src/HOL/Algebra/Group.thy
author ballarin
Tue Apr 13 09:42:40 2004 +0200 (2004-04-13)
changeset 14551 2cb6ff394bfb
parent 14286 0ae66ffb9784
child 14651 02b8f3bcf7fe
permissions -rw-r--r--
Various changes to HOL-Algebra;
Locale instantiation.
     1 (*
     2   Title:  HOL/Algebra/Group.thy
     3   Id:     $Id$
     4   Author: Clemens Ballarin, started 4 February 2003
     5 
     6 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
     7 *)
     8 
     9 header {* Groups *}
    10 
    11 theory Group = FuncSet:
    12 
    13 section {* From Magmas to Groups *}
    14 
    15 text {*
    16   Definitions follow Jacobson, Basic Algebra I, Freeman, 1985; with
    17   the exception of \emph{magma} which, following Bourbaki, is a set
    18   together with a binary, closed operation.
    19 *}
    20 
    21 subsection {* Definitions *}
    22 
    23 (* Object with a carrier set. *)
    24 
    25 record 'a partial_object =
    26   carrier :: "'a set"
    27 
    28 record 'a semigroup = "'a partial_object" +
    29   mult :: "['a, 'a] => 'a" (infixl "\<otimes>\<index>" 70)
    30 
    31 record 'a monoid = "'a semigroup" +
    32   one :: 'a ("\<one>\<index>")
    33 
    34 constdefs
    35   m_inv :: "[('a, 'm) monoid_scheme, 'a] => 'a" ("inv\<index> _" [81] 80)
    36   "m_inv G x == (THE y. y \<in> carrier G &
    37                   mult G x y = one G & mult G y x = one G)"
    38 
    39   Units :: "('a, 'm) monoid_scheme => 'a set"
    40   "Units G == {y. y \<in> carrier G &
    41                   (EX x : carrier G. mult G x y = one G & mult G y x = one G)}"
    42 
    43 consts
    44   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)
    45 
    46 defs (overloaded)
    47   nat_pow_def: "pow G a n == nat_rec (one G) (%u b. mult G b a) n"
    48   int_pow_def: "pow G a z ==
    49     let p = nat_rec (one G) (%u b. mult G b a)
    50     in if neg z then m_inv G (p (nat (-z))) else p (nat z)"
    51 
    52 locale magma = struct G +
    53   assumes m_closed [intro, simp]:
    54     "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
    55 
    56 locale semigroup = magma +
    57   assumes m_assoc:
    58     "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
    59     (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
    60 
    61 locale monoid = semigroup +
    62   assumes one_closed [intro, simp]: "\<one> \<in> carrier G"
    63     and l_one [simp]: "x \<in> carrier G ==> \<one> \<otimes> x = x"
    64     and r_one [simp]: "x \<in> carrier G ==> x \<otimes> \<one> = x"
    65 
    66 lemma monoidI:
    67   assumes m_closed:
    68       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
    69     and one_closed: "one G \<in> carrier G"
    70     and m_assoc:
    71       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
    72       mult G (mult G x y) z = mult G x (mult G y z)"
    73     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
    74     and r_one: "!!x. x \<in> carrier G ==> mult G x (one G) = x"
    75   shows "monoid G"
    76   by (fast intro!: monoid.intro magma.intro semigroup_axioms.intro
    77     semigroup.intro monoid_axioms.intro
    78     intro: prems)
    79 
    80 lemma (in monoid) Units_closed [dest]:
    81   "x \<in> Units G ==> x \<in> carrier G"
    82   by (unfold Units_def) fast
    83 
    84 lemma (in monoid) inv_unique:
    85   assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>"
    86     and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G"
    87   shows "y = y'"
    88 proof -
    89   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp
    90   also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)
    91   also from G eq have "... = y'" by simp
    92   finally show ?thesis .
