src/HOL/Algebra/Group.thy
 author ballarin Wed Jul 30 19:03:33 2008 +0200 (2008-07-30) changeset 27713 95b36bfe7fc4 parent 27698 197f0517f0bd child 27714 27b4d7c01f8b permissions -rw-r--r--
New locales for orders and lattices where the equivalence relation is not restricted to equality.
     1 (*

     2   Title:  HOL/Algebra/Group.thy

     3   Id:     $Id$

     4   Author: Clemens Ballarin, started 4 February 2003

     5

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

     7 *)

     8

     9 theory Group imports FuncSet Lattice begin

    10

    11

    12 section {* Monoids and Groups *}

    13

    14 subsection {* Definitions *}

    15

    16 text {*

    17   Definitions follow \cite{Jacobson:1985}.

    18 *}

    19

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

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

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

    23

    24 constdefs (structure G)

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

    26   "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"

    27

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

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

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

    31

    32 consts

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

    34

    35 defs (overloaded)

    36   nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"

    37   int_pow_def: "pow G a z ==

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

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

    40

    41 locale monoid =

    42   fixes G (structure)

    43   assumes m_closed [intro, simp]:

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

    45       and m_assoc:

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

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

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

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

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

    51

    52 lemma monoidI:

    53   fixes G (structure)

    54   assumes m_closed:

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

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

    57     and m_assoc:

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

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

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

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

    62   shows "monoid G"

    63   by (fast intro!: monoid.intro intro: prems)

    64

    65 lemma (in monoid) Units_closed [dest]:

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

    67   by (unfold Units_def) fast

    68

    69 lemma (in monoid) inv_unique:

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

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

    72   shows "y = y'"

    73 proof -

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

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

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

    77   finally show ?thesis .

    78 qed

    79

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

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

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

    83 proof -

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

    85     unfolding Units_def by fast

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

    87     unfolding Units_def by fast

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

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

    90   moreover note x y

    91   ultimately show ?thesis unfolding Units_def

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

    93     apply (rule_tac CollectI)

    94     apply (rule)

    95     apply (fast)

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

    97     apply (auto simp: m_assoc)

    98     done

    99 qed

   100

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

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

   103   by (unfold Units_def) auto

   104

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

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

   107   apply (unfold Units_def m_inv_def, auto)

   108   apply (rule theI2, fast)

   109    apply (fast intro: inv_unique, fast)

   110   done

   111

   112 lemma (in monoid) Units_l_inv_ex:

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

   114   by (unfold Units_def) auto

   115

   116 lemma (in monoid) Units_r_inv_ex:

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

   118   by (unfold Units_def) auto

   119

   120 lemma (in monoid) Units_l_inv [simp]:

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

   122   apply (unfold Units_def m_inv_def, auto)

   123   apply (rule theI2, fast)

   124    apply (fast intro: inv_unique, fast)

   125   done

   126

   127 lemma (in monoid) Units_r_inv [simp]:

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

   129   apply (unfold Units_def m_inv_def, auto)

   130   apply (rule theI2, fast)

   131    apply (fast intro: inv_unique, fast)

   132   done

   133

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

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

   136 proof -

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

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

   139     by (auto simp add: Units_def

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

   141 qed

   142

   143 lemma (in monoid) Units_l_cancel [simp]:

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

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

   146 proof

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

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

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

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

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

   152 next

   153   assume eq: "y = z"

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

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

   156 qed

   157

   158 lemma (in monoid) Units_inv_inv [simp]:

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

   160 proof -

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

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

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

   164 qed

   165

   166 lemma (in monoid) inv_inj_on_Units:

   167   "inj_on (m_inv G) (Units G)"

   168 proof (rule inj_onI)

   169   fix x y

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

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

   172   with G show "x = y" by simp

   173 qed

   174

   175 lemma (in monoid) Units_inv_comm:

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

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

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

   179 proof -

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

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

   182 qed

   183

   184 text {* Power *}

   185

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

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

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

   189

   190 lemma (in monoid) nat_pow_0 [simp]:

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

   192   by (simp add: nat_pow_def)

   193

   194 lemma (in monoid) nat_pow_Suc [simp]:

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

   196   by (simp add: nat_pow_def)

   197

   198 lemma (in monoid) nat_pow_one [simp]:

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

   200   by (induct n) simp_all

   201

   202 lemma (in monoid) nat_pow_mult:

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

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

   205

   206 lemma (in monoid) nat_pow_pow:

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

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

   209

   210

   211 (* Jacobson defines submonoid here. *)

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

   213

   214

   215 subsection {* Groups *}

   216

   217 text {*

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

   219 *}

   220

   221 locale group = monoid +

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

   223

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

   225

   226 theorem groupI:

   227   fixes G (structure)

   228   assumes m_closed [simp]:

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

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

   231     and m_assoc:

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

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

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

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

   236   shows "group G"

   237 proof -

   238   have l_cancel [simp]:

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

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

   241   proof

   242     fix x y z

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

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

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

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

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

   248       by (simp add: m_assoc)

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

   250   next

   251     fix x y z

   252     assume eq: "y = z"

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

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

   255   qed

   256   have r_one:

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

   258   proof -

   259     fix x

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

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

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

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

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

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

   266   qed

   267   have inv_ex:

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

   269   proof -

   270     fix x

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

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

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

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

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

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

   277       by simp

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

   279       by (fast intro: l_inv r_inv)

   280   qed

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

   282     by (unfold Units_def) fast

   283   show ?thesis by unfold_locales (auto simp: r_one m_assoc carrier_subset_Units)

   284 qed

   285

   286 lemma (in monoid) group_l_invI:

   287   assumes l_inv_ex:

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

   289   shows "group G"

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

   291

   292 lemma (in group) Units_eq [simp]:

   293   "Units G = carrier G"

   294 proof

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

   296 next

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

   298 qed

   299

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

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

   302   using Units_inv_closed by simp

   303

   304 lemma (in group) l_inv_ex [simp]:

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

   306   using Units_l_inv_ex by simp

   307

   308 lemma (in group) r_inv_ex [simp]:

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

   310   using Units_r_inv_ex by simp

   311

   312 lemma (in group) l_inv [simp]:

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

   314   using Units_l_inv by simp

   315

   316

   317 subsection {* Cancellation Laws and Basic Properties *}

   318

   319 lemma (in group) l_cancel [simp]:

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

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

   322   using Units_l_inv by simp

   323

   324 lemma (in group) r_inv [simp]:

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

   326 proof -

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

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

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

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

   331 qed

   332

   333 lemma (in group) r_cancel [simp]:

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

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

   336 proof

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

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

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

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

   341   with G show "y = z" by simp

   342 next

   343   assume eq: "y = z"

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

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

   346 qed

   347

   348 lemma (in group) inv_one [simp]:

   349   "inv \<one> = \<one>"

   350 proof -

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

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

   353   finally show ?thesis .

   354 qed

   355

   356 lemma (in group) inv_inv [simp]:

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

   358   using Units_inv_inv by simp

   359

   360 lemma (in group) inv_inj:

   361   "inj_on (m_inv G) (carrier G)"

   362   using inv_inj_on_Units by simp

   363

   364 lemma (in group) inv_mult_group:

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

   366 proof -

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

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

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

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

   371 qed

   372

   373 lemma (in group) inv_comm:

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

   375   by (rule Units_inv_comm) auto

   376

   377 lemma (in group) inv_equality:

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

   379 apply (simp add: m_inv_def)

   380 apply (rule the_equality)

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

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

   383 done

   384

   385 text {* Power *}

   386

   387 lemma (in group) int_pow_def2:

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

   389   by (simp add: int_pow_def nat_pow_def Let_def)

   390

   391 lemma (in group) int_pow_0 [simp]:

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

   393   by (simp add: int_pow_def2)

   394

   395 lemma (in group) int_pow_one [simp]:

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

   397   by (simp add: int_pow_def2)

   398

   399

   400 subsection {* Subgroups *}

   401

   402 locale subgroup =

   403   fixes H and G (structure)

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

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

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

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

   408

   409 lemma (in subgroup) is_subgroup:

   410   "subgroup H G" by (rule subgroup_axioms)

