src/HOL/Algebra/Ring.thy
 author wenzelm Thu Mar 26 20:08:55 2009 +0100 (2009-03-26) changeset 30729 461ee3e49ad3 parent 29237 e90d9d51106b child 35054 a5db9779b026 permissions -rw-r--r--
interpretation/interpret: prefixes are mandatory by default;
1 (*
2   Title:     The algebraic hierarchy of rings
3   Author:    Clemens Ballarin, started 9 December 1996
4   Copyright: Clemens Ballarin
5 *)
7 theory Ring
8 imports FiniteProduct
9 uses ("ringsimp.ML") begin
12 section {* The Algebraic Hierarchy of Rings *}
14 subsection {* Abelian Groups *}
16 record 'a ring = "'a monoid" +
17   zero :: 'a ("\<zero>\<index>")
18   add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65)
20 text {* Derived operations. *}
22 constdefs (structure R)
23   a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _"  80)
24   "a_inv R == m_inv (| carrier = carrier R, mult = add R, one = zero R |)"
26   a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
27   "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus> y == x \<oplus> (\<ominus> y)"
29 locale abelian_monoid =
30   fixes G (structure)
31   assumes a_comm_monoid:
32      "comm_monoid (| carrier = carrier G, mult = add G, one = zero G |)"
35 text {*
36   The following definition is redundant but simple to use.
37 *}
39 locale abelian_group = abelian_monoid +
40   assumes a_comm_group:
41      "comm_group (| carrier = carrier G, mult = add G, one = zero G |)"
44 subsection {* Basic Properties *}
46 lemma abelian_monoidI:
47   fixes R (structure)
48   assumes a_closed:
49       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
50     and zero_closed: "\<zero> \<in> carrier R"
51     and a_assoc:
52       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
53       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
54     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
55     and a_comm:
56       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
57   shows "abelian_monoid R"
58   by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)
60 lemma abelian_groupI:
61   fixes R (structure)
62   assumes a_closed:
63       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
64     and zero_closed: "zero R \<in> carrier R"
65     and a_assoc:
66       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
67       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
68     and a_comm:
69       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
70     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
71     and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> x = \<zero>"
72   shows "abelian_group R"
73   by (auto intro!: abelian_group.intro abelian_monoidI
74       abelian_group_axioms.intro comm_monoidI comm_groupI
75     intro: assms)
77 lemma (in abelian_monoid) a_monoid:
78   "monoid (| carrier = carrier G, mult = add G, one = zero G |)"
79 by (rule comm_monoid.axioms, rule a_comm_monoid)
81 lemma (in abelian_group) a_group:
82   "group (| carrier = carrier G, mult = add G, one = zero G |)"
83   by (simp add: group_def a_monoid)
84     (simp add: comm_group.axioms group.axioms a_comm_group)
86 lemmas monoid_record_simps = partial_object.simps monoid.simps
88 lemma (in abelian_monoid) a_closed [intro, simp]:
89   "\<lbrakk> x \<in> carrier G; y \<in> carrier G \<rbrakk> \<Longrightarrow> x \<oplus> y \<in> carrier G"
90   by (rule monoid.m_closed [OF a_monoid, simplified monoid_record_simps])
92 lemma (in abelian_monoid) zero_closed [intro, simp]:
93   "\<zero> \<in> carrier G"
94   by (rule monoid.one_closed [OF a_monoid, simplified monoid_record_simps])
96 lemma (in abelian_group) a_inv_closed [intro, simp]:
97   "x \<in> carrier G ==> \<ominus> x \<in> carrier G"
98   by (simp add: a_inv_def
99     group.inv_closed [OF a_group, simplified monoid_record_simps])
