src/HOL/Algebra/Group.thy
 author wenzelm Wed Dec 29 17:34:41 2010 +0100 (2010-12-29) changeset 41413 64cd30d6b0b8 parent 35850 dd2636f0f608 child 41528 276078f01ada permissions -rw-r--r--
explicit file specifications -- avoid secondary load path;
     1 (*  Title:      HOL/Algebra/Group.thy

     2     Author:     Clemens Ballarin, started 4 February 2003

     3

     4 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.

     5 *)

     6

     7 theory Group

     8 imports Lattice "~~/src/HOL/Library/FuncSet"

     9 begin

    10

    11 section {* Monoids and Groups *}

    12

    13 subsection {* Definitions *}

    14

    15 text {*

    16   Definitions follow \cite{Jacobson:1985}.

    17 *}

    18

    19 record 'a monoid =  "'a partial_object" +

    20   mult    :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)

    21   one     :: 'a ("\<one>\<index>")

    22

    23 definition

    24   m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _"  80)

    25   where "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G & x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)"

    26

    27 definition

    28   Units :: "_ => 'a set"

    29   --{*The set of invertible elements*}

    30   where "Units G = {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> & y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)}"

    31

    32 consts

    33   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a"  (infixr "'(^')\<index>" 75)

    34

    35 overloading nat_pow == "pow :: [_, 'a, nat] => 'a"

    36 begin

    37   definition "nat_pow G a n = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"

    38 end

    39

    40 overloading int_pow == "pow :: [_, 'a, int] => 'a"

    41 begin

    42   definition "int_pow G a z =

    43    (let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)

    44     in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))"

    45 end

    46

    47 locale monoid =

    48   fixes G (structure)

    49   assumes m_closed [intro, simp]:

    50          "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"

    51       and m_assoc:

    52          "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk>

    53           \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"

    54       and one_closed [intro, simp]: "\<one> \<in> carrier G"

    55       and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x"

    56       and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x"

    57

    58 lemma monoidI:

    59   fixes G (structure)

    60   assumes m_closed:

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

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

    63     and m_assoc:

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

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

    66     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

    67     and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"

    68   shows "monoid G"

    69   by (fast intro!: monoid.intro intro: assms)

    70

    71 lemma (in monoid) Units_closed [dest]:

    72   "x \<in> Units G ==> x \<in> carrier G"

    73   by (unfold Units_def) fast

    74

    75 lemma (in monoid) inv_unique:

    76   assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"

    77     and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"

    78   shows "y = y'"

    79 proof -

    80   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp

    81   also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)

    82   also from G eq have "... = y'" by simp

    83   finally show ?thesis .

    84 qed

    85

    86 lemma (in monoid) Units_m_closed [intro, simp]:

    87   assumes x: "x \<in> Units G" and y: "y \<in> Units G"

    88   shows "x \<otimes> y \<in> Units G"

    89 proof -

    90   from x obtain x' where x: "x \<in> carrier G" "x' \<in> carrier G" and xinv: "x \<otimes> x' = \<one>" "x' \<otimes> x = \<one>"

    91     unfolding Units_def by fast

    92   from y obtain y' where y: "y \<in> carrier G" "y' \<in> carrier G" and yinv: "y \<otimes> y' = \<one>" "y' \<otimes> y = \<one>"

    93     unfolding Units_def by fast

    94   from x y xinv yinv have "y' \<otimes> (x' \<otimes> x) \<otimes> y = \<one>" by simp

    95   moreover from x y xinv yinv have "x \<otimes> (y \<otimes> y') \<otimes> x' = \<one>" by simp

    96   moreover note x y

    97   ultimately show ?thesis unfolding Units_def

    98     -- "Must avoid premature use of @{text hyp_subst_tac}."

    99     apply (rule_tac CollectI)

   100     apply (rule)

   101     apply (fast)

   102     apply (rule bexI [where x = "y' \<otimes> x'"])

   103     apply (auto simp: m_assoc)

   104     done

   105 qed

   106

   107 lemma (in monoid) Units_one_closed [intro, simp]:

   108   "\<one> \<in> Units G"

   109   by (unfold Units_def) auto

   110

   111 lemma (in monoid) Units_inv_closed [intro, simp]:

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

   113   apply (unfold Units_def m_inv_def, auto)

   114   apply (rule theI2, fast)

   115    apply (fast intro: inv_unique, fast)

   116   done

   117

   118 lemma (in monoid) Units_l_inv_ex:

   119   "x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   120   by (unfold Units_def) auto

   121

   122 lemma (in monoid) Units_r_inv_ex:

   123   "x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

   124   by (unfold Units_def) auto

   125

   126 lemma (in monoid) Units_l_inv [simp]:

   127   "x \<in> Units G ==> inv x \<otimes> x = \<one>"

   128   apply (unfold Units_def m_inv_def, auto)

   129   apply (rule theI2, fast)

   130    apply (fast intro: inv_unique, fast)

   131   done

   132

   133 lemma (in monoid) Units_r_inv [simp]:

   134   "x \<in> Units G ==> x \<otimes> inv x = \<one>"

   135   apply (unfold Units_def m_inv_def, auto)

   136   apply (rule theI2, fast)

   137    apply (fast intro: inv_unique, fast)

   138   done

   139

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

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

   142 proof -

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

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

   145     by (auto simp add: Units_def

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

   147 qed

   148

   149 lemma (in monoid) Units_l_cancel [simp]:

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

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

   152 proof

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

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

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

   156     by (simp add: m_assoc Units_closed del: Units_l_inv)

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

   158 next

   159   assume eq: "y = z"

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

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

   162 qed

   163

   164 lemma (in monoid) Units_inv_inv [simp]:

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

   166 proof -

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

   168   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by simp

   169   with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv)

   170 qed

   171

   172 lemma (in monoid) inv_inj_on_Units:

   173   "inj_on (m_inv G) (Units G)"

   174 proof (rule inj_onI)

   175   fix x y

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

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

   178   with G show "x = y" by simp

   179 qed

   180

   181 lemma (in monoid) Units_inv_comm:

   182   assumes inv: "x \<otimes> y = \<one>"

   183     and G: "x \<in> Units G"  "y \<in> Units G"

   184   shows "y \<otimes> x = \<one>"

   185 proof -

   186   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)

   187   with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)

   188 qed

   189

   190 text {* Power *}

   191

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

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

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

   195

   196 lemma (in monoid) nat_pow_0 [simp]:

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

   198   by (simp add: nat_pow_def)

   199

   200 lemma (in monoid) nat_pow_Suc [simp]:

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

   202   by (simp add: nat_pow_def)

   203

   204 lemma (in monoid) nat_pow_one [simp]:

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

   206   by (induct n) simp_all

   207

   208 lemma (in monoid) nat_pow_mult:

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

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

   211

   212 lemma (in monoid) nat_pow_pow:

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

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

   215

   216

   217 (* Jacobson defines submonoid here. *)

   218 (* Jacobson defines the order of a monoid here. *)

   219

   220

   221 subsection {* Groups *}

   222

   223 text {*

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

   225 *}

   226

   227 locale group = monoid +

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

   229

   230 lemma (in group) is_group: "group G" by (rule group_axioms)

   231

   232 theorem groupI:

   233   fixes G (structure)

   234   assumes m_closed [simp]:

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

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

   237     and m_assoc:

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

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

   240     and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

   241     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   242   shows "group G"

   243 proof -

   244   have l_cancel [simp]:

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

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

   247   proof

   248     fix x y z

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

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

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

   252       and l_inv: "x_inv \<otimes> x = \<one>" by fast

   253     from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"

   254       by (simp add: m_assoc)

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

   256   next

   257     fix x y z

   258     assume eq: "y = z"

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

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

   261   qed

   262   have r_one:

   263     "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"

   264   proof -

   265     fix x

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

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

   268       and l_inv: "x_inv \<otimes> x = \<one>" by fast

   269     from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"

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

   271     with x xG show "x \<otimes> \<one> = x" by simp

   272   qed

   273   have inv_ex:

   274     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   275   proof -

   276     fix x

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

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

   279       and l_inv: "y \<otimes> x = \<one>" by fast

   280     from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"

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

   282     with x y have r_inv: "x \<otimes> y = \<one>"

   283       by simp

   284     from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"

   285       by (fast intro: l_inv r_inv)

   286   qed

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

   288     by (unfold Units_def) fast

   289   show ?thesis proof qed (auto simp: r_one m_assoc carrier_subset_Units)

   290 qed

   291

   292 lemma (in monoid) group_l_invI:

   293   assumes l_inv_ex:

   294     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   295   shows "group G"

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

   297

   298 lemma (in group) Units_eq [simp]:

   299   "Units G = carrier G"

   300 proof

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

   302 next

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

   304 qed

   305

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

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

   308   using Units_inv_closed by simp

   309

   310 lemma (in group) l_inv_ex [simp]:

   311   "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   312   using Units_l_inv_ex by simp

   313

   314 lemma (in group) r_inv_ex [simp]:

   315   "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"

   316   using Units_r_inv_ex by simp

   317

   318 lemma (in group) l_inv [simp]:

