src/HOL/Algebra/Group.thy
 author ballarin Thu Aug 03 14:57:26 2006 +0200 (2006-08-03) changeset 20318 0e0ea63fe768 parent 19984 29bb4659f80a child 21041 60e418260b4d permissions -rw-r--r--
Restructured algebra library, added ideals and quotient rings.
1 (*
2   Title:  HOL/Algebra/Group.thy
3   Id:     $Id$
4   Author: Clemens Ballarin, started 4 February 2003
6 Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
7 *)
9 theory Group imports FuncSet Lattice begin
12 section {* Monoids and Groups *}
14 subsection {* Definitions *}
16 text {*
17   Definitions follow \cite{Jacobson:1985}.
18 *}
20 record 'a monoid =  "'a partial_object" +
21   mult    :: "['a, 'a] \<Rightarrow> 'a" (infixl "\<otimes>\<index>" 70)
22   one     :: 'a ("\<one>\<index>")
24 constdefs (structure G)
25   m_inv :: "('a, 'b) monoid_scheme => 'a => 'a" ("inv\<index> _"  80)
26   "inv x == (THE y. y \<in> carrier G & x \<otimes> y = \<one> & y \<otimes> x = \<one>)"
28   Units :: "_ => 'a set"
29   --{*The set of invertible elements*}
30   "Units G == {y. y \<in> carrier G & (\<exists>x \<in> carrier G. x \<otimes> y = \<one> & y \<otimes> x = \<one>)}"
32 consts
33   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)
36   nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
37   int_pow_def: "pow G a z ==
38     let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
39     in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"
41 locale monoid =
42   fixes G (structure)
43   assumes m_closed [intro, simp]:
44          "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> carrier G"
45       and m_assoc:
46          "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk>
47           \<Longrightarrow> (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
48       and one_closed [intro, simp]: "\<one> \<in> carrier G"
49       and l_one [simp]: "x \<in> carrier G \<Longrightarrow> \<one> \<otimes> x = x"
50       and r_one [simp]: "x \<in> carrier G \<Longrightarrow> x \<otimes> \<one> = x"
52 lemma monoidI:
53   fixes G (structure)
54   assumes m_closed:
55       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
56     and one_closed: "\<one> \<in> carrier G"
57     and m_assoc:
58       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
59       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
60     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
61     and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
62   shows "monoid G"
63   by (fast intro!: monoid.intro intro: prems)
65 lemma (in monoid) Units_closed [dest]:
66   "x \<in> Units G ==> x \<in> carrier G"
67   by (unfold Units_def) fast
69 lemma (in monoid) inv_unique:
70   assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"
71     and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
72   shows "y = y'"
73 proof -
74   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp
75   also from G have "... = (y \<otimes> x) \<otimes> y'" by (simp add: m_assoc)
76   also from G eq have "... = y'" by simp
77   finally show ?thesis .
78 qed
80 lemma (in monoid) Units_one_closed [intro, simp]:
81   "\<one> \<in> Units G"
82   by (unfold Units_def) auto
84 lemma (in monoid) Units_inv_closed [intro, simp]:
85   "x \<in> Units G ==> inv x \<in> carrier G"
86   apply (unfold Units_def m_inv_def, auto)
87   apply (rule theI2, fast)
88    apply (fast intro: inv_unique, fast)
89   done
91 lemma (in monoid) Units_l_inv_ex:
92   "x \<in> Units G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
