src/HOL/Algebra/Group.thy
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More on Algebra by Paulo and Martin
 wenzelm@35849  1 (* Title: HOL/Algebra/Group.thy  wenzelm@35849  2  Author: Clemens Ballarin, started 4 February 2003  ballarin@13813  3 ballarin@13813  4 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.  lp15@68445  5 With additional contributions from Martin Baillon and Paulo Emílio de Vilhena.  ballarin@13813  6 *)  ballarin@13813  7 haftmann@28823  8 theory Group  immler@68188  9 imports Complete_Lattice "HOL-Library.FuncSet"  haftmann@28823  10 begin  ballarin@13813  11 wenzelm@61382  12 section \Monoids and Groups\  ballarin@13936  13 wenzelm@61382  14 subsection \Definitions\  ballarin@20318  15 wenzelm@61382  16 text \  wenzelm@58622  17  Definitions follow @{cite "Jacobson:1985"}.  wenzelm@61382  18 \  ballarin@13813  19 paulson@14963  20 record 'a monoid = "'a partial_object" +  paulson@14963  21  mult :: "['a, 'a] \ 'a" (infixl "\\" 70)  paulson@14963  22  one :: 'a ("\\")  ballarin@13817  23 wenzelm@35847  24 definition  paulson@14852  25  m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\ _" [81] 80)  wenzelm@67091  26  where "inv\<^bsub>G\<^esub> x = (THE y. y \ carrier G \ x \\<^bsub>G\<^esub> y = \\<^bsub>G\<^esub> \ y \\<^bsub>G\<^esub> x = \\<^bsub>G\<^esub>)"  ballarin@13936  27 wenzelm@35847  28 definition  wenzelm@14651  29  Units :: "_ => 'a set"  wenzelm@67443  30  \ \The set of invertible elements\  wenzelm@67091  31  where "Units G = {y. y \ carrier G \ (\x \ carrier G. x \\<^bsub>G\<^esub> y = \\<^bsub>G\<^esub> \ y \\<^bsub>G\<^esub> x = \\<^bsub>G\<^esub>)}"  ballarin@13936  32 ballarin@13936  33 consts  nipkow@67341  34  pow :: "[('a, 'm) monoid_scheme, 'a, 'b::semiring_1] => 'a" (infixr "[^]\" 75)  wenzelm@35850  35 wenzelm@35850  36 overloading nat_pow == "pow :: [_, 'a, nat] => 'a"  wenzelm@35850  37 begin  blanchet@55415  38  definition "nat_pow G a n = rec_nat \\<^bsub>G\<^esub> (%u b. b \\<^bsub>G\<^esub> a) n"  wenzelm@35850  39 end  ballarin@13936  40 wenzelm@35850  41 overloading int_pow == "pow :: [_, 'a, int] => 'a"  wenzelm@35850  42 begin  wenzelm@35850  43  definition "int_pow G a z =  blanchet@55415  44  (let p = rec_nat \\<^bsub>G\<^esub> (%u b. b \\<^bsub>G\<^esub> a)  huffman@46559  45  in if z < 0 then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z))"  wenzelm@35850  46 end  ballarin@13813  47 nipkow@67341  48 lemma int_pow_int: "x [^]\<^bsub>G\<^esub> (int n) = x [^]\<^bsub>G\<^esub> n"  Andreas@61628  49 by(simp add: int_pow_def nat_pow_def)  Andreas@61628  50 ballarin@19783  51 locale monoid =  ballarin@19783  52  fixes G (structure)  ballarin@13813  53  assumes m_closed [intro, simp]:  paulson@14963  54  "\x \ carrier G; y \ carrier G\ \ x \ y \ carrier G"  paulson@14963  55  and m_assoc:  lp15@68445  56  "\x \ carrier G; y \ carrier G; z \ carrier G\  paulson@14963  57  \ (x \ y) \ z = x \ (y \ z)"  paulson@14963  58  and one_closed [intro, simp]: "\ \ carrier G"  paulson@14963  59  and l_one [simp]: "x \ carrier G \ \ \ x = x"  paulson@14963  60  and r_one [simp]: "x \ carrier G \ x \ \ = x"  ballarin@13817  61 ballarin@13936  62 lemma monoidI:  ballarin@19783  63  fixes G (structure)  ballarin@13936  64  assumes m_closed:  wenzelm@14693  65  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y \ carrier G"  wenzelm@14693  66  and one_closed: "\ \ carrier G"  ballarin@13936  67  and m_assoc:  ballarin@13936  68  "!!x y z. [| x \ carrier G; y \ carrier G; z \ carrier G |] ==>  wenzelm@14693  69  (x \ y) \ z = x \ (y \ z)"  wenzelm@14693  70  and l_one: "!!x. x \ carrier G ==> \ \ x = x"  wenzelm@14693  71  and r_one: "!!x. x \ carrier G ==> x \ \ = x"  ballarin@13936  72  shows "monoid G"  ballarin@27714  73  by (fast intro!: monoid.intro intro: assms)  ballarin@13936  74 ballarin@13936  75 lemma (in monoid) Units_closed [dest]:  ballarin@13936  76  "x \ Units G ==> x \ carrier G"  ballarin@13936  77  by (unfold Units_def) fast  ballarin@13936  78 lp15@68443  79 lemma (in monoid) one_unique:  lp15@68443  80  assumes "u \ carrier G"  lp15@68443  81  and "\x. x \ carrier G \ u \ x = x"  lp15@68443  82  shows "u = \"  lp15@68443  83  using assms(2)[OF one_closed] r_one[OF assms(1)] by simp  lp15@68443  84 ballarin@13936  85 lemma (in monoid) inv_unique:  wenzelm@14693  86  assumes eq: "y \ x = \" "x \ y' = \"  wenzelm@14693  87  and G: "x \ carrier G" "y \ carrier G" "y' \ carrier G"  ballarin@13936  88  shows "y = y'"  ballarin@13936  89 proof -  ballarin@13936  90  from G eq have "y = y \ (x \ y')" by simp  ballarin@13936  91  also from G have "... = (y \ x) \ y'" by (simp add: m_assoc)  ballarin@13936  92  also from G eq have "... = y'" by simp  ballarin@13936  93  finally show ?thesis .  ballarin@13936  94 qed  ballarin@13936  95 lp15@68443  96 lemma (in monoid) Units_m_closed [simp, intro]:  ballarin@27698  97  assumes x: "x \ Units G" and y: "y \ Units G"  ballarin@27698  98  shows "x \ y \ Units G"  ballarin@27698  99 proof -  ballarin@27698  100  from x obtain x' where x: "x \ carrier G" "x' \ carrier G" and xinv: "x \ x' = \" "x' \ x = \"  ballarin@27698  101  unfolding Units_def by fast  ballarin@27698  102  from y obtain y' where y: "y \ carrier G" "y' \ carrier G" and yinv: "y \ y' = \" "y' \ y = \"  ballarin@27698  103  unfolding Units_def by fast  ballarin@27698  104  from x y xinv yinv have "y' \ (x' \ x) \ y = \" by simp  ballarin@27698  105  moreover from x y xinv yinv have "x \ (y \ y') \ x' = \" by simp  ballarin@27698  106  moreover note x y  ballarin@27698  107  ultimately show ?thesis unfolding Units_def  lp15@68445  108  by simp (metis m_assoc m_closed)  ballarin@27698  109 qed  ballarin@27698  110 ballarin@13940  111 lemma (in monoid) Units_one_closed [intro, simp]:  ballarin@13940  112  "\ \ Units G"  ballarin@13940  113  by (unfold Units_def) auto  ballarin@13940  114 ballarin@13936  115 lemma (in monoid) Units_inv_closed [intro, simp]:  ballarin@13936  116  "x \ Units G ==> inv x \ carrier G"  paulson@13943  117  apply (unfold Units_def m_inv_def, auto)  ballarin@13936  118  apply (rule theI2, fast)  paulson@13943  119  apply (fast intro: inv_unique, fast)  ballarin@13936  120  done  ballarin@13936  121 ballarin@19981  122 lemma (in monoid) Units_l_inv_ex:  ballarin@19981  123  "x \ Units G ==> \y \ carrier G. y \ x = \"  ballarin@19981  124  by (unfold Units_def) auto  ballarin@19981  125 ballarin@19981  126 lemma (in monoid) Units_r_inv_ex:  ballarin@19981  127  "x \ Units G ==> \y \ carrier G. x \ y = \"  ballarin@19981  128  by (unfold Units_def) auto  ballarin@19981  129 ballarin@27698  130 lemma (in monoid) Units_l_inv [simp]:  ballarin@13936  131  "x \ Units G ==> inv x \ x = \"  paulson@13943  132  apply (unfold Units_def m_inv_def, auto)  ballarin@13936  133  apply (rule theI2, fast)  paulson@13943  134  apply (fast intro: inv_unique, fast)  ballarin@13936  135  done  ballarin@13936  136 ballarin@27698  137 lemma (in monoid) Units_r_inv [simp]:  ballarin@13936  138  "x \ Units G ==> x \ inv x = \"  lp15@68458  139  by (metis (full_types) Units_closed Units_inv_closed Units_l_inv Units_r_inv_ex inv_unique)  ballarin@13936  140 lp15@68445  141 lemma (in monoid) inv_one [simp]:  lp15@68445  142  "inv \ = \"  lp15@68445  143  by (metis Units_one_closed Units_r_inv l_one monoid.Units_inv_closed monoid_axioms)  lp15@68445  144 ballarin@13936  145 lemma (in monoid) Units_inv_Units [intro, simp]:  ballarin@13936  146  "x \ Units G ==> inv x \ Units G"  ballarin@13936  147 proof -  ballarin@13936  148  assume x: "x \ Units G"  ballarin@13936  149  show "inv x \ Units G"  ballarin@13936  150  by (auto simp add: Units_def  ballarin@13936  151  intro: Units_l_inv Units_r_inv x Units_closed [OF x])  ballarin@13936  152 qed  ballarin@13936  153 ballarin@13936  154 lemma (in monoid) Units_l_cancel [simp]:  ballarin@13936  155  "[| x \ Units G; y \ carrier G; z \ carrier G |] ==>  ballarin@13936  156  (x \ y = x \ z) = (y = z)"  ballarin@13936  157 proof  ballarin@13936  158  assume eq: "x \ y = x \ z"  wenzelm@14693  159  and G: "x \ Units G" "y \ carrier G" "z \ carrier G"  ballarin@13936  160  then have "(inv x \ x) \ y = (inv x \ x) \ z"  ballarin@27698  161  by (simp add: m_assoc Units_closed del: Units_l_inv)  wenzelm@44472  162  with G show "y = z" by simp  ballarin@13936  163 next  ballarin@13936  164  assume eq: "y = z"  wenzelm@14693  165  and G: "x \ Units G" "y \ carrier G" "z \ carrier G"  ballarin@13936  166  then show "x \ y = x \ z" by simp  ballarin@13936  167 qed  ballarin@13936  168 ballarin@13936  169 lemma (in monoid) Units_inv_inv [simp]:  ballarin@13936  170  "x \ Units G ==> inv (inv x) = x"  ballarin@13936  171 proof -  ballarin@13936  172  assume x: "x \ Units G"  ballarin@27698  173  then have "inv x \ inv (inv x) = inv x \ x" by simp  ballarin@27698  174  with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv)  ballarin@13936  175 qed  ballarin@13936  176 ballarin@13936  177 lemma (in monoid) inv_inj_on_Units:  ballarin@13936  178  "inj_on (m_inv G) (Units G)"  ballarin@13936  179 proof (rule inj_onI)  ballarin@13936  180  fix x y  wenzelm@14693  181  assume G: "x \ Units G" "y \ Units G" and eq: "inv x = inv y"  ballarin@13936  182  then have "inv (inv x) = inv (inv y)" by simp  ballarin@13936  183  with G show "x = y" by simp  ballarin@13936  184 qed  ballarin@13936  185 ballarin@13940  186 lemma (in monoid) Units_inv_comm:  ballarin@13940  187  assumes inv: "x \ y = \"  wenzelm@14693  188  and G: "x \ Units G" "y \ Units G"  ballarin@13940  189  shows "y \ x = \"  