src/HOL/Algebra/Ring.thy
 author wenzelm Tue Oct 10 19:23:03 2017 +0200 (2017-10-10) changeset 66831 29ea2b900a05 parent 63167 0909deb8059b child 67091 1393c2340eec permissions -rw-r--r--
tuned: each session has at most one defining entry;
1 (*  Title:      HOL/Algebra/Ring.thy
2     Author:     Clemens Ballarin, started 9 December 1996
3     Copyright:  Clemens Ballarin
4 *)
6 theory Ring
7 imports FiniteProduct
8 begin
10 section \<open>The Algebraic Hierarchy of Rings\<close>
12 subsection \<open>Abelian Groups\<close>
14 record 'a ring = "'a monoid" +
15   zero :: 'a ("\<zero>\<index>")
16   add :: "['a, 'a] => 'a" (infixl "\<oplus>\<index>" 65)
18 text \<open>Derived operations.\<close>
20 definition
21   a_inv :: "[('a, 'm) ring_scheme, 'a ] => 'a" ("\<ominus>\<index> _"  80)
22   where "a_inv R = m_inv \<lparr>carrier = carrier R, mult = add R, one = zero R\<rparr>"
24 definition
25   a_minus :: "[('a, 'm) ring_scheme, 'a, 'a] => 'a" (infixl "\<ominus>\<index>" 65)
26   where "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<ominus>\<^bsub>R\<^esub> y = x \<oplus>\<^bsub>R\<^esub> (\<ominus>\<^bsub>R\<^esub> y)"
28 locale abelian_monoid =
29   fixes G (structure)
30   assumes a_comm_monoid:
31      "comm_monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
33 definition
34   finsum :: "[('b, 'm) ring_scheme, 'a => 'b, 'a set] => 'b" where
35   "finsum G = finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
37 syntax
38   "_finsum" :: "index => idt => 'a set => 'b => 'b"
39       ("(3\<Oplus>__\<in>_. _)" [1000, 0, 51, 10] 10)
40 translations
41   "\<Oplus>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finsum G (%i. b) A"
42   \<comment> \<open>Beware of argument permutation!\<close>
45 locale abelian_group = abelian_monoid +
46   assumes a_comm_group:
47      "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
50 subsection \<open>Basic Properties\<close>
52 lemma abelian_monoidI:
53   fixes R (structure)
54   assumes a_closed:
55       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
56     and zero_closed: "\<zero> \<in> carrier R"
57     and a_assoc:
58       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
59       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
60     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
61     and a_comm:
62       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
63   shows "abelian_monoid R"
64   by (auto intro!: abelian_monoid.intro comm_monoidI intro: assms)
66 lemma abelian_groupI:
67   fixes R (structure)
68   assumes a_closed:
69       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y \<in> carrier R"
70     and zero_closed: "zero R \<in> carrier R"
71     and a_assoc:
72       "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |] ==>
73       (x \<oplus> y) \<oplus> z = x \<oplus> (y \<oplus> z)"
74     and a_comm:
75       "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==> x \<oplus> y = y \<oplus> x"
76     and l_zero: "!!x. x \<in> carrier R ==> \<zero> \<oplus> x = x"
77     and l_inv_ex: "!!x. x \<in> carrier R ==> EX y : carrier R. y \<oplus> x = \<zero>"
78   shows "abelian_group R"
79   by (auto intro!: abelian_group.intro abelian_monoidI
80       abelian_group_axioms.intro comm_monoidI comm_groupI
81     intro: assms)
83 lemma (in abelian_monoid) a_monoid:
84   "monoid \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
85 by (rule comm_monoid.axioms, rule a_comm_monoid)
87 lemma (in abelian_group) a_group:
88   "group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
89   by (simp add: group_def a_monoid)
90     (simp add: comm_group.axioms group.axioms a_comm_group)
92 lemmas monoid_record_simps = partial_object.simps monoid.simps
94 text \<open>Transfer facts from multiplicative structures via interpretation.\<close>
96 sublocale abelian_monoid <
97   add: monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
98   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
99     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
100     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
101   by (rule a_monoid) auto
103 context abelian_monoid begin