93   by (unfold Units_def) auto
95 lemma (in monoid) Units_r_inv_ex:
96   "x \<in> Units G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
97   by (unfold Units_def) auto
99 lemma (in monoid) Units_l_inv:
100   "x \<in> Units G ==> inv x \<otimes> x = \<one>"
101   apply (unfold Units_def m_inv_def, auto)
102   apply (rule theI2, fast)
103    apply (fast intro: inv_unique, fast)
104   done
106 lemma (in monoid) Units_r_inv:
107   "x \<in> Units G ==> x \<otimes> inv x = \<one>"
108   apply (unfold Units_def m_inv_def, auto)
109   apply (rule theI2, fast)
110    apply (fast intro: inv_unique, fast)
111   done
113 lemma (in monoid) Units_inv_Units [intro, simp]:
114   "x \<in> Units G ==> inv x \<in> Units G"
115 proof -
116   assume x: "x \<in> Units G"
117   show "inv x \<in> Units G"
118     by (auto simp add: Units_def
119       intro: Units_l_inv Units_r_inv x Units_closed [OF x])
120 qed
122 lemma (in monoid) Units_l_cancel [simp]:
123   "[| x \<in> Units G; y \<in> carrier G; z \<in> carrier G |] ==>
124    (x \<otimes> y = x \<otimes> z) = (y = z)"
125 proof
126   assume eq: "x \<otimes> y = x \<otimes> z"
127     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
128   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"
129     by (simp add: m_assoc Units_closed)
130   with G show "y = z" by (simp add: Units_l_inv)
131 next
132   assume eq: "y = z"
133     and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
134   then show "x \<otimes> y = x \<otimes> z" by simp
135 qed
137 lemma (in monoid) Units_inv_inv [simp]:
138   "x \<in> Units G ==> inv (inv x) = x"
139 proof -
140   assume x: "x \<in> Units G"
141   then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x"
142     by (simp add: Units_l_inv Units_r_inv)
143   with x show ?thesis by (simp add: Units_closed)
144 qed
146 lemma (in monoid) inv_inj_on_Units:
147   "inj_on (m_inv G) (Units G)"
148 proof (rule inj_onI)
149   fix x y
150   assume G: "x \<in> Units G"  "y \<in> Units G" and eq: "inv x = inv y"
151   then have "inv (inv x) = inv (inv y)" by simp
152   with G show "x = y" by simp
153 qed
155 lemma (in monoid) Units_inv_comm:
156   assumes inv: "x \<otimes> y = \<one>"
157     and G: "x \<in> Units G"  "y \<in> Units G"
158   shows "y \<otimes> x = \<one>"
159 proof -
160   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
161   with G show ?thesis by (simp del: r_one add: m_assoc Units_closed)
162 qed
164 text {* Power *}
166 lemma (in monoid) nat_pow_closed [intro, simp]:
167   "x \<in> carrier G ==> x (^) (n::nat) \<in> carrier G"
168   by (induct n) (simp_all add: nat_pow_def)
170 lemma (in monoid) nat_pow_0 [simp]:
171   "x (^) (0::nat) = \<one>"
172   by (simp add: nat_pow_def)
174 lemma (in monoid) nat_pow_Suc [simp]:
175   "x (^) (Suc n) = x (^) n \<otimes> x"
176   by (simp add: nat_pow_def)
178 lemma (in monoid) nat_pow_one [simp]:
179   "\<one> (^) (n::nat) = \<one>"
180   by (induct n) simp_all
182 lemma (in monoid) nat_pow_mult:
183   "x \<in> carrier G ==> x (^) (n::nat) \<otimes> x (^) m = x (^) (n + m)"
184   by (induct m) (simp_all add: m_assoc [THEN sym])
186 lemma (in monoid) nat_pow_pow:
187   "x \<in> carrier G ==> (x (^) n) (^) m = x (^) (n * m::nat)"
188   by (induct m) (simp, simp add: nat_pow_mult add_commute)
190 text {*
191   A group is a monoid all of whose elements are invertible.
192 *}
194 locale group = monoid +
195   assumes Units: "carrier G <= Units G"
198 lemma (in group) is_group: "group G" .
