src/HOL/Algebra/Group.thy
 author wenzelm Mon May 22 22:29:18 2006 +0200 (2006-05-22) changeset 19699 1ecda5544e88 parent 19684 6101fbebda1d child 19783 82f365a14960 permissions -rw-r--r--
removed unchecked'';
     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 = struct G +

    44   assumes m_closed [intro, simp]:

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

    46       and m_assoc:

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

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

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

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

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

    52

    53 lemma monoidI:

    54   includes struct G

    55   assumes m_closed:

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

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

    58     and m_assoc:

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

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

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

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

    63   shows "monoid G"

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

    65

    66 lemma (in monoid) Units_closed [dest]:

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

    68   by (unfold Units_def) fast

    69

    70 lemma (in monoid) inv_unique:

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

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

    73   shows "y = y'"

    74 proof -

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

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

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

    78   finally show ?thesis .

    79 qed

    80

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

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

    83   by (unfold Units_def) auto

    84

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

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

    87   apply (unfold Units_def m_inv_def, auto)

    88   apply (rule theI2, fast)

    89    apply (fast intro: inv_unique, fast)

    90   done

    91

    92 lemma (in monoid) Units_l_inv:

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

    94   apply (unfold Units_def m_inv_def, auto)

    95   apply (rule theI2, fast)

    96    apply (fast intro: inv_unique, fast)

    97   done

    98

    99 lemma (in monoid) Units_r_inv:

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

   101   apply (unfold Units_def m_inv_def, auto)

   102   apply (rule theI2, fast)

   103    apply (fast intro: inv_unique, fast)

   104   done

   105

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

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

   108 proof -

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

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

   111     by (auto simp add: Units_def

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

   113 qed

   114

   115 lemma (in monoid) Units_l_cancel [simp]:

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

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

   118 proof

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

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

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

   122     by (simp add: m_assoc Units_closed)

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

   124 next

   125   assume eq: "y = z"

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

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

   128 qed

   129

   130 lemma (in monoid) Units_inv_inv [simp]:

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

   132 proof -

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

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

   135     by (simp add: Units_l_inv Units_r_inv)

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

   137 qed

   138

   139 lemma (in monoid) inv_inj_on_Units:

   140   "inj_on (m_inv G) (Units G)"

   141 proof (rule inj_onI)

   142   fix x y

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

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

   145   with G show "x = y" by simp

   146 qed

   147

   148 lemma (in monoid) Units_inv_comm:

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

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

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

   152 proof -

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

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

   155 qed

   156

   157 text {* Power *}

   158

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

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

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

   162

   163 lemma (in monoid) nat_pow_0 [simp]:

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

   165   by (simp add: nat_pow_def)

   166

   167 lemma (in monoid) nat_pow_Suc [simp]:

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

   169   by (simp add: nat_pow_def)

   170

   171 lemma (in monoid) nat_pow_one [simp]:

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

   173   by (induct n) simp_all

   174

   175 lemma (in monoid) nat_pow_mult:

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

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

   178

   179 lemma (in monoid) nat_pow_pow:

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

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

   182

   183 text {*

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

   185 *}

   186

   187 locale group = monoid +

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

   189

   190

   191 lemma (in group) is_group: "group G"

   192   by (rule group.intro [OF prems])

   193

   194 theorem groupI:

   195   includes struct G

   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 = var H + struct G +

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

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

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

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

   367

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

   369

   370 lemma (in subgroup) mem_carrier [simp]:

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

   372   using subset by blast

   373

   374 lemma subgroup_imp_subset:

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

   376   by (rule subgroup.subset)

   377

   378 lemma (in subgroup) subgroup_is_group [intro]:

   379   includes group G

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

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

   382

   383 text {*

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

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

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

   387 *}

   388

   389 lemma (in group) one_in_subset:

   390   "[| 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 |]

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

   392 by (force simp add: l_inv)

   393

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

   395

   396 lemma (in group) subgroupI:

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

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

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

   400   shows "subgroup H G"

   401 proof (simp add: subgroup_def prems)

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

   403 qed

   404

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

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

   407

   408 lemma subgroup_nonempty:

   409   "~ subgroup {} G"

   410   by (blast dest: subgroup.one_closed)

   411

   412 lemma (in subgroup) finite_imp_card_positive:

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

   414 proof (rule classical)

