src/HOL/Algebra/Group.thy
 author ballarin Wed Nov 04 08:13:49 2015 +0100 (2015-11-04) changeset 61565 352c73a689da parent 61384 9f5145281888 child 61628 8dd2bd4fe30b permissions -rw-r--r--
Qualifiers in locale expressions default to mandatory regardless of the command.
     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 \<open>Monoids and Groups\<close>

    12

    13 subsection \<open>Definitions\<close>

    14

    15 text \<open>

    16   Definitions follow @{cite "Jacobson:1985"}.

    17 \<close>

    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   --\<open>The set of invertible elements\<close>

    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::semiring_1] => 'a"  (infixr "'(^')\<index>" 75)

    34

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

    36 begin

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

    44     in if z < 0 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

   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 \<open>Power\<close>

   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 \<open>Groups\<close>

   222

   223 text \<open>

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

   225 \<close>

   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

   290     by standard (auto simp: r_one m_assoc carrier_subset_Units)

   291 qed

   292

   293 lemma (in monoid) group_l_invI:

   294   assumes l_inv_ex:

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

   296   shows "group G"

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

   298

   299 lemma (in group) Units_eq [simp]:

   300   "Units G = carrier G"

   301 proof

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

   303 next

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

   305 qed

   306

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

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

   309   using Units_inv_closed by simp

   310

   311 lemma (in group) l_inv_ex [simp]:

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

   313   using Units_l_inv_ex by simp

   314

   315 lemma (in group) r_inv_ex [simp]:

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

   317   using Units_r_inv_ex by simp

   318

   319 lemma (in group) l_inv [simp]:

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

   321   using Units_l_inv by simp

   322

   323

   324 subsection \<open>Cancellation Laws and Basic Properties\<close>

   325

   326 lemma (in group) l_cancel [simp]:

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

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

   329   using Units_l_inv by simp

   330

   331 lemma (in group) r_inv [simp]:

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

   333 proof -

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

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

   336     by (simp add: m_assoc [symmetric])

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

   338 qed

   339

   340 lemma (in group) r_cancel [simp]:

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

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

   343 proof

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

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

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

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

   348   with G show "y = z" by simp

   349 next

   350   assume eq: "y = z"

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

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

   353 qed

   354

   355 lemma (in group) inv_one [simp]:

   356   "inv \<one> = \<one>"

   357 proof -

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

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

   360   finally show ?thesis .

   361 qed

   362

   363 lemma (in group) inv_inv [simp]:

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

   365   using Units_inv_inv by simp

   366

   367 lemma (in group) inv_inj:

   368   "inj_on (m_inv G) (carrier G)"

   369   using inv_inj_on_Units by simp

   370

   371 lemma (in group) inv_mult_group:

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

   373 proof -

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

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

   376     by (simp add: m_assoc) (simp add: m_assoc [symmetric])

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

   378 qed

   379

   380 lemma (in group) inv_comm:

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

   382   by (rule Units_inv_comm) auto

   383

   384 lemma (in group) inv_equality:

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

   386 apply (simp add: m_inv_def)

   387 apply (rule the_equality)

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

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

   390 done

   391

   392 (* Contributed by Joachim Breitner *)

   393 lemma (in group) inv_solve_left:

   394   "\<lbrakk> a \<in> carrier G; b \<in> carrier G; c \<in> carrier G \<rbrakk> \<Longrightarrow> a = inv b \<otimes> c \<longleftrightarrow> c = b \<otimes> a"

   395   by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

   396 lemma (in group) inv_solve_right:

   397   "\<lbrakk> a \<in> carrier G; b \<in> carrier G; c \<in> carrier G \<rbrakk> \<Longrightarrow> a = b \<otimes> inv c \<longleftrightarrow> b = a \<otimes> c"

   398   by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

   399

   400 text \<open>Power\<close>

   401

   402 lemma (in group) int_pow_def2:

   403   "a (^) (z::int) = (if z < 0 then inv (a (^) (nat (-z))) else a (^) (nat z))"

   404   by (simp add: int_pow_def nat_pow_def Let_def)

   405

   406 lemma (in group) int_pow_0 [simp]:

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

   408   by (simp add: int_pow_def2)

   409

   410 lemma (in group) int_pow_one [simp]:

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

   412   by (simp add: int_pow_def2)

   413

   414 (* The following are contributed by Joachim Breitner *)

   415

   416 lemma (in group) int_pow_closed [intro, simp]:

   417   "x \<in> carrier G ==> x (^) (i::int) \<in> carrier G"

   418   by (simp add: int_pow_def2)

   419

   420 lemma (in group) int_pow_1 [simp]:

   421   "x \<in> carrier G \<Longrightarrow> x (^) (1::int) = x"

   422   by (simp add: int_pow_def2)

   423

   424 lemma (in group) int_pow_neg:

   425   "x \<in> carrier G \<Longrightarrow> x (^) (-i::int) = inv (x (^) i)"