ballarin@13940  190 proof -  ballarin@13940  191  from G have "x \ y \ x = x \ \" by (auto simp add: inv Units_closed)  ballarin@13940  192  with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)  ballarin@13940  193 qed  ballarin@13940  194 Andreas@61628  195 lemma (in monoid) carrier_not_empty: "carrier G \ {}"  Andreas@61628  196 by auto  Andreas@61628  197 wenzelm@61382  198 text \Power\  ballarin@13936  199 ballarin@13936  200 lemma (in monoid) nat_pow_closed [intro, simp]:  nipkow@67341  201  "x \ carrier G ==> x [^] (n::nat) \ carrier G"  ballarin@13936  202  by (induct n) (simp_all add: nat_pow_def)  ballarin@13936  203 ballarin@13936  204 lemma (in monoid) nat_pow_0 [simp]:  nipkow@67341  205  "x [^] (0::nat) = \"  ballarin@13936  206  by (simp add: nat_pow_def)  ballarin@13936  207 ballarin@13936  208 lemma (in monoid) nat_pow_Suc [simp]:  nipkow@67341  209  "x [^] (Suc n) = x [^] n \ x"  ballarin@13936  210  by (simp add: nat_pow_def)  ballarin@13936  211 ballarin@13936  212 lemma (in monoid) nat_pow_one [simp]:  nipkow@67341  213  "\ [^] (n::nat) = \"  ballarin@13936  214  by (induct n) simp_all  ballarin@13936  215 ballarin@13936  216 lemma (in monoid) nat_pow_mult:  nipkow@67341  217  "x \ carrier G ==> x [^] (n::nat) \ x [^] m = x [^] (n + m)"  ballarin@13936  218  by (induct m) (simp_all add: m_assoc [THEN sym])  ballarin@13936  219 lp15@68443  220 lemma (in monoid) nat_pow_comm:  lp15@68443  221  "x \ carrier G \ (x [^] (n::nat)) \ (x [^] (m :: nat)) = (x [^] m) \ (x [^] n)"  lp15@68445  222  using nat_pow_mult[of x n m] nat_pow_mult[of x m n] by (simp add: add.commute)  lp15@68443  223 lp15@68443  224 lemma (in monoid) nat_pow_Suc2:  lp15@68443  225  "x \ carrier G \ x [^] (Suc n) = x \ (x [^] n)"  lp15@68443  226  using nat_pow_mult[of x 1 n] Suc_eq_plus1[of n]  lp15@68445  227  by (metis One_nat_def Suc_eq_plus1_left l_one nat.rec(1) nat_pow_Suc nat_pow_def)  lp15@68443  228 ballarin@13936  229 lemma (in monoid) nat_pow_pow:  nipkow@67341  230  "x \ carrier G ==> (x [^] n) [^] m = x [^] (n * m::nat)"  haftmann@57512  231  by (induct m) (simp, simp add: nat_pow_mult add.commute)  ballarin@13936  232 lp15@68443  233 lemma (in monoid) nat_pow_consistent:  lp15@68443  234  "x [^] (n :: nat) = x [^]\<^bsub>(G \ carrier := H \)\<^esub> n"  lp15@68443  235  unfolding nat_pow_def by simp  lp15@68445  236 ballarin@27698  237 ballarin@27698  238 (* Jacobson defines submonoid here. *)  ballarin@27698  239 (* Jacobson defines the order of a monoid here. *)  ballarin@27698  240 ballarin@27698  241 wenzelm@61382  242 subsection \Groups\  ballarin@27698  243 wenzelm@61382  244 text \  ballarin@13936  245  A group is a monoid all of whose elements are invertible.  wenzelm@61382  246 \  ballarin@13936  247 ballarin@13936  248 locale group = monoid +  ballarin@13936  249  assumes Units: "carrier G <= Units G"  ballarin@13936  250 wenzelm@26199  251 lemma (in group) is_group: "group G" by (rule group_axioms)  paulson@14761  252 ballarin@13936  253 theorem groupI:  ballarin@19783  254  fixes G (structure)  ballarin@13936  255  assumes m_closed [simp]:  wenzelm@14693  256  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y \ carrier G"  wenzelm@14693  257  and one_closed [simp]: "\ \ carrier G"  ballarin@13936  258  and m_assoc:  ballarin@13936  259  "!!x y z. [| x \ carrier G; y \ carrier G; z \ carrier G |] ==>  wenzelm@14693  260  (x \ y) \ z = x \ (y \ z)"  wenzelm@14693  261  and l_one [simp]: "!!x. x \ carrier G ==> \ \ x = x"  paulson@14963  262  and l_inv_ex: "!!x. x \ carrier G ==> \y \ carrier G. y \ x = \"  ballarin@13936  263  shows "group G"  ballarin@13936  264 proof -  ballarin@13936  265  have l_cancel [simp]:  ballarin@13936  266  "!!x y z. [| x \ carrier G; y \ carrier G; z \ carrier G |] ==>  wenzelm@14693  267  (x \ y = x \ z) = (y = z)"  ballarin@13936  268  proof  ballarin@13936  269  fix x y z  wenzelm@14693  270  assume eq: "x \ y = x \ z"  wenzelm@14693  271  and G: "x \ carrier G" "y \ carrier G" "z \ carrier G"  ballarin@13936  272  with l_inv_ex obtain x_inv where xG: "x_inv \ carrier G"  wenzelm@14693  273  and l_inv: "x_inv \ x = \" by fast  wenzelm@14693  274  from G eq xG have "(x_inv \ x) \ y = (x_inv \ x) \ z"  ballarin@13936  275  by (simp add: m_assoc)  ballarin@13936  276  with G show "y = z" by (simp add: l_inv)  ballarin@13936  277  next  ballarin@13936  278  fix x y z  ballarin@13936  279  assume eq: "y = z"  wenzelm@14693  280  and G: "x \ carrier G" "y \ carrier G" "z \ carrier G"  wenzelm@14693  281  then show "x \ y = x \ z" by simp  ballarin@13936  282  qed  ballarin@13936  283  have r_one:  wenzelm@14693  284  "!!x. x \ carrier G ==> x \ \ = x"  ballarin@13936  285  proof -  ballarin@13936  286  fix x  ballarin@13936  287  assume x: "x \ carrier G"  ballarin@13936  288  with l_inv_ex obtain x_inv where xG: "x_inv \ carrier G"  wenzelm@14693  289  and l_inv: "x_inv \ x = \" by fast  wenzelm@14693  290  from x xG have "x_inv \ (x \ \) = x_inv \ x"  ballarin@13936  291  by (simp add: m_assoc [symmetric] l_inv)  wenzelm@14693  292  with x xG show "x \ \ = x" by simp  ballarin@13936  293  qed  ballarin@13936  294  have inv_ex:  wenzelm@67091  295  "\x. x \ carrier G \ \y \ carrier G. y \ x = \ \ x \ y = \"  ballarin@13936  296  proof -  ballarin@13936  297  fix x  ballarin@13936  298  assume x: "x \ carrier G"  ballarin@13936  299  with l_inv_ex obtain y where y: "y \ carrier G"  wenzelm@14693  300  and l_inv: "y \ x = \" by fast  wenzelm@14693  301  from x y have "y \ (x \ y) = y \ \"  ballarin@13936  302  by (simp add: m_assoc [symmetric] l_inv r_one)  wenzelm@14693  303  with x y have r_inv: "x \ y = \"  ballarin@13936  304  by simp  wenzelm@67091  305  from x y show "\y \ carrier G. y \ x = \ \ x \ y = \"  ballarin@13936  306  by (fast intro: l_inv r_inv)  ballarin@13936  307  qed  wenzelm@67091  308  then have carrier_subset_Units: "carrier G \ Units G"  ballarin@13936  309  by (unfold Units_def) fast  wenzelm@61169  310  show ?thesis  wenzelm@61169  311  by standard (auto simp: r_one m_assoc carrier_subset_Units)  ballarin@13936  312 qed  ballarin@13936  313 ballarin@27698  314 lemma (in monoid) group_l_invI:  ballarin@13936  315  assumes l_inv_ex:  paulson@14963  316  "!!x. x \ carrier G ==> \y \ carrier G. y \ x = \"  ballarin@13936  317  shows "group G"  ballarin@13936  318  by (rule groupI) (auto intro: m_assoc l_inv_ex)  ballarin@13936  319 ballarin@13936  320 lemma (in group) Units_eq [simp]:  ballarin@13936  321  "Units G = carrier G"  ballarin@13936  322 proof  wenzelm@67091  323  show "Units G \ carrier G" by fast  ballarin@13936  324 next  wenzelm@67091  325  show "carrier G \ Units G" by (rule Units)  ballarin@13936  326 qed  ballarin@13936  327 ballarin@13936  328 lemma (in group) inv_closed [intro, simp]:  ballarin@13936  329  "x \ carrier G ==> inv x \ carrier G"  ballarin@13936  330  using Units_inv_closed by simp  ballarin@13936  331 ballarin@19981  332 lemma (in group) l_inv_ex [simp]:  ballarin@19981  333  "x \ carrier G ==> \y \ carrier G. y \ x = \"  ballarin@19981  334  using Units_l_inv_ex by simp  ballarin@19981  335 ballarin@19981  336 lemma (in group) r_inv_ex [simp]:  ballarin@19981  337  "x \ carrier G ==> \y \ carrier G. x \ y = \"  ballarin@19981  338  using Units_r_inv_ex by simp  ballarin@19981  339 paulson@14963  340 lemma (in group) l_inv [simp]:  ballarin@13936  341  "x \ carrier G ==> inv x \ x = \"  lp15@68399  342  by simp  ballarin@13813  343 ballarin@20318  344 wenzelm@61382  345 subsection \Cancellation Laws and Basic Properties\  ballarin@13813  346 paulson@14963  347 lemma (in group) r_inv [simp]:  ballarin@13813  348  "x \ carrier G ==> x \ inv x = \"  lp15@68399  349  by simp  ballarin@13813  350 lp15@68399  351 lemma (in group) right_cancel [simp]:  ballarin@13813  352  "[| x \ carrier G; y \ carrier G; z \ carrier G |] ==>  ballarin@13813  353  (y \ x = z \ x) = (y = z)"  lp15@68399  354  by (metis inv_closed m_assoc r_inv r_one)  ballarin@13813  355 ballarin@13813  356 lemma (in group) inv_inv [simp]:  ballarin@13813  357  "x \ carrier G ==> inv (inv x) = x"  ballarin@13936  358  using Units_inv_inv by simp  ballarin@13936  359 ballarin@13936  360 lemma (in group) inv_inj:  ballarin@13936  361  "inj_on (m_inv G) (carrier G)"  ballarin@13936  362  using inv_inj_on_Units by simp  ballarin@13813  363 ballarin@13854  364 lemma (in group) inv_mult_group:  ballarin@13813  365  "[| x \ carrier G; y \ carrier G |] ==> inv (x \ y) = inv y \ inv x"  ballarin@13813  366 proof -  wenzelm@14693  367  assume G: "x \ carrier G" "y \ carrier G"  ballarin@13813  368  then have "inv (x \ y) \ (x \ y) = (inv y \ inv x) \ (x \ y)"  wenzelm@44472  369  by (simp add: m_assoc) (simp add: m_assoc [symmetric])  ballarin@27698  370  with G show ?thesis by (simp del: l_inv Units_l_inv)  ballarin@13813  371 qed  ballarin@13813  372 ballarin@13940  373 lemma (in group) inv_comm:  ballarin@13940  374  "[| x \ y = \; x \ carrier G; y \ carrier G |] ==> y \ x = \"  wenzelm@14693  375  by (rule Units_inv_comm) auto  ballarin@13940  376 paulson@13944  377 lemma (in group) inv_equality:  paulson@13943  378  "[|y \ x = \; x \ carrier G; y \ carrier G|] ==> inv x = y"  lp15@68399  379  using inv_unique r_inv by blast  paulson@13943  380 ballarin@57271  381 (* Contributed by Joachim Breitner *)  ballarin@57271  382 lemma (in group) inv_solve_left:  ballarin@57271  383  "\ a \ carrier G; b \ carrier G; c \ carrier G \ \ a = inv b \ c \ c = b \ a"  ballarin@57271  384  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)  ballarin@57271  