src/HOL/Algebra/Group.thy
 author ballarin Tue Jun 20 15:53:44 2006 +0200 (2006-06-20) changeset 19931 fb32b43e7f80 parent 19783 82f365a14960 child 19981 c0f124a0d385 permissions -rw-r--r--
Restructured locales with predicates: import is now an interpretation.
New method intro_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:

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

    95   apply (unfold Units_def m_inv_def, auto)

    96   apply (rule theI2, fast)

    97    apply (fast intro: inv_unique, fast)

    98   done

    99

   100 lemma (in monoid) Units_r_inv:

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

   102   apply (unfold Units_def m_inv_def, auto)

   103   apply (rule theI2, fast)

   104    apply (fast intro: inv_unique, fast)

   105   done

   106

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

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

   109 proof -

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

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

   112     by (auto simp add: Units_def

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

   114 qed

   115

   116 lemma (in monoid) Units_l_cancel [simp]:

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

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

   119 proof

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

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

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

   123     by (simp add: m_assoc Units_closed)

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

   125 next

   126   assume eq: "y = z"

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

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

   129 qed

   130

   131 lemma (in monoid) Units_inv_inv [simp]:

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

   133 proof -

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

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

   136     by (simp add: Units_l_inv Units_r_inv)

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

   138 qed

   139

   140 lemma (in monoid) inv_inj_on_Units:

   141   "inj_on (m_inv G) (Units G)"

   142 proof (rule inj_onI)

   143   fix x y

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

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

   146   with G show "x = y" by simp

   147 qed

   148

   149 lemma (in monoid) Units_inv_comm:

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

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

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

   153 proof -

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

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

   156 qed

   157

   158 text {* Power *}

   159

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

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

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

   163

   164 lemma (in monoid) nat_pow_0 [simp]:

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

   166   by (simp add: nat_pow_def)

   167

   168 lemma (in monoid) nat_pow_Suc [simp]:

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

   170   by (simp add: nat_pow_def)

   171

   172 lemma (in monoid) nat_pow_one [simp]:

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

   174   by (induct n) simp_all

   175

   176 lemma (in monoid) nat_pow_mult:

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

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

   179

   180 lemma (in monoid) nat_pow_pow:

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

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

   183

   184 text {*

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

   186 *}

   187

   188 locale group = monoid +

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

   190

   191

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

   193

   194 theorem groupI:

   195   fixes G (structure)

   196   assumes m_closed [simp]:

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

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

   199     and m_assoc:

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

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

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

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

   204   shows "group G"

   205 proof -

   206   have l_cancel [simp]:

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

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

   209   proof

   210     fix x y z

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

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

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

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

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

   216       by (simp add: m_assoc)

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

   218   next

   219     fix x y z

   220     assume eq: "y = z"

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

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

   223   qed

   224   have r_one:

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

   226   proof -

   227     fix x

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

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

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

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

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

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

   234   qed

   235   have inv_ex:

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

   237   proof -

   238     fix x

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

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

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

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

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

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

   245       by simp

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

   247       by (fast intro: l_inv r_inv)

   248   qed

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

   250     by (unfold Units_def) fast

   251   show ?thesis

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

   253       carrier_subset_Units intro: prems r_one)

   254 qed

   255

   256 lemma (in monoid) monoid_groupI:

   257   assumes l_inv_ex:

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

   259   shows "group G"

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

   261

   262 lemma (in group) Units_eq [simp]:

   263   "Units G = carrier G"

   264 proof

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

   266 next

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

   268 qed

   269

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

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

   272   using Units_inv_closed by simp

   273

   274 lemma (in group) l_inv [simp]:

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

   276   using Units_l_inv by simp

   277

   278 subsection {* Cancellation Laws and Basic Properties *}

   279

   280 lemma (in group) l_cancel [simp]:

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

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

   283   using Units_l_inv by simp

   284

   285 lemma (in group) r_inv [simp]:

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

   287 proof -

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

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

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

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

   292 qed

   293

   294 lemma (in group) r_cancel [simp]:

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

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

   297 proof

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

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

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

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

   302   with G show "y = z" by simp

   303 next

   304   assume eq: "y = z"

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

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

   307 qed

   308

   309 lemma (in group) inv_one [simp]:

   310   "inv \<one> = \<one>"

   311 proof -

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

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

   314   finally show ?thesis .

