linear arithmetic splits certain operators (e.g. min, max, abs)
2 Title: HOL/Algebra/UnivPoly.thy
4 Author: Clemens Ballarin, started 9 December 1996
5 Copyright: Clemens Ballarin
8 header {* Univariate Polynomials *}
10 theory UnivPoly imports Module begin
13 Polynomials are formalised as modules with additional operations for
14 extracting coefficients from polynomials and for obtaining monomials
15 from coefficients and exponents (record @{text "up_ring"}). The
16 carrier set is a set of bounded functions from Nat to the
17 coefficient domain. Bounded means that these functions return zero
18 above a certain bound (the degree). There is a chapter on the
19 formalisation of polynomials in the PhD thesis \cite{Ballarin:1999},
20 which was implemented with axiomatic type classes. This was later
25 subsection {* The Constructor for Univariate Polynomials *}
28 Functions with finite support.
35 assumes bound: "!!m. n < m \<Longrightarrow> f m = z"
37 declare bound.intro [intro!]
38 and bound.bound [dest]
41 assumes bound: "bound z m f" and nonzero: "f n \<noteq> z" shows "n \<le> m"
42 proof (rule classical)
44 then have "m < n" by arith
45 with bound have "f n = z" ..
46 with nonzero show ?thesis by contradiction
49 record ('a, 'p) up_ring = "('a, 'p) module" +
50 monom :: "['a, nat] => 'p"
51 coeff :: "['p, nat] => 'a"
53 constdefs (structure R)
54 up :: "('a, 'm) ring_scheme => (nat => 'a) set"
55 "up R == {f. f \<in> UNIV -> carrier R & (EX n. bound \<zero> n f)}"
56 UP :: "('a, 'm) ring_scheme => ('a, nat => 'a) up_ring"
59 mult = (%p:up R. %q:up R. %n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)),
60 one = (%i. if i=0 then \<one> else \<zero>),
62 add = (%p:up R. %q:up R. %i. p i \<oplus> q i),
63 smult = (%a:carrier R. %p:up R. %i. a \<otimes> p i),
64 monom = (%a:carrier R. %n i. if i=n then a else \<zero>),
65 coeff = (%p:up R. %n. p n) |)"
68 Properties of the set of polynomials @{term up}.
71 lemma mem_upI [intro]:
72 "[| !!n. f n \<in> carrier R; EX n. bound (zero R) n f |] ==> f \<in> up R"
73 by (simp add: up_def Pi_def)
76 "f \<in> up R ==> f n \<in> carrier R"
77 by (simp add: up_def Pi_def)
79 lemma (in cring) bound_upD [dest]:
80 "f \<in> up R ==> EX n. bound \<zero> n f"
83 lemma (in cring) up_one_closed:
84 "(%n. if n = 0 then \<one> else \<zero>) \<in> up R"
87 lemma (in cring) up_smult_closed:
88 "[| a \<in> carrier R; p \<in> up R |] ==> (%i. a \<otimes> p i) \<in> up R"
91 lemma (in cring) up_add_closed:
92 "[| p \<in> up R; q \<in> up R |] ==> (%i. p i \<oplus> q i) \<in> up R"
95 assume "p \<in> up R" and "q \<in> up R"
96 then show "p n \<oplus> q n \<in> carrier R"
99 assume UP: "p \<in> up R" "q \<in> up R"
100 show "EX n. bound \<zero> n (%i. p i \<oplus> q i)"
102 from UP obtain n where boundn: "bound \<zero> n p" by fast
103 from UP obtain m where boundm: "bound \<zero> m q" by fast
104 have "bound \<zero> (max n m) (%i. p i \<oplus> q i)"
108 with boundn and boundm and UP show "p i \<oplus> q i = \<zero>" by fastsimp
114 lemma (in cring) up_a_inv_closed:
115 "p \<in> up R ==> (%i. \<ominus> (p i)) \<in> up R"
117 assume R: "p \<in> up R"
118 then obtain n where "bound \<zero> n p" by auto
119 then have "bound \<zero> n (%i. \<ominus> p i)" by auto
120 then show "EX n. bound \<zero> n (%i. \<ominus> p i)" by auto
123 lemma (in cring) up_mult_closed:
124 "[| p \<in> up R; q \<in> up R |] ==>
125 (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> up R"