    93 qed
    94 
    95 lemma (in monoid) Units_one_closed [intro, simp]:
    96   "\<one> \<in> Units G"
    97   by (unfold Units_def) auto
    98 
    99 lemma (in monoid) Units_inv_closed [intro, simp]:
   100   "x \<in> Units G ==> inv x \<in> carrier G"
   101   apply (unfold Units_def m_inv_def, auto)
   102   apply (rule theI2, fast)
   103    apply (fast intro: inv_unique, fast)
   104   done
   105 
   106 lemma (in monoid) Units_l_inv:
   107   "x \<in> Units G ==> inv x \<otimes> x = \<one>"
   108   apply (unfold Units_def m_inv_def, auto)
   109   apply (rule theI2, fast)
   110    apply (fast intro: inv_unique, fast)
   111   done
   112 
   113 lemma (in monoid) Units_r_inv:
   114   "x \<in> Units G ==> x \<otimes> inv x = \<one>"
   115   apply (unfold Units_def m_inv_def, auto)
   116   apply (rule theI2, fast)
   117    apply (fast intro: inv_unique, fast)
   118   done
   119 
   120 lemma (in monoid) Units_inv_Units [intro, simp]:
   121   "x \<in> Units G ==> inv x \<in> Units G"
   122 proof -
   123   assume x: "x \<in> Units G"
   124   show "inv x \<in> Units G"
   125     by (auto simp add: Units_def
   126       intro: Units_l_inv Units_r_inv x Units_closed [OF x])
   127 qed
   128 
   129 lemma (in monoid) Units_l_cancel [simp]:
   130   "[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==>
   131    (x \<otimes> y = x \<otimes> z) = (y = z)"
   132 proof
   133   assume eq: "x \<otimes> y = x \<otimes> z"
   134     and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G"
   135   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"
   136     by (simp add: m_assoc Units_closed)
   137   with G show "y = z" by (simp add: Units_l_inv)
   138 next
   139   assume eq: "y = z"
   140     and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G"
   141   then show "x \<otimes> y = x \<otimes> z" by simp
   142 qed
   143 
   144 lemma (in monoid) Units_inv_inv [simp]:
   145   "x \<in> Units G ==> inv (inv x) = x"
   146 proof -
   147   assume x: "x \<in> Units G"
   148   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x"
   149     by (simp add: Units_l_inv Units_r_inv)
   150   with x show ?thesis by (simp add: Units_closed)
   151 qed
   152 
   153 lemma (in monoid) inv_inj_on_Units:
   154   "inj_on (m_inv G) (Units G)"
   155 proof (rule inj_onI)
   156   fix x y
   157   assume G: "x \<in> Units G" "y \<in> Units G" and eq: "inv x = inv y"
   158   then have "inv (inv x) = inv (inv y)" by simp
   159   with G show "x = y" by simp
   160 qed
   161 
   162 lemma (in monoid) Units_inv_comm:
   163   assumes inv: "x \<otimes> y = \<one>"
   164     and G: "x \<in> Units G" "y \<in> Units G"
   165   shows "y \<otimes> x = \<one>"
   166 proof -
   167   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
   168   with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
   169 qed
   170 
   171 text {* Power *}
   172 
   173 lemma (in monoid) nat_pow_closed [intro, simp]:
   174   "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"
   175   by (induct n) (simp_all add: nat_pow_def)
   176 
   177 lemma (in monoid) nat_pow_0 [simp]:
   178   "x (^) (0::nat) = \<one>"
   179   by (simp add: nat_pow_def)
   180 
   181 lemma (in monoid) nat_pow_Suc [simp]:
   182   "x (^) (Suc n) = x (^) n \<otimes> x"
   183   by (simp add: nat_pow_def)
   184 
   185 lemma (in monoid) nat_pow_one [simp]:
   186   "\<one> (^) (n::nat) = \<one>"
   187   by (induct n) simp_all
   188 
   189 lemma (in monoid) nat_pow_mult:
   190   "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"
   191   by (induct m) (simp_all add: m_assoc [THEN sym])
   192 
   193 lemma (in monoid) nat_pow_pow:
   194   "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
   195   by (induct m) (simp, simp add: nat_pow_mult add_commute)