105 lemmas a_closed = add.m_closed
106 lemmas zero_closed = add.one_closed
107 lemmas a_assoc = add.m_assoc
108 lemmas l_zero = add.l_one
109 lemmas r_zero = add.r_one
110 lemmas minus_unique = add.inv_unique
112 end
114 sublocale abelian_monoid <
115   add: comm_monoid "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
116   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
117     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
118     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
119     and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
120   by (rule a_comm_monoid) (auto simp: finsum_def)
122 context abelian_monoid begin
124 lemmas a_comm = add.m_comm
125 lemmas a_lcomm = add.m_lcomm
126 lemmas a_ac = a_assoc a_comm a_lcomm
128 lemmas finsum_empty = add.finprod_empty
129 lemmas finsum_insert = add.finprod_insert
130 lemmas finsum_zero = add.finprod_one
131 lemmas finsum_closed = add.finprod_closed
132 lemmas finsum_Un_Int = add.finprod_Un_Int
133 lemmas finsum_Un_disjoint = add.finprod_Un_disjoint
135 lemmas finsum_cong' = add.finprod_cong'
136 lemmas finsum_0 = add.finprod_0
137 lemmas finsum_Suc = add.finprod_Suc
138 lemmas finsum_Suc2 = add.finprod_Suc2
140 lemmas finsum_infinite = add.finprod_infinite
142 lemmas finsum_cong = add.finprod_cong
143 text \<open>Usually, if this rule causes a failed congruence proof error,
144    the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
145    Adding @{thm [source] Pi_def} to the simpset is often useful.\<close>
147 lemmas finsum_reindex = add.finprod_reindex
149 (* The following would be wrong.  Needed is the equivalent of (^) for addition,
150   or indeed the canonical embedding from Nat into the monoid.
152 lemma finsum_const:
153   assumes fin [simp]: "finite A"
154       and a [simp]: "a : carrier G"
155     shows "finsum G (%x. a) A = a (^) card A"
156   using fin apply induct
157   apply force
158   apply (subst finsum_insert)
159   apply auto
160   apply (force simp add: Pi_def)
161   apply (subst m_comm)
162   apply auto
163 done
164 *)
166 lemmas finsum_singleton = add.finprod_singleton
168 end
170 sublocale abelian_group <
171   add: group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
172   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
173     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
174     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
175     and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
176   by (rule a_group) (auto simp: m_inv_def a_inv_def)
178 context abelian_group
179 begin
181 lemmas a_inv_closed = add.inv_closed
183 lemma minus_closed [intro, simp]:
184   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y \<in> carrier G"
185   by (simp add: a_minus_def)
187 lemmas a_l_cancel = add.l_cancel
188 lemmas a_r_cancel = add.r_cancel
189 lemmas l_neg = add.l_inv [simp del]
190 lemmas r_neg = add.r_inv [simp del]
191 lemmas minus_zero = add.inv_one
192 lemmas minus_minus = add.inv_inv
193 lemmas a_inv_inj = add.inv_inj
194 lemmas minus_equality = add.inv_equality
196 end
198 sublocale abelian_group <
199   add: comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
200   rewrites "carrier \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = carrier G"
201     and "mult \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = add G"
202     and "one \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = zero G"
203     and "m_inv \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = a_inv G"
204     and "finprod \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr> = finsum G"
205   by (rule a_comm_group) (auto simp: m_inv_def a_inv_def finsum_def)
207 lemmas (in abelian_group) minus_add = add.inv_mult
209 text \<open>Derive an \<open>abelian_group\<close> from a \<open>comm_group\<close>\<close>
211 lemma comm_group_abelian_groupI:
212   fixes G (structure)
213   assumes cg: "comm_group \<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
214   shows "abelian_group G"
215 proof -
216   interpret comm_group "\<lparr>carrier = carrier G, mult = add G, one = zero G\<rparr>"
217     by (rule cg)
218   show "abelian_group G" ..
