src/HOL/Algebra/Group.thy
author ballarin
Wed Dec 10 14:29:05 2003 +0100 (2003-12-10)
changeset 14286 0ae66ffb9784
parent 14254 342634f38451
child 14551 2cb6ff394bfb
permissions -rw-r--r--
New structure "partial_object" as common root for lattices and magmas.
     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 locale subgroup = submagma H G +
   414   assumes one_closed [intro, simp]: "\<one> \<in> H"
   415     and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H"
   416 
   417 declare (in subgroup) group.intro [intro]
   418 
   419 lemma (in subgroup) group_axiomsI [intro]:
   420   includes group G
   421   shows "group_axioms (G(| carrier := H |))"
   422   by (rule group_axioms.intro) (auto intro: l_inv r_inv simp add: Units_def)
   423 
   424 lemma (in subgroup) groupI [intro]:
   425   includes group G
   426   shows "group (G(| carrier := H |))"
   427   by (rule groupI) (auto intro: m_assoc l_inv)
   428 
   429 text {*
   430   Since @{term H} is nonempty, it contains some element @{term x}.  Since
   431   it is closed under inverse, it contains @{text "inv x"}.  Since
   432   it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
   433 *}
   434 
   435 lemma (in group) one_in_subset:
   436   "[| 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 |]
   437    ==> \<one> \<in> H"
   438 by (force simp add: l_inv)
   439 
   440 text {* A characterization of subgroups: closed, non-empty subset. *}
   441 
   442 lemma (in group) subgroupI:
   443   assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
   444     and inv: "!!a. a \<in> H ==> inv a \<in> H"
   445     and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"
   446   shows "subgroup H G"
   447 proof (rule subgroup.intro)
   448   from subset and mult show "submagma H G" by (rule submagma.intro)
   449 next
   450   have "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
   451   with inv show "subgroup_axioms H G"
   452     by (intro subgroup_axioms.intro) simp_all
   453 qed
   454 
   455 text {*
   456   Repeat facts of submagmas for subgroups.  Necessary???
   457 *}
   458 
   459 lemma (in subgroup) subset:
   460   "H \<subseteq> carrier G"
   461   ..
   462 
   463 lemma (in subgroup) m_closed:
   464   "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
   465   ..
   466 
   467 declare magma.m_closed [simp]
   468 
   469 declare monoid.one_closed [iff] group.inv_closed [simp]
   470   monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
   471 
   472 lemma subgroup_nonempty:
   473   "~ subgroup {} G"
   474   by (blast dest: subgroup.one_closed)
   475 
   476 lemma (in subgroup) finite_imp_card_positive:
   477   "finite (carrier G) ==> 0 < card H"
   478 proof (rule classical)
   479   have sub: "subgroup H G" using prems by (rule subgroup.intro)
   480   assume fin: "finite (carrier G)"
   481     and zero: "~ 0 < card H"
   482   then have "finite H" by (blast intro: finite_subset dest: subset)
   483   with zero sub have "subgroup {} G" by simp
   484   with subgroup_nonempty show ?thesis by contradiction
   485 qed
   486 
   487 (*
   488 lemma (in monoid) Units_subgroup:
   489   "subgroup (Units G) G"
   490 *)
   491 
   492 subsection {* Direct Products *}
   493 
   494 constdefs
   495   DirProdSemigroup ::
   496     "[('a, 'm) semigroup_scheme, ('b, 'n) semigroup_scheme]
   497     => ('a \<times> 'b) semigroup"
   498     (infixr "\<times>\<^sub>s" 80)
   499   "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,
   500     mult = (%(xg, xh) (yg, yh). (mult G xg yg, mult H xh yh)) |)"
   501 
   502   DirProdGroup ::
   503     "[('a, 'm) monoid_scheme, ('b, 'n) monoid_scheme] => ('a \<times> 'b) monoid"
   504     (infixr "\<times>\<^sub>g" 80)
   505   "G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>s H),
   506     mult = mult (G \<times>\<^sub>s H),
   507     one = (one G, one H) |)"
   508 
   509 lemma DirProdSemigroup_magma:
   510   includes magma G + magma H
   511   shows "magma (G \<times>\<^sub>s H)"
   512   by (rule magma.intro) (auto simp add: DirProdSemigroup_def)
   513 
   514 lemma DirProdSemigroup_semigroup_axioms:
   515   includes semigroup G + semigroup H
   516   shows "semigroup_axioms (G \<times>\<^sub>s H)"
   517   by (rule semigroup_axioms.intro)
   518     (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)
   519 
