src/HOL/Algebra/Group.thy
 author ballarin Tue Jul 04 14:47:01 2006 +0200 (2006-07-04) changeset 19984 29bb4659f80a parent 19981 c0f124a0d385 child 20318 0e0ea63fe768 permissions -rw-r--r--
Method intro_locales replaced by intro_locales and unfold_locales.
     1 (*

     2   Title:  HOL/Algebra/Group.thy

     3   Id:     $Id$

     4   Author: Clemens Ballarin, started 4 February 2003

     5

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

     7 *)

     8

     9 header {* Groups *}

    10

    11 theory Group imports FuncSet Lattice begin

    12

    13

    14 section {* Monoids and Groups *}

    15

    16 text {*

    17   Definitions follow \cite{Jacobson:1985}.

    18 *}

    19

    20 subsection {* Definitions *}

    21

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

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

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

    25

    26 constdefs (structure G)

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

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

    29

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

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

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

    33

    34 consts

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

    36

    37 defs (overloaded)

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

    39   int_pow_def: "pow G a z ==

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

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

    42

    43 locale monoid =

    44   fixes G (structure)

    45   assumes m_closed [intro, simp]:

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

    47       and m_assoc:

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

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

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

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

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

    53

    54 lemma monoidI:

    55   fixes G (structure)

    56   assumes m_closed:

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

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

    59     and m_assoc:

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

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

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

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

    64   shows "monoid G"

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

    66

    67 lemma (in monoid) Units_closed [dest]:

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

    69   by (unfold Units_def) fast

    70

    71 lemma (in monoid) inv_unique:

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

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

    74   shows "y = y'"

    75 proof -

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

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

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

    79   finally show ?thesis .

    80 qed

    81

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

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

    84   by (unfold Units_def) auto

    85

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

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

    88   apply (unfold Units_def m_inv_def, auto)

    89   apply (rule theI2, fast)

    90    apply (fast intro: inv_unique, fast)

    91   done

    92

    93 lemma (in monoid) Units_l_inv_ex:

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

    95   by (unfold Units_def) auto

    96

    97 lemma (in monoid) Units_r_inv_ex:

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

    99   by (unfold Units_def) auto

   100

   101 lemma (in monoid) Units_l_inv:

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

   103   apply (unfold Units_def m_inv_def, auto)

   104   apply (rule theI2, fast)

   105    apply (fast intro: inv_unique, fast)

   106   done

   107

   108 lemma (in monoid) Units_r_inv:

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

   110   apply (unfold Units_def m_inv_def, auto)

   111   apply (rule theI2, fast)

   112    apply (fast intro: inv_unique, fast)

   113   done

   114

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

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

   117 proof -

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

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

   120     by (auto simp add: Units_def

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

   122 qed

   123

   124 lemma (in monoid) Units_l_cancel [simp]:

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

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

   127 proof

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

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

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

   131     by (simp add: m_assoc Units_closed)

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

   133 next

   134   assume eq: "y = z"

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

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

   137 qed

   138

   139 lemma (in monoid) Units_inv_inv [simp]:

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

   141 proof -

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

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

   144     by (simp add: Units_l_inv Units_r_inv)

   145   with x show ?thesis by (simp add: Units_closed)

   146 qed

   147

   148 lemma (in monoid) inv_inj_on_Units:

   149   "inj_on (m_inv G) (Units G)"

   150 proof (rule inj_onI)

   151   fix x y

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

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

   154   with G show "x = y" by simp

   155 qed

   156

   157 lemma (in monoid) Units_inv_comm:

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

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

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

   161 proof -

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

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

   164 qed

   165

   166 text {* Power *}

   167

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

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

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

   171

   172 lemma (in monoid) nat_pow_0 [simp]:

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

   174   by (simp add: nat_pow_def)

   175

   176 lemma (in monoid) nat_pow_Suc [simp]:

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

   178   by (simp add: nat_pow_def)

   179

   180 lemma (in monoid) nat_pow_one [simp]:

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

   182   by (induct n) simp_all

   183

   184 lemma (in monoid) nat_pow_mult:

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

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

   187

   188 lemma (in monoid) nat_pow_pow:

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

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

   191

   192 text {*

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

   194 *}

   195

   196 locale group = monoid +

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

   198

   199

   200 lemma (in group) is_group: "group G" .

   201

   202 theorem groupI:

   203   fixes G (structure)

   204   assumes m_closed [simp]:

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

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

   207     and m_assoc:

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

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

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

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

   212   shows "group G"

   213 proof -

   214   have l_cancel [simp]:

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

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

   217   proof

   218     fix x y z

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

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

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

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

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

   224       by (simp add: m_assoc)

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

   226   next

   227     fix x y z

   228     assume eq: "y = z"

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

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

   231   qed

   232   have r_one:

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

   234   proof -

   235     fix x

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

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

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

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

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

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

   242   qed

   243   have inv_ex:

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

   245   proof -

   246     fix x

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

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

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

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

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

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

   253       by simp

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

   255       by (fast intro: l_inv r_inv)