   411

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

   413

   414 lemma (in subgroup) mem_carrier [simp]:

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

   416   using subset by blast

   417

   418 lemma subgroup_imp_subset:

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

   420   by (rule subgroup.subset)

   421

   422 lemma (in subgroup) subgroup_is_group [intro]:

   423   assumes "group G"

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

   425 proof -

   426   interpret group [G] by fact

   427   show ?thesis

   428     apply (rule monoid.group_l_invI)

   429     apply (unfold_locales) 

   430     apply (auto intro: m_assoc l_inv mem_carrier)

   431     done

   432 qed

   433

   434 text {*

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

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

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

   438 *}

   439

   440 lemma (in group) one_in_subset:

   441   "[| 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 |]

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

   443 by (force simp add: l_inv)

   444

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

   446

   447 lemma (in group) subgroupI:

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

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

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

   451   shows "subgroup H G"

   452 proof (simp add: subgroup_def prems)

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

   454 qed

   455

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

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

   458

   459 lemma subgroup_nonempty:

   460   "~ subgroup {} G"

   461   by (blast dest: subgroup.one_closed)

   462

   463 lemma (in subgroup) finite_imp_card_positive:

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

   465 proof (rule classical)

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

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

   468   with prems have "subgroup {} G" by simp

   469   with subgroup_nonempty show ?thesis by contradiction

   470 qed

   471

   472 (*

   473 lemma (in monoid) Units_subgroup:

   474   "subgroup (Units G) G"

   475 *)

   476

   477

   478 subsection {* Direct Products *}

   479

   480 constdefs

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

   482   "G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H,

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

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

   485

   486 lemma DirProd_monoid:

   487   assumes "monoid G" and "monoid H"

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

   489 proof -

   490   interpret G: monoid [G] by fact

   491   interpret H: monoid [H] by fact

   492   from prems

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

   494 qed

   495

   496

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

   498 lemma DirProd_group:

   499   assumes "group G" and "group H"

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

   501 proof -

   502   interpret G: group [G] by fact

   503   interpret H: group [H] by fact

   504   show ?thesis by (rule groupI)

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

   506            simp add: DirProd_def)

   507 qed

   508

   509 lemma carrier_DirProd [simp]:

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

   511   by (simp add: DirProd_def)

   512

   513 lemma one_DirProd [simp]:

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

   515   by (simp add: DirProd_def)

   516

   517 lemma mult_DirProd [simp]:

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

   519   by (simp add: DirProd_def)

   520

   521 lemma inv_DirProd [simp]:

   522   assumes "group G" and "group H"

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

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

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

   526 proof -

   527   interpret G: group [G] by fact

   528   interpret H: group [H] by fact

   529   interpret Prod: group ["G \<times>\<times> H"]

   530     by (auto intro: DirProd_group group.intro group.axioms prems)

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

   532 qed

   533

   534

   535 subsection {* Homomorphisms and Isomorphisms *}

   536

   537 constdefs (structure G and H)

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

   539   "hom G H ==

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

   541       (\<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)}"

   542

   543 lemma hom_mult:

   544   "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]

   545    ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"

   546   by (simp add: hom_def)

   547

   548 lemma hom_closed:

   549   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"

   550   by (auto simp add: hom_def funcset_mem)

   551

   552 lemma (in group) hom_compose:

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

   554 apply (auto simp add: hom_def funcset_compose)

   555 apply (simp add: compose_def funcset_mem)

   556 done

   557

   558 constdefs

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

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

   561

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

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

   564

   565 lemma (in group) iso_sym:

   566      "h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G"

   567 apply (simp add: iso_def bij_betw_Inv)

   568 apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G")

   569  prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv])

   570 apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f)

   571 done

   572

   573 lemma (in group) iso_trans:

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

   575 by (auto simp add: iso_def hom_compose bij_betw_compose)

   576

   577 lemma DirProd_commute_iso:

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

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

   580

   581 lemma DirProd_assoc_iso:

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

   583 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)

   584

   585

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

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

   588 locale group_hom = group G + group H + var h +

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

   590   notes hom_mult [simp] = hom_mult [OF homh]