101 lemma (in abelian_group) minus_closed [intro, simp]:
102   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
103   by (simp add: a_minus_def)
105 lemma (in abelian_group) a_l_cancel [simp]:
106   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
107    (x \<oplus> y = x \<oplus> z) = (y = z)"
108   by (rule group.l_cancel [OF a_group, simplified monoid_record_simps])
110 lemma (in abelian_group) a_r_cancel [simp]:
111   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
112    (y \<oplus> x = z \<oplus> x) = (y = z)"
113   by (rule group.r_cancel [OF a_group, simplified monoid_record_simps])
115 lemma (in abelian_monoid) a_assoc:
116   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
117   (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
118   by (rule monoid.m_assoc [OF a_monoid, simplified monoid_record_simps])
120 lemma (in abelian_monoid) l_zero [simp]:
121   "x \<in> carrier G ==> \<zero> \<oplus> x = x"
122   by (rule monoid.l_one [OF a_monoid, simplified monoid_record_simps])
124 lemma (in abelian_group) l_neg:
125   "x \<in> carrier G ==> \<ominus> x \<oplus> x = \<zero>"
126   by (simp add: a_inv_def
127     group.l_inv [OF a_group, simplified monoid_record_simps])
129 lemma (in abelian_monoid) a_comm:
130   "\<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x"
131   by (rule comm_monoid.m_comm [OF a_comm_monoid,
132     simplified monoid_record_simps])
134 lemma (in abelian_monoid) a_lcomm:
135   "\<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow>
136    x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
137   by (rule comm_monoid.m_lcomm [OF a_comm_monoid,
138                                 simplified monoid_record_simps])
140 lemma (in abelian_monoid) r_zero [simp]:
141   "x \<in> carrier G ==> x \<oplus> \<zero> = x"
142   using monoid.r_one [OF a_monoid]
143   by simp
145 lemma (in abelian_group) r_neg:
146   "x \<in> carrier G ==> x \<oplus> (\<ominus> x) = \<zero>"
147   using group.r_inv [OF a_group]
148   by (simp add: a_inv_def)
150 lemma (in abelian_group) minus_zero [simp]:
151   "\<ominus> \<zero> = \<zero>"
152   by (simp add: a_inv_def
153     group.inv_one [OF a_group, simplified monoid_record_simps])
155 lemma (in abelian_group) minus_minus [simp]:
156   "x \<in> carrier G ==> \<ominus> (\<ominus> x) = x"
157   using group.inv_inv [OF a_group, simplified monoid_record_simps]
158   by (simp add: a_inv_def)
160 lemma (in abelian_group) a_inv_inj:
161   "inj_on (a_inv G) (carrier G)"
162   using group.inv_inj [OF a_group, simplified monoid_record_simps]
163   by (simp add: a_inv_def)
165 lemma (in abelian_group) minus_add:
166   "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> (x \<oplus> y) = \<ominus> x \<oplus> \<ominus> y"
167   using comm_group.inv_mult [OF a_comm_group]
168   by (simp add: a_inv_def)
170 lemma (in abelian_group) minus_equality:
171   "[| x \<in> carrier G; y \<in> carrier G; y \<oplus> x = \<zero> |] ==> \<ominus> x = y"
172   using group.inv_equality [OF a_group]
173   by (auto simp add: a_inv_def)
175 lemma (in abelian_monoid) minus_unique:
176   "[| x \<in> carrier G; y \<in> carrier G; y' \<in> carrier G;
177       y \<oplus> x = \<zero>; x \<oplus> y' = \<zero> |] ==> y = y'"
178   using monoid.inv_unique [OF a_monoid]
179   by (simp add: a_inv_def)
181 lemmas (in abelian_monoid) a_ac = a_assoc a_comm a_lcomm
183 text {* Derive an @{text "abelian_group"} from a @{text "comm_group"} *}
184 lemma comm_group_abelian_groupI:
185   fixes G (structure)
186   assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
187   shows "abelian_group G"
188 proof -
189   interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
190     by (rule cg)
191   show "abelian_group G" ..