385 lemma (in group) inv_solve_right:  ballarin@57271  386  "\ a \ carrier G; b \ carrier G; c \ carrier G \ \ a = b \ inv c \ b = a \ c"  ballarin@57271  387  by (metis inv_equality l_inv_ex l_one m_assoc r_inv)  ballarin@57271  388 wenzelm@61382  389 text \Power\  ballarin@13936  390 ballarin@13936  391 lemma (in group) int_pow_def2:  nipkow@67341  392  "a [^] (z::int) = (if z < 0 then inv (a [^] (nat (-z))) else a [^] (nat z))"  ballarin@13936  393  by (simp add: int_pow_def nat_pow_def Let_def)  ballarin@13936  394 ballarin@13936  395 lemma (in group) int_pow_0 [simp]:  nipkow@67341  396  "x [^] (0::int) = \"  ballarin@13936  397  by (simp add: int_pow_def2)  ballarin@13936  398 ballarin@13936  399 lemma (in group) int_pow_one [simp]:  nipkow@67341  400  "\ [^] (z::int) = \"  ballarin@13936  401  by (simp add: int_pow_def2)  ballarin@13936  402 ballarin@57271  403 (* The following are contributed by Joachim Breitner *)  ballarin@20318  404 ballarin@57271  405 lemma (in group) int_pow_closed [intro, simp]:  nipkow@67341  406  "x \ carrier G ==> x [^] (i::int) \ carrier G"  ballarin@57271  407  by (simp add: int_pow_def2)  ballarin@57271  408 ballarin@57271  409 lemma (in group) int_pow_1 [simp]:  nipkow@67341  410  "x \ carrier G \ x [^] (1::int) = x"  ballarin@57271  411  by (simp add: int_pow_def2)  ballarin@57271  412 ballarin@57271  413 lemma (in group) int_pow_neg:  nipkow@67341  414  "x \ carrier G \ x [^] (-i::int) = inv (x [^] i)"  ballarin@57271  415  by (simp add: int_pow_def2)  ballarin@57271  416 ballarin@57271  417 lemma (in group) int_pow_mult:  nipkow@67341  418  "x \ carrier G \ x [^] (i + j::int) = x [^] i \ x [^] j"  ballarin@57271  419 proof -  ballarin@57271  420  have [simp]: "-i - j = -j - i" by simp  wenzelm@67613  421  assume "x \ carrier G" then  ballarin@57271  422  show ?thesis  ballarin@57271  423  by (auto simp add: int_pow_def2 inv_solve_left inv_solve_right nat_add_distrib [symmetric] nat_pow_mult )  ballarin@57271  424 qed  ballarin@57271  425 lp15@68443  426 lemma (in group) nat_pow_inv:  lp15@68443  427  "x \ carrier G \ (inv x) [^] (i :: nat) = inv (x [^] i)"  lp15@68443  428 proof (induction i)  lp15@68443  429  case 0 thus ?case by simp  lp15@68443  430 next  lp15@68443  431  case (Suc i)  lp15@68443  432  have "(inv x) [^] Suc i = ((inv x) [^] i) \ inv x"  lp15@68443  433  by simp  lp15@68443  434  also have " ... = (inv (x [^] i)) \ inv x"  lp15@68443  435  by (simp add: Suc.IH Suc.prems)  lp15@68443  436  also have " ... = inv (x \ (x [^] i))"  lp15@68443  437  using inv_mult_group[OF Suc.prems nat_pow_closed[OF Suc.prems, of i]] by simp  lp15@68443  438  also have " ... = inv (x [^] (Suc i))"  lp15@68443  439  using Suc.prems nat_pow_Suc2 by auto  lp15@68445  440  finally show ?case .  lp15@68443  441 qed  lp15@68443  442 lp15@68443  443 lemma (in group) int_pow_inv:  lp15@68443  444  "x \ carrier G \ (inv x) [^] (i :: int) = inv (x [^] i)"  lp15@68443  445  by (simp add: nat_pow_inv int_pow_def2)  lp15@68443  446 lp15@68443  447 lemma (in group) int_pow_pow:  lp15@68443  448  assumes "x \ carrier G"  lp15@68443  449  shows "(x [^] (n :: int)) [^] (m :: int) = x [^] (n * m :: int)"  lp15@68443  450 proof (cases)  lp15@68443  451  assume n_ge: "n \ 0" thus ?thesis  lp15@68443  452  proof (cases)  lp15@68443  453  assume m_ge: "m \ 0" thus ?thesis  lp15@68443  454  using n_ge nat_pow_pow[OF assms, of "nat n" "nat m"] int_pow_def2  lp15@68443  455  by (simp add: mult_less_0_iff nat_mult_distrib)  lp15@68443  456  next  lp15@68443  457  assume m_lt: "\ m \ 0" thus ?thesis  lp15@68443  458  using n_ge int_pow_def2 nat_pow_pow[OF assms, of "nat n" "nat (- m)"]  lp15@68443  459  by (smt assms group.int_pow_neg is_group mult_minus_right nat_mult_distrib split_mult_neg_le)  lp15@68443  460  qed  lp15@68443  461 next  lp15@68443  462  assume n_lt: "\ n \ 0" thus ?thesis  lp15@68443  463  proof (cases)  lp15@68443  464  assume m_ge: "m \ 0" thus ?thesis  lp15@68443  465  using n_lt nat_pow_pow[OF assms, of "nat (- n)" "nat m"]  lp15@68443  466  nat_pow_inv[of "x [^] nat (- n)" "nat m"] int_pow_def2  lp15@68443  467  by (smt assms group.int_pow_closed group.int_pow_neg is_group mult_minus_right  lp15@68443  468  mult_nonpos_nonpos nat_mult_distrib_neg)  lp15@68443  469  next  lp15@68443  470  assume m_lt: "\ m \ 0" thus ?thesis  lp15@68443  471  using n_lt nat_pow_pow[OF assms, of "nat (- n)" "nat (- m)"]  lp15@68443  472  nat_pow_inv[of "x [^] nat (- n)" "nat (- m)"] int_pow_def2  lp15@68443  473  by (smt assms inv_inv mult_nonpos_nonpos nat_mult_distrib_neg nat_pow_closed)  lp15@68443  474  qed  lp15@68443  475 qed  lp15@68443  476 Andreas@61628  477 lemma (in group) int_pow_diff:  nipkow@67341  478  "x \ carrier G \ x [^] (n - m :: int) = x [^] n \ inv (x [^] m)"  Andreas@61628  479 by(simp only: diff_conv_add_uminus int_pow_mult int_pow_neg)  Andreas@61628  480 Andreas@61628  481 lemma (in group) inj_on_multc: "c \ carrier G \ inj_on (\x. x \ c) (carrier G)"  Andreas@61628  482 by(simp add: inj_on_def)  Andreas@61628  483 Andreas@61628  484 lemma (in group) inj_on_cmult: "c \ carrier G \ inj_on (\x. c \ x) (carrier G)"  Andreas@61628  485 by(simp add: inj_on_def)  Andreas@61628  486 lp15@68443  487 (*Following subsection contributed by Martin Baillon*)  lp15@68443  488 subsection \Submonoids\  lp15@68443  489 lp15@68443  490 locale submonoid =  lp15@68443  491  fixes H and G (structure)  lp15@68443  492  assumes subset: "H \ carrier G"  lp15@68443  493  and m_closed [intro, simp]: "\x \ H; y \ H\ \ x \ y \ H"  lp15@68443  494  and one_closed [simp]: "\ \ H"  lp15@68443  495 lp15@68443  496 lemma (in submonoid) is_submonoid:  lp15@68443  497  "submonoid H G" by (rule submonoid_axioms)  lp15@68443  498 lp15@68443  499 lemma (in submonoid) mem_carrier [simp]:  lp15@68443  500  "x \ H \ x \ carrier G"  lp15@68443  501  using subset by blast  lp15@68443  502 lp15@68443  503 lemma (in submonoid) submonoid_is_monoid [intro]:  lp15@68443  504  assumes "monoid G"  lp15@68443  505  shows "monoid (G\carrier := H\)"  lp15@68443  506 proof -  lp15@68443  507  interpret monoid G by fact  lp15@68443  508  show ?thesis  lp15@68443  509  by (simp add: monoid_def m_assoc)  lp15@68443  510 qed  lp15@68443  511 lp15@68443  512 lemma submonoid_nonempty:  lp15@68443  513  "~ submonoid {} G"  lp15@68443  514  by (blast dest: submonoid.one_closed)  lp15@68443  515 lp15@68443  516 lemma (in submonoid) finite_monoid_imp_card_positive:  lp15@68443  517  "finite (carrier G) ==> 0 < card H"  lp15@68443  518 proof (rule classical)  lp15@68443  519  assume "finite (carrier G)" and a: "~ 0 < card H"  lp15@68443  520  then have "finite H" by (blast intro: finite_subset [OF subset])  lp15@68443  521  with is_submonoid a have "submonoid {} G" by simp  lp15@68443  522  with submonoid_nonempty show ?thesis by contradiction  lp15@68443  523 qed  lp15@68443  524 lp15@68443  525 lp15@68443  526 lemma (in monoid) monoid_incl_imp_submonoid :  lp15@68443  527  assumes "H \ carrier G"  lp15@68443  528 and "monoid (G\carrier := H\)"  lp15@68443  529 shows "submonoid H G"  lp15@68443  530 proof (intro submonoid.intro[OF assms(1)])  lp15@68443  531  have ab_eq : "\ a b. a \ H \ b \ H \ a \\<^bsub>G\carrier := H\\<^esub> b = a \ b" using assms by simp  lp15@68443  532  have "\a b. a \ H \ b \ H \ a \ b \ carrier (G\carrier := H\) "  lp15@68443  533  using assms ab_eq unfolding group_def using monoid.m_closed by fastforce  lp15@68443  534  thus "\a b. a \ H \ b \ H \ a \ b \ H" by simp  lp15@68443  535  show "\ \ H " using monoid.one_closed[OF assms(2)] assms by simp  lp15@68443  536 qed  lp15@68443  537 lp15@68517  538 lemma (in monoid) inv_unique':  lp15@68517  539  assumes "x \ carrier G" "y \ carrier G"  lp15@68517  540  shows "\ x \ y = \; y \ x = \ \ \ y = inv x"  lp15@68517  541 proof -  lp15@68517  542  assume "x \ y = \" and l_inv: "y \ x = \"  lp15@68517  543  hence unit: "x \ Units G"  lp15@68517  544  using assms unfolding Units_def by auto  lp15@68517  545  show "y = inv x"  lp15@68517  546  using inv_unique[OF l_inv Units_r_inv[OF unit] assms Units_inv_closed[OF unit]] .  lp15@68517  547 qed  lp15@68517  548 lp15@68517  549 lemma (in monoid) m_inv_monoid_consistent: (* contributed by Paulo *)  lp15@68517  550  assumes "x \ Units (G \ carrier := H \)" and "submonoid H G"  lp15@68517  551  shows "inv\<^bsub>(G \ carrier := H \)\<^esub> x = inv x"  lp15@68517  552 proof -  lp15@68517  553  have monoid: "monoid (G \ carrier := H \)"  lp15@68517  554  using submonoid.submonoid_is_monoid[OF assms(2) monoid_axioms] .  