   426   by (simp add: int_pow_def2)

   427

   428 lemma (in group) int_pow_mult:

   429   "x \<in> carrier G \<Longrightarrow> x (^) (i + j::int) = x (^) i \<otimes> x (^) j"

   430 proof -

   431   have [simp]: "-i - j = -j - i" by simp

   432   assume "x : carrier G" then

   433   show ?thesis

   434     by (auto simp add: int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult )

   435 qed

   436

   437

   438 subsection \<open>Subgroups\<close>

   439

   440 locale subgroup =

   441   fixes H and G (structure)

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

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

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

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

   446

   447 lemma (in subgroup) is_subgroup:

   448   "subgroup H G" by (rule subgroup_axioms)

   449

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

   451

   452 lemma (in subgroup) mem_carrier [simp]:

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

   454   using subset by blast

   455

   456 lemma subgroup_imp_subset:

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

   458   by (rule subgroup.subset)

   459

   460 lemma (in subgroup) subgroup_is_group [intro]:

   461   assumes "group G"

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

   463 proof -

   464   interpret group G by fact

   465   show ?thesis

   466     apply (rule monoid.group_l_invI)

   467     apply (unfold_locales) 

   468     apply (auto intro: m_assoc l_inv mem_carrier)

   469     done

   470 qed

   471

   472 text \<open>

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

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

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

   476 \<close>

   477

   478 lemma (in group) one_in_subset:

   479   "[| 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 |]

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

   481 by force

   482

   483 text \<open>A characterization of subgroups: closed, non-empty subset.\<close>

   484

   485 lemma (in group) subgroupI:

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

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

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

   489   shows "subgroup H G"

   490 proof (simp add: subgroup_def assms)

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

   492 qed

   493

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

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

   496

   497 lemma subgroup_nonempty:

   498   "~ subgroup {} G"

   499   by (blast dest: subgroup.one_closed)

   500

   501 lemma (in subgroup) finite_imp_card_positive:

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

   503 proof (rule classical)

   504   assume "finite (carrier G)" and a: "~ 0 < card H"

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

   506   with is_subgroup a have "subgroup {} G" by simp

   507   with subgroup_nonempty show ?thesis by contradiction

   508 qed

   509

   510 (*

   511 lemma (in monoid) Units_subgroup:

   512   "subgroup (Units G) G"

   513 *)

   514

   515

   516 subsection \<open>Direct Products\<close>

   517

   518 definition

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

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

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

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

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

   524

   525 lemma DirProd_monoid:

   526   assumes "monoid G" and "monoid H"

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

   528 proof -

   529   interpret G: monoid G by fact

   530   interpret H: monoid H by fact

   531   from assms

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

   533 qed

   534

   535

   536 text\<open>Does not use the previous result because it's easier just to use auto.\<close>

   537 lemma DirProd_group:

   538   assumes "group G" and "group H"

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

   540 proof -

   541   interpret G: group G by fact

   542   interpret H: group H by fact

   543   show ?thesis by (rule groupI)

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

   545            simp add: DirProd_def)

   546 qed

   547

   548 lemma carrier_DirProd [simp]:

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

   550   by (simp add: DirProd_def)

   551

   552 lemma one_DirProd [simp]:

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

   554   by (simp add: DirProd_def)

   555

   556 lemma mult_DirProd [simp]:

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

   558   by (simp add: DirProd_def)

   559

   560 lemma inv_DirProd [simp]:

   561   assumes "group G" and "group H"

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

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

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

   565 proof -

   566   interpret G: group G by fact

   567   interpret H: group H by fact

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

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

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

   571 qed

   572

   573

   574 subsection \<open>Homomorphisms and Isomorphisms\<close>

   575

   576 definition

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

   578   "hom G H =

   579     {h. h \<in> carrier G \<rightarrow> carrier H &

   580       (\<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)}"

   581

   582 lemma (in group) hom_compose:

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

   584 by (fastforce simp add: hom_def compose_def)

   585

   586 definition

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

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

   589

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

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

   592

   593 lemma (in group) iso_sym:

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

   595 apply (simp add: iso_def bij_betw_inv_into)

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

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

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

   599 done

   600

   601 lemma (in group) iso_trans:

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

   603 by (auto simp add: iso_def hom_compose bij_betw_compose)

   604

   605 lemma DirProd_commute_iso:

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

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

   608

   609 lemma DirProd_assoc_iso:

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

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

   612

   613

   614 text\<open>Basis for homomorphism proofs: we assume two groups @{term G} and

   615   @{term H}, with a homomorphism @{term h} between them\<close>

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

   617   fixes h

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

   619

   620 lemma (in group_hom) hom_mult [simp]:

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

   622 proof -

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

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

   625 qed

   626

   627 lemma (in group_hom) hom_closed [simp]:

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

   629 proof -

   630   assume "x \<in> carrier G"