200 theorem groupI:
201   fixes G (structure)
202   assumes m_closed [simp]:
203       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
204     and one_closed [simp]: "\<one> \<in> carrier G"
205     and m_assoc:
206       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
207       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
208     and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
209     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
210   shows "group G"
211 proof -
212   have l_cancel [simp]:
213     "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
214     (x \<otimes> y = x \<otimes> z) = (y = z)"
215   proof
216     fix x y z
217     assume eq: "x \<otimes> y = x \<otimes> z"
218       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
219     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
220       and l_inv: "x_inv \<otimes> x = \<one>" by fast
221     from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"
222       by (simp add: m_assoc)
223     with G show "y = z" by (simp add: l_inv)
224   next
225     fix x y z
226     assume eq: "y = z"
227       and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
228     then show "x \<otimes> y = x \<otimes> z" by simp
229   qed
230   have r_one:
231     "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
232   proof -
233     fix x
234     assume x: "x \<in> carrier G"
235     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
236       and l_inv: "x_inv \<otimes> x = \<one>" by fast
237     from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"
238       by (simp add: m_assoc [symmetric] l_inv)
239     with x xG show "x \<otimes> \<one> = x" by simp
240   qed
241   have inv_ex:
242     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
243   proof -
244     fix x
245     assume x: "x \<in> carrier G"
246     with l_inv_ex obtain y where y: "y \<in> carrier G"
247       and l_inv: "y \<otimes> x = \<one>" by fast
248     from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"
249       by (simp add: m_assoc [symmetric] l_inv r_one)
250     with x y have r_inv: "x \<otimes> y = \<one>"
251       by simp
252     from x y show "\<exists>y \<in> carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
253       by (fast intro: l_inv r_inv)
254   qed
255   then have carrier_subset_Units: "carrier G <= Units G"
256     by (unfold Units_def) fast
257   show ?thesis
258     by (fast intro!: group.intro monoid.intro group_axioms.intro
259       carrier_subset_Units intro: prems r_one)
260 qed
262 lemma (in monoid) monoid_groupI:
263   assumes l_inv_ex:
264     "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
265   shows "group G"
266   by (rule groupI) (auto intro: m_assoc l_inv_ex)
268 lemma (in group) Units_eq [simp]:
269   "Units G = carrier G"
270 proof
271   show "Units G <= carrier G" by fast
272 next
273   show "carrier G <= Units G" by (rule Units)
274 qed
276 lemma (in group) inv_closed [intro, simp]:
277   "x \<in> carrier G ==> inv x \<in> carrier G"
278   using Units_inv_closed by simp
280 lemma (in group) l_inv_ex [simp]:
281   "x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
282   using Units_l_inv_ex by simp
284 lemma (in group) r_inv_ex [simp]:
285   "x \<in> carrier G ==> \<exists>y \<in> carrier G. x \<otimes> y = \<one>"
286   using Units_r_inv_ex by simp
288 lemma (in group) l_inv [simp]:
289   "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
290   using Units_l_inv by simp
293 subsection {* Cancellation Laws and Basic Properties *}
295 lemma (in group) l_cancel [simp]:
296   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
297    (x \<otimes> y = x \<otimes> z) = (y = z)"
298   using Units_l_inv by simp
300 lemma (in group) r_inv [simp]:
301   "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
302 proof -
303   assume x: "x \<in> carrier G"
304   then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
305     by (simp add: m_assoc [symmetric] l_inv)
306   with x show ?thesis by (simp del: r_one)
307 qed
309 lemma (in group) r_cancel [simp]:
310   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
311    (y \<otimes> x = z \<otimes> x) = (y = z)"
312 proof
313   assume eq: "y \<otimes> x = z \<otimes> x"
314     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
315   then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
316     by (simp add: m_assoc [symmetric] del: r_inv)
317   with G show "y = z" by simp
318 next
319   assume eq: "y = z"
320     and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
321   then show "y \<otimes> x = z \<otimes> x" by simp
322 qed
324 lemma (in group) inv_one [simp]:
325   "inv \<one> = \<one>"
326 proof -
327   have "inv \<one> = \<one> \<otimes> (inv \<one>)" by (simp del: r_inv)
328   moreover have "... = \<one>" by simp
329   finally show ?thesis .