lp15@68517  555  obtain y where y: "y \ H" "x \ y = \" "y \ x = \"  lp15@68517  556  using assms(1) unfolding Units_def by auto  lp15@68517  557  have x: "x \ H" and in_carrier: "x \ carrier G" "y \ carrier G"  lp15@68517  558  using y(1) submonoid.subset[OF assms(2)] assms(1) unfolding Units_def by auto  lp15@68517  559  show ?thesis  lp15@68517  560  using monoid.inv_unique'[OF monoid, of x y] x y  lp15@68517  561  using inv_unique'[OF in_carrier y(2-3)] by auto  lp15@68517  562 qed  lp15@68517  563 wenzelm@61382  564 subsection \Subgroups\  ballarin@13813  565 ballarin@19783  566 locale subgroup =  ballarin@19783  567  fixes H and G (structure)  paulson@14963  568  assumes subset: "H \ carrier G"  paulson@14963  569  and m_closed [intro, simp]: "\x \ H; y \ H\ \ x \ y \ H"  ballarin@20318  570  and one_closed [simp]: "\ \ H"  paulson@14963  571  and m_inv_closed [intro,simp]: "x \ H \ inv x \ H"  ballarin@13813  572 ballarin@20318  573 lemma (in subgroup) is_subgroup:  wenzelm@26199  574  "subgroup H G" by (rule subgroup_axioms)  ballarin@20318  575 ballarin@13813  576 declare (in subgroup) group.intro [intro]  ballarin@13949  577 paulson@14963  578 lemma (in subgroup) mem_carrier [simp]:  paulson@14963  579  "x \ H \ x \ carrier G"  paulson@14963  580  using subset by blast  ballarin@13813  581 paulson@14963  582 lemma (in subgroup) subgroup_is_group [intro]:  ballarin@27611  583  assumes "group G"  ballarin@27611  584  shows "group (G\carrier := H\)"  ballarin@27611  585 proof -  ballarin@29237  586  interpret group G by fact  lp15@68458  587  have "Group.monoid (G\carrier := H\)"  lp15@68458  588  by (simp add: monoid_axioms submonoid.intro submonoid.submonoid_is_monoid subset)  lp15@68458  589  then show ?thesis  lp15@68458  590  by (rule monoid.group_l_invI) (auto intro: l_inv mem_carrier)  ballarin@27611  591 qed  ballarin@13813  592 lp15@68445  593 lemma (in group) subgroup_inv_equality:  lp15@68443  594  assumes "subgroup H G" "x \ H"  lp15@68445  595  shows "m_inv (G \carrier := H\) x = inv x"  lp15@68443  596  unfolding m_inv_def apply auto  lp15@68443  597  using subgroup.m_inv_closed[OF assms] inv_equality  lp15@68443  598  by (metis (no_types, hide_lams) assms subgroup.mem_carrier)  lp15@68443  599 lp15@68443  600 lemma (in group) int_pow_consistent: (* by Paulo *)  lp15@68443  601  assumes "subgroup H G" "x \ H"  lp15@68443  602  shows "x [^] (n :: int) = x [^]\<^bsub>(G \ carrier := H \)\<^esub> n"  lp15@68443  603 proof (cases)  lp15@68443  604  assume ge: "n \ 0"  lp15@68443  605  hence "x [^] n = x [^] (nat n)"  lp15@68443  606  using int_pow_def2 by auto  lp15@68443  607  also have " ... = x [^]\<^bsub>(G \ carrier := H \)\<^esub> (nat n)"  lp15@68443  608  using nat_pow_consistent by simp  lp15@68443  609  also have " ... = x [^]\<^bsub>(G \ carrier := H \)\<^esub> n"  lp15@68443  610  using group.int_pow_def2[OF subgroup.subgroup_is_group[OF assms(1) is_group]] ge by auto  lp15@68443  611  finally show ?thesis .  lp15@68445  612 next  lp15@68443  613  assume "\ n \ 0" hence lt: "n < 0" by simp  lp15@68443  614  hence "x [^] n = inv (x [^] (nat (- n)))"  lp15@68443  615  using int_pow_def2 by auto  lp15@68443  616  also have " ... = (inv x) [^] (nat (- n))"  lp15@68443  617  by (metis assms nat_pow_inv subgroup.mem_carrier)  lp15@68443  618  also have " ... = (inv\<^bsub>(G \ carrier := H \)\<^esub> x) [^]\<^bsub>(G \ carrier := H \)\<^esub> (nat (- n))"  lp15@68445  619  using subgroup_inv_equality[OF assms] nat_pow_consistent by auto  lp15@68443  620  also have " ... = inv\<^bsub>(G \ carrier := H \)\<^esub> (x [^]\<^bsub>(G \ carrier := H \)\<^esub> (nat (- n)))"  lp15@68443  621  using group.nat_pow_inv[OF subgroup.subgroup_is_group[OF assms(1) is_group]] assms(2) by auto  lp15@68443  622  also have " ... = x [^]\<^bsub>(G \ carrier := H \)\<^esub> n"  lp15@68443  623  using group.int_pow_def2[OF subgroup.subgroup_is_group[OF assms(1) is_group]] lt by auto  lp15@68443  624  finally show ?thesis .  lp15@68443  625 qed  lp15@68443  626 wenzelm@61382  627 text \  ballarin@13813  628  Since @{term H} is nonempty, it contains some element @{term x}. Since  wenzelm@63167  629  it is closed under inverse, it contains \inv x\. Since  wenzelm@63167  630  it is closed under product, it contains \x \ inv x = \\.  wenzelm@61382  631 \  ballarin@13813  632 ballarin@13813  633 lemma (in group) one_in_subset:  ballarin@13813  634  "[| H \ carrier G; H \ {}; \a \ H. inv a \ H; \a\H. \b\H. a \ b \ H |]  ballarin@13813  635  ==> \ \ H"  wenzelm@44472  636 by force  ballarin@13813  637 wenzelm@61382  638 text \A characterization of subgroups: closed, non-empty subset.\  ballarin@13813  639 ballarin@13813  640 lemma (in group) subgroupI:  ballarin@13813  641  assumes subset: "H \ carrier G" and non_empty: "H \ {}"  paulson@14963  642  and inv: "!!a. a \ H \ inv a \ H"  paulson@14963  643  and mult: "!!a b. \a \ H; b \ H\ \ a \ b \ H"  ballarin@13813  644  shows "subgroup H G"  ballarin@27714  645 proof (simp add: subgroup_def assms)  ballarin@27714  646  show "\ \ H" by (rule one_in_subset) (auto simp only: assms)  ballarin@13813  647 qed  ballarin@13813  648 lp15@68443  649 lemma (in group) subgroupE:  lp15@68443  650  assumes "subgroup H G"  lp15@68443  651  shows "H \ carrier G"  lp15@68443  652  and "H \ {}"  lp15@68443  653  and "\a. a \ H \ inv a \ H"  lp15@68517  654  and "\a b. \ a \ H; b \ H \ \ a \ b \ H"  lp15@68517  655  using assms unfolding subgroup_def[of H G] by auto  lp15@68443  656 ballarin@13936  657 declare monoid.one_closed [iff] group.inv_closed [simp]  ballarin@13936  658  monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]  ballarin@13813  659 ballarin@13813  660 lemma subgroup_nonempty:  wenzelm@67091  661  "\ subgroup {} G"  ballarin@13813  662  by (blast dest: subgroup.one_closed)  ballarin@13813  663 lp15@68517  664 lemma (in subgroup) finite_imp_card_positive: "finite (carrier G) \ 0 < card H"  lp15@68517  665  using subset one_closed card_gt_0_iff finite_subset by blast  ballarin@13813  666 lp15@68443  667 (*Following 3 lemmas contributed by Martin Baillon*)  lp15@68443  668 lp15@68443  669 lemma (in subgroup) subgroup_is_submonoid :  lp15@68443  670  "submonoid H G"  lp15@68443  671  by (simp add: submonoid.intro subset)  lp15@68443  672 lp15@68443  673 lemma (in group) submonoid_subgroupI :  lp15@68443  674  assumes "submonoid H G"  lp15@68443  675  and "\a. a \ H \ inv a \ H"  lp15@68443  676  shows "subgroup H G"  lp15@68443  677  by (metis assms subgroup_def submonoid_def)  lp15@68443  678 lp15@68443  679 lemma (in group) group_incl_imp_subgroup:  lp15@68443  680  assumes "H \ carrier G"  lp15@68445  681  and "group (G\carrier := H\)"  lp15@68445  682  shows "subgroup H G"  lp15@68443  683 proof (intro submonoid_subgroupI[OF monoid_incl_imp_submonoid[OF assms(1)]])  lp15@68443  684  show "monoid (G\carrier := H\)" using group_def assms by blast  lp15@68443  685  have ab_eq : "\ a b. a \ H \ b \ H \ a \\<^bsub>G\carrier := H\\<^esub> b = a \ b" using assms by simp  lp15@68445  686  fix a assume aH : "a \ H"  lp15@68443  687  have " inv\<^bsub>G\carrier := H\\<^esub> a \ carrier G"  lp15@68443  688  using assms aH group.inv_closed[OF assms(2)] by auto  lp15@68443  689  moreover have "\\<^bsub>G\carrier := H\\<^esub> = \" using assms monoid.one_closed ab_eq one_def by simp  lp15@68443  690  hence "a \\<^bsub>G\carrier := H\\<^esub> inv\<^bsub>G\carrier := H\\<^esub> a= \"  lp15@68443  691  using assms ab_eq aH group.r_inv[OF assms(2)] by simp  lp15@68443  692  hence "a \ inv\<^bsub>G\carrier := H\\<^esub> a= \"  lp15@68443  693  using aH assms group.inv_closed[OF assms(2)] ab_eq by simp  lp15@68443  694  ultimately have "inv\<^bsub>G\carrier := H\\<^esub> a = inv a"  lp15@68443  695  by (smt aH assms(1) contra_subsetD group.inv_inv is_group local.inv_equality)  lp15@68443  696  moreover have "inv\<^bsub>G\carrier := H\\<^esub> a \ H" using aH group.inv_closed[OF assms(2)] by auto  lp15@68443  697  ultimately show "inv a \ H" by auto  lp15@68443  698 qed  lp15@68443  699 ballarin@13936  700 wenzelm@61382  701 subsection \Direct Products\  ballarin@13813  702 wenzelm@35848  703 definition  wenzelm@35848  704  DirProd :: "_ \ _ \ ('a \ 'b) monoid" (infixr "\\" 80) where  wenzelm@35848  705  "G \\ H =  wenzelm@35848  706  \carrier = carrier G \ carrier H,  wenzelm@35848  707  mult = ($$g, h) (g', h'). (g \\<^bsub>G\<^esub> g', h \\<^bsub>H\<^esub> h')),  wenzelm@35848  708  one = (\\<^bsub>G\<^esub>, \\<^bsub>H\<^esub>)\"  ballarin@13813  709 paulson@14963  710 lemma DirProd_monoid:  ballarin@27611  711  assumes "monoid G" and "monoid H"  paulson@14963  712  shows "monoid (G \\ H)"  paulson@14963  713 proof -  wenzelm@30729  714  interpret G: monoid G by fact  wenzelm@30729  715  interpret H: monoid H by fact  ballarin@27714  716  from assms  lp15@68445  717  show ?thesis by (unfold monoid_def DirProd_def, auto)  paulson@14963  718 qed  ballarin@13813  719 ballarin@13813  720 wenzelm@61382  721 text\Does not use the previous result because it's easier just to use auto.\  paulson@14963  722 lemma DirProd_group:  ballarin@27611  723  assumes "group G" and "group H"  paulson@14963  724  shows "group (G \\ H)"  ballarin@27611  725 proof -  wenzelm@30729  726  interpret G: group G by fact  wenzelm@30729  727  interpret H: group H by fact  ballarin@27611  728  show ?thesis by (rule groupI)  paulson@14963  729  (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv  paulson@14963  730  simp add: DirProd_def)  ballarin@27611  731 qed  ballarin@13813  732 paulson@14963  733 lemma carrier_DirProd [simp]:  paulson@14963  734  "carrier (G \\ H) = carrier G \ carrier H"  paulson@14963  735  by (simp add: DirProd_def)  paulson@13944  736 paulson@14963  737 lemma one_DirProd [simp]:  paulson@14963  738  "\\<^bsub>G \\ H\<^esub> = (\\<^bsub>G\<^esub>, \\<^bsub>H\<^esub>)"  paulson@14963  739  by (simp add: DirProd_def)  paulson@13944  740 paulson@14963  741 lemma mult_DirProd [simp]:  paulson@14963  742  "(g, h) \\<^bsub>(G \\ H)\<^esub> (g', h') = (g \\<^bsub>G\<^esub> g', h \\<^bsub>H\<^esub> h')"  paulson@14963  743  by (simp add: DirProd_def)  paulson@13944  744 lp15@68443  745 lemma DirProd_assoc :  lp15@68443  746 "(G \\ H \\ I) = (G \\ (H \\ I))"  lp15@68443  747  by auto  lp15@68443  748 paulson@14963  749 lemma inv_DirProd [simp]:  ballarin@27611  750  assumes "group G" and "group H"  paulson@13944  751  assumes g: "g \ carrier G"  paulson@13944  752  and h: "h \ carrier H"  paulson@14963  753  shows "m_inv (G \\ H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"  ballarin@27611  754 proof -  wenzelm@30729  755  interpret G: group G by fact  wenzelm@30729  756  interpret H: group H by fact  wenzelm@30729  757  interpret Prod: group "G \\ H"  ballarin@27714  758  by (auto intro: DirProd_group group.intro group.axioms assms)  paulson@14963  759  show ?thesis by (simp add: Prod.inv_equality g h)  paulson@14963  760 qed  ballarin@27698  761 lp15@68443  762 lemma DirProd_subgroups :  lp15@68443  763  assumes "group G"  lp15@68445  764  and "subgroup H G"  lp15@68445  765  and "group K"  lp15@68445  766  and "subgroup I K"  lp15@68445  767  shows "subgroup (H \ I) (G \\ K)"  lp15@68443  768 proof (intro group.group_incl_imp_subgroup[OF DirProd_group[OF assms(1)assms(3)]])  lp15@68445  769  have "H \ carrier G" "I \ carrier K" using subgroup.subset assms apply blast+.  