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

   320   using Units_l_inv by simp

   321

   322

   323 subsection {* Cancellation Laws and Basic Properties *}

   324

   325 lemma (in group) l_cancel [simp]:

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

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

   328   using Units_l_inv by simp

   329

   330 lemma (in group) r_inv [simp]:

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

   332 proof -

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

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

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

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

   337 qed

   338

   339 lemma (in group) r_cancel [simp]:

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

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

   342 proof

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

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

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

   346     by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv)

   347   with G show "y = z" by simp

   348 next

   349   assume eq: "y = z"

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

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

   352 qed

   353

   354 lemma (in group) inv_one [simp]:

   355   "inv \<one> = \<one>"

   356 proof -

   357   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv)

   358   moreover have "... = \<one>" by simp

   359   finally show ?thesis .

   360 qed

   361

   362 lemma (in group) inv_inv [simp]:

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

   364   using Units_inv_inv by simp

   365

   366 lemma (in group) inv_inj:

   367   "inj_on (m_inv G) (carrier G)"

   368   using inv_inj_on_Units by simp

   369

   370 lemma (in group) inv_mult_group:

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

   372 proof -

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

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

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

   376   with G show ?thesis by (simp del: l_inv Units_l_inv)

   377 qed

   378

   379 lemma (in group) inv_comm:

   380   "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"

   381   by (rule Units_inv_comm) auto

   382

   383 lemma (in group) inv_equality:

   384      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"

   385 apply (simp add: m_inv_def)

   386 apply (rule the_equality)

   387  apply (simp add: inv_comm [of y x])

   388 apply (rule r_cancel [THEN iffD1], auto)

   389 done

   390

   391 text {* Power *}

   392

   393 lemma (in group) int_pow_def2:

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

   395   by (simp add: int_pow_def nat_pow_def Let_def)

   396

   397 lemma (in group) int_pow_0 [simp]:

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

   399   by (simp add: int_pow_def2)

   400

   401 lemma (in group) int_pow_one [simp]:

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

   403   by (simp add: int_pow_def2)

   404

   405

   406 subsection {* Subgroups *}

   407

   408 locale subgroup =

   409   fixes H and G (structure)

   410   assumes subset: "H \<subseteq> carrier G"

   411     and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"

   412     and one_closed [simp]: "\<one> \<in> H"

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

   414

   415 lemma (in subgroup) is_subgroup:

   416   "subgroup H G" by (rule subgroup_axioms)

   417

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

   419

   420 lemma (in subgroup) mem_carrier [simp]:

   421   "x \<in> H \<Longrightarrow> x \<in> carrier G"

   422   using subset by blast

   423

   424 lemma subgroup_imp_subset:

   425   "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"

   426   by (rule subgroup.subset)

   427

   428 lemma (in subgroup) subgroup_is_group [intro]:

   429   assumes "group G"

   430   shows "group (G\<lparr>carrier := H\<rparr>)"

   431 proof -

   432   interpret group G by fact

   433   show ?thesis

   434     apply (rule monoid.group_l_invI)

   435     apply (unfold_locales) 

   436     apply (auto intro: m_assoc l_inv mem_carrier)

   437     done

   438 qed

   439

   440 text {*

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

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

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

   444 *}

   445

   446 lemma (in group) one_in_subset:

   447   "[| 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 |]

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

   449 by (force simp add: l_inv)

   450

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

   452

   453 lemma (in group) subgroupI:

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

   455     and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"

   456     and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"

   457   shows "subgroup H G"

   458 proof (simp add: subgroup_def assms)

   459   show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms)

   460 qed

   461

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

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

   464

   465 lemma subgroup_nonempty:

   466   "~ subgroup {} G"

   467   by (blast dest: subgroup.one_closed)

   468

   469 lemma (in subgroup) finite_imp_card_positive:

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

   471 proof (rule classical)

   472   assume "finite (carrier G)" "~ 0 < card H"

   473   then have "finite H" by (blast intro: finite_subset [OF subset])

   474   with prems 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

   484 subsection {* Direct Products *}

   485

   486 definition

   487   DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) where

   488   "G \<times>\<times> H =

   489     \<lparr>carrier = carrier G \<times> carrier H,

   490      mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),

   491      one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"

   492

   493 lemma DirProd_monoid:

   494   assumes "monoid G" and "monoid H"

   495   shows "monoid (G \<times>\<times> H)"

   496 proof -

   497   interpret G: monoid G by fact

   498   interpret H: monoid H by fact

   499   from assms

   500   show ?thesis by (unfold monoid_def DirProd_def, auto)

   501 qed

   502

   503

   504 text{*Does not use the previous result because it's easier just to use auto.*}

