author  nipkow 
Fri, 13 Nov 2009 14:14:04 +0100  
changeset 33657  a4179bf442d1 
parent 33057  764547b68538 
child 35416  d8d7d1b785af 
permissions  rwrr 
13813  1 
(* 
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Title: HOL/Algebra/Group.thy 

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Author: Clemens Ballarin, started 4 February 2003 

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

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*) 

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28823  8 
theory Group 
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imports Lattice FuncSet 

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begin 

13813  11 

14761  12 

14963  13 
section {* Monoids and Groups *} 
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subsection {* Definitions *} 
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13813  17 
text {* 
14963  18 
Definitions follow \cite{Jacobson:1985}. 
13813  19 
*} 
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14963  21 
record 'a monoid = "'a partial_object" + 
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mult :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70) 

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

13817  24 

14651  25 
constdefs (structure G) 
14852  26 
m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _" [81] 80) 
14651  27 
"inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)" 
13936  28 

14651  29 
Units :: "_ => 'a set" 
14852  30 
{*The set of invertible elements*} 
14963  31 
"Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}" 
13936  32 

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consts 

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

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19699  36 
defs (overloaded) 
14693  37 
nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n" 
13936  38 
int_pow_def: "pow G a z == 
14693  39 
let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) 
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in if neg z then inv\<^bsub>G\<^esub> (p (nat (z))) else p (nat z)" 

13813  41 

19783  42 
locale monoid = 
43 
fixes G (structure) 

13813  44 
assumes m_closed [intro, simp]: 
14963  45 
"\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G" 
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and m_assoc: 

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

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

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

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

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

13817  52 

13936  53 
lemma monoidI: 
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fixes G (structure) 
13936  55 
assumes m_closed: 
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"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y \<in> carrier G" 
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and one_closed: "\<one> \<in> carrier G" 

13936  58 
and m_assoc: 
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"!!x y z. [ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

14693  60 
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" 
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and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" 

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

13936  63 
shows "monoid G" 
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by (fast intro!: monoid.intro intro: assms) 
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lemma (in monoid) Units_closed [dest]: 

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

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by (unfold Units_def) fast 

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70 
lemma (in monoid) inv_unique: 

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assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>" 
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and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G" 

13936  73 
shows "y = y'" 
74 
proof  

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

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

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

78 
finally show ?thesis . 

79 
qed 

80 

27698  81 
lemma (in monoid) Units_m_closed [intro, simp]: 
82 
assumes x: "x \<in> Units G" and y: "y \<in> Units G" 

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shows "x \<otimes> y \<in> Units G" 

84 
proof  

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from x obtain x' where x: "x \<in> carrier G" "x' \<in> carrier G" and xinv: "x \<otimes> x' = \<one>" "x' \<otimes> x = \<one>" 

86 
unfolding Units_def by fast 

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from y obtain y' where y: "y \<in> carrier G" "y' \<in> carrier G" and yinv: "y \<otimes> y' = \<one>" "y' \<otimes> y = \<one>" 

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unfolding Units_def by fast 

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from x y xinv yinv have "y' \<otimes> (x' \<otimes> x) \<otimes> y = \<one>" by simp 

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moreover from x y xinv yinv have "x \<otimes> (y \<otimes> y') \<otimes> x' = \<one>" by simp 

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moreover note x y 

92 
ultimately show ?thesis unfolding Units_def 

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 "Must avoid premature use of @{text hyp_subst_tac}." 

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apply (rule_tac CollectI) 

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apply (rule) 

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apply (fast) 

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apply (rule bexI [where x = "y' \<otimes> x'"]) 

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apply (auto simp: m_assoc) 

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done 

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qed 

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lemma (in monoid) Units_one_closed [intro, simp]: 
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"\<one> \<in> Units G" 

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by (unfold Units_def) auto 

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lemma (in monoid) Units_inv_closed [intro, simp]: 
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"x \<in> Units G ==> inv x \<in> carrier G" 

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apply (unfold Units_def m_inv_def, auto) 
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apply (rule theI2, fast) 
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apply (fast intro: inv_unique, fast) 
13936  111 
done 
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lemma (in monoid) Units_l_inv_ex: 
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"x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" 

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by (unfold Units_def) auto 

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lemma (in monoid) Units_r_inv_ex: 

118 
"x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" 