lp15@68443  770  thus "(H \ I) \ carrier (G \\ K)" unfolding DirProd_def by auto  lp15@68443  771  have "Group.group ((G\carrier := H$$ \\ (K\carrier := I\))"  lp15@68443  772  using DirProd_group[OF subgroup.subgroup_is_group[OF assms(2)assms(1)]  lp15@68445  773  subgroup.subgroup_is_group[OF assms(4)assms(3)]].  lp15@68443  774  moreover have "((G\carrier := H\) \\ (K\carrier := I\)) = ((G \\ K)\carrier := H \ I\)"  lp15@68443  775  unfolding DirProd_def using assms apply simp.  lp15@68443  776  ultimately show "Group.group ((G \\ K)\carrier := H \ I\)" by simp  lp15@68443  777 qed  paulson@14963  778 wenzelm@61382  779 subsection \Homomorphisms and Isomorphisms\  ballarin@13813  780 wenzelm@35847  781 definition  wenzelm@35847  782  hom :: "_ => _ => ('a => 'b) set" where  wenzelm@35848  783  "hom G H =  wenzelm@67091  784  {h. h \ carrier G \ carrier H \  wenzelm@14693  785  (\x \ carrier G. \y \ carrier G. h (x \\<^bsub>G\<^esub> y) = h x \\<^bsub>H\<^esub> h y)}"  ballarin@13813  786 paulson@14761  787 lemma (in group) hom_compose:  nipkow@31754  788  "[|h \ hom G H; i \ hom H I|] ==> compose (carrier G) i h \ hom G I"  nipkow@44890  789 by (fastforce simp add: hom_def compose_def)  paulson@13943  790 wenzelm@35848  791 definition  lp15@68445  792  iso :: "_ => _ => ('a => 'b) set"  lp15@68443  793  where "iso G H = {h. h \ hom G H \ bij_betw h (carrier G) (carrier H)}"  lp15@68443  794 lp15@68443  795 definition  lp15@68443  796  is_iso :: "_ \ _ \ bool" (infixr "\" 60)  lp15@68445  797  where "G \ H = (iso G H \ {})"  paulson@14761  798 lp15@68443  799 lemma iso_set_refl: "(\x. x) \ iso G G"  lp15@68443  800  by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)  paulson@14761  801 lp15@68443  802 corollary iso_refl : "G \ G"  lp15@68443  803  using iso_set_refl unfolding is_iso_def by auto  lp15@68443  804 lp15@68443  805 lemma (in group) iso_set_sym:  lp15@68458  806  assumes "h \ iso G H"  lp15@68458  807  shows "inv_into (carrier G) h \ iso H G"  lp15@68458  808 proof -  lp15@68458  809  have h: "h \ hom G H" "bij_betw h (carrier G) (carrier H)"  lp15@68458  810  using assms by (auto simp add: iso_def bij_betw_inv_into)  lp15@68458  811  then have HG: "bij_betw (inv_into (carrier G) h) (carrier H) (carrier G)"  lp15@68458  812  by (simp add: bij_betw_inv_into)  lp15@68458  813  have "inv_into (carrier G) h \ hom H G"  lp15@68458  814  unfolding hom_def  lp15@68458  815  proof safe  lp15@68458  816  show *: "\x. x \ carrier H \ inv_into (carrier G) h x \ carrier G"  lp15@68458  817  by (meson HG bij_betwE)  lp15@68458  818  show "inv_into (carrier G) h (x \\<^bsub>H\<^esub> y) = inv_into (carrier G) h x \ inv_into (carrier G) h y"  lp15@68458  819  if "x \ carrier H" "y \ carrier H" for x y  lp15@68458  820  proof (rule inv_into_f_eq)  lp15@68458  821  show "inj_on h (carrier G)"  lp15@68458  822  using bij_betw_def h(2) by blast  lp15@68458  823  show "inv_into (carrier G) h x \ inv_into (carrier G) h y \ carrier G"  lp15@68458  824  by (simp add: * that)  lp15@68458  825  show "h (inv_into (carrier G) h x \ inv_into (carrier G) h y) = x \\<^bsub>H\<^esub> y"  lp15@68458  826  using h bij_betw_inv_into_right [of h] unfolding hom_def by (simp add: "*" that)  lp15@68458  827  qed  lp15@68458  828  qed  lp15@68458  829  then show ?thesis  lp15@68458  830  by (simp add: Group.iso_def bij_betw_inv_into h)  lp15@68458  831 qed  paulson@14761  832 lp15@68458  833 lp15@68458  834 corollary (in group) iso_sym: "G \ H \ H \ G"  lp15@68443  835  using iso_set_sym unfolding is_iso_def by auto  lp15@68443  836 lp15@68445  837 lemma (in group) iso_set_trans:  lp15@68443  838  "[|h \ iso G H; i \ iso H I|] ==> (compose (carrier G) i h) \ iso G I"  paulson@14761  839 by (auto simp add: iso_def hom_compose bij_betw_compose)  paulson@14761  840 lp15@68458  841 corollary (in group) iso_trans: "\G \ H ; H \ I\ \ G \ I"  lp15@68443  842  using iso_set_trans unfolding is_iso_def by blast  lp15@68443  843 lp15@68445  844 (* Next four lemmas contributed by Paulo. *)  lp15@68443  845 lp15@68443  846 lemma (in monoid) hom_imp_img_monoid:  lp15@68443  847  assumes "h \ hom G H"  lp15@68443  848  shows "monoid (H \ carrier := h  (carrier G), one := h \\<^bsub>G\<^esub> \)" (is "monoid ?h_img")  lp15@68443  849 proof (rule monoidI)  lp15@68443  850  show "\\<^bsub>?h_img\<^esub> \ carrier ?h_img"  lp15@68443  851  by auto  lp15@68443  852 next  lp15@68443  853  fix x y z assume "x \ carrier ?h_img" "y \ carrier ?h_img" "z \ carrier ?h_img"  lp15@68443  854  then obtain g1 g2 g3  lp15@68443  855  where g1: "g1 \ carrier G" "x = h g1"  lp15@68443  856  and g2: "g2 \ carrier G" "y = h g2"  lp15@68443  857  and g3: "g3 \ carrier G" "z = h g3"  lp15@68443  858  using image_iff[where ?f = h and ?A = "carrier G"] by auto  lp15@68443  859  have aux_lemma:  lp15@68443  860  "\a b. \ a \ carrier G; b \ carrier G \ \ h a \\<^bsub>(?h_img)\<^esub> h b = h (a \ b)"  lp15@68443  861  using assms unfolding hom_def by auto  lp15@68443  862 lp15@68443  863  show "x \\<^bsub>(?h_img)\<^esub> \\<^bsub>(?h_img)\<^esub> = x"  lp15@68443  864  using aux_lemma[OF g1(1) one_closed] g1(2) r_one[OF g1(1)] by simp  lp15@68443  865 lp15@68443  866  show "\\<^bsub>(?h_img)\<^esub> \\<^bsub>(?h_img)\<^esub> x = x"  lp15@68443  867  using aux_lemma[OF one_closed g1(1)] g1(2) l_one[OF g1(1)] by simp  lp15@68443  868 lp15@68443  869  have "x \\<^bsub>(?h_img)\<^esub> y = h (g1 \ g2)"  lp15@68443  870  using aux_lemma g1 g2 by auto  lp15@68443  871  thus "x \\<^bsub>(?h_img)\<^esub> y \ carrier ?h_img"  lp15@68443  872  using g1(1) g2(1) by simp  lp15@68443  873 lp15@68443  874  have "(x \\<^bsub>(?h_img)\<^esub> y) \\<^bsub>(?h_img)\<^esub> z = h ((g1 \ g2) \ g3)"  lp15@68443  875  using aux_lemma g1 g2 g3 by auto  lp15@68443  876  also have " ... = h (g1 \ (g2 \ g3))"  lp15@68443  877  using m_assoc[OF g1(1) g2(1) g3(1)] by simp  lp15@68443  878  also have " ... = x \\<^bsub>(?h_img)\<^esub> (y \\<^bsub>(?h_img)\<^esub> z)"  lp15@68443  879  using aux_lemma g1 g2 g3 by auto  lp15@68443  880  finally show "(x \\<^bsub>(?h_img)\<^esub> y) \\<^bsub>(?h_img)\<^esub> z = x \\<^bsub>(?h_img)\<^esub> (y \\<^bsub>(?h_img)\<^esub> z)" .  lp15@68443  881 qed  lp15@68443  882 lp15@68443  883 lemma (in group) hom_imp_img_group:  lp15@68443  884  assumes "h \ hom G H"  lp15@68443  885  shows "group (H \ carrier := h  (carrier G), one := h \\<^bsub>G\<^esub> \)" (is "group ?h_img")  lp15@68443  886 proof -  lp15@68443  887  interpret monoid ?h_img  lp15@68443  888  using hom_imp_img_monoid[OF assms] .  lp15@68443  889 lp15@68443  890  show ?thesis  lp15@68443  891  proof (unfold_locales)  lp15@68443  892  show "carrier ?h_img \ Units ?h_img"  lp15@68443  893  proof (auto simp add: Units_def)  lp15@68443  894  have aux_lemma:  lp15@68443  895  "\g1 g2. \ g1 \ carrier G; g2 \ carrier G \ \ h g1 \\<^bsub>H\<^esub> h g2 = h (g1 \ g2)"  lp15@68443  896  using assms unfolding hom_def by auto  lp15@68443  897 lp15@68443  898  fix g1 assume g1: "g1 \ carrier G"  lp15@68443  899  thus "\g2 \ carrier G. (h g2) \\<^bsub>H\<^esub> (h g1) = h \ \ (h g1) \\<^bsub>H\<^esub> (h g2) = h \"  lp15@68443  900  using aux_lemma[OF g1 inv_closed[OF g1]]  lp15@68443  901  aux_lemma[OF inv_closed[OF g1] g1]  lp15@68443  902  inv_closed by auto  lp15@68443  903  qed  lp15@68443  904  qed  lp15@68443  905 qed  lp15@68443  906 lp15@68443  907 lemma (in group) iso_imp_group:  lp15@68443  908  assumes "G \ H" and "monoid H"  lp15@68443  909  shows "group H"  lp15@68443  910 proof -  lp15@68443  911  obtain \ where phi: "\ \ iso G H" "inv_into (carrier G) \ \ iso H G"  lp15@68443  912  using iso_set_sym assms unfolding is_iso_def by blast  lp15@68443  913  define \ where psi_def: "\ = inv_into (carrier G) \"  lp15@68445  914 lp15@68443  915  from phi  lp15@68443  916  have surj: "\  (carrier G) = (carrier H)" "\  (carrier H) = (carrier G)"  lp15@68443  917  and inj: "inj_on \ (carrier G)" "inj_on \ (carrier H)"  lp15@68443  918  and phi_hom: "\g1 g2. \ g1 \ carrier G; g2 \ carrier G \ \ \ (g1 \ g2) = (\ g1) \\<^bsub>H\<^esub> (\ g2)"  lp15@68443  919  and psi_hom: "\h1 h2. \ h1 \ carrier H; h2 \ carrier H \ \ \ (h1 \\<^bsub>H\<^esub> h2) = (\ h1) \ (\ h2)"  lp15@68443  920  using psi_def unfolding iso_def bij_betw_def hom_def by auto  lp15@68443  921 lp15@68443  922  have phi_one: "\ \ = \\<^bsub>H\<^esub>"  lp15@68443  923  proof -  lp15@68443  924  have "(\ \) \\<^bsub>H\<^esub> \\<^bsub>H\<^esub> = (\ \) \\<^bsub>H\<^esub> (\ \)"  lp15@68443  925  by (metis assms(2) image_eqI monoid.r_one one_closed phi_hom r_one surj(1))  lp15@68443  926  thus ?thesis  lp15@68443  927  by (metis (no_types, hide_lams) Units_eq Units_one_closed assms(2) f_inv_into_f imageI  lp15@68443  