192 qed
195 subsection {* Sums over Finite Sets *}
197 text {*
198   This definition makes it easy to lift lemmas from @{term finprod}.
199 *}
201 constdefs
202   finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b"
203   "finsum G f A == finprod (| carrier = carrier G,
204      mult = add G, one = zero G |) f A"
206 syntax
207   "_finsum" :: "index => idt => 'a set => 'b => 'b"
208       ("(3\<Oplus>__:_. _)" [1000, 0, 51, 10] 10)
209 syntax (xsymbols)
210   "_finsum" :: "index => idt => 'a set => 'b => 'b"
211       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
212 syntax (HTML output)
213   "_finsum" :: "index => idt => 'a set => 'b => 'b"
214       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
215 translations
216   "\<Oplus>\<index>i:A. b" == "finsum \<struct>\<index> (%i. b) A"
217   -- {* Beware of argument permutation! *}
219 context abelian_monoid begin
221 (*
222   lemmas finsum_empty [simp] =
223     comm_monoid.finprod_empty [OF a_comm_monoid, simplified]
224   is dangeous, because attributes (like simplified) are applied upon opening
225   the locale, simplified refers to the simpset at that time!!!
227   lemmas finsum_empty [simp] =
228     abelian_monoid.finprod_empty [OF a_abelian_monoid, folded finsum_def,
229       simplified monoid_record_simps]
230   makes the locale slow, because proofs are repeated for every
231   "lemma (in abelian_monoid)" command.
232   When lemma is used time in UnivPoly.thy from beginning to UP_cring goes down
233   from 110 secs to 60 secs.
234 *)
236 lemma finsum_empty [simp]:
237   "finsum G f {} = \<zero>"
238   by (rule comm_monoid.finprod_empty [OF a_comm_monoid,
239     folded finsum_def, simplified monoid_record_simps])
241 lemma finsum_insert [simp]:
242   "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |]
243   ==> finsum G f (insert a F) = f a \<oplus> finsum G f F"
244   by (rule comm_monoid.finprod_insert [OF a_comm_monoid,
245     folded finsum_def, simplified monoid_record_simps])
247 lemma finsum_zero [simp]:
248   "finite A ==> (\<Oplus>i\<in>A. \<zero>) = \<zero>"
249   by (rule comm_monoid.finprod_one [OF a_comm_monoid, folded finsum_def,
250     simplified monoid_record_simps])
252 lemma finsum_closed [simp]:
253   fixes A
254   assumes fin: "finite A" and f: "f \<in> A -> carrier G"
255   shows "finsum G f A \<in> carrier G"
256   apply (rule comm_monoid.finprod_closed [OF a_comm_monoid,
257     folded finsum_def, simplified monoid_record_simps])
258    apply (rule fin)
259   apply (rule f)
260   done
262 lemma finsum_Un_Int:
263   "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
264      finsum G g (A Un B) \<oplus> finsum G g (A Int B) =
265      finsum G g A \<oplus> finsum G g B"
266   by (rule comm_monoid.finprod_Un_Int [OF a_comm_monoid,
267     folded finsum_def, simplified monoid_record_simps])
269 lemma finsum_Un_disjoint:
270   "[| finite A; finite B; A Int B = {};
271       g \<in> A -> carrier G; g \<in> B -> carrier G |]
272    ==> finsum G g (A Un B) = finsum G g A \<oplus> finsum G g B"
273   by (rule comm_monoid.finprod_Un_disjoint [OF a_comm_monoid,
274     folded finsum_def, simplified monoid_record_simps])
277   "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
278    finsum G (%x. f x \<oplus> g x) A = (finsum G f A \<oplus> finsum G g A)"
279   by (rule comm_monoid.finprod_multf [OF a_comm_monoid,
280     folded finsum_def, simplified monoid_record_simps])
282 lemma finsum_cong':
283   "[| A = B; g : B -> carrier G;
284       !!i. i : B ==> f i = g i |] ==> finsum G f A = finsum G g B"
285   by (rule comm_monoid.finprod_cong' [OF a_comm_monoid,
286     folded finsum_def, simplified monoid_record_simps]) auto
288 lemma finsum_0 [simp]:
289   "f : {0::nat} -> carrier G ==> finsum G f {..0} = f 0"
290   by (rule comm_monoid.finprod_0 [OF a_comm_monoid, folded finsum_def,
291     simplified monoid_record_simps])