219 qed
222 subsection \<open>Rings: Basic Definitions\<close>
224 locale semiring = abelian_monoid R + monoid R for R (structure) +
225   assumes l_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
226       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
227     and r_distr: "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
228       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
229     and l_null[simp]: "x \<in> carrier R ==> \<zero> \<otimes> x = \<zero>"
230     and r_null[simp]: "x \<in> carrier R ==> x \<otimes> \<zero> = \<zero>"
232 locale ring = abelian_group R + monoid R for R (structure) +
233   assumes "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
234       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
235     and "[| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
236       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
238 locale cring = ring + comm_monoid R
240 locale "domain" = cring +
241   assumes one_not_zero [simp]: "\<one> ~= \<zero>"
242     and integral: "[| a \<otimes> b = \<zero>; a \<in> carrier R; b \<in> carrier R |] ==>
243                   a = \<zero> | b = \<zero>"
245 locale field = "domain" +
246   assumes field_Units: "Units R = carrier R - {\<zero>}"
249 subsection \<open>Rings\<close>
251 lemma ringI:
252   fixes R (structure)
253   assumes abelian_group: "abelian_group R"
254     and monoid: "monoid R"
255     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
256       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
257     and r_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
258       ==> z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
259   shows "ring R"
260   by (auto intro: ring.intro
261     abelian_group.axioms ring_axioms.intro assms)
263 context ring begin
265 lemma is_abelian_group: "abelian_group R" ..
267 lemma is_monoid: "monoid R"
268   by (auto intro!: monoidI m_assoc)
270 lemma is_ring: "ring R"
271   by (rule ring_axioms)
273 end
275 lemmas ring_record_simps = monoid_record_simps ring.simps
277 lemma cringI:
278   fixes R (structure)
279   assumes abelian_group: "abelian_group R"
280     and comm_monoid: "comm_monoid R"
281     and l_distr: "!!x y z. [| x \<in> carrier R; y \<in> carrier R; z \<in> carrier R |]
282       ==> (x \<oplus> y) \<otimes> z = x \<otimes> z \<oplus> y \<otimes> z"
283   shows "cring R"
284 proof (intro cring.intro ring.intro)
285   show "ring_axioms R"
286     \<comment> \<open>Right-distributivity follows from left-distributivity and
287           commutativity.\<close>
288   proof (rule ring_axioms.intro)
289     fix x y z
290     assume R: "x \<in> carrier R" "y \<in> carrier R" "z \<in> carrier R"
291     note [simp] = comm_monoid.axioms [OF comm_monoid]
292       abelian_group.axioms [OF abelian_group]
293       abelian_monoid.a_closed
295     from R have "z \<otimes> (x \<oplus> y) = (x \<oplus> y) \<otimes> z"
296       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
297     also from R have "... = x \<otimes> z \<oplus> y \<otimes> z" by (simp add: l_distr)
298     also from R have "... = z \<otimes> x \<oplus> z \<otimes> y"
299       by (simp add: comm_monoid.m_comm [OF comm_monoid.intro])
300     finally show "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y" .
301   qed (rule l_distr)
302 qed (auto intro: cring.intro
303   abelian_group.axioms comm_monoid.axioms ring_axioms.intro assms)
305 (*
306 lemma (in cring) is_comm_monoid:
307   "comm_monoid R"
308   by (auto intro!: comm_monoidI m_assoc m_comm)
309 *)
311 lemma (in cring) is_cring:
312   "cring R" by (rule cring_axioms)
315 subsubsection \<open>Normaliser for Rings\<close>
317 lemma (in abelian_group) r_neg2:
318   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<oplus> (\<ominus> x \<oplus> y) = y"
319 proof -
320   assume G: "x \<in> carrier G" "y \<in> carrier G"
321   then have "(x \<oplus> \<ominus> x) \<oplus> y = y"
322     by (simp only: r_neg l_zero)
323   with G show ?thesis
324     by (simp add: a_ac)
325 qed
327 lemma (in abelian_group) r_neg1:
328   "[| x \<in> carrier G; y \<in> carrier G |] ==> \<ominus> x \<oplus> (x \<oplus> y) = y"
329 proof -
330   assume G: "x \<in> carrier G" "y \<in> carrier G"
331   then have "(\<ominus> x \<oplus> x) \<oplus> y = y"
332     by (simp only: l_neg l_zero)
333   with G show ?thesis by (simp add: a_ac)