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by (unfold Units_def) auto 

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lemma (in monoid) Units_l_inv [simp]: 
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"x \<in> Units G ==> inv x \<otimes> x = \<one>" 
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apply (unfold Units_def m_inv_def, auto) 
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apply (rule theI2, fast) 
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apply (fast intro: inv_unique, fast) 
13936  126 
done 
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27698  128 
lemma (in monoid) Units_r_inv [simp]: 
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"x \<in> Units G ==> x \<otimes> inv x = \<one>" 
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apply (unfold Units_def m_inv_def, auto) 
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apply (rule theI2, fast) 
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apply (fast intro: inv_unique, fast) 
13936  133 
done 
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135 
lemma (in monoid) Units_inv_Units [intro, simp]: 

136 
"x \<in> Units G ==> inv x \<in> Units G" 

137 
proof  

138 
assume x: "x \<in> Units G" 

139 
show "inv x \<in> Units G" 

140 
by (auto simp add: Units_def 

141 
intro: Units_l_inv Units_r_inv x Units_closed [OF x]) 

142 
qed 

143 

144 
lemma (in monoid) Units_l_cancel [simp]: 

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

146 
(x \<otimes> y = x \<otimes> z) = (y = z)" 

147 
proof 

148 
assume eq: "x \<otimes> y = x \<otimes> z" 

14693  149 
and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" 
13936  150 
then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" 
27698  151 
by (simp add: m_assoc Units_closed del: Units_l_inv) 
13936  152 
with G show "y = z" by (simp add: Units_l_inv) 
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next 

154 
assume eq: "y = z" 

14693  155 
and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G" 
13936  156 
then show "x \<otimes> y = x \<otimes> z" by simp 
157 
qed 

158 

159 
lemma (in monoid) Units_inv_inv [simp]: 

160 
"x \<in> Units G ==> inv (inv x) = x" 

161 
proof  

162 
assume x: "x \<in> Units G" 

27698  163 
then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by simp 
164 
with x show ?thesis by (simp add: Units_closed del: Units_l_inv Units_r_inv) 

13936  165 
qed 
166 

167 
lemma (in monoid) inv_inj_on_Units: 

168 
"inj_on (m_inv G) (Units G)" 

169 
proof (rule inj_onI) 

170 
fix x y 

14693  171 
assume G: "x \<in> Units G" "y \<in> Units G" and eq: "inv x = inv y" 
13936  172 
then have "inv (inv x) = inv (inv y)" by simp 
173 
with G show "x = y" by simp 

174 
qed 

175 

13940  176 
lemma (in monoid) Units_inv_comm: 
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assumes inv: "x \<otimes> y = \<one>" 

14693  178 
and G: "x \<in> Units G" "y \<in> Units G" 
13940  179 
shows "y \<otimes> x = \<one>" 
180 
proof  

181 
from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed) 

182 
with G show ?thesis by (simp del: r_one add: m_assoc Units_closed) 

183 
qed 

184 

13936  185 
text {* Power *} 
186 

187 
lemma (in monoid) nat_pow_closed [intro, simp]: 

188 
"x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G" 

189 
by (induct n) (simp_all add: nat_pow_def) 

190 

191 
lemma (in monoid) nat_pow_0 [simp]: 

192 
"x (^) (0::nat) = \<one>" 

193 
by (simp add: nat_pow_def) 

194 

195 
lemma (in monoid) nat_pow_Suc [simp]: 

196 
"x (^) (Suc n) = x (^) n \<otimes> x" 

197 
by (simp add: nat_pow_def) 

198 

199 
lemma (in monoid) nat_pow_one [simp]: 

200 
"\<one> (^) (n::nat) = \<one>" 

201 
by (induct n) simp_all 

202 

203 
lemma (in monoid) nat_pow_mult: 

204 
"x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)" 

205 
by (induct m) (simp_all add: m_assoc [THEN sym]) 

206 

207 
lemma (in monoid) nat_pow_pow: 

208 
"x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)" 

209 
by (induct m) (simp, simp add: nat_pow_mult add_commute) 

210 

27698  211 

212 
(* Jacobson defines submonoid here. *) 

213 
(* Jacobson defines the order of a monoid here. *) 

214 

215 

216 
subsection {* Groups *} 

217 

13936  218 
text {* 
219 
A group is a monoid all of whose elements are invertible. 