928  monoid.l_one monoid.one_closed phi_hom psi_def r_one surj)  lp15@68443  929  qed  lp15@68443  930 lp15@68443  931  have "carrier H \ Units H"  lp15@68443  932  proof  lp15@68443  933  fix h assume h: "h \ carrier H"  lp15@68443  934  let ?inv_h = "\ (inv (\ h))"  lp15@68443  935  have "h \\<^bsub>H\<^esub> ?inv_h = \ (\ h) \\<^bsub>H\<^esub> ?inv_h"  lp15@68443  936  by (simp add: f_inv_into_f h psi_def surj(1))  lp15@68443  937  also have " ... = \ ((\ h) \ inv (\ h))"  lp15@68443  938  by (metis h imageI inv_closed phi_hom surj(2))  lp15@68443  939  also have " ... = \ \"  lp15@68443  940  by (simp add: h inv_into_into psi_def surj(1))  lp15@68443  941  finally have 1: "h \\<^bsub>H\<^esub> ?inv_h = \\<^bsub>H\<^esub>"  lp15@68443  942  using phi_one by simp  lp15@68443  943 lp15@68443  944  have "?inv_h \\<^bsub>H\<^esub> h = ?inv_h \\<^bsub>H\<^esub> \ (\ h)"  lp15@68443  945  by (simp add: f_inv_into_f h psi_def surj(1))  lp15@68443  946  also have " ... = \ (inv (\ h) \ (\ h))"  lp15@68443  947  by (metis h imageI inv_closed phi_hom surj(2))  lp15@68443  948  also have " ... = \ \"  lp15@68443  949  by (simp add: h inv_into_into psi_def surj(1))  lp15@68443  950  finally have 2: "?inv_h \\<^bsub>H\<^esub> h = \\<^bsub>H\<^esub>"  lp15@68443  951  using phi_one by simp  lp15@68443  952 lp15@68443  953  thus "h \ Units H" unfolding Units_def using 1 2 h surj by fastforce  lp15@68443  954  qed  lp15@68443  955  thus ?thesis unfolding group_def group_axioms_def using assms(2) by simp  lp15@68443  956 qed  lp15@68443  957 lp15@68443  958 corollary (in group) iso_imp_img_group:  lp15@68443  959  assumes "h \ iso G H"  lp15@68443  960  shows "group (H \ one := h \ \)"  lp15@68443  961 proof -  lp15@68443  962  let ?h_img = "H \ carrier := h  (carrier G), one := h \ \"  lp15@68443  963  have "h \ iso G ?h_img"  lp15@68443  964  using assms unfolding iso_def hom_def bij_betw_def by auto  lp15@68443  965  hence "G \ ?h_img"  lp15@68443  966  unfolding is_iso_def by auto  lp15@68443  967  hence "group ?h_img"  lp15@68443  968  using iso_imp_group[of ?h_img] hom_imp_img_monoid[of h H] assms unfolding iso_def by simp  lp15@68443  969  moreover have "carrier H = carrier ?h_img"  lp15@68443  970  using assms unfolding iso_def bij_betw_def by simp  lp15@68443  971  hence "H \ one := h \ \ = ?h_img"  lp15@68443  972  by simp  lp15@68443  973  ultimately show ?thesis by simp  lp15@68443  974 qed  lp15@68443  975 lp15@68443  976 lemma DirProd_commute_iso_set:  lp15@68443  977  shows "($$x,y). (y,x)) \ iso (G \\ H) (H \\ G)"  lp15@68443  978  by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)  lp15@68443  979 lp15@68443  980 corollary DirProd_commute_iso :  lp15@68443  981 "(G \\ H) \ (H \\ G)"  lp15@68443  982  using DirProd_commute_iso_set unfolding is_iso_def by blast  lp15@68443  983 lp15@68443  984 lemma DirProd_assoc_iso_set:  lp15@68443  985  shows "(\(x,y,z). (x,(y,z))) \ iso (G \\ H \\ I) (G \\ (H \\ I))"  nipkow@31754  986 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def)  paulson@14761  987 lp15@68445  988 lemma (in group) DirProd_iso_set_trans:  lp15@68443  989  assumes "g \ iso G G2"  lp15@68443  990  and "h \ iso H I"  lp15@68443  991  shows "(\(x,y). (g x, h y)) \ iso (G \\ H) (G2 \\ I)"  lp15@68443  992 proof-  lp15@68443  993  have "(\(x,y). (g x, h y)) \ hom (G \\ H) (G2 \\ I)"  lp15@68443  994  using assms unfolding iso_def hom_def by auto  lp15@68443  995  moreover have " inj_on (\(x,y). (g x, h y)) (carrier (G \\ H))"  lp15@68443  996  using assms unfolding iso_def DirProd_def bij_betw_def inj_on_def by auto  lp15@68443  997  moreover have "(\(x, y). (g x, h y))  carrier (G \\ H) = carrier (G2 \\ I)"  lp15@68443  998  using assms unfolding iso_def bij_betw_def image_def DirProd_def by fastforce  lp15@68443  999  ultimately show "(\(x,y). (g x, h y)) \ iso (G \\ H) (G2 \\ I)"  lp15@68443  1000  unfolding iso_def bij_betw_def by auto  lp15@68443  1001 qed  lp15@68443  1002 lp15@68443  1003 corollary (in group) DirProd_iso_trans :  lp15@68443  1004  assumes "G \ G2"  lp15@68443  1005  and "H \ I"  lp15@68443  1006  shows "G \\ H \ G2 \\ I"  lp15@68443  1007  using DirProd_iso_set_trans assms unfolding is_iso_def by blast  paulson@14761  1008 paulson@14761  1009 wenzelm@61382  1010 text\Basis for homomorphism proofs: we assume two groups @{term G} and  wenzelm@61382  1011  @{term H}, with a homomorphism @{term h} between them\  ballarin@61565  1012 locale group_hom = G?: group G + H?: group H for G (structure) and H (structure) +  ballarin@29237  1013  fixes h  ballarin@13813  1014  assumes homh: "h \ hom G H"  ballarin@29240  1015 ballarin@29240  1016 lemma (in group_hom) hom_mult [simp]:  ballarin@29240  1017  "[| x \ carrier G; y \ carrier G |] ==> h (x \\<^bsub>G\<^esub> y) = h x \\<^bsub>H\<^esub> h y"  ballarin@29240  1018 proof -  ballarin@29240  1019  assume "x \ carrier G" "y \ carrier G"  ballarin@29240  1020  with homh [unfolded hom_def] show ?thesis by simp  ballarin@29240  1021 qed  ballarin@29240  1022 ballarin@29240  1023 lemma (in group_hom) hom_closed [simp]:  ballarin@29240  1024  "x \ carrier G ==> h x \ carrier H"  ballarin@29240  1025 proof -  ballarin@29240  1026  assume "x \ carrier G"  nipkow@31754  1027  with homh [unfolded hom_def] show ?thesis by auto  ballarin@29240  1028 qed  ballarin@13813  1029 ballarin@13813  1030 lemma (in group_hom) one_closed [simp]:  ballarin@13813  1031  "h \ \ carrier H"  ballarin@13813  1032  by simp  ballarin@13813  1033 ballarin@13813  1034 lemma (in group_hom) hom_one [simp]:  wenzelm@14693  1035  "h \ = \\<^bsub>H\<^esub>"  ballarin@13813  1036 proof -  ballarin@15076  1037  have "h \ \\<^bsub>H\<^esub> \\<^bsub>H\<^esub> = h \ \\<^bsub>H\<^esub> h \"  ballarin@13813  1038  by (simp add: hom_mult [symmetric] del: hom_mult)  ballarin@13813  1039  then show ?thesis by (simp del: r_one)  ballarin@13813  1040 qed  ballarin@13813  1041 ballarin@13813  1042 lemma (in group_hom) inv_closed [simp]:  ballarin@13813  1043  "x \ carrier G ==> h (inv x) \ carrier H"  ballarin@13813  1044  by simp  ballarin@13813  1045 ballarin@13813  1046 lemma (in group_hom) hom_inv [simp]:  wenzelm@14693  1047  "x \ carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"  ballarin@13813  1048 proof -  ballarin@13813  1049  assume x: "x \ carrier G"  wenzelm@14693  1050  then have "h x \\<^bsub>H\<^esub> h (inv x) = \\<^bsub>H\<^esub>"  paulson@14963  1051  by (simp add: hom_mult [symmetric] del: hom_mult)  wenzelm@14693  1052  also from x have "... = h x \\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"  paulson@14963  1053  by (simp add: hom_mult [symmetric] del: hom_mult)  wenzelm@14693  1054  finally have "h x \\<^bsub>H\<^esub> h (inv x) = h x \\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .  ballarin@27698  1055  with x show ?thesis by (simp del: H.r_inv H.Units_r_inv)  ballarin@13813  1056 qed  ballarin@13813  1057 ballarin@57271  1058 (* Contributed by Joachim Breitner *)  ballarin@57271  1059 lemma (in group) int_pow_is_hom:  nipkow@67399  1060  "x \ carrier G \ (([^]) x) \ hom \ carrier = UNIV, mult = (+), one = 0::int \ G "  ballarin@57271  1061  unfolding hom_def by (simp add: int_pow_mult)  ballarin@57271  1062 lp15@68445  1063 (* Next six lemmas contributed by Paulo. *)  lp15@68443  1064 lp15@68443  1065 lemma (in group_hom) img_is_subgroup: "subgroup (h  (carrier G)) H"  lp15@68443  1066  apply (rule subgroupI)  lp15@68443  1067  apply (auto simp add: image_subsetI)  lp15@68443  1068  apply (metis (no_types, hide_lams) G.inv_closed hom_inv image_iff)  lp15@68443  1069  apply (smt G.monoid_axioms hom_mult image_iff monoid.m_closed)  lp15@68443  1070  done  lp15@68443  1071 lp15@68443  1072 lemma (in group_hom) subgroup_img_is_subgroup:  lp15@68443  1073  assumes "subgroup I G"  lp15@68443  1074  shows "subgroup (h  I) H"  lp15@68443  1075 proof -  lp15@68443  1076  have "h \ hom (G \ carrier := I$$ H"  lp15@68443  1077  using G.subgroupE[OF assms] subgroup.mem_carrier[OF assms] homh  lp15@68443  1078  unfolding hom_def by auto  lp15@68443  1079  hence "group_hom (G \ carrier := I \) H h"  lp15@68443  1080  using subgroup.subgroup_is_group[OF assms G.is_group] is_group  lp15@68443  1081  unfolding group_hom_def group_hom_axioms_def by simp  lp15@68443  1082  thus ?thesis  lp15@68443  1083  using group_hom.img_is_subgroup[of "G \ carrier := I \" H h] by simp  lp15@68443  1084 qed  lp15@68443  1085 lp15@68443  1086 lemma (in group_hom) induced_group_hom:  lp15@68443  1087  assumes "subgroup I G"  lp15@68443  1088  shows "group_hom (G \ carrier := I \) (H \ carrier := h  I \) h"  lp15@68443  1089 proof -  lp15@68443  1090  have "h \ hom (G \ carrier := I \) (H \ carrier := h  I \)"  lp15@68443  1091  using homh subgroup.mem_carrier[OF assms] unfolding hom_def by auto  lp15@68443  1092  thus ?thesis  lp15@68443  1093  unfolding group_hom_def group_hom_axioms_def  lp15@68443  1094  using subgroup.subgroup_is_group[OF assms G.is_group]  lp15@68443  1095  subgroup.subgroup_is_group[OF subgroup_img_is_subgroup[OF assms] is_group] by simp  lp15@68443  1096 qed  lp15@68443  1097 lp15@68443  1098 lemma (in group) canonical_inj_is_hom:  lp15@68443  1099  assumes "subgroup H G"  lp15@68443  1100  shows "group_hom (G \ carrier := H \) G id"  lp15@68443  1101  unfolding group_hom_def group_hom_axioms_def hom_def  lp15@68443  1102  using subgroup.subgroup_is_group[OF assms is_group]  lp15@68445  1103  is_group subgroup.subset[OF assms] by auto  lp15@68443  1104 lp15@68443  1105 lemma (in group_hom) nat_pow_hom:  lp15@68443  1106  "x \ carrier G \ h (x [^] (n :: nat)) = (h x) [^]\<^bsub>H\<^esub> n"  lp15@68443  1107  by (induction n) auto  lp15@68443  1108 lp15@68443  1109 lemma (in group_hom) int_pow_hom:  lp15@68443  1110  "x \ carrier G \ h (x [^] (n :: int)) = (h x) [^]\<^bsub>H\<^esub> n"  lp15@68443  1111  using int_pow_def2 nat_pow_hom by (simp add: G.int_pow_def2)  lp15@68443  1112 ballarin@20318  1113 wenzelm@61382  1114 subsection \Commutative Structures\  ballarin@13936  1115 wenzelm@61382  1116 text \  ballarin@13936  1117  Naming convention: multiplicative structures that are commutative  ballarin@13936  1118  are called \emph{commutative}, additive structures are called  ballarin@13936  1119  \emph{Abelian}.  