src/HOL/Algebra/Group.thy
 author ballarin Thu Feb 27 15:12:29 2003 +0100 (2003-02-27) changeset 13835 12b2ffbe543a parent 13817 7e031a968443 child 13854 91c9ab25fece permissions -rw-r--r--
Change to meta simplifier: congruence rules may now have frees as head of term.
     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 {* Algebraic Structures up to Abelian Groups *}

    10

    11 theory Group = FuncSet:

    12

    13 text {*

    14   Definitions follow Jacobson, Basic Algebra I, Freeman, 1985; with

    15   the exception of \emph{magma} which, following Bourbaki, is a set

    16   together with a binary, closed operation.

    17 *}

    18

    19 section {* From Magmas to Groups *}

    20

    21 subsection {* Definitions *}

    22

    23 (* The following may be unnecessary. *)

    24 text {*

    25   We write groups additively.  This simplifies notation for rings.

    26   All rings have an additive inverse, only fields have a

    27   multiplicative one.  This definitions reduces the need for

    28   qualifiers.

    29 *}

    30

    31 record 'a semigroup =

    32   carrier :: "'a set"

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

    34

    35 record 'a monoid = "'a semigroup" +

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

    37

    38 record 'a group = "'a monoid" +

    39   m_inv :: "'a => 'a" ("inv\<index> _"  80)

    40

    41 locale magma = struct G +

    42   assumes m_closed [intro, simp]:

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

    44

    45 locale semigroup = magma +

    46   assumes m_assoc:

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

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

    49

    50 locale l_one = struct G +

    51   assumes one_closed [intro, simp]: "\<one> \<in> carrier G"

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

    53

    54 locale group = semigroup + l_one +

    55   assumes inv_closed [intro, simp]: "x \<in> carrier G ==> inv x \<in> carrier G"

    56     and l_inv: "x \<in> carrier G ==> inv x \<otimes> x = \<one>"

    57

    58 subsection {* Cancellation Laws and Basic Properties *}

    59

    60 lemma (in group) l_cancel [simp]:

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

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

    63 proof

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

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

    66   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" by (simp add: m_assoc)

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

    68 next

    69   assume eq: "y = z"

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

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

    72 qed

    73

    74 lemma (in group) r_one [simp]:

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

    76 proof -

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

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

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

    80   with x show ?thesis by simp

    81 qed

    82

    83 lemma (in group) r_inv:

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

    85 proof -

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

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

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

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

    90 qed

    91

    92 lemma (in group) r_cancel [simp]:

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

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

    95 proof

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

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

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

    99     by (simp add: m_assoc [symmetric])

   100   with G show "y = z" by (simp add: r_inv)

   101 next

   102   assume eq: "y = z"

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

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

   105 qed

   106

   107 lemma (in group) inv_inv [simp]:

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

   109 proof -

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

   111   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by (simp add: l_inv r_inv)

   112   with x show ?thesis by simp

   113 qed

   114

   115 lemma (in group) inv_mult:

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

   117 proof -

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

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

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

   121   with G show ?thesis by simp

   122 qed

   123

   124 subsection {* Substructures *}

   125

   126 locale submagma = var H + struct G +

   127   assumes subset [intro, simp]: "H \<subseteq> carrier G"

   128     and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"

   129

   130 declare (in submagma) magma.intro [intro] semigroup.intro [intro]

   131

   132 (*

   133 alternative definition of submagma

   134

   135 locale submagma = var H + struct G +

   136   assumes subset [intro, simp]: "carrier H \<subseteq> carrier G"

   137     and m_equal [simp]: "mult H = mult G"

   138     and m_closed [intro, simp]:

   139       "[| x \<in> carrier H; y \<in> carrier H |] ==> x \<otimes> y \<in> carrier H"

   140 *)

   141

   142 lemma submagma_imp_subset:

   143   "submagma H G ==> H \<subseteq> carrier G"

   144   by (rule submagma.subset)

   145

   146 lemma (in submagma) subsetD [dest, simp]:

   147   "x \<in> H ==> x \<in> carrier G"

   148   using subset by blast

   149

   150 lemma (in submagma) magmaI [intro]:

   151   includes magma G

   152   shows "magma (G(| carrier := H |))"

   153   by rule simp

   154

   155 lemma (in submagma) semigroup_axiomsI [intro]:

   156   includes semigroup G

   157   shows "semigroup_axioms (G(| carrier := H |))"

   158     by rule (simp add: m_assoc)

   159

   160 lemma (in submagma) semigroupI [intro]:

   161   includes semigroup G

   162   shows "semigroup (G(| carrier := H |))"

   163   using prems by fast

   164

   165 locale subgroup = submagma H G +

   166   assumes one_closed [intro, simp]: "\<one> \<in> H"

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

   168

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

   170

   171 lemma (in subgroup) l_oneI [intro]:

   172   includes l_one G

   173   shows "l_one (G(| carrier := H |))"

   174   by rule simp_all

   175

   176 lemma (in subgroup) group_axiomsI [intro]:

   177   includes group G

   178   shows "group_axioms (G(| carrier := H |))"

   179   by rule (simp_all add: l_inv)

   180

   181 lemma (in subgroup) groupI [intro]:

   182   includes group G

   183   shows "group (G(| carrier := H |))"

   184   using prems by fast

   185

   186 text {*

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

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

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

   190 *}

   191

   192 lemma (in group) one_in_subset:

   193   "[| 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 |]

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

   195 by (force simp add: l_inv)

   196

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

   198

   199 lemma (in group) subgroupI:

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

   201     and inv: "!!a. a \<in> H ==> inv a \<in> H"

   202     and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"

   203   shows "subgroup H G"

   204 proof

   205   from subset and mult show "submagma H G" ..

   206 next

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

   208   with inv show "subgroup_axioms H G"

   209     by (intro subgroup_axioms.intro) simp_all

   210 qed

   211

   212 text {*

   213   Repeat facts of submagmas for subgroups.  Necessary???

   214 *}

   215

   216 lemma (in subgroup) subset:

   217   "H \<subseteq> carrier G"

   218   ..

   219

   220 lemma (in subgroup) m_closed:

   221   "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"

   222   ..

   223

   224 declare magma.m_closed [simp]

   225

   226 declare l_one.one_closed [iff] group.inv_closed [simp]

   227   l_one.l_one [simp] group.r_one [simp] group.inv_inv [simp]

   228

   229 lemma subgroup_nonempty:

   230   "~ subgroup {} G"

   231   by (blast dest: subgroup.one_closed)

   232

   233 lemma (in subgroup) finite_imp_card_positive:

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

   235 proof (rule classical)

   236   have sub: "subgroup H G" using prems ..

   237   assume fin: "finite (carrier G)"

   238     and zero: "~ 0 < card H"

   239   then have "finite H" by (blast intro: finite_subset dest: subset)

   240   with zero sub have "subgroup {} G" by simp

   241   with subgroup_nonempty show ?thesis by contradiction

   242 qed

   243

   244 subsection {* Direct Products *}

   245

   246 constdefs

   247   DirProdSemigroup ::

   248     "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme]

   249     => ('a \<times> 'b) semigroup"

   250     (infixr "\<times>\<^sub>s" 80)

   251   "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,

   252     mult = (%(xg, xh) (yg, yh). (mult G xg yg, mult H xh yh)) |)"

   253

   254   DirProdMonoid ::

   255     "[('a, 'c) monoid_scheme, ('b, 'd) monoid_scheme] => ('a \<times> 'b) monoid"

   256     (infixr "\<times>\<^sub>m" 80)

   257   "G \<times>\<^sub>m H == (| carrier = carrier (G \<times>\<^sub>s H),

   258     mult = mult (G \<times>\<^sub>s H),

   259     one = (one G, one H) |)"

   260

   261   DirProdGroup ::

   262     "[('a, 'c) group_scheme, ('b, 'd) group_scheme] => ('a \<times> 'b) group"

   263     (infixr "\<times>\<^sub>g" 80)

   264   "G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>m H),

   265     mult = mult (G \<times>\<^sub>m H),

   266     one = one (G \<times>\<^sub>m H),

   267     m_inv = (%(g, h). (m_inv G g, m_inv H h)) |)"

   268

   269 lemma DirProdSemigroup_magma:

   270   includes magma G + magma H

   271   shows "magma (G \<times>\<^sub>s H)"

   272   by rule (auto simp add: DirProdSemigroup_def)

   273

   274 lemma DirProdSemigroup_semigroup_axioms:

   275   includes semigroup G + semigroup H

   276   shows "semigroup_axioms (G \<times>\<^sub>s H)"

   277   by rule (auto simp add: DirProdSemigroup_def G.m_assoc H.m_assoc)

   278

   279 lemma DirProdSemigroup_semigroup:

   280   includes semigroup G + semigroup H

   281   shows "semigroup (G \<times>\<^sub>s H)"

   282   using prems

   283   by (fast intro: semigroup.intro

   284     DirProdSemigroup_magma DirProdSemigroup_semigroup_axioms)

   285

   286 lemma DirProdGroup_magma:

   287   includes magma G + magma H

   288   shows "magma (G \<times>\<^sub>g H)"

   289   by rule

   290     (auto simp add: DirProdGroup_def DirProdMonoid_def DirProdSemigroup_def)

   291

   292 lemma DirProdGroup_semigroup_axioms:

   293   includes semigroup G + semigroup H

   294   shows "semigroup_axioms (G \<times>\<^sub>g H)"

   295   by rule

   296     (auto simp add: DirProdGroup_def DirProdMonoid_def DirProdSemigroup_def

   297       G.m_assoc H.m_assoc)

   298

   299 lemma DirProdGroup_semigroup:

   300   includes semigroup G + semigroup H

   301   shows "semigroup (G \<times>\<^sub>g H)"

   302   using prems

   303   by (fast intro: semigroup.intro

   304     DirProdGroup_magma DirProdGroup_semigroup_axioms)

   305

   306 (* ... and further lemmas for group ... *)

   307

   308 lemma DirProdGroup_group:

   309   includes group G + group H

   310   shows "group (G \<times>\<^sub>g H)"

   311 by rule

   312   (auto intro: magma.intro l_one.intro

   313       semigroup_axioms.intro group_axioms.intro

   314     simp add: DirProdGroup_def DirProdMonoid_def DirProdSemigroup_def

   315       G.m_assoc H.m_assoc G.l_inv H.l_inv)

   316

   317 subsection {* Homomorphisms *}

   318

   319 constdefs

   320   hom :: "[('a, 'c) semigroup_scheme, ('b, 'd) semigroup_scheme]

   321     => ('a => 'b)set"

   322   "hom G H ==

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

   324       (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}"

   325

   326 lemma (in semigroup) hom:

   327   includes semigroup G

   328   shows "semigroup (| carrier = hom G G, mult = op o |)"

   329 proof

   330   show "magma (| carrier = hom G G, mult = op o |)"

   331     by rule (simp add: Pi_def hom_def)

   332 next

   333   show "semigroup_axioms (| carrier = hom G G, mult = op o |)"

   334     by rule (simp add: o_assoc)

   335 qed

   336

   337 lemma hom_mult:

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

   339    ==> h (mult G x y) = mult H (h x) (h y)"

   340   by (simp add: hom_def)

   341

   342 lemma hom_closed:

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

   344   by (auto simp add: hom_def funcset_mem)

   345

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

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

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

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

   350

   351 lemma (in group_hom) one_closed [simp]:

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

   353   by simp

   354

   355 lemma (in group_hom) hom_one [simp]:

   356   "h \<one> = \<one>\<^sub>2"

   357 proof -

   358   have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"

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

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

   361 qed

   362

   363 lemma (in group_hom) inv_closed [simp]:

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

   365   by simp

   366

   367 lemma (in group_hom) hom_inv [simp]:

   368   "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"

   369 proof -

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

   371   then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"

   372     by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)

   373   also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"

   374     by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)

   375   finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .

   376   with x show ?thesis by simp

   377 qed

   378

   379 section {* Abelian Structures *}

   380

   381 subsection {* Definition *}

   382

   383 locale abelian_semigroup = semigroup +

   384   assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"

   385

   386 lemma (in abelian_semigroup) m_lcomm:

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

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

   389 proof -

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

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

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

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

   394   finally show ?thesis .

   395 qed

   396

   397 lemmas (in abelian_semigroup) ac = m_assoc m_comm m_lcomm

   398

   399 locale abelian_monoid = abelian_semigroup + l_one

   400

   401 lemma (in abelian_monoid) l_one [simp]:

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

   403 proof -

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

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

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

   407   finally show ?thesis .

   408 qed

   409

   410 locale abelian_group = abelian_monoid + group

   411

   412 end