293 lemma finsum_Suc [simp]:
294   "f : {..Suc n} -> carrier G ==>
295    finsum G f {..Suc n} = (f (Suc n) \<oplus> finsum G f {..n})"
296   by (rule comm_monoid.finprod_Suc [OF a_comm_monoid, folded finsum_def,
297     simplified monoid_record_simps])
299 lemma finsum_Suc2:
300   "f : {..Suc n} -> carrier G ==>
301    finsum G f {..Suc n} = (finsum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
302   by (rule comm_monoid.finprod_Suc2 [OF a_comm_monoid, folded finsum_def,
303     simplified monoid_record_simps])
305 lemma finsum_add [simp]:
306   "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
307      finsum G (%i. f i \<oplus> g i) {..n::nat} =
308      finsum G f {..n} \<oplus> finsum G g {..n}"
309   by (rule comm_monoid.finprod_mult [OF a_comm_monoid, folded finsum_def,
310     simplified monoid_record_simps])
312 lemma finsum_cong:
313   "[| A = B; f : B -> carrier G;
314       !!i. i : B =simp=> f i = g i |] ==> finsum G f A = finsum G g B"
315   by (rule comm_monoid.finprod_cong [OF a_comm_monoid, folded finsum_def,
316     simplified monoid_record_simps]) (auto simp add: simp_implies_def)
318 text {*Usually, if this rule causes a failed congruence proof error,
319    the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
320    Adding @{thm [source] Pi_def} to the simpset is often useful. *}
322 lemma finsum_reindex:
323   assumes fin: "finite A"
324     shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow>
325         inj_on h A ==> finsum G f (h ` A) = finsum G (%x. f (h x)) A"
326   using fin apply induct
327   apply (auto simp add: finsum_insert Pi_def)
328 done
330 (* The following is wrong.  Needed is the equivalent of (^) for addition,
331   or indeed the canonical embedding from Nat into the monoid.
333 lemma finsum_const:
334   assumes fin [simp]: "finite A"
335       and a [simp]: "a : carrier G"
336     shows "finsum G (%x. a) A = a (^) card A"
337   using fin apply induct
338   apply force
339   apply (subst finsum_insert)
340   apply auto
341   apply (force simp add: Pi_def)
342   apply (subst m_comm)
343   apply auto
344 done
345 *)
347 (* By Jesus Aransay. *)
349 lemma finsum_singleton:
350   assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
351   shows "(\<Oplus>j\<in>A. if i = j then f j else \<zero>) = f i"
352   using i_in_A finsum_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<zero>)"]
353     fin_A f_Pi finsum_zero [of "A - {i}"]
354     finsum_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<zero>)" "(\<lambda>i. \<zero>)"]
355   unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
357 end
360 subsection {* Rings: Basic Definitions *}
362 locale ring = abelian_group R + monoid R for R (structure) +
363   assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
364       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
365     and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
366       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
368 locale cring = ring + comm_monoid R
370 locale "domain" = cring +
371   assumes one_not_zero [simp]: "\<one> ~= \<zero>"
372     and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
373                   a = \<zero> | b = \<zero>"
375 locale field = "domain" +
376   assumes field_Units: "Units R = carrier R - {\<zero>}"
379 subsection {* Rings *}
381 lemma ringI:
382   fixes R (structure)
383   assumes abelian_group: "abelian_group R"
384     and monoid: "monoid R"
385     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
386       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
387     and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
388       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
389   shows "ring R"
390   by (auto intro: ring.intro
391     abelian_group.axioms ring_axioms.intro assms)
393 lemma (in ring) is_abelian_group:
394   "abelian_group R"
395   ..