330 qed
332 lemma (in group) inv_inv [simp]:
333   "x \<in> carrier G ==> inv (inv x) = x"
334   using Units_inv_inv by simp
336 lemma (in group) inv_inj:
337   "inj_on (m_inv G) (carrier G)"
338   using inv_inj_on_Units by simp
340 lemma (in group) inv_mult_group:
341   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
342 proof -
343   assume G: "x \<in> carrier G"  "y \<in> carrier G"
344   then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
345     by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric])
346   with G show ?thesis by (simp del: l_inv)
347 qed
349 lemma (in group) inv_comm:
350   "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
351   by (rule Units_inv_comm) auto
353 lemma (in group) inv_equality:
354      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
355 apply (simp add: m_inv_def)
356 apply (rule the_equality)
357  apply (simp add: inv_comm [of y x])
358 apply (rule r_cancel [THEN iffD1], auto)
359 done
361 text {* Power *}
363 lemma (in group) int_pow_def2:
364   "a (^) (z::int) = (if neg z then inv (a (^) (nat (-z))) else a (^) (nat z))"
365   by (simp add: int_pow_def nat_pow_def Let_def)
367 lemma (in group) int_pow_0 [simp]:
368   "x (^) (0::int) = \<one>"
369   by (simp add: int_pow_def2)
371 lemma (in group) int_pow_one [simp]:
372   "\<one> (^) (z::int) = \<one>"
373   by (simp add: int_pow_def2)
376 subsection {* Subgroups *}
378 locale subgroup =
379   fixes H and G (structure)
380   assumes subset: "H \<subseteq> carrier G"
381     and m_closed [intro, simp]: "\<lbrakk>x \<in> H; y \<in> H\<rbrakk> \<Longrightarrow> x \<otimes> y \<in> H"
382     and one_closed [simp]: "\<one> \<in> H"
383     and m_inv_closed [intro,simp]: "x \<in> H \<Longrightarrow> inv x \<in> H"
385 lemma (in subgroup) is_subgroup:
386   "subgroup H G" .
388 declare (in subgroup) group.intro [intro]
390 lemma (in subgroup) mem_carrier [simp]:
391   "x \<in> H \<Longrightarrow> x \<in> carrier G"
392   using subset by blast
394 lemma subgroup_imp_subset:
395   "subgroup H G \<Longrightarrow> H \<subseteq> carrier G"
396   by (rule subgroup.subset)
398 lemma (in subgroup) subgroup_is_group [intro]:
399   includes group G
400   shows "group (G\<lparr>carrier := H\<rparr>)"
401   by (rule groupI) (auto intro: m_assoc l_inv mem_carrier)
403 text {*
404   Since @{term H} is nonempty, it contains some element @{term x}.  Since
405   it is closed under inverse, it contains @{text "inv x"}.  Since
406   it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
407 *}
409 lemma (in group) one_in_subset:
410   "[| 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 |]
411    ==> \<one> \<in> H"
412 by (force simp add: l_inv)
414 text {* A characterization of subgroups: closed, non-empty subset. *}
416 lemma (in group) subgroupI:
417   assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
418     and inv: "!!a. a \<in> H \<Longrightarrow> inv a \<in> H"
419     and mult: "!!a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes> b \<in> H"
420   shows "subgroup H G"
421 proof (simp add: subgroup_def prems)
422   show "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
423 qed
425 declare monoid.one_closed [iff] group.inv_closed [simp]
426   monoid.l_one [simp] monoid.r_one [simp] group.inv_inv [simp]
428 lemma subgroup_nonempty:
429   "~ subgroup {} G"
430   by (blast dest: subgroup.one_closed)
432 lemma (in subgroup) finite_imp_card_positive:
433   "finite (carrier G) ==> 0 < card H"
434 proof (rule classical)
435   assume "finite (carrier G)" "~ 0 < card H"
436   then have "finite H" by (blast intro: finite_subset [OF subset])
437   with prems have "subgroup {} G" by simp
438   with subgroup_nonempty show ?thesis by contradiction
439 qed
441 (*
442 lemma (in monoid) Units_subgroup:
443   "subgroup (Units G) G"
444 *)
447 subsection {* Direct Products *}
449 constdefs