   256   qed

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

   258     by (unfold Units_def) fast

   259   show ?thesis

   260     by (fast intro!: group.intro monoid.intro group_axioms.intro

   261       carrier_subset_Units intro: prems r_one)

   262 qed

   263

   264 lemma (in monoid) monoid_groupI:

   265   assumes l_inv_ex:

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

   267   shows "group G"

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

   269

   270 lemma (in group) Units_eq [simp]:

   271   "Units G = carrier G"

   272 proof

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

   274 next

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

   276 qed

   277

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

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

   280   using Units_inv_closed by simp

   281

   282 lemma (in group) l_inv_ex [simp]:

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

   284   using Units_l_inv_ex by simp

   285

   286 lemma (in group) r_inv_ex [simp]:

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

   288   using Units_r_inv_ex by simp

   289

   290 lemma (in group) l_inv [simp]:

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

   292   using Units_l_inv by simp

   293

   294 subsection {* Cancellation Laws and Basic Properties *}

   295

   296 lemma (in group) l_cancel [simp]:

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

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

   299   using Units_l_inv by simp

   300

   301 lemma (in group) r_inv [simp]:

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

   303 proof -

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

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

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

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

   308 qed

   309

   310 lemma (in group) r_cancel [simp]:

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

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

   313 proof

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

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

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

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

   318   with G show "y = z" by simp

   319 next

   320   assume eq: "y = z"

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

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

   323 qed

   324

   325 lemma (in group) inv_one [simp]:

   326   "inv \<one> = \<one>"

   327 proof -

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

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

   330   finally show ?thesis .

   331 qed

   332

   333 lemma (in group) inv_inv [simp]:

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

   335   using Units_inv_inv by simp

   336

   337 lemma (in group) inv_inj:

   338   "inj_on (m_inv G) (carrier G)"

   339   using inv_inj_on_Units by simp

   340

   341 lemma (in group) inv_mult_group:

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

   343 proof -

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

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

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

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

   348 qed

   349

   350 lemma (in group) inv_comm:

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

   352   by (rule Units_inv_comm) auto

   353

   354 lemma (in group) inv_equality:

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

   356 apply (simp add: m_inv_def)

   357 apply (rule the_equality)

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

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

   360 done

   361

   362 text {* Power *}

   363

   364 lemma (in group) int_pow_def2:

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

   366   by (simp add: int_pow_def nat_pow_def Let_def)

   367

   368 lemma (in group) int_pow_0 [simp]:

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

   370   by (simp add: int_pow_def2)

   371

   372 lemma (in group) int_pow_one [simp]:

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

   374   by (simp add: int_pow_def2)

   375

   376 subsection {* Subgroups *}

   377

   378 locale subgroup =

   379   fixes H and G (structure)

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

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

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

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

   384

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

   386

   387 lemma (in subgroup) mem_carrier [simp]:

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

   389   using subset by blast

   390

   391 lemma subgroup_imp_subset:

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

   393   by (rule subgroup.subset)

   394

   395 lemma (in subgroup) subgroup_is_group [intro]:

   396   includes group G

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

   398   by (rule groupI) (auto intro: m_assoc l_inv mem_carrier)

   399

   400 text {*

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

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

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

   404 *}

   405

   406 lemma (in group) one_in_subset:

   407   "[| 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 |]

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

   409 by (force simp add: l_inv)

   410

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

   412

   413 lemma (in group) subgroupI:

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

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

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

   417   shows "subgroup H G"

   418 proof (simp add: subgroup_def prems)

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

   420 qed

   421

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

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

   424

   425 lemma subgroup_nonempty:

   426   "~ subgroup {} G"

   427   by (blast dest: subgroup.one_closed)

   428

   429 lemma (in subgroup) finite_imp_card_positive:

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

   431 proof (rule classical)

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

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

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

   435   with subgroup_nonempty show ?thesis by contradiction

   436 qed

   437

   438 (*

   439 lemma (in monoid) Units_subgroup:

   440   "subgroup (Units G) G"

   441 *)

   442

   443 subsection {* Direct Products *}

   444

   445 constdefs

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

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

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

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

   450

   451 lemma DirProd_monoid:

   452   includes monoid G + monoid H

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

   454 proof -

   455   from prems

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

   457 qed

   458

   459

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

   461 lemma DirProd_group:

   462   includes group G + group H

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

   464   by (rule groupI)

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

   466            simp add: DirProd_def)

   467

   468 lemma carrier_DirProd [simp]:

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

   470   by (simp add: DirProd_def)

   471

   472 lemma one_DirProd [simp]:

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

   474   by (simp add: DirProd_def)

   475

   476 lemma mult_DirProd [simp]:

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

   478   by (simp add: DirProd_def)

   479

   480 lemma inv_DirProd [simp]:

   481   includes group G + group H

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

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

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

   485   apply (rule group.inv_equality [OF DirProd_group])

   486   apply (simp_all add: prems group.l_inv)

   487   done

   488

   489 text{*This alternative proof of the previous result demonstrates interpret.