397 lemma (in ring) is_monoid:
398   "monoid R"
399   by (auto intro!: monoidI m_assoc)
401 lemma (in ring) is_ring:
402   "ring R"
403   by (rule ring_axioms)
405 lemmas ring_record_simps = monoid_record_simps ring.simps
407 lemma cringI:
408   fixes R (structure)
409   assumes abelian_group: "abelian_group R"
410     and comm_monoid: "comm_monoid R"
411     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
412       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
413   shows "cring R"
414 proof (intro cring.intro ring.intro)
415   show "ring_axioms R"
416     -- {* Right-distributivity follows from left-distributivity and
417           commutativity. *}
418   proof (rule ring_axioms.intro)
419     fix x y z
420     assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
421     note [simp] = comm_monoid.axioms [OF comm_monoid]
422       abelian_group.axioms [OF abelian_group]
423       abelian_monoid.a_closed
425     from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
426       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
427     also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
428     also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
429       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
430     finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
431   qed (rule l_distr)
432 qed (auto intro: cring.intro
433   abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)
435 (*
436 lemma (in cring) is_comm_monoid:
437   "comm_monoid R"
438   by (auto intro!: comm_monoidI m_assoc m_comm)
439 *)
441 lemma (in cring) is_cring:
442   "cring R" by (rule cring_axioms)
445 subsubsection {* Normaliser for Rings *}
447 lemma (in abelian_group) r_neg2:
448   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
449 proof -
450   assume G: "x \<in> carrier G" "y \<in> carrier G"
451   then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
452     by (simp only: r_neg l_zero)
453   with G show ?thesis
454     by (simp add: a_ac)
455 qed
457 lemma (in abelian_group) r_neg1:
458   "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
459 proof -
460   assume G: "x \<in> carrier G" "y \<in> carrier G"
461   then have "(\<ominus> x \<oplus> x) \<oplus> y = y"
462     by (simp only: l_neg l_zero)
463   with G show ?thesis by (simp add: a_ac)
464 qed
466 text {*
467   The following proofs are from Jacobson, Basic Algebra I, pp.~88--89
468 *}
470 lemma (in ring) l_null [simp]:
471   "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
472 proof -
473   assume R: "x \<in> carrier R"
474   then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
475     by (simp add: l_distr del: l_zero r_zero)
476   also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
477   finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
478   with R show ?thesis by (simp del: r_zero)
479 qed
481 lemma (in ring) r_null [simp]:
482   "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
483 proof -
484   assume R: "x \<in> carrier R"
485   then have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
486     by (simp add: r_distr del: l_zero r_zero)
487   also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
488   finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
489   with R show ?thesis by (simp del: r_zero)
490 qed
492 lemma (in ring) l_minus:
493   "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
494 proof -
495   assume R: "x \<in> carrier R" "y \<in> carrier R"
496   then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
497   also from R have "... = \<zero>" by (simp add: l_neg l_null)
498   finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
499   with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
500   with R show ?thesis by (simp add: a_assoc r_neg)
501 qed
503 lemma (in ring) r_minus:
504   "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
505 proof -
506   assume R: "x \<in> carrier R" "y \<in> carrier R"
507   then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
508   also from R have "... = \<zero>" by (simp add: l_neg r_null)
509   finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
510   with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
511   with R show ?thesis by (simp add: a_assoc r_neg )
512 qed
514 lemma (in abelian_group) minus_eq:
515   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
516   by (simp only: a_minus_def)
518 text {* Setup algebra method:
519   compute distributive normal form in locale contexts *}
521 use "ringsimp.ML"
523 setup Algebra.setup
525 lemmas (in ring) ring_simprules
526   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
527   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
528   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
529   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
530   a_lcomm r_distr l_null r_null l_minus r_minus
532 lemmas (in cring)
533   [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
534   _
536 lemmas (in cring) cring_simprules
537   [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
538   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
539   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
540   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
541   a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
544 lemma (in cring) nat_pow_zero:
545   "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
546   by (induct n) simp_all
548 lemma (in ring) one_zeroD:
549   assumes onezero: "\<one> = \<zero>"
550   shows "carrier R = {\<zero>}"
551 proof (rule, rule)
552   fix x
553   assume xcarr: "x \<in> carrier R"
554   from xcarr
555       have "x = x \<otimes> \<one>" by simp
556   from this and onezero
557       have "x = x \<otimes> \<zero>" by simp
558   from this and xcarr
559       have "x = \<zero>" by simp
560   thus "x \<in> {\<zero>}" by fast
561 qed fast
563 lemma (in ring) one_zeroI:
564   assumes carrzero: "carrier R = {\<zero>}"
565   shows "\<one> = \<zero>"
566 proof -
567   from one_closed and carrzero
568       show "\<one> = \<zero>" by simp
569 qed
571 lemma (in ring) carrier_one_zero:
572   shows "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
573   by (rule, erule one_zeroI, erule one_zeroD)
575 lemma (in ring) carrier_one_not_zero:
576   shows "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
577   by (simp add: carrier_one_zero)
579 text {* Two examples for use of method algebra *}
581 lemma
582   fixes R (structure) and S (structure)
583   assumes "ring R" "cring S"
584   assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
585   shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
586 proof -
587   interpret ring R by fact
588   interpret cring S by fact
589 ML_val {* Algebra.print_structures @{context} *}
590   from RS show ?thesis by algebra
591 qed
593 lemma
594   fixes R (structure)
595   assumes "ring R"
596   assumes R: "a \<in> carrier R" "b \<in> carrier R"
597   shows "a \<ominus> (a \<ominus> b) = b"
598 proof -
599   interpret ring R by fact
600   from R show ?thesis by algebra
601 qed
603 subsubsection {* Sums over Finite Sets *}
605 lemma (in ring) finsum_ldistr:
606   "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
607    finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
608 proof (induct set: finite)
609   case empty then show ?case by simp
610 next
611   case (insert x F) then show ?case by (simp add: Pi_def l_distr)
612 qed
614 lemma (in ring) finsum_rdistr:
615   "[| finite A; a \<in> carrier R; f \<in> A -> carrier R |] ==>
616    a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
617 proof (induct set: finite)
618   case empty then show ?case by simp
619 next
620   case (insert x F) then show ?case by (simp add: Pi_def r_distr)
621 qed
624 subsection {* Integral Domains *}
626 lemma (in "domain") zero_not_one [simp]:
627   "\<zero> ~= \<one>"
628   by (rule not_sym) simp
630 lemma (in "domain") integral_iff: (* not by default a simp rule! *)
631   "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
632 proof
633   assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
634   then show "a = \<zero> | b = \<zero>" by (simp add: integral)
635 next
636   assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>"
637   then show "a \<otimes> b = \<zero>" by auto
638 qed
640 lemma (in "domain") m_lcancel:
641   assumes prem: "a ~= \<zero>"
642     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
643   shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
644 proof
645   assume eq: "a \<otimes> b = a \<otimes> c"
646   with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
647   with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
648   with prem and R have "b \<ominus> c = \<zero>" by auto
649   with R have "b = b \<ominus> (b \<ominus> c)" by algebra
650   also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
651   finally show "b = c" .
652 next
653   assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
654 qed
656 lemma (in "domain") m_rcancel:
657   assumes prem: "a ~= \<zero>"
658     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
659   shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
660 proof -
661   from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
662   with R show ?thesis by algebra
663 qed
666 subsection {* Fields *}
668 text {* Field would not need to be derived from domain, the properties
669   for domain follow from the assumptions of field *}
670 lemma (in cring) cring_fieldI:
671   assumes field_Units: "Units R = carrier R - {\<zero>}"
672   shows "field R"
673 proof
674   from field_Units
675   have a: "\<zero> \<notin> Units R" by fast
676   have "\<one> \<in> Units R" by fast
677   from this and a
678   show "\<one> \<noteq> \<zero>" by force
679 next
680   fix a b
681   assume acarr: "a \<in> carrier R"
682     and bcarr: "b \<in> carrier R"
683     and ab: "a \<otimes> b = \<zero>"
684   show "a = \<zero> \<or> b = \<zero>"
685   proof (cases "a = \<zero>", simp)
686     assume "a \<noteq> \<zero>"
687     from this and field_Units and acarr
688     have aUnit: "a \<in> Units R" by fast
689     from bcarr
690     have "b = \<one> \<otimes> b" by algebra
691     also from aUnit acarr
692     have "... = (inv a \<otimes> a) \<otimes> b" by (simp add: Units_l_inv)
693     also from acarr bcarr aUnit[THEN Units_inv_closed]
694     have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
695     also from ab and acarr bcarr aUnit
696     have "... = (inv a) \<otimes> \<zero>" by simp
697     also from aUnit[THEN Units_inv_closed]
698     have "... = \<zero>" by algebra
699     finally
700     have "b = \<zero>" .