450   DirProd :: "_ \<Rightarrow> _ \<Rightarrow> ('a \<times> 'b) monoid"  (infixr "\<times>\<times>" 80)
451   "G \<times>\<times> H \<equiv> \<lparr>carrier = carrier G \<times> carrier H,
452                 mult = (\<lambda>(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')),
453                 one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)\<rparr>"
455 lemma DirProd_monoid:
456   includes monoid G + monoid H
457   shows "monoid (G \<times>\<times> H)"
458 proof -
459   from prems
460   show ?thesis by (unfold monoid_def DirProd_def, auto)
461 qed
464 text{*Does not use the previous result because it's easier just to use auto.*}
465 lemma DirProd_group:
466   includes group G + group H
467   shows "group (G \<times>\<times> H)"
468   by (rule groupI)
469      (auto intro: G.m_assoc H.m_assoc G.l_inv H.l_inv
470            simp add: DirProd_def)
472 lemma carrier_DirProd [simp]:
473      "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
474   by (simp add: DirProd_def)
476 lemma one_DirProd [simp]:
477      "\<one>\<^bsub>G \<times>\<times> H\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
478   by (simp add: DirProd_def)
480 lemma mult_DirProd [simp]:
481      "(g, h) \<otimes>\<^bsub>(G \<times>\<times> H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
482   by (simp add: DirProd_def)
484 lemma inv_DirProd [simp]:
485   includes group G + group H
486   assumes g: "g \<in> carrier G"
487       and h: "h \<in> carrier H"
488   shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
489   apply (rule group.inv_equality [OF DirProd_group])
490   apply (simp_all add: prems group.l_inv)
491   done
493 text{*This alternative proof of the previous result demonstrates interpret.
494    It uses @{text Prod.inv_equality} (available after @{text interpret})
495    instead of @{text "group.inv_equality [OF DirProd_group]"}. *}
496 lemma
497   includes group G + group H
498   assumes g: "g \<in> carrier G"
499       and h: "h \<in> carrier H"
500   shows "m_inv (G \<times>\<times> H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
501 proof -
502   interpret Prod: group ["G \<times>\<times> H"]
503     by (auto intro: DirProd_group group.intro group.axioms prems)
504   show ?thesis by (simp add: Prod.inv_equality g h)
505 qed
508 subsection {* Homomorphisms and Isomorphisms *}
510 constdefs (structure G and H)
511   hom :: "_ => _ => ('a => 'b) set"
512   "hom G H ==
513     {h. h \<in> carrier G -> carrier H &
514       (\<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)}"
516 lemma hom_mult:
517   "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
518    ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
519   by (simp add: hom_def)
521 lemma hom_closed:
522   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
523   by (auto simp add: hom_def funcset_mem)
525 lemma (in group) hom_compose:
526      "[|h \<in> hom G H; i \<in> hom H I|] ==> compose (carrier G) i h \<in> hom G I"
527 apply (auto simp add: hom_def funcset_compose)
528 apply (simp add: compose_def funcset_mem)
529 done
531 constdefs
532   iso :: "_ => _ => ('a => 'b) set"  (infixr "\<cong>" 60)
533   "G \<cong> H == {h. h \<in> hom G H & bij_betw h (carrier G) (carrier H)}"
535 lemma iso_refl: "(%x. x) \<in> G \<cong> G"
536 by (simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
538 lemma (in group) iso_sym:
539      "h \<in> G \<cong> H \<Longrightarrow> Inv (carrier G) h \<in> H \<cong> G"
540 apply (simp add: iso_def bij_betw_Inv)
541 apply (subgoal_tac "Inv (carrier G) h \<in> carrier H \<rightarrow> carrier G")
542  prefer 2 apply (simp add: bij_betw_imp_funcset [OF bij_betw_Inv])
543 apply (simp add: hom_def bij_betw_def Inv_f_eq funcset_mem f_Inv_f)
544 done
546 lemma (in group) iso_trans:
547      "[|h \<in> G \<cong> H; i \<in> H \<cong> I|] ==> (compose (carrier G) i h) \<in> G \<cong> I"