334 qed
336 context ring begin
338 text \<open>
339   The following proofs are from Jacobson, Basic Algebra I, pp.~88--89.
340 \<close>
342 sublocale semiring
343 proof -
344   note [simp] = ring_axioms[unfolded ring_def ring_axioms_def]
345   show "semiring R"
346   proof (unfold_locales)
347     fix x
348     assume R: "x \<in> carrier R"
349     then have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = (\<zero> \<oplus> \<zero>) \<otimes> x"
350       by (simp del: l_zero r_zero)
351     also from R have "... = \<zero> \<otimes> x \<oplus> \<zero>" by simp
352     finally have "\<zero> \<otimes> x \<oplus> \<zero> \<otimes> x = \<zero> \<otimes> x \<oplus> \<zero>" .
353     with R show "\<zero> \<otimes> x = \<zero>" by (simp del: r_zero)
354     from R have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> (\<zero> \<oplus> \<zero>)"
355       by (simp del: l_zero r_zero)
356     also from R have "... = x \<otimes> \<zero> \<oplus> \<zero>" by simp
357     finally have "x \<otimes> \<zero> \<oplus> x \<otimes> \<zero> = x \<otimes> \<zero> \<oplus> \<zero>" .
358     with R show "x \<otimes> \<zero> = \<zero>" by (simp del: r_zero)
359   qed auto
360 qed
362 lemma l_minus:
363   "[| x \<in> carrier R; y \<in> carrier R |] ==> \<ominus> x \<otimes> y = \<ominus> (x \<otimes> y)"
364 proof -
365   assume R: "x \<in> carrier R" "y \<in> carrier R"
366   then have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = (\<ominus> x \<oplus> x) \<otimes> y" by (simp add: l_distr)
367   also from R have "... = \<zero>" by (simp add: l_neg)
368   finally have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y = \<zero>" .
369   with R have "(\<ominus> x) \<otimes> y \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
370   with R show ?thesis by (simp add: a_assoc r_neg)
371 qed
373 lemma r_minus:
374   "[| x \<in> carrier R; y \<in> carrier R |] ==> x \<otimes> \<ominus> y = \<ominus> (x \<otimes> y)"
375 proof -
376   assume R: "x \<in> carrier R" "y \<in> carrier R"
377   then have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = x \<otimes> (\<ominus> y \<oplus> y)" by (simp add: r_distr)
378   also from R have "... = \<zero>" by (simp add: l_neg)
379   finally have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y = \<zero>" .
380   with R have "x \<otimes> (\<ominus> y) \<oplus> x \<otimes> y \<oplus> \<ominus> (x \<otimes> y) = \<zero> \<oplus> \<ominus> (x \<otimes> y)" by simp
381   with R show ?thesis by (simp add: a_assoc r_neg )
382 qed
384 end
386 lemma (in abelian_group) minus_eq:
387   "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<ominus> y = x \<oplus> \<ominus> y"
388   by (simp only: a_minus_def)
390 text \<open>Setup algebra method:
391   compute distributive normal form in locale contexts\<close>
393 ML_file "ringsimp.ML"
395 attribute_setup algebra = \<open>
396   Scan.lift ((Args.add >> K true || Args.del >> K false) --| Args.colon || Scan.succeed true)
397     -- Scan.lift Args.name -- Scan.repeat Args.term
398     >> (fn ((b, n), ts) => if b then Ringsimp.add_struct (n, ts) else Ringsimp.del_struct (n, ts))
399 \<close> "theorems controlling algebra method"
401 method_setup algebra = \<open>
402   Scan.succeed (SIMPLE_METHOD' o Ringsimp.algebra_tac)
403 \<close> "normalisation of algebraic structure"
405 lemmas (in semiring) semiring_simprules
406   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
407   a_closed zero_closed  m_closed one_closed
408   a_assoc l_zero  a_comm m_assoc l_one l_distr r_zero
409   a_lcomm r_distr l_null r_null
411 lemmas (in ring) ring_simprules
412   [algebra ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
413   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
414   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr minus_eq
415   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
416   a_lcomm r_distr l_null r_null l_minus r_minus
418 lemmas (in cring)
419   [algebra del: ring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
420   _
422 lemmas (in cring) cring_simprules