   196 
   197 text {*
   198   A group is a monoid all of whose elements are invertible.
   199 *}
   200 
   201 locale group = monoid +
   202   assumes Units: "carrier G <= Units G"
   203 
   204 theorem groupI:
   205   assumes m_closed [simp]:
   206       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
   207     and one_closed [simp]: "one G \<in> carrier G"
   208     and m_assoc:
   209       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   210       mult G (mult G x y) z = mult G x (mult G y z)"
   211     and l_one [simp]: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
   212     and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
   213   shows "group G"
   214 proof -
   215   have l_cancel [simp]:
   216     "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   217     (mult G x y = mult G x z) = (y = z)"
   218   proof
   219     fix x y z
   220     assume eq: "mult G x y = mult G x z"
   221       and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
   222     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
   223       and l_inv: "mult G x_inv x = one G" by fast
   224     from G eq xG have "mult G (mult G x_inv x) y = mult G (mult G x_inv x) z"
   225       by (simp add: m_assoc)
   226     with G show "y = z" by (simp add: l_inv)
   227   next
   228     fix x y z
   229     assume eq: "y = z"
   230       and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
   231     then show "mult G x y = mult G x z" by simp
   232   qed
   233   have r_one:
   234     "!!x. x \<in> carrier G ==> mult G x (one G) = x"
   235   proof -
   236     fix x
   237     assume x: "x \<in> carrier G"
   238     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
   239       and l_inv: "mult G x_inv x = one G" by fast
   240     from x xG have "mult G x_inv (mult G x (one G)) = mult G x_inv x"
   241       by (simp add: m_assoc [symmetric] l_inv)
   242     with x xG show "mult G x (one G) = x" by simp 
   243   qed
   244   have inv_ex:
   245     "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G &
   246       mult G x y = one G"
   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: "mult G y x = one G" by fast
   252     from x y have "mult G y (mult G x y) = mult G y (one G)"
   253       by (simp add: m_assoc [symmetric] l_inv r_one)
   254     with x y have r_inv: "mult G x y = one G"
   255       by simp
   256     from x y show "EX y : carrier G. mult G y x = one G &
   257       mult G x y = one G"
   258       by (fast intro: l_inv r_inv)
   259   qed
   260   then have carrier_subset_Units: "carrier G <= Units G"
   261     by (unfold Units_def) fast
   262   show ?thesis
   263     by (fast intro!: group.intro magma.intro semigroup_axioms.intro
   264       semigroup.intro monoid_axioms.intro group_axioms.intro
   265       carrier_subset_Units intro: prems r_one)
   266 qed
   267 
   268 lemma (in monoid) monoid_groupI:
   269   assumes l_inv_ex:
   270     "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
   271   shows "group G"
   272   by (rule groupI) (auto intro: m_assoc l_inv_ex)
   273 
   274 lemma (in group) Units_eq [simp]:
   275   "Units G = carrier G"
   276 proof
   277   show "Units G <= carrier G" by fast
   278 next
   279   show "carrier G <= Units G" by (rule Units)
   280 qed
   281 
   282 lemma (in group) inv_closed [intro, simp]:
   283   "x \<in> carrier G ==> inv x \<in> carrier G"
   284   using Units_inv_closed by simp
   285 
   286 lemma (in group) l_inv:
   287   "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
   288   using Units_l_inv by simp
   289 
   290 subsection {* Cancellation Laws and Basic Properties *}
   291 
   292 lemma (in group) l_cancel [simp]:
   293   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   294    (x \<otimes> y = x \<otimes> z) = (y = z)"
   295   using Units_l_inv by simp
   296 
   297 lemma (in group) r_inv:
   298   "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
   299 proof -
   300   assume x: "x \<in> carrier G"
   301   then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
   302     by (simp add: m_assoc [symmetric] l_inv)
   303   with x show ?thesis by (simp del: r_one)
   304 qed
   305 
   306 lemma (in group) r_cancel [simp]:
   307   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   308    (y \<otimes> x = z \<otimes> x) = (y = z)"
   309 proof
   310   assume eq: "y \<otimes> x = z \<otimes> x"
   311     and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
   312   then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
   313     by (simp add: m_assoc [symmetric])
   314   with G show "y = z" by (simp add: r_inv)
   315 next
   316   assume eq: "y = z"
   317     and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
   318   then show "y \<otimes> x = z \<otimes> x" by simp
   319 qed
   320 
   321 lemma (in group) inv_one [simp]:
   322   "inv \<one> = \<one>"
   323 proof -
   324   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by simp
   325   moreover have "... = \<one>" by (simp add: r_inv)
   326   finally show ?thesis .