   315 qed

   316

   317 lemma (in group) inv_inv [simp]:

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

   319   using Units_inv_inv by simp

   320

   321 lemma (in group) inv_inj:

   322   "inj_on (m_inv G) (carrier G)"

   323   using inv_inj_on_Units by simp

   324

   325 lemma (in group) inv_mult_group:

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

   327 proof -

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

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

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

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

   332 qed

   333

   334 lemma (in group) inv_comm:

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

   336   by (rule Units_inv_comm) auto

   337

   338 lemma (in group) inv_equality:

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

   340 apply (simp add: m_inv_def)

   341 apply (rule the_equality)

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

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

   344 done

   345

   346 text {* Power *}

   347

   348 lemma (in group) int_pow_def2:

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

   350   by (simp add: int_pow_def nat_pow_def Let_def)

   351

   352 lemma (in group) int_pow_0 [simp]:

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

   354   by (simp add: int_pow_def2)

   355

   356 lemma (in group) int_pow_one [simp]:

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

   358   by (simp add: int_pow_def2)

   359

   360 subsection {* Subgroups *}

   361

   362 locale subgroup =

   363   fixes H and G (structure)

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

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

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

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

   368

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

   370

   371 lemma (in subgroup) mem_carrier [simp]:

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

   373   using subset by blast

   374

   375 lemma subgroup_imp_subset:

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

   377   by (rule subgroup.subset)

   378

   379 lemma (in subgroup) subgroup_is_group [intro]:

   380   includes group G

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

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

   383

   384 text {*

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

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

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

   388 *}

   389

   390 lemma (in group) one_in_subset:

   391   "[| 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 |]

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

   393 by (force simp add: l_inv)

   394

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

   396

   397 lemma (in group) subgroupI:

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

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

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

   401   shows "subgroup H G"

   402 proof (simp add: subgroup_def prems)

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

   404 qed

   405

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

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

   408

   409 lemma subgroup_nonempty:

   410   "~ subgroup {} G"

   411   by (blast dest: subgroup.one_closed)

   412

   413 lemma (in subgroup) finite_imp_card_positive:

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

   415 proof (rule classical)

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

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

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

   419   with subgroup_nonempty show ?thesis by contradiction

   420 qed

   421

   422 (*

   423 lemma (in monoid) Units_subgroup:

   424   "subgroup (Units G) G"

   425 *)

   426

   427 subsection {* Direct Products *}

   428

   429 constdefs

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

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

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

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

   434

   435 lemma DirProd_monoid:

   436   includes monoid G + monoid H

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

   438 proof -

   439   from prems

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

   441 qed

   442

   443

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

   445 lemma DirProd_group:

   446   includes group G + group H

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

   448   by (rule groupI)

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

   450            simp add: DirProd_def)

   451

   452 lemma carrier_DirProd [simp]:

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

   454   by (simp add: DirProd_def)

   455

   456 lemma one_DirProd [simp]:

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

   458   by (simp add: DirProd_def)

   459

   460 lemma mult_DirProd [simp]:

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

   462   by (simp add: DirProd_def)

   463

   464 lemma inv_DirProd [simp]:

   465   includes group G + group H

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

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

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

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

   470   apply (simp_all add: prems group.l_inv)

   471   done

   472

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

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

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

   476 lemma

   477   includes group G + group H

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

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

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

   481 proof -

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

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

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

   485 qed

   486

   487

   488 subsection {* Homomorphisms and Isomorphisms *}

   489

   490 constdefs (structure G and H)

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

   492   "hom G H ==

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

   494       (\<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)}"

   495

   496 lemma hom_mult:

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

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

   499   by (simp add: hom_def)

   500

   501 lemma hom_closed:

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

   503   by (auto simp add: hom_def funcset_mem)

   504

   505 lemma (in group) hom_compose:

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

   507 apply (auto simp add: hom_def funcset_compose)

   508 apply (simp add: compose_def funcset_mem)

   509 done

   510

   511

   512 subsection {* Isomorphisms *}

   513

   514 constdefs

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

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

   517

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

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

   520

   521 lemma (in group) iso_sym:

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

   523 apply (simp add: iso_def bij_betw_Inv)

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

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

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

   527 done

   528

   529 lemma (in group) iso_trans:

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

   531 by (auto simp add: iso_def hom_compose bij_betw_compose)

   532

   533 lemma DirProd_commute_iso:

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

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

   536

   537 lemma DirProd_assoc_iso:

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

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

   540

   541

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

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

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

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

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

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

   548

   549 lemma (in group_hom) one_closed [simp]:

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

   551   by simp

   552

   553 lemma (in group_hom) hom_one [simp]:

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

   555 proof -

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

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

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

   559 qed

   560

   561 lemma (in group_hom) inv_closed [simp]:

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

   563   by simp

   564

   565 lemma (in group_hom) hom_inv [simp]:

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

   567 proof -

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

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

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

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

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

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

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

   575 qed

   576

   577 subsection {* Commutative Structures *}

   578

   579 text {*

   580   Naming convention: multiplicative structures that are commutative

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

   582   \emph{Abelian}.

   583 *}

   584

   585 subsection {* Definition *}

   586

   587 locale comm_monoid = monoid +

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

   589

   590 lemma (in comm_monoid) m_lcomm:

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

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

   593 proof -

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

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

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

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

   598   finally show ?thesis .

   599 qed

   600

   601 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   602

   603 lemma comm_monoidI:

   604   fixes G (structure)

   605   assumes m_closed:

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

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

   608     and m_assoc:

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

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

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

   612     and m_comm:

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

   614   shows "comm_monoid G"

   615   using l_one

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

   617              intro: prems simp: m_closed one_closed m_comm)

   618

   619 lemma (in monoid) monoid_comm_monoidI:

   620   assumes m_comm:

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

   622   shows "comm_monoid G"

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

   624

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

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

   627 proof -

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

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

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

   631   finally show ?thesis .

   632 qed*)

   633

   634 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   638

   639 locale comm_group = comm_monoid + group

   640

   641 lemma (in group) group_comm_groupI:

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

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

   644   shows "comm_group G"

   645   by intro_locales (simp_all add: m_comm)

   646

   647 lemma comm_groupI:

   648   fixes G (structure)

   649   assumes m_closed:

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

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

   652     and m_assoc:

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

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

   655     and m_comm:

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

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

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

   659   shows "comm_group G"

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

   661

   662 lemma (in comm_group) inv_mult:

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

   664   by (simp add: m_ac inv_mult_group)

   665

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

   667

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

   669

   670 theorem (in group) subgroups_partial_order:

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

   672   by (rule partial_order.intro) simp_all

   673

   674 lemma (in group) subgroup_self:

   675   "subgroup (carrier G) G"

   676   by (rule subgroupI) auto

   677

   678 lemma (in group) subgroup_imp_group:

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

   680   by (rule subgroup.subgroup_is_group)

   681

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

   683   "monoid G"

   684   by (auto intro: monoid.intro m_assoc)

   685

   686 lemma (in group) subgroup_inv_equality:

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

   688 apply (rule_tac inv_equality [THEN sym])

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

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

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

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

   693 done

   694

   695 theorem (in group) subgroups_Inter:

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

   697     and not_empty: "A ~= {}"

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

   699 proof (rule subgroupI)

   700   from subgr [THEN subgroup.subset] and not_empty

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

   702 next

   703   from subgr [THEN subgroup.one_closed]

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

   705 next

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

   707   with subgr [THEN subgroup.m_inv_closed]

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

   709 next

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

   711   with subgr [THEN subgroup.m_closed]

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

   713 qed

   714

   715 theorem (in group) subgroups_complete_lattice:

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

   717     (is "complete_lattice ?L")

   718 proof (rule partial_order.complete_lattice_criterion1)

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

   720 next

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

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

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

   724 next

   725   fix A

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

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

   728     by (fastsimp intro: subgroups_Inter)

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

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

   731   proof (rule greatest_LowerI)

   732     fix H

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

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

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

   736       by (rule subgroup_imp_group)

   737     from groupH have monoidH: "monoid ?H"

   738       by (rule group.is_monoid)

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

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

   741   next

   742     fix H

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

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

   745   next

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

   747   next

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

   749   qed

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

   751 qed

   752

   753 end