220 
*} 

221 

222 
locale group = monoid + 

223 
assumes Units: "carrier G <= Units G" 

224 

26199  225 
lemma (in group) is_group: "group G" by (rule group_axioms) 
14761  226 

13936  227 
theorem groupI: 
19783  228 
fixes G (structure) 
13936  229 
assumes m_closed [simp]: 
14693  230 
"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y \<in> carrier G" 
231 
and one_closed [simp]: "\<one> \<in> carrier G" 

13936  232 
and m_assoc: 
233 
"!!x y z. [ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

14693  234 
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" 
235 
and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" 

14963  236 
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" 
13936  237 
shows "group G" 
238 
proof  

239 
have l_cancel [simp]: 

240 
"!!x y z. [ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

14693  241 
(x \<otimes> y = x \<otimes> z) = (y = z)" 
13936  242 
proof 
243 
fix x y z 

14693  244 
assume eq: "x \<otimes> y = x \<otimes> z" 
245 
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" 

13936  246 
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" 
14693  247 
and l_inv: "x_inv \<otimes> x = \<one>" by fast 
248 
from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z" 

13936  249 
by (simp add: m_assoc) 
250 
with G show "y = z" by (simp add: l_inv) 

251 
next 

252 
fix x y z 

253 
assume eq: "y = z" 

14693  254 
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" 
255 
then show "x \<otimes> y = x \<otimes> z" by simp 

13936  256 
qed 
257 
have r_one: 

14693  258 
"!!x. x \<in> carrier G ==> x \<otimes> \<one> = x" 
13936  259 
proof  
260 
fix x 

261 
assume x: "x \<in> carrier G" 

262 
with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G" 

14693  263 
and l_inv: "x_inv \<otimes> x = \<one>" by fast 
264 
from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x" 

13936  265 
by (simp add: m_assoc [symmetric] l_inv) 
14693  266 
with x xG show "x \<otimes> \<one> = x" by simp 
13936  267 
qed 
268 
have inv_ex: 

14963  269 
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" 
13936  270 
proof  
271 
fix x 

272 
assume x: "x \<in> carrier G" 

273 
with l_inv_ex obtain y where y: "y \<in> carrier G" 

14693  274 
and l_inv: "y \<otimes> x = \<one>" by fast 
275 
from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>" 

13936  276 
by (simp add: m_assoc [symmetric] l_inv r_one) 
14693  277 
with x y have r_inv: "x \<otimes> y = \<one>" 
13936  278 
by simp 
14963  279 
from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>" 
13936  280 
by (fast intro: l_inv r_inv) 
281 
qed 

282 
then have carrier_subset_Units: "carrier G <= Units G" 

283 
by (unfold Units_def) fast 

28823  284 
show ?thesis proof qed (auto simp: r_one m_assoc carrier_subset_Units) 
13936  285 
qed 
286 

27698  287 
lemma (in monoid) group_l_invI: 
13936  288 
assumes l_inv_ex: 
14963  289 
"!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" 
13936  290 
shows "group G" 
291 
by (rule groupI) (auto intro: m_assoc l_inv_ex) 

292 

293 
lemma (in group) Units_eq [simp]: 

294 
"Units G = carrier G" 

295 
proof 

296 
show "Units G <= carrier G" by fast 

297 
next 

298 
show "carrier G <= Units G" by (rule Units) 

299 
qed 

300 

301 
lemma (in group) inv_closed [intro, simp]: 

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

303 
using Units_inv_closed by simp 

304 

19981  305 
lemma (in group) l_inv_ex [simp]: 
306 
"x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" 

307 
using Units_l_inv_ex by simp 

308 

309 
lemma (in group) r_inv_ex [simp]: 

310 
"x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>" 

311 
using Units_r_inv_ex by simp 

312 

14963  313 
lemma (in group) l_inv [simp]: 
13936  314 
"x \<in> carrier G ==> inv x \<otimes> x = \<one>" 
315 
using Units_l_inv by simp 

13813  316 

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13813  318 
subsection {* Cancellation Laws and Basic Properties *} 
319 

320 
lemma (in group) l_cancel [simp]: 

321 
"[ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

322 
(x \<otimes> y = x \<otimes> z) = (y = z)" 

13936  323 
using Units_l_inv by simp 
13940  324 

14963  325 
lemma (in group) r_inv [simp]: 
13813  326 
"x \<in> carrier G ==> x \<otimes> inv x = \<one>" 
327 
proof  

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

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

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

331 
with x show ?thesis by (simp del: r_one) 

332 
qed 

333 

334 
lemma (in group) r_cancel [simp]: 