548 by (auto simp add: iso_def hom_compose bij_betw_compose)
550 lemma DirProd_commute_iso:
551   shows "(\<lambda>(x,y). (y,x)) \<in> (G \<times>\<times> H) \<cong> (H \<times>\<times> G)"
552 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
554 lemma DirProd_assoc_iso:
555   shows "(\<lambda>(x,y,z). (x,(y,z))) \<in> (G \<times>\<times> H \<times>\<times> I) \<cong> (G \<times>\<times> (H \<times>\<times> I))"
556 by (auto simp add: iso_def hom_def inj_on_def bij_betw_def Pi_def)
559 text{*Basis for homomorphism proofs: we assume two groups @{term G} and
560   @{term H}, with a homomorphism @{term h} between them*}
561 locale group_hom = group G + group H + var h +
562   assumes homh: "h \<in> hom G H"
563   notes hom_mult [simp] = hom_mult [OF homh]
564     and hom_closed [simp] = hom_closed [OF homh]
566 lemma (in group_hom) one_closed [simp]:
567   "h \<one> \<in> carrier H"
568   by simp
570 lemma (in group_hom) hom_one [simp]:
571   "h \<one> = \<one>\<^bsub>H\<^esub>"
572 proof -
573   have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^bsub>H\<^esub> h \<one>"
574     by (simp add: hom_mult [symmetric] del: hom_mult)
575   then show ?thesis by (simp del: r_one)
576 qed
578 lemma (in group_hom) inv_closed [simp]:
579   "x \<in> carrier G ==> h (inv x) \<in> carrier H"
580   by simp
582 lemma (in group_hom) hom_inv [simp]:
583   "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
584 proof -
585   assume x: "x \<in> carrier G"
586   then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
587     by (simp add: hom_mult [symmetric] del: hom_mult)
588   also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
589     by (simp add: hom_mult [symmetric] del: hom_mult)
590   finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .
591   with x show ?thesis by (simp del: H.r_inv)
592 qed
595 subsection {* Commutative Structures *}
597 text {*
598   Naming convention: multiplicative structures that are commutative
599   are called \emph{commutative}, additive structures are called
600   \emph{Abelian}.
601 *}
603 locale comm_monoid = monoid +
604   assumes m_comm: "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes> y = y \<otimes> x"
606 lemma (in comm_monoid) m_lcomm:
607   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
608    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
609 proof -
610   assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
611   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
612   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
613   also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
614   finally show ?thesis .
615 qed
617 lemmas (in comm_monoid) m_ac = m_assoc m_comm m_lcomm
619 lemma comm_monoidI:
620   fixes G (structure)
621   assumes m_closed:
622       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
623     and one_closed: "\<one> \<in> carrier G"
624     and m_assoc:
625       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
626       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
627     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
628     and m_comm:
629       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
630   shows "comm_monoid G"
631   using l_one
632     by (auto intro!: comm_monoid.intro comm_monoid_axioms.intro monoid.intro
633              intro: prems simp: m_closed one_closed m_comm)
635 lemma (in monoid) monoid_comm_monoidI:
636   assumes m_comm:
637       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
638   shows "comm_monoid G"
639   by (rule comm_monoidI) (auto intro: m_assoc m_comm)
641 (*lemma (in comm_monoid) r_one [simp]:
642   "x \<in> carrier G ==> x \<otimes> \<one> = x"
643 proof -
644   assume G: "x \<in> carrier G"
645   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
646   also from G have "... = x" by simp
647   finally show ?thesis .