   591     and hom_closed [simp] = hom_closed [OF homh]

   592

   593 lemma (in group_hom) one_closed [simp]:

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

   595   by simp

   596

   597 lemma (in group_hom) hom_one [simp]:

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

   599 proof -

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

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

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

   603 qed

   604

   605 lemma (in group_hom) inv_closed [simp]:

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

   607   by simp

   608

   609 lemma (in group_hom) hom_inv [simp]:

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

   611 proof -

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

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

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

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

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

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

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

   619 qed

   620

   621

   622 subsection {* Commutative Structures *}

   623

   624 text {*

   625   Naming convention: multiplicative structures that are commutative

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

   627   \emph{Abelian}.

   628 *}

   629

   630 locale comm_monoid = monoid +

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

   632

   633 lemma (in comm_monoid) m_lcomm:

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

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

   636 proof -

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

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

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

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

   641   finally show ?thesis .

   642 qed

   643

   644 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   645

   646 lemma comm_monoidI:

   647   fixes G (structure)

   648   assumes m_closed:

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

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

   651     and m_assoc:

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

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

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

   655     and m_comm:

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

   657   shows "comm_monoid G"

   658   using l_one

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

   660              intro: prems simp: m_closed one_closed m_comm)

   661

   662 lemma (in monoid) monoid_comm_monoidI:

   663   assumes m_comm:

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

   665   shows "comm_monoid G"

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

   667

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

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

   670 proof -

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

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

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

   674   finally show ?thesis .

   675 qed*)

   676

   677 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   681

   682 locale comm_group = comm_monoid + group

   683

   684 lemma (in group) group_comm_groupI:

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

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

   687   shows "comm_group G"

   688   by unfold_locales (simp_all add: m_comm)

   689

   690 lemma comm_groupI:

   691   fixes G (structure)

   692   assumes m_closed:

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

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

   695     and m_assoc:

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

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

   698     and m_comm:

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

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

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

   702   shows "comm_group G"

   703   by (fast intro: group.group_comm_groupI groupI prems)

   704

   705 lemma (in comm_group) inv_mult:

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

   707   by (simp add: m_ac inv_mult_group)

   708

   709

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

   711

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

   713

   714 theorem (in group) subgroups_partial_order:

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

   716   by unfold_locales simp_all

   717

   718 lemma (in group) subgroup_self:

   719   "subgroup (carrier G) G"

   720   by (rule subgroupI) auto

   721

   722 lemma (in group) subgroup_imp_group:

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

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

   725

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

   727   "monoid G"

   728   by (auto intro: monoid.intro m_assoc)

   729

   730 lemma (in group) subgroup_inv_equality:

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

   732 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   737 done

   738

   739 theorem (in group) subgroups_Inter:

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

   741     and not_empty: "A ~= {}"

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

   743 proof (rule subgroupI)

   744   from subgr [THEN subgroup.subset] and not_empty

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

   746 next

   747   from subgr [THEN subgroup.one_closed]

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

   749 next

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

   751   with subgr [THEN subgroup.m_inv_closed]

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

   753 next

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

   755   with subgr [THEN subgroup.m_closed]

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

   757 qed

   758

   759 theorem (in group) subgroups_complete_lattice:

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

   761     (is "complete_lattice ?L")

   762 proof (rule partial_order.complete_lattice_criterion1)

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

   764 next

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

   766   proof

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

   768       by (unfold greatest_def)

   769         (simp add: subgroup.subset subgroup_self)

   770   qed

   771 next

   772   fix A

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

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

   775     by (fastsimp intro: subgroups_Inter)

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

   777   proof

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

   779       (is "greatest _ ?Int _")

   780     proof (rule greatest_LowerI)

   781       fix H

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

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

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

   785 	by (rule subgroup_imp_group)

   786       from groupH have monoidH: "monoid ?H"

   787 	by (rule group.is_monoid)

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

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

   790     next

   791       fix H

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

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

   794 	by (fastsimp simp: Lower_def intro: Inter_greatest)

   795     next

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

   797     next

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

   799     qed

   800   qed

   801 qed

   802

   803 end