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

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

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

   418   with subgroup_nonempty show ?thesis by contradiction

   419 qed

   420

   421 (*

   422 lemma (in monoid) Units_subgroup:

   423   "subgroup (Units G) G"

   424 *)

   425

   426 subsection {* Direct Products *}

   427

   428 constdefs

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

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

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

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

   433

   434 lemma DirProd_monoid:

   435   includes monoid G + monoid H

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

   437 proof -

   438   from prems

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

   440 qed

   441

   442

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

   444 lemma DirProd_group:

   445   includes group G + group H

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

   447   by (rule groupI)

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

   449            simp add: DirProd_def)

   450

   451 lemma carrier_DirProd [simp]:

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

   453   by (simp add: DirProd_def)

   454

   455 lemma one_DirProd [simp]:

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

   457   by (simp add: DirProd_def)

   458

   459 lemma mult_DirProd [simp]:

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

   461   by (simp add: DirProd_def)

   462

   463 lemma inv_DirProd [simp]:

   464   includes group G + group H

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

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

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

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

   469   apply (simp_all add: prems group_def group.l_inv)

   470   done

   471

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

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

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

   475 lemma

   476   includes group G + group H

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

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

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

   480 proof -

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

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

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

   484 qed

   485

   486

   487 subsection {* Homomorphisms and Isomorphisms *}

   488

   489 constdefs (structure G and H)

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

   491   "hom G H ==

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

   493       (\<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)}"

   494

   495 lemma hom_mult:

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

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

   498   by (simp add: hom_def)

   499

   500 lemma hom_closed:

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

   502   by (auto simp add: hom_def funcset_mem)

   503

   504 lemma (in group) hom_compose:

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

   506 apply (auto simp add: hom_def funcset_compose)

   507 apply (simp add: compose_def funcset_mem)

   508 done

   509

   510

   511 subsection {* Isomorphisms *}

   512

   513 constdefs

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

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

   516

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

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

   519

   520 lemma (in group) iso_sym:

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

   522 apply (simp add: iso_def bij_betw_Inv)

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

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

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

   526 done

   527

   528 lemma (in group) iso_trans:

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

   530 by (auto simp add: iso_def hom_compose bij_betw_compose)

   531

   532 lemma DirProd_commute_iso:

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

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

   535

   536 lemma DirProd_assoc_iso:

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

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

   539

   540

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

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

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

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

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

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

   547

   548 lemma (in group_hom) one_closed [simp]:

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

   550   by simp

   551

   552 lemma (in group_hom) hom_one [simp]:

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

   554 proof -

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

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

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

   558 qed

   559

   560 lemma (in group_hom) inv_closed [simp]:

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

   562   by simp

   563

   564 lemma (in group_hom) hom_inv [simp]:

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

   566 proof -

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

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

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

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

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

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

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

   574 qed

   575

   576 subsection {* Commutative Structures *}

   577

   578 text {*

   579   Naming convention: multiplicative structures that are commutative

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

   581   \emph{Abelian}.

   582 *}

   583

   584 subsection {* Definition *}

   585

   586 locale comm_monoid = monoid +

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

   588

   589 lemma (in comm_monoid) m_lcomm:

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

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

   592 proof -

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

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

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

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

   597   finally show ?thesis .

   598 qed

   599

   600 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm

   601

   602 lemma comm_monoidI:

   603   includes struct G

   604   assumes m_closed:

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

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

   607     and m_assoc:

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

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

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

   611     and m_comm:

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

   613   shows "comm_monoid G"

   614   using l_one

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

   616              intro: prems simp: m_closed one_closed m_comm)

   617

   618 lemma (in monoid) monoid_comm_monoidI:

   619   assumes m_comm:

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

   621   shows "comm_monoid G"

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

   623

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

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

   626 proof -

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

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

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

   630   finally show ?thesis .

   631 qed*)

   632

   633 lemma (in comm_monoid) nat_pow_distr:

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

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

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

   637

   638 locale comm_group = comm_monoid + group

   639

   640 lemma (in group) group_comm_groupI:

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

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

   643   shows "comm_group G"

   644   by (fast intro: comm_group.intro comm_monoid_axioms.intro

   645                   is_group prems)

   646

   647 lemma comm_groupI:

   648   includes struct G

   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   using subgroup.subgroup_is_group [OF _ group.intro] .

   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