423   [algebra add: cring "zero R" "add R" "a_inv R" "a_minus R" "one R" "mult R"] =
424   a_closed zero_closed a_inv_closed minus_closed m_closed one_closed
425   a_assoc l_zero l_neg a_comm m_assoc l_one l_distr m_comm minus_eq
426   r_zero r_neg r_neg2 r_neg1 minus_add minus_minus minus_zero
427   a_lcomm m_lcomm r_distr l_null r_null l_minus r_minus
429 lemma (in semiring) nat_pow_zero:
430   "(n::nat) ~= 0 ==> \<zero> (^) n = \<zero>"
431   by (induct n) simp_all
433 context semiring begin
435 lemma one_zeroD:
436   assumes onezero: "\<one> = \<zero>"
437   shows "carrier R = {\<zero>}"
438 proof (rule, rule)
439   fix x
440   assume xcarr: "x \<in> carrier R"
441   from xcarr have "x = x \<otimes> \<one>" by simp
442   with onezero have "x = x \<otimes> \<zero>" by simp
443   with xcarr have "x = \<zero>" by simp
444   then show "x \<in> {\<zero>}" by fast
445 qed fast
447 lemma one_zeroI:
448   assumes carrzero: "carrier R = {\<zero>}"
449   shows "\<one> = \<zero>"
450 proof -
451   from one_closed and carrzero
452       show "\<one> = \<zero>" by simp
453 qed
455 lemma carrier_one_zero: "(carrier R = {\<zero>}) = (\<one> = \<zero>)"
456   apply rule
457    apply (erule one_zeroI)
458   apply (erule one_zeroD)
459   done
461 lemma carrier_one_not_zero: "(carrier R \<noteq> {\<zero>}) = (\<one> \<noteq> \<zero>)"
462   by (simp add: carrier_one_zero)
464 end
466 text \<open>Two examples for use of method algebra\<close>
468 lemma
469   fixes R (structure) and S (structure)
470   assumes "ring R" "cring S"
471   assumes RS: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier S" "d \<in> carrier S"
472   shows "a \<oplus> \<ominus> (a \<oplus> \<ominus> b) = b & c \<otimes>\<^bsub>S\<^esub> d = d \<otimes>\<^bsub>S\<^esub> c"
473 proof -
474   interpret ring R by fact
475   interpret cring S by fact
476   from RS show ?thesis by algebra
477 qed
479 lemma
480   fixes R (structure)
481   assumes "ring R"
482   assumes R: "a \<in> carrier R" "b \<in> carrier R"
483   shows "a \<ominus> (a \<ominus> b) = b"
484 proof -
485   interpret ring R by fact
486   from R show ?thesis by algebra
487 qed
490 subsubsection \<open>Sums over Finite Sets\<close>
492 lemma (in semiring) finsum_ldistr:
493   "[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==>
494    finsum R f A \<otimes> a = finsum R (%i. f i \<otimes> a) A"
495 proof (induct set: finite)
496   case empty then show ?case by simp
497 next
498   case (insert x F) then show ?case by (simp add: Pi_def l_distr)
499 qed
501 lemma (in semiring) finsum_rdistr:
502   "[| finite A; a \<in> carrier R; f \<in> A \<rightarrow> carrier R |] ==>
503    a \<otimes> finsum R f A = finsum R (%i. a \<otimes> f i) A"
504 proof (induct set: finite)
505   case empty then show ?case by simp
506 next
507   case (insert x F) then show ?case by (simp add: Pi_def r_distr)
508 qed
511 subsection \<open>Integral Domains\<close>
513 context "domain" begin
515 lemma zero_not_one [simp]:
516   "\<zero> ~= \<one>"
517   by (rule not_sym) simp
519 lemma integral_iff: (* not by default a simp rule! *)
520   "[| a \<in> carrier R; b \<in> carrier R |] ==> (a \<otimes> b = \<zero>) = (a = \<zero> | b = \<zero>)"
521 proof
522   assume "a \<in> carrier R" "b \<in> carrier R" "a \<otimes> b = \<zero>"
523   then show "a = \<zero> | b = \<zero>" by (simp add: integral)
524 next
525   assume "a \<in> carrier R" "b \<in> carrier R" "a = \<zero> | b = \<zero>"
526   then show "a \<otimes> b = \<zero>" by auto
527 qed
529 lemma m_lcancel:
530   assumes prem: "a ~= \<zero>"
531     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
532   shows "(a \<otimes> b = a \<otimes> c) = (b = c)"
533 proof
534   assume eq: "a \<otimes> b = a \<otimes> c"
535   with R have "a \<otimes> (b \<ominus> c) = \<zero>" by algebra
536   with R have "a = \<zero> | (b \<ominus> c) = \<zero>" by (simp add: integral_iff)
537   with prem and R have "b \<ominus> c = \<zero>" by auto
538   with R have "b = b \<ominus> (b \<ominus> c)" by algebra
539   also from R have "b \<ominus> (b \<ominus> c) = c" by algebra
540   finally show "b = c" .