335 
"[ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

336 
(y \<otimes> x = z \<otimes> x) = (y = z)" 

337 
proof 

338 
assume eq: "y \<otimes> x = z \<otimes> x" 

14693  339 
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" 
13813  340 
then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)" 
27698  341 
by (simp add: m_assoc [symmetric] del: r_inv Units_r_inv) 
14963  342 
with G show "y = z" by simp 
13813  343 
next 
344 
assume eq: "y = z" 

14693  345 
and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" 
13813  346 
then show "y \<otimes> x = z \<otimes> x" by simp 
347 
qed 

348 

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lemma (in group) inv_one [simp]: 
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"inv \<one> = \<one>" 
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proof  
27698  352 
have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv Units_r_inv) 
14963  353 
moreover have "... = \<one>" by simp 
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finally show ?thesis . 
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qed 
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13813  357 
lemma (in group) inv_inv [simp]: 
358 
"x \<in> carrier G ==> inv (inv x) = x" 

13936  359 
using Units_inv_inv by simp 
360 

361 
lemma (in group) inv_inj: 

362 
"inj_on (m_inv G) (carrier G)" 

363 
using inv_inj_on_Units by simp 

13813  364 

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lemma (in group) inv_mult_group: 
13813  366 
"[ x \<in> carrier G; y \<in> carrier G ] ==> inv (x \<otimes> y) = inv y \<otimes> inv x" 
367 
proof  

14693  368 
assume G: "x \<in> carrier G" "y \<in> carrier G" 
13813  369 
then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)" 
14963  370 
by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric]) 
27698  371 
with G show ?thesis by (simp del: l_inv Units_l_inv) 
13813  372 
qed 
373 

13940  374 
lemma (in group) inv_comm: 
375 
"[ x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G ] ==> y \<otimes> x = \<one>" 

14693  376 
by (rule Units_inv_comm) auto 
13940  377 

13944  378 
lemma (in group) inv_equality: 
13943  379 
"[y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G] ==> inv x = y" 
380 
apply (simp add: m_inv_def) 

381 
apply (rule the_equality) 

14693  382 
apply (simp add: inv_comm [of y x]) 
383 
apply (rule r_cancel [THEN iffD1], auto) 

13943  384 
done 
385 

13936  386 
text {* Power *} 
387 

388 
lemma (in group) int_pow_def2: 

389 
"a (^) (z::int) = (if neg z then inv (a (^) (nat (z))) else a (^) (nat z))" 

390 
by (simp add: int_pow_def nat_pow_def Let_def) 

391 

392 
lemma (in group) int_pow_0 [simp]: 

393 
"x (^) (0::int) = \<one>" 

394 
by (simp add: int_pow_def2) 

395 

396 
lemma (in group) int_pow_one [simp]: 

397 
"\<one> (^) (z::int) = \<one>" 

398 
by (simp add: int_pow_def2) 

399 

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14963  401 
subsection {* Subgroups *} 
13813  402 

19783  403 
locale subgroup = 
404 
fixes H and G (structure) 

14963  405 
assumes subset: "H \<subseteq> carrier G" 
406 
and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H" 

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and one_closed [simp]: "\<one> \<in> H" 
14963  408 
and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H" 
13813  409 

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lemma (in subgroup) is_subgroup: 
26199  411 
"subgroup H G" by (rule subgroup_axioms) 
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412 

13813  413 
declare (in subgroup) group.intro [intro] 
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414 

14963  415 
lemma (in subgroup) mem_carrier [simp]: 
416 
"x \<in> H \<Longrightarrow> x \<in> carrier G" 

417 
using subset by blast 

13813  418 

14963  419 
lemma subgroup_imp_subset: 
420 
"subgroup H G \<Longrightarrow> H \<subseteq> carrier G" 

421 
by (rule subgroup.subset) 

422 

423 
lemma (in subgroup) subgroup_is_group [intro]: 

27611  424 
assumes "group G" 
425 
shows "group (G\<lparr>carrier := H\<rparr>)" 

426 
proof  

29237  427 
interpret group G by fact 
27611  428 
show ?thesis 
27698  429 
apply (rule monoid.group_l_invI) 
430 
apply (unfold_locales) [1] 

431 
apply (auto intro: m_assoc l_inv mem_carrier) 

432 
done 

27611  433 
qed 
13813  434 

435 
text {* 

436 
Since @{term H} is nonempty, it contains some element @{term x}. Since 

437 
it is closed under inverse, it contains @{text "inv x"}. Since 

438 
it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}. 