wenzelm@61382  1120 \  ballarin@13813  1121 paulson@14963  1122 locale comm_monoid = monoid +  paulson@14963  1123  assumes m_comm: "\x \ carrier G; y \ carrier G\ \ x \ y = y \ x"  ballarin@13813  1124 paulson@14963  1125 lemma (in comm_monoid) m_lcomm:  paulson@14963  1126  "\x \ carrier G; y \ carrier G; z \ carrier G\ \  ballarin@13813  1127  x \ (y \ z) = y \ (x \ z)"  ballarin@13813  1128 proof -  wenzelm@14693  1129  assume xyz: "x \ carrier G" "y \ carrier G" "z \ carrier G"  ballarin@13813  1130  from xyz have "x \ (y \ z) = (x \ y) \ z" by (simp add: m_assoc)  ballarin@13813  1131  also from xyz have "... = (y \ x) \ z" by (simp add: m_comm)  ballarin@13813  1132  also from xyz have "... = y \ (x \ z)" by (simp add: m_assoc)  ballarin@13813  1133  finally show ?thesis .  ballarin@13813  1134 qed  ballarin@13813  1135 paulson@14963  1136 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm  ballarin@13813  1137 ballarin@13936  1138 lemma comm_monoidI:  ballarin@19783  1139  fixes G (structure)  ballarin@13936  1140  assumes m_closed:  wenzelm@14693  1141  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y \ carrier G"  wenzelm@14693  1142  and one_closed: "\ \ carrier G"  ballarin@13936  1143  and m_assoc:  ballarin@13936  1144  "!!x y z. [| x \ carrier G; y \ carrier G; z \ carrier G |] ==>  wenzelm@14693  1145  (x \ y) \ z = x \ (y \ z)"  wenzelm@14693  1146  and l_one: "!!x. x \ carrier G ==> \ \ x = x"  ballarin@13936  1147  and m_comm:  wenzelm@14693  1148  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y = y \ x"  ballarin@13936  1149  shows "comm_monoid G"  ballarin@13936  1150  using l_one  lp15@68445  1151  by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro  ballarin@27714  1152  intro: assms simp: m_closed one_closed m_comm)  ballarin@13817  1153 ballarin@13936  1154 lemma (in monoid) monoid_comm_monoidI:  ballarin@13936  1155  assumes m_comm:  wenzelm@14693  1156  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y = y \ x"  ballarin@13936  1157  shows "comm_monoid G"  ballarin@13936  1158  by (rule comm_monoidI) (auto intro: m_assoc m_comm)  paulson@14963  1159 ballarin@13936  1160 lemma (in comm_monoid) nat_pow_distr:  ballarin@13936  1161  "[| x \ carrier G; y \ carrier G |] ==>  nipkow@67341  1162  (x \ y) [^] (n::nat) = x [^] n \ y [^] n"  ballarin@13936  1163  by (induct n) (simp, simp add: m_ac)  ballarin@13936  1164 lp15@68443  1165 lemma (in comm_monoid) submonoid_is_comm_monoid :  lp15@68443  1166  assumes "submonoid H G"  lp15@68443  1167  shows "comm_monoid (G\carrier := H\)"  lp15@68443  1168 proof (intro monoid.monoid_comm_monoidI)  lp15@68443  1169  show "monoid (G\carrier := H\)"  lp15@68443  1170  using submonoid.submonoid_is_monoid assms comm_monoid_axioms comm_monoid_def by blast  lp15@68443  1171  show "\x y. x \ carrier (G\carrier := H\) \ y \ carrier (G\carrier := H\)  lp15@68443  1172  \ x \\<^bsub>G\carrier := H\\<^esub> y = y \\<^bsub>G\carrier := H\\<^esub> x" apply simp  lp15@68443  1173  using assms comm_monoid_axioms_def submonoid.mem_carrier  lp15@68443  1174  by (metis m_comm)  lp15@68443  1175 qed  lp15@68443  1176 ballarin@13936  1177 locale comm_group = comm_monoid + group  ballarin@13936  1178 ballarin@13936  1179 lemma (in group) group_comm_groupI:  ballarin@13936  1180  assumes m_comm: "!!x y. [| x \ carrier G; y \ carrier G |] ==>  wenzelm@14693  1181  x \ y = y \ x"  ballarin@13936  1182  shows "comm_group G"  wenzelm@61169  1183  by standard (simp_all add: m_comm)  ballarin@13817  1184 ballarin@13936  1185 lemma comm_groupI:  ballarin@19783  1186  fixes G (structure)  ballarin@13936  1187  assumes m_closed:  wenzelm@14693  1188  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y \ carrier G"  wenzelm@14693  1189  and one_closed: "\ \ carrier G"  ballarin@13936  1190  and m_assoc:  ballarin@13936  1191  "!!x y z. [| x \ carrier G; y \ carrier G; z \ carrier G |] ==>  wenzelm@14693  1192  (x \ y) \ z = x \ (y \ z)"  ballarin@13936  1193  and m_comm:  wenzelm@14693  1194  "!!x y. [| x \ carrier G; y \ carrier G |] ==> x \ y = y \ x"  wenzelm@14693  1195  and l_one: "!!x. x \ carrier G ==> \ \ x = x"  paulson@14963  1196  and l_inv_ex: "!!x. x \ carrier G ==> \y \ carrier G. y \ x = \"  ballarin@13936  1197  shows "comm_group G"  ballarin@27714  1198  by (fast intro: group.group_comm_groupI groupI assms)  ballarin@13936  1199 lp15@68443  1200 lemma comm_groupE:  lp15@68443  1201  fixes G (structure)  lp15@68443  1202  assumes "comm_group G"  lp15@68443  1203  shows "\x y. \ x \ carrier G; y \ carrier G \ \ x \ y \ carrier G"  lp15@68443  1204  and "\ \ carrier G"  lp15@68443  1205  and "\x y z. \ x \ carrier G; y \ carrier G; z \ carrier G \ \ (x \ y) \ z = x \ (y \ z)"  lp15@68443  1206  and "\x y. \ x \ carrier G; y \ carrier G \ \ x \ y = y \ x"  lp15@68443  1207  and "\x. x \ carrier G \ \ \ x = x"  lp15@68443  1208  and "\x. x \ carrier G \ \y \ carrier G. y \ x = \"  lp15@68443  1209  apply (simp_all add: group.axioms assms comm_group.axioms comm_monoid.m_comm comm_monoid.m_ac(1))  lp15@68443  1210  by (simp_all add: Group.group.axioms(1) assms comm_group.axioms(2) monoid.m_closed group.r_inv_ex)  lp15@68443  1211 ballarin@13936  1212 lemma (in comm_group) inv_mult:  ballarin@13854  1213  "[| x \ carrier G; y \ carrier G |] ==> inv (x \ y) = inv x \ inv y"  ballarin@13936  1214  by (simp add: m_ac inv_mult_group)  ballarin@13854  1215 lp15@68445  1216 (* Next three lemmas contributed by Paulo. *)  lp15@68443  1217 lp15@68443  1218 lemma (in comm_monoid) hom_imp_img_comm_monoid:  lp15@68443  1219  assumes "h \ hom G H"  lp15@68443  1220  shows "comm_monoid (H \ carrier := h  (carrier G), one := h \\<^bsub>G\<^esub> \)" (is "comm_monoid ?h_img")  lp15@68443  1221 proof (rule monoid.monoid_comm_monoidI)  lp15@68443  1222  show "monoid ?h_img"  lp15@68443  1223  using hom_imp_img_monoid[OF assms] .  lp15@68443  1224 next  lp15@68443  1225  fix x y assume "x \ carrier ?h_img" "y \ carrier ?h_img"  lp15@68443  1226  then obtain g1 g2  lp15@68443  1227  where g1: "g1 \ carrier G" "x = h g1"  lp15@68443  1228  and g2: "g2 \ carrier G" "y = h g2"  lp15@68443  1229  by auto  lp15@68443  1230  have "x \\<^bsub>(?h_img)\<^esub> y = h (g1 \ g2)"  lp15@68443  1231  using g1 g2 assms unfolding hom_def by auto  lp15@68443  1232  also have " ... = h (g2 \ g1)"  lp15@68443  1233  using m_comm[OF g1(1) g2(1)] by simp  lp15@68443  1234  also have " ... = y \\<^bsub>(?h_img)\<^esub> x"  lp15@68443  1235  using g1 g2 assms unfolding hom_def by auto  lp15@68443  1236  finally show "x \\<^bsub>(?h_img)\<^esub> y = y \\<^bsub>(?h_img)\<^esub> x" .  lp15@68443  1237 qed  lp15@68443  1238 lp15@68517  1239 lemma (in comm_group) hom_imp_img_comm_group:  lp15@68517  1240  assumes "h \ hom G H"  lp15@68517  1241  shows "comm_group (H \ carrier := h  (carrier G), one := h \\<^bsub>G\<^esub> \)"  lp15@68517  1242  unfolding comm_group_def  lp15@68517  1243  using hom_imp_img_group[OF assms] hom_imp_img_comm_monoid[OF assms] by simp  lp15@68517  1244 lp15@68443  1245 lemma (in comm_group) iso_imp_img_comm_group:  lp15@68443  1246  assumes "h \ iso G H"  lp15@68443  1247  shows "comm_group (H \ one := h \\<^bsub>G\<^esub> \)"  lp15@68443  1248 proof -  lp15@68443  1249  let ?h_img = "H \ carrier := h  (carrier G), one := h \ \"  lp15@68517  1250  have "comm_group ?h_img"  lp15@68517  1251  using hom_imp_img_comm_group[of h H] assms unfolding iso_def by auto  lp15@68443  1252  moreover have "carrier H = carrier ?h_img"  lp15@68443  1253  using assms unfolding iso_def bij_betw_def by simp  lp15@68443  1254  hence "H \ one := h \ \ = ?h_img"  lp15@68443  1255  by simp  lp15@68517  1256  ultimately show ?thesis by simp  lp15@68443  1257 qed  lp15@68443  1258 lp15@68443  1259 lemma (in comm_group) iso_imp_comm_group:  lp15@68443  1260  assumes "G \ H" "monoid H"  lp15@68443  1261  shows "comm_group H"  lp15@68443  1262 proof -  lp15@68443  1263  obtain h where h: "h \ iso G H"  lp15@68443  1264  using assms(1) unfolding is_iso_def by auto  lp15@68443  1265  hence comm_gr: "comm_group (H \ one := h \ \)"  lp15@68443  1266  using iso_imp_img_comm_group[of h H] by simp  lp15@68443  1267  hence "\x. x \ carrier H \ h \ \\<^bsub>H\<^esub> x = x"  lp15@68443  1268  using monoid.l_one[of "H \ one := h \ \"] unfolding comm_group_def comm_monoid_def by simp  lp15@68443  1269  moreover have "h \ \ carrier H"  lp15@68443  1270  using h one_closed unfolding iso_def hom_def by auto  lp15@68443  1271  ultimately have "h \ = \\<^bsub>H\<^esub>"  lp15@68443  1272  using monoid.one_unique[OF assms(2), of "h \"] by simp  lp15@68443  1273  hence "H = H \ one := h \ \"  lp15@68443  1274  by simp  lp15@68443  1275  thus ?thesis  lp15@68443  1276  using comm_gr by simp  lp15@68443  1277 qed  lp15@68443  1278 lp15@68445  1279 (*A subgroup of a subgroup is a subgroup of the group*)  lp15@68445  1280 lemma (in group) incl_subgroup:  lp15@68445  1281  assumes "subgroup J G"  lp15@68445  1282  and "subgroup I (G\carrier:=J\)"  lp15@68445  1283  shows "subgroup I G" unfolding subgroup_def  lp15@68445  1284 proof  lp15@68452  1285  have H1: "I \ carrier (G\carrier:=J\)" using assms(2) subgroup.subset by blast  lp15@68445  1286  also have H2: "...