541 next
542   assume "b = c" then show "a \<otimes> b = a \<otimes> c" by simp
543 qed
545 lemma m_rcancel:
546   assumes prem: "a ~= \<zero>"
547     and R: "a \<in> carrier R" "b \<in> carrier R" "c \<in> carrier R"
548   shows conc: "(b \<otimes> a = c \<otimes> a) = (b = c)"
549 proof -
550   from prem and R have "(a \<otimes> b = a \<otimes> c) = (b = c)" by (rule m_lcancel)
551   with R show ?thesis by algebra
552 qed
554 end
557 subsection \<open>Fields\<close>
559 text \<open>Field would not need to be derived from domain, the properties
560   for domain follow from the assumptions of field\<close>
561 lemma (in cring) cring_fieldI:
562   assumes field_Units: "Units R = carrier R - {\<zero>}"
563   shows "field R"
564 proof
565   from field_Units have "\<zero> \<notin> Units R" by fast
566   moreover have "\<one> \<in> Units R" by fast
567   ultimately show "\<one> \<noteq> \<zero>" by force
568 next
569   fix a b
570   assume acarr: "a \<in> carrier R"
571     and bcarr: "b \<in> carrier R"
572     and ab: "a \<otimes> b = \<zero>"
573   show "a = \<zero> \<or> b = \<zero>"
574   proof (cases "a = \<zero>", simp)
575     assume "a \<noteq> \<zero>"
576     with field_Units and acarr have aUnit: "a \<in> Units R" by fast
577     from bcarr have "b = \<one> \<otimes> b" by algebra
578     also from aUnit acarr have "... = (inv a \<otimes> a) \<otimes> b" by simp
579     also from acarr bcarr aUnit[THEN Units_inv_closed]
580     have "... = (inv a) \<otimes> (a \<otimes> b)" by algebra
581     also from ab and acarr bcarr aUnit have "... = (inv a) \<otimes> \<zero>" by simp
582     also from aUnit[THEN Units_inv_closed] have "... = \<zero>" by algebra
583     finally have "b = \<zero>" .