439 
*} 

440 

441 
lemma (in group) one_in_subset: 

442 
"[ 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 ] 

443 
==> \<one> \<in> H" 

444 
by (force simp add: l_inv) 

445 

446 
text {* A characterization of subgroups: closed, nonempty subset. *} 

447 

448 
lemma (in group) subgroupI: 

449 
assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}" 

14963  450 
and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H" 
451 
and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H" 

13813  452 
shows "subgroup H G" 
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453 
proof (simp add: subgroup_def assms) 
27b4d7c01f8b
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454 
show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: assms) 
13813  455 
qed 
456 

13936  457 
declare monoid.one_closed [iff] group.inv_closed [simp] 
458 
monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp] 

13813  459 

460 
lemma subgroup_nonempty: 

461 
"~ subgroup {} G" 

462 
by (blast dest: subgroup.one_closed) 

463 

464 
lemma (in subgroup) finite_imp_card_positive: 

465 
"finite (carrier G) ==> 0 < card H" 

466 
proof (rule classical) 

14963  467 
assume "finite (carrier G)" "~ 0 < card H" 
468 
then have "finite H" by (blast intro: finite_subset [OF subset]) 

469 
with prems have "subgroup {} G" by simp 

13813  470 
with subgroup_nonempty show ?thesis by contradiction 
471 
qed 

472 

13936  473 
(* 
474 
lemma (in monoid) Units_subgroup: 

475 
"subgroup (Units G) G" 

476 
*) 

477 

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478 

13813  479 
subsection {* Direct Products *} 
480 

14963  481 
constdefs 
482 
DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid" (infixr "\<times>\<times>" 80) 

483 
"G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H, 

484 
mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')), 

485 
one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>" 

13813  486 

14963  487 
lemma DirProd_monoid: 
27611  488 
assumes "monoid G" and "monoid H" 
14963  489 
shows "monoid (G \<times>\<times> H)" 
490 
proof  

30729
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491 
interpret G: monoid G by fact 
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492 
interpret H: monoid H by fact 
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493 
from assms 
14963  494 
show ?thesis by (unfold monoid_def DirProd_def, auto) 
495 
qed 

13813  496 

497 

14963  498 
text{*Does not use the previous result because it's easier just to use auto.*} 
499 
lemma DirProd_group: 

27611  500 
assumes "group G" and "group H" 
14963  501 
shows "group (G \<times>\<times> H)" 
27611  502 
proof  
30729
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503 
interpret G: group G by fact 
461ee3e49ad3
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504 
interpret H: group H by fact 
27611  505 
show ?thesis by (rule groupI) 
14963  506 
(auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv 
507 
simp add: DirProd_def) 

27611  508 
qed 
13813  509 

14963  510 
lemma carrier_DirProd [simp]: 
511 
"carrier (G \<times>\<times> H) = carrier G \<times> carrier H" 

512 
by (simp add: DirProd_def) 

13944  513 

14963  514 
lemma one_DirProd [simp]: 
515 
"\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)" 

516 
by (simp add: DirProd_def) 

13944  517 

14963  518 
lemma mult_DirProd [simp]: 
519 
"(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')" 

520 
by (simp add: DirProd_def) 

13944  521 

14963  522 
lemma inv_DirProd [simp]: 
27611  523 
assumes "group G" and "group H" 
13944  524 
assumes g: "g \<in> carrier G" 
525 
and h: "h \<in> carrier H" 

14963  526 
shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)" 
27611  527 
proof  
30729
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528 
interpret G: group G by fact 
461ee3e49ad3
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diff
changeset

529 
interpret H: group H by fact 
461ee3e49ad3
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diff
changeset

530 
interpret Prod: group "G \<times>\<times> H" 
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531 
by (auto intro: DirProd_group group.intro group.axioms assms) 
14963  532 
show ?thesis by (simp add: Prod.inv_equality g h) 
533 
qed 

27698  534 

14963  535 

536 
subsection {* Homomorphisms and Isomorphisms *} 

13813  537 

14651  538 
constdefs (structure G and H) 
539 
hom :: "_ => _ => ('a => 'b) set" 

13813  540 
"hom G H == 
541 
{h. h \<in> carrier G > carrier H & 

14693  542 
(\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}" 
13813  543 

14761  544 
lemma (in group) hom_compose: 
31754  545 
"[h \<in> hom G H; i \<in> hom H I] ==> compose (carrier G) i h \<in> hom G I" 
546 
by (fastsimp simp add: hom_def compose_def) 