648 qed*)
650 lemma (in comm_monoid) nat_pow_distr:
651   "[| x \<in> carrier G; y \<in> carrier G |] ==>
652   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
653   by (induct n) (simp, simp add: m_ac)
655 locale comm_group = comm_monoid + group
657 lemma (in group) group_comm_groupI:
658   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
659       x \<otimes> y = y \<otimes> x"
660   shows "comm_group G"
661   by unfold_locales (simp_all add: m_comm)
663 lemma comm_groupI:
664   fixes G (structure)
665   assumes m_closed:
666       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
667     and one_closed: "\<one> \<in> carrier G"
668     and m_assoc:
669       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
670       (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
671     and m_comm:
672       "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
673     and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
674     and l_inv_ex: "!!x. x \<in> carrier G ==> \<exists>y \<in> carrier G. y \<otimes> x = \<one>"
675   shows "comm_group G"
676   by (fast intro: group.group_comm_groupI groupI prems)
678 lemma (in comm_group) inv_mult:
679   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv x \<otimes> inv y"
680   by (simp add: m_ac inv_mult_group)
683 subsection {* The Lattice of Subgroups of a Group *}
685 text_raw {* \label{sec:subgroup-lattice} *}
687 theorem (in group) subgroups_partial_order:
688   "partial_order (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
689   by (rule partial_order.intro) simp_all
691 lemma (in group) subgroup_self:
692   "subgroup (carrier G) G"
693   by (rule subgroupI) auto
695 lemma (in group) subgroup_imp_group:
696   "subgroup H G ==> group (G(| carrier := H |))"
697   by (rule subgroup.subgroup_is_group)
699 lemma (in group) is_monoid [intro, simp]:
700   "monoid G"
701   by (auto intro: monoid.intro m_assoc)
703 lemma (in group) subgroup_inv_equality:
704   "[| subgroup H G; x \<in> H |] ==> m_inv (G (| carrier := H |)) x = inv x"
705 apply (rule_tac inv_equality [THEN sym])
706   apply (rule group.l_inv [OF subgroup_imp_group, simplified], assumption+)
707  apply (rule subsetD [OF subgroup.subset], assumption+)
708 apply (rule subsetD [OF subgroup.subset], assumption)
709 apply (rule_tac group.inv_closed [OF subgroup_imp_group, simplified], assumption+)
710 done
712 theorem (in group) subgroups_Inter:
713   assumes subgr: "(!!H. H \<in> A ==> subgroup H G)"
714     and not_empty: "A ~= {}"
715   shows "subgroup (\<Inter>A) G"
716 proof (rule subgroupI)
717   from subgr [THEN subgroup.subset] and not_empty
718   show "\<Inter>A \<subseteq> carrier G" by blast
719 next
720   from subgr [THEN subgroup.one_closed]
721   show "\<Inter>A ~= {}" by blast
722 next
723   fix x assume "x \<in> \<Inter>A"
724   with subgr [THEN subgroup.m_inv_closed]
725   show "inv x \<in> \<Inter>A" by blast
726 next
727   fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
728   with subgr [THEN subgroup.m_closed]
729   show "x \<otimes> y \<in> \<Inter>A" by blast
730 qed
732 theorem (in group) subgroups_complete_lattice:
733   "complete_lattice (| carrier = {H. subgroup H G}, le = op \<subseteq> |)"
734     (is "complete_lattice ?L")
735 proof (rule partial_order.complete_lattice_criterion1)
736   show "partial_order ?L" by (rule subgroups_partial_order)
737 next
738   have "greatest ?L (carrier G) (carrier ?L)"
739     by (unfold greatest_def) (simp add: subgroup.subset subgroup_self)
740   then show "\<exists>G. greatest ?L G (carrier ?L)" ..
741 next
742   fix A
743   assume L: "A \<subseteq> carrier ?L" and non_empty: "A ~= {}"
744   then have Int_subgroup: "subgroup (\<Inter>A) G"
745     by (fastsimp intro: subgroups_Inter)
746   have "greatest ?L (\<Inter>A) (Lower ?L A)"
747     (is "greatest ?L ?Int _")
748   proof (rule greatest_LowerI)
749     fix H
750     assume H: "H \<in> A"
751     with L have subgroupH: "subgroup H G" by auto
752     from subgroupH have groupH: "group (G (| carrier := H |))" (is "group ?H")
753       by (rule subgroup_imp_group)
754     from groupH have monoidH: "monoid ?H"
755       by (rule group.is_monoid)
756     from H have Int_subset: "?Int \<subseteq> H" by fastsimp
757     then show "le ?L ?Int H" by simp
758   next
759     fix H
760     assume H: "H \<in> Lower ?L A"
761     with L Int_subgroup show "le ?L H ?Int" by (fastsimp intro: Inter_greatest)
762   next
763     show "A \<subseteq> carrier ?L" by (rule L)
764   next
765     show "?Int \<in> carrier ?L" by simp (rule Int_subgroup)
766   qed
767   then show "\<exists>I. greatest ?L I (Lower ?L A)" ..
768 qed
770 end