   327 qed
   328 
   329 lemma (in group) inv_inv [simp]:
   330   "x \<in> carrier G ==> inv (inv x) = x"
   331   using Units_inv_inv by simp
   332 
   333 lemma (in group) inv_inj:
   334   "inj_on (m_inv G) (carrier G)"
   335   using inv_inj_on_Units by simp
   336 
   337 lemma (in group) inv_mult_group:
   338   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
   339 proof -
   340   assume G: "x \<in> carrier G" "y \<in> carrier G"
   341   then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
   342     by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv)
   343   with G show ?thesis by simp
   344 qed
   345 
   346 lemma (in group) inv_comm:
   347   "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
   348   by (rule Units_inv_comm) auto                          
   349 
   350 lemma (in group) inv_equality:
   351      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
   352 apply (simp add: m_inv_def)
   353 apply (rule the_equality)
   354  apply (simp add: inv_comm [of y x]) 
   355 apply (rule r_cancel [THEN iffD1], auto) 
   356 done
   357 
   358 text {* Power *}
   359 
   360 lemma (in group) int_pow_def2:
   361   "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
   362   by (simp add: int_pow_def nat_pow_def Let_def)
   363 
   364 lemma (in group) int_pow_0 [simp]:
   365   "x (^) (0::int) = \<one>"
   366   by (simp add: int_pow_def2)
   367 
   368 lemma (in group) int_pow_one [simp]:
   369   "\<one> (^) (z::int) = \<one>"
   370   by (simp add: int_pow_def2)
   371 
   372 subsection {* Substructures *}
   373 
   374 locale submagma = var H + struct G +
   375   assumes subset [intro, simp]: "H \<subseteq> carrier G"
   376     and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
   377 
   378 declare (in submagma) magma.intro [intro] semigroup.intro [intro]
   379   semigroup_axioms.intro [intro]
   380 (*
   381 alternative definition of submagma
   382 
   383 locale submagma = var H + struct G +
   384   assumes subset [intro, simp]: "carrier H \<subseteq> carrier G"
   385     and m_equal [simp]: "mult H = mult G"
   386     and m_closed [intro, simp]:
   387       "[| x \<in> carrier H; y \<in> carrier H |] ==> x \<otimes> y \<in> carrier H"
   388 *)
   389 
   390 lemma submagma_imp_subset:
   391   "submagma H G ==> H \<subseteq> carrier G"
   392   by (rule submagma.subset)
   393 
   394 lemma (in submagma) subsetD [dest, simp]:
   395   "x \<in> H ==> x \<in> carrier G"
   396   using subset by blast
   397 
   398 lemma (in submagma) magmaI [intro]:
   399   includes magma G
   400   shows "magma (G(| carrier := H |))"
   401   by rule simp
   402 
   403 lemma (in submagma) semigroup_axiomsI [intro]:
   404   includes semigroup G
   405   shows "semigroup_axioms (G(| carrier := H |))"
   406     by rule (simp add: m_assoc)
   407 
   408 lemma (in submagma) semigroupI [intro]:
   409   includes semigroup G
   410   shows "semigroup (G(| carrier := H |))"
   411   using prems by fast
   412 
   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 group_axioms.intro) (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 (rule subgroup.intro)
   449   from subset and mult show "submagma H G" by (rule submagma.intro)
   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 by (rule subgroup.intro)
   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 magma.intro) (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 semigroup_axioms.intro)
   519     (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)
   520 
   521 lemma DirProdSemigroup_semigroup:
   522   includes semigroup G + semigroup H
   523   shows "semigroup (G \<times>\<^sub>s H)"
   524   using prems
   525   by (fast intro: semigroup.intro
   526     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)
   527 
   528 lemma DirProdGroup_magma:
   529   includes magma G + magma H
   530   shows "magma (G \<times>\<^sub>g H)"
   531   by (rule magma.intro)
   532     (auto simp add: DirProdGroup_def DirProdSemigroup_def)
   533 
   534 lemma DirProdGroup_semigroup_axioms:
   535   includes semigroup G + semigroup H
   536   shows "semigroup_axioms (G \<times>\<^sub>g H)"
   537   by (rule semigroup_axioms.intro)
   538     (auto simp add: DirProdGroup_def DirProdSemigroup_def