   520 lemma DirProdSemigroup_semigroup:
   521   includes semigroup G + semigroup H
   522   shows "semigroup (G \<times>\<^sub>s H)"
   523   using prems
   524   by (fast intro: semigroup.intro
   525     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)
   526 
   527 lemma DirProdGroup_magma:
   528   includes magma G + magma H
   529   shows "magma (G \<times>\<^sub>g H)"
   530   by (rule magma.intro)
   531     (auto simp add: DirProdGroup_def DirProdSemigroup_def)
   532 
   533 lemma DirProdGroup_semigroup_axioms:
   534   includes semigroup G + semigroup H
   535   shows "semigroup_axioms (G \<times>\<^sub>g H)"
   536   by (rule semigroup_axioms.intro)
   537     (auto simp add: DirProdGroup_def DirProdSemigroup_def
   538       G.m_assoc H.m_assoc)
   539 
   540 lemma DirProdGroup_semigroup:
   541   includes semigroup G + semigroup H
   542   shows "semigroup (G \<times>\<^sub>g H)"
   543   using prems
   544   by (fast intro: semigroup.intro
   545     DirProdGroup_magma DirProdGroup_semigroup_axioms)
   546 
   547 (* ... and further lemmas for group ... *)
   548 
   549 lemma DirProdGroup_group:
   550   includes group G + group H
   551   shows "group (G \<times>\<^sub>g H)"
   552   by (rule groupI)
   553     (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
   554       simp add: DirProdGroup_def DirProdSemigroup_def)
   555 
   556 lemma carrier_DirProdGroup [simp]:
   557      "carrier (G \<times>\<^sub>g H) = carrier G \<times> carrier H"
   558   by (simp add: DirProdGroup_def DirProdSemigroup_def)
   559 
   560 lemma one_DirProdGroup [simp]:
   561      "one (G \<times>\<^sub>g H) = (one G, one H)"
   562   by (simp add: DirProdGroup_def DirProdSemigroup_def);
   563 
   564 lemma mult_DirProdGroup [simp]:
   565      "mult (G \<times>\<^sub>g H) (g, h) (g', h') = (mult G g g', mult H h h')"
   566   by (simp add: DirProdGroup_def DirProdSemigroup_def)
   567 
   568 lemma inv_DirProdGroup [simp]:
   569   includes group G + group H
   570   assumes g: "g \<in> carrier G"
   571       and h: "h \<in> carrier H"
   572   shows "m_inv (G \<times>\<^sub>g H) (g, h) = (m_inv G g, m_inv H h)"
   573   apply (rule group.inv_equality [OF DirProdGroup_group])
   574   apply (simp_all add: prems group_def group.l_inv)
   575   done
   576 
   577 subsection {* Homomorphisms *}
   578 
   579 constdefs
   580   hom :: "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme]
   581     => ('a => 'b)set"
   582   "hom G H ==
   583     {h. h \<in> carrier G -> carrier H &
   584       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}"
   585 
   586 lemma (in semigroup) hom:
   587   includes semigroup G
   588   shows "semigroup (| carrier = hom G G, mult = op o |)"
   589 proof (rule semigroup.intro)
   590   show "magma (| carrier = hom G G, mult = op o |)"
   591     by (rule magma.intro) (simp add: Pi_def hom_def)
   592 next
   593   show "semigroup_axioms (| carrier = hom G G, mult = op o |)"
   594     by (rule semigroup_axioms.intro) (simp add: o_assoc)
   595 qed
   596 
   597 lemma hom_mult:
   598   "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] 
   599    ==> h (mult G x y) = mult H (h x) (h y)"
   600   by (simp add: hom_def) 
   601 
   602 lemma hom_closed:
   603   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
   604   by (auto simp add: hom_def funcset_mem)
   605 
   606 lemma compose_hom:
   607      "[|group G; h \<in> hom G G; h' \<in> hom G G; h' \<in> carrier G -> carrier G|]
   608       ==> compose (carrier G) h h' \<in> hom G G"
   609 apply (simp (no_asm_simp) add: hom_def)
   610 apply (intro conjI) 
   611  apply (force simp add: funcset_compose hom_def)
   612 apply (simp add: compose_def group.axioms hom_mult funcset_mem) 
   613 done
   614 
   615 locale group_hom = group G + group H + var h +
   616   assumes homh: "h \<in> hom G H"
   617   notes hom_mult [simp] = hom_mult [OF homh]
   618     and hom_closed [simp] = hom_closed [OF homh]
   619 
   620 lemma (in group_hom) one_closed [simp]:
   621   "h \<one> \<in> carrier H"
   622   by simp
   623 
   624 lemma (in group_hom) hom_one [simp]:
   625   "h \<one> = \<one>\<^sub>2"
   626 proof -
   627   have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"
   628     by (simp add: hom_mult [symmetric] del: hom_mult)
   629   then show ?thesis by (simp del: r_one)
   630 qed