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

   632 qed

   633

   634 lemma (in group_hom) one_closed [simp]:

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

   636   by simp

   637

   638 lemma (in group_hom) hom_one [simp]:

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

   640 proof -

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

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

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

   644 qed

   645

   646 lemma (in group_hom) inv_closed [simp]:

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

   648   by simp

   649

   650 lemma (in group_hom) hom_inv [simp]:

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

   652 proof -

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

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

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

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

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

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

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

   660 qed

   661

   662 (* Contributed by Joachim Breitner *)

   663 lemma (in group) int_pow_is_hom:

   664   "x \<in> carrier G \<Longrightarrow> (op(^) x) \<in> hom \<lparr> carrier = UNIV, mult = op +, one = 0::int \<rparr> G "

   665   unfolding hom_def by (simp add: int_pow_mult)

   666

   667

   668 subsection \<open>Commutative Structures\<close>

   669

   670 text \<open>

   671   Naming convention: multiplicative structures that are commutative

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

   673   \emph{Abelian}.

   674 \<close>

   675

   676 locale comm_monoid = monoid +

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

   678

   679 lemma (in comm_monoid) m_lcomm:

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

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

   682 proof -

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

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

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

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

   687   finally show ?thesis .

   688 qed

   689

   690 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   691

   692 lemma comm_monoidI:

   693   fixes G (structure)

   694   assumes m_closed:

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

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

   697     and m_assoc:

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

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

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

   701     and m_comm:

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

   703   shows "comm_monoid G"

   704   using l_one

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

   706              intro: assms simp: m_closed one_closed m_comm)

   707

   708 lemma (in monoid) monoid_comm_monoidI:

   709   assumes m_comm:

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

   711   shows "comm_monoid G"

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

   713

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

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

   716 proof -

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

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

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

   720   finally show ?thesis .

   721 qed*)

   722

   723 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   727

   728 locale comm_group = comm_monoid + group

   729

   730 lemma (in group) group_comm_groupI:

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

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

   733   shows "comm_group G"

   734   by standard (simp_all add: m_comm)

   735

   736 lemma comm_groupI:

   737   fixes G (structure)

   738   assumes m_closed:

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

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

   741     and m_assoc:

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

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

   744     and m_comm:

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

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

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

   748   shows "comm_group G"

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

   750

   751 lemma (in comm_group) inv_mult:

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

   753   by (simp add: m_ac inv_mult_group)

   754

   755

   756 subsection \<open>The Lattice of Subgroups of a Group\<close>

   757

   758 text_raw \<open>\label{sec:subgroup-lattice}\<close>

   759

   760 theorem (in group) subgroups_partial_order:

   761   "partial_order \<lparr>carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq>\<rparr>"

   762   by standard simp_all

   763

   764 lemma (in group) subgroup_self:

   765   "subgroup (carrier G) G"

   766   by (rule subgroupI) auto

   767

   768 lemma (in group) subgroup_imp_group:

   769   "subgroup H G ==> group (G\<lparr>carrier := H\<rparr>)"

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

   771

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

   773   "monoid G"

   774   by (auto intro: monoid.intro m_assoc)

   775

   776 lemma (in group) subgroup_inv_equality:

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

   778 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   783 done

   784

   785 theorem (in group) subgroups_Inter:

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

   787     and not_empty: "A ~= {}"

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

   789 proof (rule subgroupI)

   790   from subgr [THEN subgroup.subset] and not_empty

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

   792 next

   793   from subgr [THEN subgroup.one_closed]

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

   795 next

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

   797   with subgr [THEN subgroup.m_inv_closed]

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

   799 next

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

   801   with subgr [THEN subgroup.m_closed]

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

   803 qed

   804

   805 theorem (in group) subgroups_complete_lattice:

   806   "complete_lattice \<lparr>carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq>\<rparr>"

   807     (is "complete_lattice ?L")

   808 proof (rule partial_order.complete_lattice_criterion1)

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

   810 next

   811   have "greatest ?L (carrier G) (carrier ?L)"

   812     by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)

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

   814 next

   815   fix A

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

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

   818     by (fastforce intro: subgroups_Inter)

   819   have "greatest ?L (\<Inter>A) (Lower ?L A)" (is "greatest _ ?Int _")

   820   proof (rule greatest_LowerI)

   821     fix H

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

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

   824     from subgroupH have groupH: "group (G \<lparr>carrier := H\<rparr>)" (is "group ?H")

   825       by (rule subgroup_imp_group)

   826     from groupH have monoidH: "monoid ?H"

   827       by (rule group.is_monoid)

   828     from H have Int_subset: "?Int \<subseteq> H" by fastforce

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

   830   next

   831     fix H

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

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

   834       by (fastforce simp: Lower_def intro: Inter_greatest)

   835   next

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

   837   next

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

   839   qed

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

   841 qed

   842

   843 end