13943  547 

14803  548 
constdefs 
549 
iso :: "_ => _ => ('a => 'b) set" (infixr "\<cong>" 60) 

550 
"G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}" 

14761  551 

14803  552 
lemma iso_refl: "(%x. x) \<in> G \<cong> G" 
31727  553 
by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def) 
14761  554 

555 
lemma (in group) iso_sym: 

33057  556 
"h \<in> G \<cong> H \<Longrightarrow> inv_into (carrier G) h \<in> H \<cong> G" 
557 
apply (simp add: iso_def bij_betw_inv_into) 

558 
apply (subgoal_tac "inv_into (carrier G) h \<in> carrier H \<rightarrow> carrier G") 

559 
prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_inv_into]) 

560 
apply (simp add: hom_def bij_betw_def inv_into_f_eq f_inv_into_f Pi_def) 

14761  561 
done 
562 

563 
lemma (in group) iso_trans: 

14803  564 
"[h \<in> G \<cong> H; i \<in> H \<cong> I] ==> (compose (carrier G) i h) \<in> G \<cong> I" 
14761  565 
by (auto simp add: iso_def hom_compose bij_betw_compose) 
566 

14963  567 
lemma DirProd_commute_iso: 
568 
shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)" 

31754  569 
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) 
14761  570 

14963  571 
lemma DirProd_assoc_iso: 
572 
shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))" 

31727  573 
by (auto simp add: iso_def hom_def inj_on_def bij_betw_def) 
14761  574 

575 

14963  576 
text{*Basis for homomorphism proofs: we assume two groups @{term G} and 
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577 
@{term H}, with a homomorphism @{term h} between them*} 
29237  578 
locale group_hom = G: group G + H: group H for G (structure) and H (structure) + 
579 
fixes h 

13813  580 
assumes homh: "h \<in> hom G H" 
29240  581 

582 
lemma (in group_hom) hom_mult [simp]: 

583 
"[ x \<in> carrier G; y \<in> carrier G ] ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" 

584 
proof  

585 
assume "x \<in> carrier G" "y \<in> carrier G" 

586 
with homh [unfolded hom_def] show ?thesis by simp 

587 
qed 

588 

589 
lemma (in group_hom) hom_closed [simp]: 

590 
"x \<in> carrier G ==> h x \<in> carrier H" 

591 
proof  

592 
assume "x \<in> carrier G" 

31754  593 
with homh [unfolded hom_def] show ?thesis by auto 
29240  594 
qed 
13813  595 

596 
lemma (in group_hom) one_closed [simp]: 

597 
"h \<one> \<in> carrier H" 

598 
by simp 

599 

600 
lemma (in group_hom) hom_one [simp]: 

14693  601 
"h \<one> = \<one>\<^bsub>H\<^esub>" 
13813  602 
proof  
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New prover for transitive and reflexivetransitive closure of relations.
ballarin
parents:
14963
diff
changeset

603 
have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>" 
13813  604 
by (simp add: hom_mult [symmetric] del: hom_mult) 
605 
then show ?thesis by (simp del: r_one) 

606 
qed 

607 

608 
lemma (in group_hom) inv_closed [simp]: 

609 
"x \<in> carrier G ==> h (inv x) \<in> carrier H" 

610 
by simp 

611 

612 
lemma (in group_hom) hom_inv [simp]: 

14693  613 
"x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)" 
13813  614 
proof  
615 
assume x: "x \<in> carrier G" 

14693  616 
then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>" 
14963  617 
by (simp add: hom_mult [symmetric] del: hom_mult) 
14693  618 
also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" 
14963  619 
by (simp add: hom_mult [symmetric] del: hom_mult) 
14693  620 
finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" . 
27698  621 
with x show ?thesis by (simp del: H.r_inv H.Units_r_inv) 
13813  622 
qed 
623 

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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset

624 

13949
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ballarin
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13944
diff
changeset

625 
subsection {* Commutative Structures *} 
13936  626 

627 
text {* 

628 
Naming convention: multiplicative structures that are commutative 

629 
are called \emph{commutative}, additive structures are called 

630 
\emph{Abelian}. 