   631 
   632 lemma (in group_hom) inv_closed [simp]:
   633   "x \<in> carrier G ==> h (inv x) \<in> carrier H"
   634   by simp
   635 
   636 lemma (in group_hom) hom_inv [simp]:
   637   "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"
   638 proof -
   639   assume x: "x \<in> carrier G"
   640   then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"
   641     by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
   642   also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"
   643     by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
   644   finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .
   645   with x show ?thesis by simp
   646 qed
   647 
   648 subsection {* Commutative Structures *}
   649 
   650 text {*
   651   Naming convention: multiplicative structures that are commutative
   652   are called \emph{commutative}, additive structures are called
   653   \emph{Abelian}.
   654 *}
   655 
   656 subsection {* Definition *}
   657 
   658 locale comm_semigroup = semigroup +
   659   assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   660 
   661 lemma (in comm_semigroup) m_lcomm:
   662   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   663    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
   664 proof -
   665   assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
   666   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
   667   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
   668   also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
   669   finally show ?thesis .
   670 qed
   671 
   672 lemmas (in comm_semigroup) m_ac = m_assoc m_comm m_lcomm
   673 
   674 locale comm_monoid = comm_semigroup + monoid
   675 
   676 lemma comm_monoidI:
   677   assumes m_closed:
   678       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
   679     and one_closed: "one G \<in> carrier G"
   680     and m_assoc:
   681       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   682       mult G (mult G x y) z = mult G x (mult G y z)"
   683     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
   684     and m_comm:
   685       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
   686   shows "comm_monoid G"
   687   using l_one
   688   by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro
   689     comm_semigroup_axioms.intro monoid_axioms.intro
   690     intro: prems simp: m_closed one_closed m_comm)
   691 
   692 lemma (in monoid) monoid_comm_monoidI:
   693   assumes m_comm:
   694       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
   695   shows "comm_monoid G"
   696   by (rule comm_monoidI) (auto intro: m_assoc m_comm)
   697 (*
   698 lemma (in comm_monoid) r_one [simp]:
   699   "x \<in> carrier G ==> x \<otimes> \<one> = x"
   700 proof -
   701   assume G: "x \<in> carrier G"
   702   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
   703   also from G have "... = x" by simp
   704   finally show ?thesis .
   705 qed
   706 *)
   707 
   708 lemma (in comm_monoid) nat_pow_distr:
   709   "[| x \<in> carrier G; y \<in> carrier G |] ==>
   710   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
   711   by (induct n) (simp, simp add: m_ac)
   712 
   713 locale comm_group = comm_monoid + group
   714 
   715 lemma (in group) group_comm_groupI:
   716   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
   717       mult G x y = mult G y x"
   718   shows "comm_group G"
   719   by (fast intro: comm_group.intro comm_semigroup_axioms.intro
   720     group.axioms prems)
   721 
   722 lemma comm_groupI:
   723   assumes m_closed:
   724       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
   725     and one_closed: "one G \<in> carrier G"
   726     and m_assoc:
   727       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
   728       mult G (mult G x y) z = mult G x (mult G y z)"
   729     and m_comm:
   730       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
   731     and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
   732     and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
   733   shows "comm_group G"
   734   by (fast intro: group.group_comm_groupI groupI prems)
   735 
   736 lemma (in comm_group) inv_mult:
   737   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
   738   by (simp add: m_ac inv_mult_group)
   739 
   740 end