src/HOL/Algebra/Group.thy
 author wenzelm Sat Nov 04 15:24:40 2017 +0100 (20 months ago) changeset 67003 49850a679c2c parent 66579 2db3fe23fdaf child 67091 1393c2340eec permissions -rw-r--r--
more robust sorted_entries;
     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 Complete_Lattice "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   \<comment>\<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 lemma int_pow_int: "x (^)\<^bsub>G\<^esub> (int n) = x (^)\<^bsub>G\<^esub> n"

    48 by(simp add: int_pow_def nat_pow_def)

    49

    50 locale monoid =

    51   fixes G (structure)

    52   assumes m_closed [intro, simp]:

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

    54       and m_assoc:

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

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

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

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

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

    60

    61 lemma monoidI:

    62   fixes G (structure)

    63   assumes m_closed:

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

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

    66     and m_assoc:

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

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

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

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

    71   shows "monoid G"

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

    73

    74 lemma (in monoid) Units_closed [dest]:

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

    76   by (unfold Units_def) fast

    77

    78 lemma (in monoid) inv_unique:

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

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

    81   shows "y = y'"

    82 proof -

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

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

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

    86   finally show ?thesis .

    87 qed

    88

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

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

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

    92 proof -

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

    94     unfolding Units_def by fast

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

    96     unfolding Units_def by fast

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

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

    99   moreover note x y

   100   ultimately show ?thesis unfolding Units_def

   101     \<comment> "Must avoid premature use of \<open>hyp_subst_tac\<close>."

   102     apply (rule_tac CollectI)

   103     apply (rule)

   104     apply (fast)

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

   106     apply (auto simp: m_assoc)

   107     done

   108 qed

   109

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

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

   112   by (unfold Units_def) auto

   113

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

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

   116   apply (unfold Units_def m_inv_def, auto)

   117   apply (rule theI2, fast)

   118    apply (fast intro: inv_unique, fast)

   119   done

   120

   121 lemma (in monoid) Units_l_inv_ex:

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

   123   by (unfold Units_def) auto

   124

   125 lemma (in monoid) Units_r_inv_ex:

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

   127   by (unfold Units_def) auto

   128

   129 lemma (in monoid) Units_l_inv [simp]:

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

   131   apply (unfold Units_def m_inv_def, auto)

   132   apply (rule theI2, fast)

   133    apply (fast intro: inv_unique, fast)

   134   done

   135

   136 lemma (in monoid) Units_r_inv [simp]:

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

   138   apply (unfold Units_def m_inv_def, auto)

   139   apply (rule theI2, fast)

   140    apply (fast intro: inv_unique, fast)

   141   done

   142

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

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

   145 proof -

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

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

   148     by (auto simp add: Units_def

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

   150 qed

   151

   152 lemma (in monoid) Units_l_cancel [simp]:

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

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

   155 proof

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

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

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

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

   160   with G show "y = z" by simp

   161 next

   162   assume eq: "y = z"

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

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

   165 qed

   166

   167 lemma (in monoid) Units_inv_inv [simp]:

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

   169 proof -

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

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

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

   173 qed

   174

   175 lemma (in monoid) inv_inj_on_Units:

   176   "inj_on (m_inv G) (Units G)"

   177 proof (rule inj_onI)

   178   fix x y

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

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

   181   with G show "x = y" by simp

   182 qed

   183

   184 lemma (in monoid) Units_inv_comm:

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

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

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

   188 proof -

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

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

   191 qed

   192

   193 lemma (in monoid) carrier_not_empty: "carrier G \<noteq> {}"

   194 by auto

   195

   196 text \<open>Power\<close>

   197

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

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

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

   201

   202 lemma (in monoid) nat_pow_0 [simp]:

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

   204   by (simp add: nat_pow_def)

   205

   206 lemma (in monoid) nat_pow_Suc [simp]:

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

   208   by (simp add: nat_pow_def)

   209

   210 lemma (in monoid) nat_pow_one [simp]:

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

   212   by (induct n) simp_all

   213

   214 lemma (in monoid) nat_pow_mult:

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

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

   217

   218 lemma (in monoid) nat_pow_pow:

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

   220   by (induct m) (simp, simp add: nat_pow_mult add.commute)

   221

   222

   223 (* Jacobson defines submonoid here. *)

