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