128 assume "p \<in> up R" "q \<in> up R"
129 then show "(\<Oplus>i \<in> {..n}. p i \<otimes> q (n-i)) \<in> carrier R"
130 by (simp add: mem_upD funcsetI)
132 assume UP: "p \<in> up R" "q \<in> up R"
133 show "EX n. bound \<zero> n (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n-i))"
135 from UP obtain n where boundn: "bound \<zero> n p" by fast
136 from UP obtain m where boundm: "bound \<zero> m q" by fast
137 have "bound \<zero> (n + m) (%n. \<Oplus>i \<in> {..n}. p i \<otimes> q (n - i))"
139 fix k assume bound: "n + m < k"
142 have "p i \<otimes> q (k-i) = \<zero>"
143 proof (cases "n < i")
145 with boundn have "p i = \<zero>" by auto
146 moreover from UP have "q (k-i) \<in> carrier R" by auto
147 ultimately show ?thesis by simp
150 with bound have "m < k-i" by arith
151 with boundm have "q (k-i) = \<zero>" by auto
152 moreover from UP have "p i \<in> carrier R" by auto
153 ultimately show ?thesis by simp
156 then show "(\<Oplus>i \<in> {..k}. p i \<otimes> q (k-i)) = \<zero>"
157 by (simp add: Pi_def)
159 then show ?thesis by fast
164 subsection {* Effect of operations on coefficients *}
167 fixes R (structure) and P (structure)
168 defines P_def: "P == UP R"
170 locale UP_cring = UP + cring R
172 locale UP_domain = UP_cring + "domain" R
175 Temporarily declare @{thm [locale=UP] P_def} as simp rule.
178 declare (in UP) P_def [simp]
180 lemma (in UP_cring) coeff_monom [simp]:
181 "a \<in> carrier R ==>
182 coeff P (monom P a m) n = (if m=n then a else \<zero>)"
184 assume R: "a \<in> carrier R"
185 then have "(%n. if n = m then a else \<zero>) \<in> up R"
186 using up_def by force
187 with R show ?thesis by (simp add: UP_def)
190 lemma (in UP_cring) coeff_zero [simp]:
191 "coeff P \<zero>\<^bsub>P\<^esub> n = \<zero>"
192 by (auto simp add: UP_def)
194 lemma (in UP_cring) coeff_one [simp]:
195 "coeff P \<one>\<^bsub>P\<^esub> n = (if n=0 then \<one> else \<zero>)"
196 using up_one_closed by (simp add: UP_def)
198 lemma (in UP_cring) coeff_smult [simp]:
199 "[| a \<in> carrier R; p \<in> carrier P |] ==>
200 coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
201 by (simp add: UP_def up_smult_closed)
203 lemma (in UP_cring) coeff_add [simp]:
204 "[| p \<in> carrier P; q \<in> carrier P |] ==>
205 coeff P (p \<oplus>\<^bsub>P\<^esub> q) n = coeff P p n \<oplus> coeff P q n"
206 by (simp add: UP_def up_add_closed)
208 lemma (in UP_cring) coeff_mult [simp]:
209 "[| p \<in> carrier P; q \<in> carrier P |] ==>
210 coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = (\<Oplus>i \<in> {..n}. coeff P p i \<otimes> coeff P q (n-i))"
211 by (simp add: UP_def up_mult_closed)
213 lemma (in UP) up_eqI:
214 assumes prem: "!!n. coeff P p n = coeff P q n"
215 and R: "p \<in> carrier P" "q \<in> carrier P"
219 from prem and R show "p x = q x" by (simp add: UP_def)
222 subsection {* Polynomials form a commutative ring. *}
224 text {* Operations are closed over @{term P}. *}
226 lemma (in UP_cring) UP_mult_closed [simp]:
227 "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<otimes>\<^bsub>P\<^esub> q \<in> carrier P"
228 by (simp add: UP_def up_mult_closed)
230 lemma (in UP_cring) UP_one_closed [simp]:
231 "\<one>\<^bsub>P\<^esub> \<in> carrier P"
232 by (simp add: UP_def up_one_closed)
234 lemma (in UP_cring) UP_zero_closed [intro, simp]:
235 "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
236 by (auto simp add: UP_def)