   505 lemma DirProd_group:

   506   assumes "group G" and "group H"

   507   shows "group (G \<times>\<times> H)"

   508 proof -

   509   interpret G: group G by fact

   510   interpret H: group H by fact

   511   show ?thesis by (rule groupI)

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

   513            simp add: DirProd_def)

   514 qed

   515

   516 lemma carrier_DirProd [simp]:

   517      "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"

   518   by (simp add: DirProd_def)

   519

   520 lemma one_DirProd [simp]:

   521      "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"

   522   by (simp add: DirProd_def)

   523

   524 lemma mult_DirProd [simp]:

   525      "(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"

   526   by (simp add: DirProd_def)

   527

   528 lemma inv_DirProd [simp]:

   529   assumes "group G" and "group H"

   530   assumes g: "g \<in> carrier G"

   531       and h: "h \<in> carrier H"

   532   shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"

   533 proof -

   534   interpret G: group G by fact

   535   interpret H: group H by fact

   536   interpret Prod: group "G \<times>\<times> H"

   537     by (auto intro: DirProd_group group.intro group.axioms assms)

   538   show ?thesis by (simp add: Prod.inv_equality g h)

   539 qed

   540

   541

   542 subsection {* Homomorphisms and Isomorphisms *}

   543

   544 definition

   545   hom :: "_ => _ => ('a => 'b) set" where

   546   "hom G H =

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

   548       (\<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)}"

   549

   550 lemma (in group) hom_compose:

   551   "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"

   552 by (fastsimp simp add: hom_def compose_def)

   553

   554 definition

   555   iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60)

   556   where "G \<cong> H = {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"

   557

   558 lemma iso_refl: "(%x. x) \<in> G \<cong> G"

   559 by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

   560

   561 lemma (in group) iso_sym:

   562      "h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G"

   563 apply (simp add: iso_def bij_betw_inv_into)

   564 apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G")

   565  prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into])

   566 apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def)

   567 done

   568

   569 lemma (in group) iso_trans:

   570      "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"

   571 by (auto simp add: iso_def hom_compose bij_betw_compose)

   572

   573 lemma DirProd_commute_iso:

   574   shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"

   575 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

   576

   577 lemma DirProd_assoc_iso:

   578   shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"

   579 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)

   580

   581

   582 text{*Basis for homomorphism proofs: we assume two groups @{term G} and

   583   @{term H}, with a homomorphism @{term h} between them*}

   584 locale group_hom = G: group G + H: group H for G (structure) and H (structure) +

   585   fixes h

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

   587

   588 lemma (in group_hom) hom_mult [simp]:

   589   "[| x \<in> carrier G; y \<in> carrier G |] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"

   590 proof -

   591   assume "x \<in> carrier G" "y \<in> carrier G"

   592   with homh [unfolded hom_def] show ?thesis by simp

   593 qed

   594

   595 lemma (in group_hom) hom_closed [simp]:

   596   "x \<in> carrier G ==> h x \<in> carrier H"

   597 proof -

   598   assume "x \<in> carrier G"

   599   with homh [unfolded hom_def] show ?thesis by auto

   600 qed

   601

   602 lemma (in group_hom) one_closed [simp]:

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

   604   by simp

   605

   606 lemma (in group_hom) hom_one [simp]:

   607   "h \<one> = \<one>\<^bsub>H\<^esub>"

   608 proof -

   609   have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"

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

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

   612 qed

   613

   614 lemma (in group_hom) inv_closed [simp]:

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

   616   by simp

   617

   618 lemma (in group_hom) hom_inv [simp]:

   619   "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"

   620 proof -

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

   622   then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"

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

   624   also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"

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

   626   finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .

   627   with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)

   628 qed

   629

   630

   631 subsection {* Commutative Structures *}

   632

   633 text {*

   634   Naming convention: multiplicative structures that are commutative

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

   636   \emph{Abelian}.

   637 *}

   638

   639 locale comm_monoid = monoid +

   640   assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"

   641

   642 lemma (in comm_monoid) m_lcomm:

   643   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>

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

   645 proof -

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

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

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

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

   650   finally show ?thesis .

   651 qed

   652

   653 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   654

   655 lemma comm_monoidI:

   656   fixes G (structure)

   657   assumes m_closed:

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

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

   660     and m_assoc:

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

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

   663     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

   664     and m_comm:

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

   666   shows "comm_monoid G"

   667   using l_one

   668     by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro

   669              intro: assms simp: m_closed one_closed m_comm)

   670

   671 lemma (in monoid) monoid_comm_monoidI:

   672   assumes m_comm:

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

   674   shows "comm_monoid G"

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

   676

   677 (*lemma (in comm_monoid) r_one [simp]:

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

   679 proof -

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

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

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

   683   finally show ?thesis .