   490    It uses @{text Prod.inv_equality} (available after @{text interpret})

   491    instead of @{text "group.inv_equality [OF DirProd_group]"}. *}

   492 lemma

   493   includes group G + group H

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

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

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

   497 proof -

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

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

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

   501 qed

   502

   503

   504 subsection {* Homomorphisms and Isomorphisms *}

   505

   506 constdefs (structure G and H)

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

   508   "hom G H ==

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

   510       (\<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)}"

   511

   512 lemma hom_mult:

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

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

   515   by (simp add: hom_def)

   516

   517 lemma hom_closed:

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

   519   by (auto simp add: hom_def funcset_mem)

   520

   521 lemma (in group) hom_compose:

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

   523 apply (auto simp add: hom_def funcset_compose)

   524 apply (simp add: compose_def funcset_mem)

   525 done

   526

   527

   528 subsection {* Isomorphisms *}

   529

   530 constdefs

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

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

   533

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

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

   536

   537 lemma (in group) iso_sym:

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

   539 apply (simp add: iso_def bij_betw_Inv)

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

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

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

   543 done

   544

   545 lemma (in group) iso_trans:

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

   547 by (auto simp add: iso_def hom_compose bij_betw_compose)

   548

   549 lemma DirProd_commute_iso:

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

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

   552

   553 lemma DirProd_assoc_iso:

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

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

   556

   557

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

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

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

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

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

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

   564

   565 lemma (in group_hom) one_closed [simp]:

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

   567   by simp

   568

   569 lemma (in group_hom) hom_one [simp]:

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

   571 proof -

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

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

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

   575 qed

   576

   577 lemma (in group_hom) inv_closed [simp]:

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

   579   by simp

   580

   581 lemma (in group_hom) hom_inv [simp]:

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

   583 proof -

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

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

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

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

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

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

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

   591 qed

   592

   593 subsection {* Commutative Structures *}

   594

   595 text {*

   596   Naming convention: multiplicative structures that are commutative

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

   598   \emph{Abelian}.

   599 *}

   600

   601 subsection {* Definition *}

   602

   603 locale comm_monoid = monoid +

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

   605

   606 lemma (in comm_monoid) m_lcomm:

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

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

   609 proof -

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

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

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

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

   614   finally show ?thesis .

   615 qed

   616

   617 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   618

   619 lemma comm_monoidI:

   620   fixes G (structure)

   621   assumes m_closed:

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

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

   624     and m_assoc:

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

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

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

   628     and m_comm:

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

   630   shows "comm_monoid G"

   631   using l_one

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

   633              intro: prems simp: m_closed one_closed m_comm)

   634

   635 lemma (in monoid) monoid_comm_monoidI:

   636   assumes m_comm:

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

   638   shows "comm_monoid G"

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

   640

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

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

   643 proof -

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

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

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

   647   finally show ?thesis .

   648 qed*)

   649

   650 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   654

   655 locale comm_group = comm_monoid + group

   656

   657 lemma (in group) group_comm_groupI:

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

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

   660   shows "comm_group G"

   661   by unfold_locales (simp_all add: m_comm)

   662

   663 lemma comm_groupI:

   664   fixes G (structure)

   665   assumes m_closed:

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

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

   668     and m_assoc:

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

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

   671     and m_comm:

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

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

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

   675   shows "comm_group G"

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

   677

   678 lemma (in comm_group) inv_mult:

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

   680   by (simp add: m_ac inv_mult_group)

   681

   682 subsection {* Lattice of subgroups of a group *}

   683

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

   685

   686 theorem (in group) subgroups_partial_order:

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

   688   by (rule partial_order.intro) simp_all

   689

   690 lemma (in group) subgroup_self:

   691   "subgroup (carrier G) G"

   692   by (rule subgroupI) auto

   693

   694 lemma (in group) subgroup_imp_group:

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

   696   by (rule subgroup.subgroup_is_group)

   697

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

   699   "monoid G"

   700   by (auto intro: monoid.intro m_assoc)

   701

   702 lemma (in group) subgroup_inv_equality:

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

   704 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   709 done

   710

   711 theorem (in group) subgroups_Inter:

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

   713     and not_empty: "A ~= {}"

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

   715 proof (rule subgroupI)

   716   from subgr [THEN subgroup.subset] and not_empty

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

   718 next

   719   from subgr [THEN subgroup.one_closed]

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

   721 next

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

   723   with subgr [THEN subgroup.m_inv_closed]

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

   725 next

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

   727   with subgr [THEN subgroup.m_closed]

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

   729 qed

   730

   731 theorem (in group) subgroups_complete_lattice:

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

   733     (is "complete_lattice ?L")

   734 proof (rule partial_order.complete_lattice_criterion1)

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

   736 next

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

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

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

   740 next

   741   fix A

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

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

   744     by (fastsimp intro: subgroups_Inter)

   745   have "greatest ?L (\<Inter>A) (Lower ?L A)"

   746     (is "greatest ?L ?Int _")

   747   proof (rule greatest_LowerI)

   748     fix H

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

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

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

   752       by (rule subgroup_imp_group)

   753     from groupH have monoidH: "monoid ?H"

   754       by (rule group.is_monoid)

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

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

   757   next

   758     fix H

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

   760     with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)

   761   next

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

   763   next

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

   765   qed

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

   767 qed

   768

   769 end