631 
*} 

13813  632 

14963  633 
locale comm_monoid = monoid + 
634 
assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x" 

13813  635 

14963  636 
lemma (in comm_monoid) m_lcomm: 
637 
"\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> 

13813  638 
x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)" 
639 
proof  

14693  640 
assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G" 
13813  641 
from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc) 
642 
also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm) 

643 
also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc) 

644 
finally show ?thesis . 

645 
qed 

646 

14963  647 
lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm 
13813  648 

13936  649 
lemma comm_monoidI: 
19783  650 
fixes G (structure) 
13936  651 
assumes m_closed: 
14693  652 
"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y \<in> carrier G" 
653 
and one_closed: "\<one> \<in> carrier G" 

13936  654 
and m_assoc: 
655 
"!!x y z. [ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

14693  656 
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" 
657 
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" 

13936  658 
and m_comm: 
14693  659 
"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y = y \<otimes> x" 
13936  660 
shows "comm_monoid G" 
661 
using l_one 

14963  662 
by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro 
27714
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changeset

663 
intro: assms simp: m_closed one_closed m_comm) 
13817  664 

13936  665 
lemma (in monoid) monoid_comm_monoidI: 
666 
assumes m_comm: 

14693  667 
"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y = y \<otimes> x" 
13936  668 
shows "comm_monoid G" 
669 
by (rule comm_monoidI) (auto intro: m_assoc m_comm) 

14963  670 

14693  671 
(*lemma (in comm_monoid) r_one [simp]: 
13817  672 
"x \<in> carrier G ==> x \<otimes> \<one> = x" 
673 
proof  

674 
assume G: "x \<in> carrier G" 

675 
then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm) 

676 
also from G have "... = x" by simp 

677 
finally show ?thesis . 

14693  678 
qed*) 
14963  679 

13936  680 
lemma (in comm_monoid) nat_pow_distr: 
681 
"[ x \<in> carrier G; y \<in> carrier G ] ==> 

682 
(x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n" 

683 
by (induct n) (simp, simp add: m_ac) 

684 

685 
locale comm_group = comm_monoid + group 

686 

687 
lemma (in group) group_comm_groupI: 

688 
assumes m_comm: "!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> 

14693  689 
x \<otimes> y = y \<otimes> x" 
13936  690 
shows "comm_group G" 
28823  691 
proof qed (simp_all add: m_comm) 
13817  692 

13936  693 
lemma comm_groupI: 
19783  694 
fixes G (structure) 
13936  695 
assumes m_closed: 
14693  696 
"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y \<in> carrier G" 
697 
and one_closed: "\<one> \<in> carrier G" 

13936  698 
and m_assoc: 
699 
"!!x y z. [ x \<in> carrier G; y \<in> carrier G; z \<in> carrier G ] ==> 

14693  700 
(x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)" 
13936  701 
and m_comm: 
14693  702 
"!!x y. [ x \<in> carrier G; y \<in> carrier G ] ==> x \<otimes> y = y \<otimes> x" 
703 
and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x" 

14963  704 
and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>" 
13936  705 
shows "comm_group G" 
27714
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ballarin
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27713
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changeset

706 
by (fast intro: group.group_comm_groupI groupI assms) 
13936  707 

708 
lemma (in comm_group) inv_mult: 

13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset

709 
"[ x \<in> carrier G; y \<in> carrier G ] ==> inv (x \<otimes> y) = inv x \<otimes> inv y" 
13936  710 
by (simp add: m_ac inv_mult_group) 
13854
91c9ab25fece
First distributed version of Group and Ring theory.
ballarin
parents:
13835
diff
changeset

711 

20318
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Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset

712 

0e0ea63fe768
Restructured algebra library, added ideals and quotient rings.
ballarin
parents:
19984
diff
changeset

713 
subsection {* The Lattice of Subgroups of a Group *} 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

714 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

715 
text_raw {* \label{sec:subgrouplattice} *} 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

716 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

717 
theorem (in group) subgroups_partial_order: 
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27698
diff
changeset

718 
"partial_order ( carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> )" 
28823  719 
proof qed simp_all 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

720 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

721 
lemma (in group) subgroup_self: 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

722 
"subgroup (carrier G) G" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

723 
by (rule subgroupI) auto 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

724 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

725 
lemma (in group) subgroup_imp_group: 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

726 
"subgroup H G ==> group (G( carrier := H ))" 
26199  727 
by (erule subgroup.subgroup_is_group) (rule group_axioms) 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

728 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

729 
lemma (in group) is_monoid [intro, simp]: 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

730 
"monoid G" 
14963  731 
by (auto intro: monoid.intro m_assoc) 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