   684 qed*)

   685

   686 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   690

   691 locale comm_group = comm_monoid + group

   692

   693 lemma (in group) group_comm_groupI:

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

   695       x \<otimes> y = y \<otimes> x"

   696   shows "comm_group G"

   697   proof qed (simp_all add: m_comm)

   698

   699 lemma comm_groupI:

   700   fixes G (structure)

   701   assumes m_closed:

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

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

   704     and m_assoc:

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

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

   707     and m_comm:

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

   709     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"

   710     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"

   711   shows "comm_group G"

   712   by (fast intro: group.group_comm_groupI groupI assms)

   713

   714 lemma (in comm_group) inv_mult:

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

   716   by (simp add: m_ac inv_mult_group)

   717

   718

   719 subsection {* The Lattice of Subgroups of a Group *}

   720

   721 text_raw {* \label{sec:subgroup-lattice} *}

   722

   723 theorem (in group) subgroups_partial_order:

   724   "partial_order (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"

   725   proof qed simp_all

   726

   727 lemma (in group) subgroup_self:

   728   "subgroup (carrier G) G"

   729   by (rule subgroupI) auto

   730

   731 lemma (in group) subgroup_imp_group:

   732   "subgroup H G ==> group (G(| carrier := H |))"

   733   by (erule subgroup.subgroup_is_group) (rule group_axioms)

   734

   735 lemma (in group) is_monoid [intro, simp]:

   736   "monoid G"

   737   by (auto intro: monoid.intro m_assoc)

   738

   739 lemma (in group) subgroup_inv_equality:

   740   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"

   741 apply (rule_tac inv_equality [THEN sym])

   742   apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)

   743  apply (rule subsetD [OF subgroup.subset], assumption+)

   744 apply (rule subsetD [OF subgroup.subset], assumption)

   745 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)

   746 done

   747

   748 theorem (in group) subgroups_Inter:

   749   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"

   750     and not_empty: "A ~= {}"

   751   shows "subgroup (\<Inter>A) G"

   752 proof (rule subgroupI)

   753   from subgr [THEN subgroup.subset] and not_empty

   754   show "\<Inter>A \<subseteq> carrier G" by blast

   755 next

   756   from subgr [THEN subgroup.one_closed]

   757   show "\<Inter>A ~= {}" by blast

   758 next

   759   fix x assume "x \<in> \<Inter>A"

   760   with subgr [THEN subgroup.m_inv_closed]

   761   show "inv x \<in> \<Inter>A" by blast

   762 next

   763   fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"

   764   with subgr [THEN subgroup.m_closed]

   765   show "x \<otimes> y \<in> \<Inter>A" by blast

   766 qed

   767

   768 theorem (in group) subgroups_complete_lattice:

   769   "complete_lattice (| carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> |)"

   770     (is "complete_lattice ?L")

   771 proof (rule partial_order.complete_lattice_criterion1)

   772   show "partial_order ?L" by (rule subgroups_partial_order)

   773 next

   774   show "\<exists>G. greatest ?L G (carrier ?L)"

   775   proof

   776     show "greatest ?L (carrier G) (carrier ?L)"

   777       by (unfold greatest_def)

   778         (simp add: subgroup.subset subgroup_self)

   779   qed

   780 next

   781   fix A

   782   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"

   783   then have Int_subgroup: "subgroup (\<Inter>A) G"

   784     by (fastsimp intro: subgroups_Inter)

   785   show "\<exists>I. greatest ?L I (Lower ?L A)"

   786   proof

   787     show "greatest ?L (\<Inter>A) (Lower ?L A)"

   788       (is "greatest _ ?Int _")

   789     proof (rule greatest_LowerI)

   790       fix H

   791       assume H: "H \<in> A"

   792       with L have subgroupH: "subgroup H G" by auto

   793       from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")

   794         by (rule subgroup_imp_group)

   795       from groupH have monoidH: "monoid ?H"

   796         by (rule group.is_monoid)

   797       from H have Int_subset: "?Int \<subseteq> H" by fastsimp

   798       then show "le ?L ?Int H" by simp

   799     next

   800       fix H

   801       assume H: "H \<in> Lower ?L A"

   802       with L Int_subgroup show "le ?L H ?Int"

   803         by (fastsimp simp: Lower_def intro: Inter_greatest)

   804     next

   805       show "A \<subseteq> carrier ?L" by (rule L)

   806     next

   807       show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)

   808     qed

   809   qed

   810 qed

   811

   812 end