show "EX I. greatest ?L I (Lower ?L A)" ..
   835 qed
   836 
   837 end