732 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

733 
lemma (in group) subgroup_inv_equality: 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

734 
"[ subgroup H G; x \<in> H ] ==> m_inv (G ( carrier := H )) x = inv x" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

735 
apply (rule_tac inv_equality [THEN sym]) 
14761  736 
apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+) 
737 
apply (rule subsetD [OF subgroup.subset], assumption+) 

738 
apply (rule subsetD [OF subgroup.subset], assumption) 

739 
apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+) 

14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

740 
done 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

741 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

742 
theorem (in group) subgroups_Inter: 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

743 
assumes subgr: "(!!H. H \<in> A ==> subgroup H G)" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

744 
and not_empty: "A ~= {}" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

745 
shows "subgroup (\<Inter>A) G" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

746 
proof (rule subgroupI) 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

747 
from subgr [THEN subgroup.subset] and not_empty 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

748 
show "\<Inter>A \<subseteq> carrier G" by blast 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

749 
next 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

750 
from subgr [THEN subgroup.one_closed] 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

751 
show "\<Inter>A ~= {}" by blast 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

752 
next 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

753 
fix x assume "x \<in> \<Inter>A" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

754 
with subgr [THEN subgroup.m_inv_closed] 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

755 
show "inv x \<in> \<Inter>A" by blast 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

756 
next 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

757 
fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

758 
with subgr [THEN subgroup.m_closed] 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

759 
show "x \<otimes> y \<in> \<Inter>A" by blast 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

760 
qed 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

761 

0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

762 
theorem (in group) subgroups_complete_lattice: 
27713
95b36bfe7fc4
New locales for orders and lattices where the equivalence relation is not restricted to equality.
ballarin
parents:
27698
diff
changeset

763 
"complete_lattice ( carrier = {H. subgroup H G}, eq = op =, le = op \<subseteq> )" 
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset

764 
(is "complete_lattice ?L") 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

765 
proof (rule partial_order.complete_lattice_criterion1) 
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset

766 
show "partial_order ?L" by (rule subgroups_partial_order) 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

767 
next 
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

768 
show "\<exists>G. greatest ?L G (carrier ?L)" 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

769 
proof 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

770 
show "greatest ?L (carrier G) (carrier ?L)" 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

771 
by (unfold greatest_def) 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

772 
(simp add: subgroup.subset subgroup_self) 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

773 
qed 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

774 
next 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

775 
fix A 
22063
717425609192
Reverted to structure representation with records.
ballarin
parents:
21041
diff
changeset

776 
assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}" 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

777 
then have Int_subgroup: "subgroup (\<Inter>A) G" 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

778 
by (fastsimp intro: subgroups_Inter) 
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

779 
show "\<exists>I. greatest ?L I (Lower ?L A)" 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

780 
proof 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

781 
show "greatest ?L (\<Inter>A) (Lower ?L A)" 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

782 
(is "greatest _ ?Int _") 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

783 
proof (rule greatest_LowerI) 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

784 
fix H 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

785 
assume H: "H \<in> A" 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

786 
with L have subgroupH: "subgroup H G" by auto 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

787 
from subgroupH have groupH: "group (G ( carrier := H ))" (is "group ?H") 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31754
diff
changeset

788 
by (rule subgroup_imp_group) 
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

789 
from groupH have monoidH: "monoid ?H" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31754
diff
changeset

790 
by (rule group.is_monoid) 
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

791 
from H have Int_subset: "?Int \<subseteq> H" by fastsimp 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

792 
then show "le ?L ?Int H" by simp 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

793 
next 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

794 
fix H 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

795 
assume H: "H \<in> Lower ?L A" 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

796 
with L Int_subgroup show "le ?L H ?Int" 
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tabwidth;
wenzelm
parents:
31754
diff
changeset

797 
by (fastsimp simp: Lower_def intro: Inter_greatest) 
26805
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

798 
next 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

799 
show "A \<subseteq> carrier ?L" by (rule L) 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

800 
next 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

801 
show "?Int \<in> carrier ?L" by simp (rule Int_subgroup) 
27941d7d9a11
Replaced forward proofs of existential statements by backward proofs
berghofe
parents:
26199
diff
changeset

802 
qed 
14751
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

803 
qed 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

804 
qed 
0d7850e27fed
Change of theory hierarchy: Group is now based in Lattice.
ballarin
parents:
14706
diff
changeset

805 

13813  806 
end 