701     thus "a = \<zero> \<or> b = \<zero>" by simp
702   qed
703 qed (rule field_Units)
705 text {* Another variant to show that something is a field *}
706 lemma (in cring) cring_fieldI2:
707   assumes notzero: "\<zero> \<noteq> \<one>"
708   and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
709   shows "field R"
710   apply (rule cring_fieldI, simp add: Units_def)
711   apply (rule, clarsimp)
712   apply (simp add: notzero)
713 proof (clarsimp)
714   fix x
715   assume xcarr: "x \<in> carrier R"
716     and "x \<noteq> \<zero>"
717   from this
718   have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
719   from this
720   obtain y
721     where ycarr: "y \<in> carrier R"
722     and xy: "x \<otimes> y = \<one>"
723     by fast
724   from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
725   from ycarr and this and xy
726   show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
727 qed
730 subsection {* Morphisms *}
732 constdefs (structure R S)
733   ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
734   "ring_hom R S == {h. h \<in> carrier R -> carrier S &
735       (ALL x y. x \<in> carrier R & y \<in> carrier R -->
736         h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y) &
737       h \<one> = \<one>\<^bsub>S\<^esub>}"
739 lemma ring_hom_memI:
740   fixes R (structure) and S (structure)
741   assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
742     and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
743       h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
744     and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
745       h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
746     and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
747   shows "h \<in> ring_hom R S"
748   by (auto simp add: ring_hom_def assms Pi_def)
750 lemma ring_hom_closed:
751   "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
752   by (auto simp add: ring_hom_def funcset_mem)
754 lemma ring_hom_mult:
755   fixes R (structure) and S (structure)
756   shows
757     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
758     h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
759     by (simp add: ring_hom_def)
762   fixes R (structure) and S (structure)
763   shows
764     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
765     h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
766     by (simp add: ring_hom_def)
768 lemma ring_hom_one:
769   fixes R (structure) and S (structure)
770   shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
771   by (simp add: ring_hom_def)
773 locale ring_hom_cring = R: cring R + S: cring S
774     for R (structure) and S (structure) +
775   fixes h
776   assumes homh [simp, intro]: "h \<in> ring_hom R S"
777   notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
778     and hom_mult [simp] = ring_hom_mult [OF homh]
779     and hom_add [simp] = ring_hom_add [OF homh]
780     and hom_one [simp] = ring_hom_one [OF homh]
782 lemma (in ring_hom_cring) hom_zero [simp]:
783   "h \<zero> = \<zero>\<^bsub>S\<^esub>"
784 proof -
785   have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
787   then show ?thesis by (simp del: S.r_zero)
788 qed
790 lemma (in ring_hom_cring) hom_a_inv [simp]:
791   "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
792 proof -
793   assume R: "x \<in> carrier R"
794   then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)"
795     by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
796   with R show ?thesis by simp
797 qed
799 lemma (in ring_hom_cring) hom_finsum [simp]:
800   "[| finite A; f \<in> A -> carrier R |] ==>
801   h (finsum R f A) = finsum S (h o f) A"
802 proof (induct set: finite)
803   case empty then show ?case by simp
804 next
805   case insert then show ?case by (simp add: Pi_def)
806 qed
808 lemma (in ring_hom_cring) hom_finprod:
809   "[| finite A; f \<in> A -> carrier R |] ==>
810   h (finprod R f A) = finprod S (h o f) A"
811 proof (induct set: finite)
812   case empty then show ?case by simp
813 next
814   case insert then show ?case by (simp add: Pi_def)
815 qed
817 declare ring_hom_cring.hom_finprod [simp]
819 lemma id_ring_hom [simp]:
820   "id \<in> ring_hom R R"
821   by (auto intro!: ring_hom_memI)
823 end