   539       G.m_assoc H.m_assoc)
   540 
   541 lemma DirProdGroup_semigroup:
   542   includes semigroup G + semigroup H
   543   shows "semigroup (G \<times>\<^sub>g H)"
   544   using prems
   545   by (fast intro: semigroup.intro
   546     DirProdGroup_magma DirProdGroup_semigroup_axioms)
   547 
   548 (* ... and further lemmas for group ... *)
   549 
   550 lemma DirProdGroup_group:
   551   includes group G + group H
   552   shows "group (G \<times>\<^sub>g H)"
   553   by (rule groupI)
   554     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
   555       simp add: DirProdGroup_def DirProdSemigroup_def)
   556 
   557 lemma carrier_DirProdGroup [simp]:
   558      "carrier (G \<times>\<^sub>g H) = carrier G \<times> carrier H"
   559   by (simp add: DirProdGroup_def DirProdSemigroup_def)
   560 
   561 lemma one_DirProdGroup [simp]:
   562      "one (G \<times>\<^sub>g H) = (one G, one H)"
   563   by (simp add: DirProdGroup_def DirProdSemigroup_def);
   564 
   565 lemma mult_DirProdGroup [simp]:
   566      "mult (G \<times>\<^sub>g H) (g, h) (g', h') = (mult G g g', mult H h h')"
   567   by (simp add: DirProdGroup_def DirProdSemigroup_def)
   568 
   569 lemma inv_DirProdGroup [simp]:
   570   includes group G + group H
   571   assumes g: "g \<in> carrier G"
   572       and h: "h \<in> carrier H"
   573   shows "m_inv (G \<times>\<^sub>g H) (g, h) = (m_inv G g, m_inv H h)"
   574   apply (rule group.inv_equality [OF DirProdGroup_group])
   575   apply (simp_all add: prems group_def group.l_inv)
   576   done
   577 
   578 subsection {* Homomorphisms *}
   579 
   580 constdefs
   581   hom :: "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme]
   582     => ('a => 'b)set"
   583   "hom G H ==
   584     {h. h \<in> carrier G -> carrier H &
   585       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}"
   586 
   587 lemma (in semigroup) hom:
   588   includes semigroup G
   589   shows "semigroup (| carrier = hom G G, mult = op o |)"
   590 proof (rule semigroup.intro)
   591   show "magma (| carrier = hom G G, mult = op o |)"
   592     by (rule magma.intro) (simp add: Pi_def hom_def)
   593 next
   594   show "semigroup_axioms (| carrier = hom G G, mult = op o |)"
   595     by (rule semigroup_axioms.intro) (simp add: o_assoc)
   596 qed
   597 
   598 lemma hom_mult:
   599   "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] 
   600    ==> h (mult G x y) = mult H (h x) (h y)"
   601   by (simp add: hom_def) 
   602 
   603 lemma hom_closed:
   604   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
   605   by (auto simp add: hom_def funcset_mem)
   606 
   607 lemma compose_hom:
   608      "[|group G; h \<in> hom G G; h' \<in> hom G G; h' \<in> carrier G -> carrier G|]
   609       ==> compose (carrier G) h h' \<in> hom G G"
   610 apply (simp (no_asm_simp) add: hom_def)
   611 apply (intro conjI) 
   612  apply (force simp add: funcset_compose hom_def)
   613 apply (simp add: compose_def group.axioms hom_mult funcset_mem) 
   614 done
   615 
   616 locale group_hom = group G + group H + var h +
   617   assumes homh: "h \<in> hom G H"
   618   notes hom_mult [simp] = hom_mult [OF homh]
   619     and hom_closed [simp] = hom_closed [OF homh]
   620 
   621 lemma (in group_hom) one_closed [simp]:
   622   "h \<one> \<in> carrier H"
   623   by simp
   624 
   625 lemma (in group_hom) hom_one [simp]:
   626   "h \<one> = \<one>\<^sub>2"
   627 proof -
   628   have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"
   629     by (simp add: hom_mult [symmetric] del: hom_mult)
   630   then show ?thesis by (simp del: r_one)
   631 qed
   632 
   633 lemma (in group_hom) inv_closed [simp]:
   634   "x \<in> carrier G ==> h (inv x) \<in> carrier H"
   635   by simp
   636 
   637 lemma (in group_hom) hom_inv [simp]:
   638   "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"
   639 proof -
   640   assume x: "x \<in> carrier G"
   641   then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"
   642     by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
   643   also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"
   644     by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
   645   finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .
   646   with x show ?thesis by simp
   647 qed
   648 
   649 subsection {* Commutative Structures *}
   650 
   651 text {*
   652   Naming convention: multiplicative structures that are commutative
   653   are called \emph{commutative}, additive structures are called
   654   \emph{Abelian}.
   655 *}
   656 
   657 subsection {* Definition *}
   658 
   659 locale comm_semigroup = semigroup +
   660   assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   661 
   662 lemma (in comm_semigroup) m_lcomm:
   663   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   664    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
   665 proof -
   666   assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
   667   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
   668   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
   669   also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
   670   finally show ?thesis .
   671 qed
   672 
   673 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm
   674 
   675 locale comm_monoid = comm_semigroup + monoid
   676 
   677 lemma comm_monoidI:
   678   assumes m_closed:
   679       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
   680     and one_closed: "one G \<in> carrier G"
   681     and m_assoc:
   682       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   683       mult G (mult G x y) z = mult G x (mult G y z)"
   684     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
   685     and m_comm:
   686       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
   687   shows "comm_monoid G"
   688   using l_one
   689   by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro
   690     comm_semigroup_axioms.intro monoid_axioms.intro
   691     intro: prems simp: m_closed one_closed m_comm)
   692 
   693 lemma (in monoid) monoid_comm_monoidI:
   694   assumes m_comm:
   695       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
   696   shows "comm_monoid G"
   697   by (rule comm_monoidI) (auto intro: m_assoc m_comm)
   698 (*
   699 lemma (in comm_monoid) r_one [simp]:
   700   "x \<in> carrier G ==> x \<otimes> \<one> = x"
   701 proof -
   702   assume G: "x \<in> carrier G"
   703   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
   704   also from G have "... = x" by simp
   705   finally show ?thesis .
   706 qed
   707 *)
   708 
   709 lemma (in comm_monoid) nat_pow_distr:
   710   "[| x \<in> carrier G; y \<in> carrier G |] ==>
   711   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
   712   by (induct n) (simp, simp add: m_ac)
   713 
   714 locale comm_group = comm_monoid + group
   715 
   716 lemma (in group) group_comm_groupI:
   717   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
   718       mult G x y = mult G y x"
   719   shows "comm_group G"
   720   by (fast intro: comm_group.intro comm_semigroup_axioms.intro
   721     group.axioms prems)
   722 
   723 lemma comm_groupI:
   724   assumes m_closed:
   725       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
   726     and one_closed: "one G \<in> carrier G"
   727     and m_assoc:
   728       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   729       mult G (mult G x y) z = mult G x (mult G y z)"
   730     and m_comm:
   731       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
   732     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
   733     and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
   734   shows "comm_group G"
   735   by (fast intro: group.group_comm_groupI groupI prems)
   736 
   737 lemma (in comm_group) inv_mult:
   738   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
   739   by (simp add: m_ac inv_mult_group)
   740 
   741 end