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

   225

   226

   227 subsection \<open>Groups\<close>

   228

   229 text \<open>

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

   231 \<close>

   232

   233 locale group = monoid +

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

   235

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

   237

   238 theorem groupI:

   239   fixes G (structure)

   240   assumes m_closed [simp]:

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

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

   243     and m_assoc:

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

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

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

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

   248   shows "group G"

   249 proof -

   250   have l_cancel [simp]:

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

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

   253   proof

   254     fix x y z

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

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

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

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

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

   260       by (simp add: m_assoc)

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

   262   next

   263     fix x y z

   264     assume eq: "y = z"

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

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

   267   qed

   268   have r_one:

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

   270   proof -

   271     fix x

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

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

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

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

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

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

   278   qed

   279   have inv_ex:

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

   281   proof -

   282     fix x

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

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

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

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

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

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

   289       by simp

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

   291       by (fast intro: l_inv r_inv)

   292   qed

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

   294     by (unfold Units_def) fast

   295   show ?thesis

   296     by standard (auto simp: r_one m_assoc carrier_subset_Units)

   297 qed

   298

   299 lemma (in monoid) group_l_invI:

   300   assumes l_inv_ex:

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

   302   shows "group G"

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

   304

   305 lemma (in group) Units_eq [simp]:

   306   "Units G = carrier G"

   307 proof

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

   309 next

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

   311 qed

   312

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

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

   315   using Units_inv_closed by simp

   316

   317 lemma (in group) l_inv_ex [simp]:

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

   319   using Units_l_inv_ex by simp

   320

   321 lemma (in group) r_inv_ex [simp]:

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

   323   using Units_r_inv_ex by simp

   324

   325 lemma (in group) l_inv [simp]:

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

   327   using Units_l_inv by simp

   328

   329

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

   331

   332 lemma (in group) l_cancel [simp]:

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

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

   335   using Units_l_inv by simp

   336

   337 lemma (in group) r_inv [simp]:

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

   339 proof -

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

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

   342     by (simp add: m_assoc [symmetric])

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

   344 qed

   345

   346 lemma (in group) r_cancel [simp]:

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

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

   349 proof

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

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

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

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

   354   with G show "y = z" by simp

   355 next

   356   assume eq: "y = z"

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

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

   359 qed

   360

   361 lemma (in group) inv_one [simp]:

   362   "inv \<one> = \<one>"

   363 proof -

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

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

   366   finally show ?thesis .

   367 qed

   368

   369 lemma (in group) inv_inv [simp]:

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

   371   using Units_inv_inv by simp

   372

   373 lemma (in group) inv_inj:

   374   "inj_on (m_inv G) (carrier G)"

   375   using inv_inj_on_Units by simp

   376

   377 lemma (in group) inv_mult_group:

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

   379 proof -

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

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

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

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

   384 qed

   385

   386 lemma (in group) inv_comm:

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

   388   by (rule Units_inv_comm) auto

   389

   390 lemma (in group) inv_equality:

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

   392 apply (simp add: m_inv_def)

   393 apply (rule the_equality)

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

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

   396 done

   397

   398 (* Contributed by Joachim Breitner *)

   399 lemma (in group) inv_solve_left:

   400   "\<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"

   401   by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

   402 lemma (in group) inv_solve_right:

   403   "\<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"

   404   by (metis inv_equality l_inv_ex l_one m_assoc r_inv)

   405

   406 text \<open>Power\<close>

   407

   408 lemma (in group) int_pow_def2:

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

   410   by (simp add: int_pow_def nat_pow_def Let_def)

   411

   412 lemma (in group) int_pow_0 [simp]:

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

   414   by (simp add: int_pow_def2)

   415

   416 lemma (in group) int_pow_one [simp]:

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

   418   by (simp add: int_pow_def2)

   419

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

   421

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

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

   424   by (simp add: int_pow_def2)

   425

   426 lemma (in group) int_pow_1 [simp]:

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

   428   by (simp add: int_pow_def2)

   429

   430 lemma (in group) int_pow_neg:

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

   432   by (simp add: int_pow_def2)

   433

   434 lemma (in group) int_pow_mult:

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

   436 proof -

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

   438   assume "x : carrier G" then

   439   show ?thesis

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

   441 qed

   442

   443 lemma (in group) int_pow_diff:

   444   "x \<in> carrier G \<Longrightarrow> x (^) (n - m :: int) = x (^) n \<otimes> inv (x (^) m)"

   445 by(simp only: diff_conv_add_uminus int_pow_mult int_pow_neg)

   446

   447 lemma (in group) inj_on_multc: "c \<in> carrier G \<Longrightarrow> inj_on (\<lambda>x. x \<otimes> c) (carrier G)"

   448 by(simp add: inj_on_def)

   449

   450 lemma (in group) inj_on_cmult: "c \<in> carrier G \<Longrightarrow> inj_on (\<lambda>x. c \<otimes> x) (carrier G)"

   451 by(simp add: inj_on_def)

   452

   453 subsection \<open>Subgroups\<close>

   454

   455 locale subgroup =

   456   fixes H and G (structure)

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

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

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

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

   461

   462 lemma (in subgroup) is_subgroup:

   463   "subgroup H G" by (rule subgroup_axioms)

   464

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

   466

   467 lemma (in subgroup) mem_carrier [simp]:

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

   469   using subset by blast

   470

   471 lemma subgroup_imp_subset:

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

   473   by (rule subgroup.subset)

   474

   475 lemma (in subgroup) subgroup_is_group [intro]:

   476   assumes "group G"

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

   478 proof -

   479   interpret group G by fact

   480   show ?thesis

   481     apply (rule monoid.group_l_invI)

   482     apply (unfold_locales) 

   483     apply (auto intro: m_assoc l_inv mem_carrier)

   484     done

   485 qed

   486

   487 text \<open>

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

   489   it is closed under inverse, it contains \<open>inv x\<close>.  Since

   490   it is closed under product, it contains \<open>x \<otimes> inv x = \<one>\<close>.

   491 \<close>

   492

   493 lemma (in group) one_in_subset:

   494   "[| 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 |]

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

   496 by force

   497

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

   499

   500 lemma (in group) subgroupI:

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

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

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

   504   shows "subgroup H G"

   505 proof (simp add: subgroup_def assms)

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

   507 qed

   508

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

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

   511

   512 lemma subgroup_nonempty:

   513   "~ subgroup {} G"

   514   by (blast dest: subgroup.one_closed)

   515

   516 lemma (in subgroup) finite_imp_card_positive:

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

   518 proof (rule classical)

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

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

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

   522   with subgroup_nonempty show ?thesis by contradiction

   523 qed

   524

   525 (*

   526 lemma (in monoid) Units_subgroup:

   527   "subgroup (Units G) G"

   528 *)

   529

   530

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

   532

   533 definition

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

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

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

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

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

   539

   540 lemma DirProd_monoid:

   541   assumes "monoid G" and "monoid H"

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

   543 proof -

   544   interpret G: monoid G by fact

   545   interpret H: monoid H by fact

   546   from assms

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

   548 qed

   549

   550

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

   552 lemma DirProd_group:

   553   assumes "group G" and "group H"

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

   555 proof -

   556   interpret G: group G by fact

   557   interpret H: group H by fact

   558   show ?thesis by (rule groupI)

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

   560            simp add: DirProd_def)

   561 qed

   562

   563 lemma carrier_DirProd [simp]:

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

   565   by (simp add: DirProd_def)

   566

   567 lemma one_DirProd [simp]:

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

   569   by (simp add: DirProd_def)

   570

   571 lemma mult_DirProd [simp]:

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

   573   by (simp add: DirProd_def)

   574

   575 lemma inv_DirProd [simp]:

   576   assumes "group G" and "group H"

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

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

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

   580 proof -

   581   interpret G: group G by fact

   582   interpret H: group H by fact

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

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

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

   586 qed

   587

   588

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

   590

   591 definition

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

   593   "hom G H =

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

   595       (\<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)}"

   596

   597 lemma (in group) hom_compose:

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

   599 by (fastforce simp add: hom_def compose_def)

   600

   601 definition

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

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

   604

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

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

   607

   608 lemma (in group) iso_sym:

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

   610 apply (simp add: iso_def bij_betw_inv_into)

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

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

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

   614 done

   615

   616 lemma (in group) iso_trans:

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

   618 by (auto simp add: iso_def hom_compose bij_betw_compose)

   619

   620 lemma DirProd_commute_iso:

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

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

   623

   624 lemma DirProd_assoc_iso:

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

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

   627

   628

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

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

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

   632   fixes h

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

   634

   635 lemma (in group_hom) hom_mult [simp]:

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

   637 proof -

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

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

   640 qed

   641

   642 lemma (in group_hom) hom_closed [simp]:

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

   644 proof -

   645   assume "x \<in> carrier G"

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

   647 qed

   648

   649 lemma (in group_hom) one_closed [simp]:

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

   651   by simp

   652

   653 lemma (in group_hom) hom_one [simp]:

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

   655 proof -

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

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

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

   659 qed

   660

   661 lemma (in group_hom) inv_closed [simp]:

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

   663   by simp

   664

   665 lemma (in group_hom) hom_inv [simp]:

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

   667 proof -

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

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

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

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

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

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

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

   675 qed

   676

   677 (* Contributed by Joachim Breitner *)

   678 lemma (in group) int_pow_is_hom:

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

   680   unfolding hom_def by (simp add: int_pow_mult)

   681

   682

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

   684

   685 text \<open>

   686   Naming convention: multiplicative structures that are commutative

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

   688   \emph{Abelian}.

   689 \<close>

   690

   691 locale comm_monoid = monoid +

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

   693

   694 lemma (in comm_monoid) m_lcomm:

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

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

   697 proof -

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

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

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

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

   702   finally show ?thesis .

   703 qed

   704

   705 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   706

   707 lemma comm_monoidI:

   708   fixes G (structure)

   709   assumes m_closed:

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

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

   712     and m_assoc:

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

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

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

   716     and m_comm:

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

   718   shows "comm_monoid G"

   719   using l_one

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

   721              intro: assms simp: m_closed one_closed m_comm)

   722

   723 lemma (in monoid) monoid_comm_monoidI:

   724   assumes m_comm:

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

   726   shows "comm_monoid G"

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

   728

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

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

   731 proof -

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

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

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

   735   finally show ?thesis .

   736 qed*)

   737

   738 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   742

   743 locale comm_group = comm_monoid + group

   744

   745 lemma (in group) group_comm_groupI:

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

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

   748   shows "comm_group G"

   749   by standard (simp_all add: m_comm)

   750

   751 lemma comm_groupI:

   752   fixes G (structure)

   753   assumes m_closed:

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

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

   756     and m_assoc:

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

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

   759     and m_comm:

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

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

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

   763   shows "comm_group G"

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

   765

   766 lemma (in comm_group) inv_mult:

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

   768   by (simp add: m_ac inv_mult_group)

   769

   770

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

   772

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

   774

   775 theorem (in group) subgroups_partial_order:

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

   777   by standard simp_all

   778

   779 lemma (in group) subgroup_self:

   780   "subgroup (carrier G) G"

   781   by (rule subgroupI) auto

   782

   783 lemma (in group) subgroup_imp_group:

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

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

   786

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

   788   "monoid G"

   789   by (auto intro: monoid.intro m_assoc)

   790

   791 lemma (in group) subgroup_inv_equality:

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

   793 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   798 done

   799

   800 theorem (in group) subgroups_Inter:

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

   802     and not_empty: "A ~= {}"

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

   804 proof (rule subgroupI)

   805   from subgr [THEN subgroup.subset] and not_empty

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

   807 next

   808   from subgr [THEN subgroup.one_closed]

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

   810 next

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

   812   with subgr [THEN subgroup.m_inv_closed]

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

   814 next

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

   816   with subgr [THEN subgroup.m_closed]

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

   818 qed

   819

   820 theorem (in group) subgroups_complete_lattice:

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

   822     (is "complete_lattice ?L")

   823 proof (rule partial_order.complete_lattice_criterion1)

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

   825 next

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

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

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

   829 next

   830   fix A

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

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

   833     by (fastforce intro: subgroups_Inter)

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

   835   proof (rule greatest_LowerI)

   836     fix H

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

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

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

   840       by (rule subgroup_imp_group)

   841     from groupH have monoidH: "monoid ?H"

   842       by (rule group.is_monoid)

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

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

   845   next

   846     fix H

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

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

   849       by (fastforce simp: Lower_def intro: Inter_greatest)

   850   next

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

   852   next

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

   854   qed

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

   856 qed

   857

   858 end