584     then show "a = \<zero> \<or> b = \<zero>" by simp
585   qed
586 qed (rule field_Units)
588 text \<open>Another variant to show that something is a field\<close>
589 lemma (in cring) cring_fieldI2:
590   assumes notzero: "\<zero> \<noteq> \<one>"
591   and invex: "\<And>a. \<lbrakk>a \<in> carrier R; a \<noteq> \<zero>\<rbrakk> \<Longrightarrow> \<exists>b\<in>carrier R. a \<otimes> b = \<one>"
592   shows "field R"
593   apply (rule cring_fieldI, simp add: Units_def)
594   apply (rule, clarsimp)
595   apply (simp add: notzero)
596 proof (clarsimp)
597   fix x
598   assume xcarr: "x \<in> carrier R"
599     and "x \<noteq> \<zero>"
600   then have "\<exists>y\<in>carrier R. x \<otimes> y = \<one>" by (rule invex)
601   then obtain y where ycarr: "y \<in> carrier R" and xy: "x \<otimes> y = \<one>" by fast
602   from xy xcarr ycarr have "y \<otimes> x = \<one>" by (simp add: m_comm)
603   with ycarr and xy show "\<exists>y\<in>carrier R. y \<otimes> x = \<one> \<and> x \<otimes> y = \<one>" by fast
604 qed
607 subsection \<open>Morphisms\<close>
609 definition
610   ring_hom :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme] => ('a => 'b) set"
611   where "ring_hom R S =
612     {h. h \<in> carrier R \<rightarrow> carrier S &
613       (ALL x y. x \<in> carrier R & y \<in> carrier R -->
614         h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y & h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) &
615       h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
617 lemma ring_hom_memI:
618   fixes R (structure) and S (structure)
619   assumes hom_closed: "!!x. x \<in> carrier R ==> h x \<in> carrier S"
620     and hom_mult: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
621       h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
622     and hom_add: "!!x y. [| x \<in> carrier R; y \<in> carrier R |] ==>
623       h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
624     and hom_one: "h \<one> = \<one>\<^bsub>S\<^esub>"
625   shows "h \<in> ring_hom R S"
626   by (auto simp add: ring_hom_def assms Pi_def)
628 lemma ring_hom_closed:
629   "[| h \<in> ring_hom R S; x \<in> carrier R |] ==> h x \<in> carrier S"
630   by (auto simp add: ring_hom_def funcset_mem)
632 lemma ring_hom_mult:
633   fixes R (structure) and S (structure)
634   shows
635     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
636     h (x \<otimes> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
637     by (simp add: ring_hom_def)
640   fixes R (structure) and S (structure)
641   shows
642     "[| h \<in> ring_hom R S; x \<in> carrier R; y \<in> carrier R |] ==>
643     h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
644     by (simp add: ring_hom_def)
646 lemma ring_hom_one:
647   fixes R (structure) and S (structure)
648   shows "h \<in> ring_hom R S ==> h \<one> = \<one>\<^bsub>S\<^esub>"
649   by (simp add: ring_hom_def)
651 locale ring_hom_cring = R?: cring R + S?: cring S
652     for R (structure) and S (structure) +
653   fixes h
654   assumes homh [simp, intro]: "h \<in> ring_hom R S"
655   notes hom_closed [simp, intro] = ring_hom_closed [OF homh]
656     and hom_mult [simp] = ring_hom_mult [OF homh]
657     and hom_add [simp] = ring_hom_add [OF homh]
658     and hom_one [simp] = ring_hom_one [OF homh]
660 lemma (in ring_hom_cring) hom_zero [simp]:
661   "h \<zero> = \<zero>\<^bsub>S\<^esub>"
662 proof -
663   have "h \<zero> \<oplus>\<^bsub>S\<^esub> h \<zero> = h \<zero> \<oplus>\<^bsub>S\<^esub> \<zero>\<^bsub>S\<^esub>"
665   then show ?thesis by (simp del: S.r_zero)
666 qed
668 lemma (in ring_hom_cring) hom_a_inv [simp]:
669   "x \<in> carrier R ==> h (\<ominus> x) = \<ominus>\<^bsub>S\<^esub> h x"
670 proof -
671   assume R: "x \<in> carrier R"
672   then have "h x \<oplus>\<^bsub>S\<^esub> h (\<ominus> x) = h x \<oplus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> h x)"
673     by (simp add: hom_add [symmetric] R.r_neg S.r_neg del: hom_add)
674   with R show ?thesis by simp
675 qed
677 lemma (in ring_hom_cring) hom_finsum [simp]:
678   "f \<in> A \<rightarrow> carrier R ==>
679   h (finsum R f A) = finsum S (h o f) A"
680   by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
682 lemma (in ring_hom_cring) hom_finprod:
683   "f \<in> A \<rightarrow> carrier R ==>
684   h (finprod R f A) = finprod S (h o f) A"
685   by (induct A rule: infinite_finite_induct, auto simp: Pi_def)
687 declare ring_hom_cring.hom_finprod [simp]
689 lemma id_ring_hom [simp]:
690   "id \<in> ring_hom R R"
691   by (auto intro!: ring_hom_memI)
693 end