\J" by simp  lp15@68452  1287  also have "...$$carrier G)" by (simp add: assms(1) subgroup.subset)  lp15@68445  1288  finally have H: "I \ carrier G" by simp  lp15@68445  1289  have "(\x y. \x \ I ; y \ I\ \ x \ y \ I)" using assms(2) by (auto simp add: subgroup_def)  lp15@68445  1290  thus "I \ carrier G \ (\x y. x \ I \ y \ I \ x \ y \ I)" using H by blast  lp15@68445  1291  have K: "\ \ I" using assms(2) by (auto simp add: subgroup_def)  lp15@68445  1292  have "(\x. x \ I \ inv x \ I)" using assms subgroup.m_inv_closed H  lp15@68445  1293  by (metis H1 H2 subgroup_inv_equality subsetCE)  lp15@68445  1294  thus "\ \ I \ (\x. x \ I \ inv x \ I)" using K by blast  lp15@68445  1295 qed  lp15@68445  1296 lp15@68445  1297 (*A subgroup included in another subgroup is a subgroup of the subgroup*)  lp15@68445  1298 lemma (in group) subgroup_incl:  lp15@68445  1299  assumes "subgroup I G"  lp15@68445  1300  and "subgroup J G"  lp15@68445  1301  and "I\J"  lp15@68445  1302  shows "subgroup I (G\carrier:=J$$"using assms subgroup_inv_equality  lp15@68445  1303  by (auto simp add: subgroup_def)  lp15@68445  1304 lp15@68443  1305 ballarin@20318  1306 wenzelm@61382  1307 subsection \The Lattice of Subgroups of a Group\  ballarin@14751  1308 wenzelm@61382  1309 text_raw \\label{sec:subgroup-lattice}\  ballarin@14751  1310 ballarin@14751  1311 theorem (in group) subgroups_partial_order:  nipkow@67399  1312  "partial_order \carrier = {H. subgroup H G}, eq = (=), le = (\)\"  wenzelm@61169  1313  by standard simp_all  ballarin@14751  1314 ballarin@14751  1315 lemma (in group) subgroup_self:  ballarin@14751  1316  "subgroup (carrier G) G"  ballarin@14751  1317  by (rule subgroupI) auto  ballarin@14751  1318 ballarin@14751  1319 lemma (in group) subgroup_imp_group:  wenzelm@55926  1320  "subgroup H G ==> group (G\carrier := H\)"  wenzelm@26199  1321  by (erule subgroup.subgroup_is_group) (rule group_axioms)  ballarin@14751  1322 ballarin@14751  1323 lemma (in group) is_monoid [intro, simp]:  ballarin@14751  1324  "monoid G"  lp15@68445  1325  by (auto intro: monoid.intro m_assoc)  ballarin@14751  1326 lp15@68443  1327 lemma (in group) subgroup_mult_equality:  lp15@68443  1328  "\ subgroup H G; h1 \ H; h2 \ H \ \ h1 \\<^bsub>G \ carrier := H \\<^esub> h2 = h1 \ h2"  lp15@68443  1329  unfolding subgroup_def by simp  lp15@68443  1330 ballarin@14751  1331 theorem (in group) subgroups_Inter:  wenzelm@67091  1332  assumes subgr: "(\H. H \ A \ subgroup H G)"  wenzelm@67091  1333  and not_empty: "A \ {}"  ballarin@14751  1334  shows "subgroup (\A) G"  ballarin@14751  1335 proof (rule subgroupI)  ballarin@14751  1336  from subgr [THEN subgroup.subset] and not_empty  ballarin@14751  1337  show "\A \ carrier G" by blast  ballarin@14751  1338 next  ballarin@14751  1339  from subgr [THEN subgroup.one_closed]  wenzelm@67091  1340  show "\A \ {}" by blast  ballarin@14751  1341 next  ballarin@14751  1342  fix x assume "x \ \A"  ballarin@14751  1343  with subgr [THEN subgroup.m_inv_closed]  ballarin@14751  1344  show "inv x \ \A" by blast  ballarin@14751  1345 next  ballarin@14751  1346  fix x y assume "x \ \A" "y \ \A"  ballarin@14751  1347  with subgr [THEN subgroup.m_closed]  ballarin@14751  1348  show "x \ y \ \A" by blast  ballarin@14751  1349 qed  ballarin@14751  1350 lp15@68443  1351 lemma (in group) subgroups_Inter_pair :  lp15@68445  1352  assumes "subgroup I G"  lp15@68443  1353  and "subgroup J G"  lp15@68443  1354  shows "subgroup (I\J) G" using subgroups_Inter[ where ?A = "{I,J}"] assms by auto  lp15@68443  1355 ballarin@66579  1356 theorem (in group) subgroups_complete_lattice:  nipkow@67399  1357  "complete_lattice \carrier = {H. subgroup H G}, eq = (=), le = (\)\"  ballarin@66579  1358  (is "complete_lattice ?L")  ballarin@66579  1359 proof (rule partial_order.complete_lattice_criterion1)  ballarin@66579  1360  show "partial_order ?L" by (rule subgroups_partial_order)  ballarin@66579  1361 next  ballarin@66579  1362  have "greatest ?L (carrier G) (carrier ?L)"  ballarin@66579  1363  by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)  ballarin@66579  1364  then show "\G. greatest ?L G (carrier ?L)" ..  ballarin@66579  1365 next  ballarin@66579  1366  fix A  wenzelm@67091  1367  assume L: "A \ carrier ?L" and non_empty: "A \ {}"  ballarin@66579  1368  then have Int_subgroup: "subgroup (\A) G"  ballarin@66579  1369  by (fastforce intro: subgroups_Inter)  ballarin@66579  1370  have "greatest ?L (\A) (Lower ?L A)" (is "greatest _ ?Int _")  ballarin@66579  1371  proof (rule greatest_LowerI)  ballarin@66579  1372  fix H  ballarin@66579  1373  assume H: "H \ A"  ballarin@66579  1374  with L have subgroupH: "subgroup H G" by auto  ballarin@66579  1375  from subgroupH have groupH: "group (G \carrier := H\)" (is "group ?H")  ballarin@66579  1376  by (rule subgroup_imp_group)  ballarin@66579  1377  from groupH have monoidH: "monoid ?H"  ballarin@66579  1378  by (rule group.is_monoid)  ballarin@66579  1379  from H have Int_subset: "?Int \ H" by fastforce  ballarin@66579  1380  then show "le ?L ?Int H" by simp  ballarin@66579  1381  next  ballarin@66579  1382  fix H  ballarin@66579  1383  assume H: "H \ Lower ?L A"  ballarin@66579  1384  with L Int_subgroup show "le ?L H ?Int"  ballarin@66579  1385  by (fastforce simp: Lower_def intro: Inter_greatest)  ballarin@66579  1386  next  ballarin@66579  1387  show "A \ carrier ?L" by (rule L)  ballarin@66579  1388  next  ballarin@66579  1389  show "?Int \ carrier ?L" by simp (rule Int_subgroup)  ballarin@66579  1390  qed  ballarin@66579  1391  then show "\I. greatest ?L I (Lower ?L A)" ..  ballarin@66579  1392 qed  ballarin@66579  1393 lp15@68445  1394 subsection\Jeremy Avigad's @{text"More_Group"} material\  lp15@68445  1395 lp15@68445  1396 text \  lp15@68445  1397  Show that the units in any monoid give rise to a group.  lp15@68445  1398 lp15@68445  1399  The file Residues.thy provides some infrastructure to use  lp15@68445  1400  facts about the unit group within the ring locale.  lp15@68445  1401 \  lp15@68445  1402 lp15@68445  1403 definition units_of :: "('a, 'b) monoid_scheme \ 'a monoid"  lp15@68445  1404  where "units_of G =  lp15@68445  1405  \carrier = Units G, Group.monoid.mult = Group.monoid.mult G, one = one G\"  lp15@68445  1406 lp15@68445  1407 lemma (in monoid) units_group: "group (units_of G)"  lp15@68458  1408 proof -  lp15@68458  1409  have "\x y z. \x \ Units G; y \ Units G; z \ Units G\ \ x \ y \ z = x \ (y \ z)"  lp15@68458  1410  by (simp add: Units_closed m_assoc)  lp15@68458  1411  moreover have "\x. x \ Units G \ \y\Units G. y \ x = \"  lp15@68458  1412  using Units_l_inv by blast  lp15@68458  1413  ultimately show ?thesis  lp15@68458  1414  unfolding units_of_def  lp15@68458  1415  by (force intro!: groupI)  lp15@68458  1416 qed  lp15@68445  1417 lp15@68445  1418 lemma (in comm_monoid) units_comm_group: "comm_group (units_of G)"  lp15@68458  1419 proof -  lp15@68458  1420  have "\x y. \x \ carrier (units_of G); y \ carrier (units_of G)\  lp15@68458  1421  \ x \\<^bsub>units_of G\<^esub> y = y \\<^bsub>units_of G\<^esub> x"  lp15@68458  1422  by (simp add: Units_closed m_comm units_of_def)  lp15@68458  1423  then show ?thesis  lp15@68458  1424  by (rule group.group_comm_groupI [OF units_group]) auto  lp15@68458  1425 qed  lp15@68445  1426 lp15@68445  1427 lemma units_of_carrier: "carrier (units_of G) = Units G"  lp15@68445  1428  by (auto simp: units_of_def)  lp15@68445  1429 lp15@68445  1430 lemma units_of_mult: "mult (units_of G) = mult G"  lp15@68445  1431  by (auto simp: units_of_def)  lp15@68445  1432 lp15@68445  1433 lemma units_of_one: "one (units_of G) = one G"  lp15@68445  1434  by (auto simp: units_of_def)  lp15@68445  1435 lp15@68458  1436 lemma (in monoid) units_of_inv:  lp15@68458  1437  assumes "x \ Units G"  lp15@68458  1438  shows "m_inv (units_of G) x = m_inv G x"  lp15@68458  1439  by (simp add: assms group.inv_equality units_group units_of_carrier units_of_mult units_of_one)  lp15@68445  1440 lp15@68551  1441 lemma units_of_units [simp] : "Units (units_of G) = Units G"  lp15@68551  1442  unfolding units_of_def Units_def by force  lp15@68551  1443 lp15@68445  1444 lemma (in group) surj_const_mult: "a \ carrier G \ (\x. a \ x)  carrier G = carrier G"  lp15@68445  1445  apply (auto simp add: image_def)  lp15@68458  1446  by (metis inv_closed inv_solve_left m_closed)  lp15@68445  1447 lp15@68445  1448 lemma (in group) l_cancel_one [simp]: "x \ carrier G \ a \ carrier G \ x \ a = x \ a = one G"  lp15@68445  1449  by (metis Units_eq Units_l_cancel monoid.r_one monoid_axioms one_closed)  lp15@68445  1450 lp15@68445  1451 lemma (in group) r_cancel_one [simp]: "x \ carrier G \ a \ carrier G \ a \ x = x \ a = one G"  lp15@68445  1452  by (metis monoid.l_one monoid_axioms one_closed right_cancel)  lp15@68445  1453 lp15@68445  1454 lemma (in group) l_cancel_one' [simp]: "x \ carrier G \ a \ carrier G \ x = x \ a \ a = one G"  lp15@68445  1455  using l_cancel_one by fastforce  lp15@68445  1456 lp15@68445  1457 lemma (in group) r_cancel_one' [simp]: "x \ carrier G \ a \ carrier G \ x = a \ x \ a = one G"  lp15@68445  1458  using r_cancel_one by fastforce  lp15@68445  1459 ballarin@13813  1460 end