238 lemma (in UP_cring) UP_a_closed [intro, simp]:
239 "[| p \<in> carrier P; q \<in> carrier P |] ==> p \<oplus>\<^bsub>P\<^esub> q \<in> carrier P"
240 by (simp add: UP_def up_add_closed)
242 lemma (in UP_cring) monom_closed [simp]:
243 "a \<in> carrier R ==> monom P a n \<in> carrier P"
244 by (auto simp add: UP_def up_def Pi_def)
246 lemma (in UP_cring) UP_smult_closed [simp]:
247 "[| a \<in> carrier R; p \<in> carrier P |] ==> a \<odot>\<^bsub>P\<^esub> p \<in> carrier P"
248 by (simp add: UP_def up_smult_closed)
250 lemma (in UP) coeff_closed [simp]:
251 "p \<in> carrier P ==> coeff P p n \<in> carrier R"
252 by (auto simp add: UP_def)
254 declare (in UP) P_def [simp del]
256 text {* Algebraic ring properties *}
258 lemma (in UP_cring) UP_a_assoc:
259 assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
260 shows "(p \<oplus>\<^bsub>P\<^esub> q) \<oplus>\<^bsub>P\<^esub> r = p \<oplus>\<^bsub>P\<^esub> (q \<oplus>\<^bsub>P\<^esub> r)"
261 by (rule up_eqI, simp add: a_assoc R, simp_all add: R)
263 lemma (in UP_cring) UP_l_zero [simp]:
264 assumes R: "p \<in> carrier P"
265 shows "\<zero>\<^bsub>P\<^esub> \<oplus>\<^bsub>P\<^esub> p = p"
266 by (rule up_eqI, simp_all add: R)
268 lemma (in UP_cring) UP_l_neg_ex:
269 assumes R: "p \<in> carrier P"
270 shows "EX q : carrier P. q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
272 let ?q = "%i. \<ominus> (p i)"
273 from R have closed: "?q \<in> carrier P"
274 by (simp add: UP_def P_def up_a_inv_closed)
275 from R have coeff: "!!n. coeff P ?q n = \<ominus> (coeff P p n)"
276 by (simp add: UP_def P_def up_a_inv_closed)
279 show "?q \<oplus>\<^bsub>P\<^esub> p = \<zero>\<^bsub>P\<^esub>"
280 by (auto intro!: up_eqI simp add: R closed coeff R.l_neg)
284 lemma (in UP_cring) UP_a_comm:
285 assumes R: "p \<in> carrier P" "q \<in> carrier P"
286 shows "p \<oplus>\<^bsub>P\<^esub> q = q \<oplus>\<^bsub>P\<^esub> p"
287 by (rule up_eqI, simp add: a_comm R, simp_all add: R)
289 lemma (in UP_cring) UP_m_assoc:
290 assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
291 shows "(p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
295 fix k and a b c :: "nat=>'a"
296 assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
297 "c \<in> UNIV -> carrier R"
298 then have "k <= n ==>
299 (\<Oplus>j \<in> {..k}. (\<Oplus>i \<in> {..j}. a i \<otimes> b (j-i)) \<otimes> c (n-j)) =
300 (\<Oplus>j \<in> {..k}. a j \<otimes> (\<Oplus>i \<in> {..k-j}. b i \<otimes> c (n-j-i)))"
301 (is "_ \<Longrightarrow> ?eq k")
303 case 0 then show ?case by (simp add: Pi_def m_assoc)
306 then have "k <= n" by arith
307 then have "?eq k" by (rule Suc)
309 by (simp cong: finsum_cong
310 add: Suc_diff_le Pi_def l_distr r_distr m_assoc)
311 (simp cong: finsum_cong add: Pi_def a_ac finsum_ldistr m_assoc)
314 with R show "coeff P ((p \<otimes>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r) n = coeff P (p \<otimes>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)) n"
315 by (simp add: Pi_def)
316 qed (simp_all add: R)
318 lemma (in UP_cring) UP_l_one [simp]:
319 assumes R: "p \<in> carrier P"
320 shows "\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p = p"
323 show "coeff P (\<one>\<^bsub>P\<^esub> \<otimes>\<^bsub>P\<^esub> p) n = coeff P p n"
325 case 0 with R show ?thesis by simp
327 case Suc with R show ?thesis
328 by (simp del: finsum_Suc add: finsum_Suc2 Pi_def)
330 qed (simp_all add: R)
332 lemma (in UP_cring) UP_l_distr:
333 assumes R: "p \<in> carrier P" "q \<in> carrier P" "r \<in> carrier P"
334 shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = (p \<otimes>\<^bsub>P\<^esub> r) \<oplus>\<^bsub>P\<^esub> (q \<otimes>\<^bsub>P\<^esub> r)"
335 by (rule up_eqI) (simp add: l_distr R Pi_def, simp_all add: R)
337 lemma (in UP_cring) UP_m_comm:
338 assumes R: "p \<in> carrier P" "q \<in> carrier P"
339 shows "p \<otimes>\<^bsub>P\<^esub> q = q \<otimes>\<^bsub>P\<^esub> p"
343 fix k and a b :: "nat=>'a"
344 assume R: "a \<in> UNIV -> carrier R" "b \<in> UNIV -> carrier R"
345 then have "k <= n ==>
346 (\<Oplus>i \<in> {..k}. a i \<otimes> b (n-i)) =
347 (\<Oplus>i \<in> {..k}. a (k-i) \<otimes> b (i+n-k))"
348 (is "_ \<Longrightarrow> ?eq k")
350 case 0 then show ?case by (simp add: Pi_def)
352 case (Suc k) then show ?case
353 by (subst (2) finsum_Suc2) (simp add: Pi_def a_comm)+
357 from R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) n = coeff P (q \<otimes>\<^bsub>P\<^esub> p) n"
358 apply (simp add: Pi_def)
360 apply (auto simp add: Pi_def)
361 apply (simp add: m_comm)
363 qed (simp_all add: R)
365 theorem (in UP_cring) UP_cring:
367 by (auto intro!: cringI abelian_groupI comm_monoidI UP_a_assoc UP_l_zero
368 UP_l_neg_ex UP_a_comm UP_m_assoc UP_l_one UP_m_comm UP_l_distr)
370 lemma (in UP_cring) UP_ring:
372 we want "UP_ring R P ==> ring P", not "UP_cring R P ==> ring P" *)
374 by (auto intro: ring.intro cring.axioms UP_cring)
376 lemma (in UP_cring) UP_a_inv_closed [intro, simp]:
377 "p \<in> carrier P ==> \<ominus>\<^bsub>P\<^esub> p \<in> carrier P"
378 by (rule abelian_group.a_inv_closed
379 [OF ring.is_abelian_group [OF UP_ring]])
381 lemma (in UP_cring) coeff_a_inv [simp]:
382 assumes R: "p \<in> carrier P"
383 shows "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> (coeff P p n)"
385 from R coeff_closed UP_a_inv_closed have
386 "coeff P (\<ominus>\<^bsub>P\<^esub> p) n = \<ominus> coeff P p n \<oplus> (coeff P p n \<oplus> coeff P (\<ominus>\<^bsub>P\<^esub> p) n)"
388 also from R have "... = \<ominus> (coeff P p n)"
389 by (simp del: coeff_add add: coeff_add [THEN sym]
390 abelian_group.r_neg [OF ring.is_abelian_group [OF UP_ring]])
391 finally show ?thesis .
395 Interpretation of lemmas from @{term cring}. Saves lifting 43
399 interpretation UP_cring < cring P
401 (rule cring.axioms ring.axioms abelian_group.axioms comm_monoid.axioms UP_cring)+
404 subsection {* Polynomials form an Algebra *}
406 lemma (in UP_cring) UP_smult_l_distr:
407 "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
408 (a \<oplus> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> b \<odot>\<^bsub>P\<^esub> p"
409 by (rule up_eqI) (simp_all add: R.l_distr)
411 lemma (in UP_cring) UP_smult_r_distr:
412 "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
413 a \<odot>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = a \<odot>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> q"
414 by (rule up_eqI) (simp_all add: R.r_distr)
416 lemma (in UP_cring) UP_smult_assoc1:
417 "[| a \<in> carrier R; b \<in> carrier R; p \<in> carrier P |] ==>
418 (a \<otimes> b) \<odot>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> p)"
419 by (rule up_eqI) (simp_all add: R.m_assoc)
421 lemma (in UP_cring) UP_smult_one [simp]:
422 "p \<in> carrier P ==> \<one> \<odot>\<^bsub>P\<^esub> p = p"
423 by (rule up_eqI) simp_all
425 lemma (in UP_cring) UP_smult_assoc2:
426 "[| a \<in> carrier R; p \<in> carrier P; q \<in> carrier P |] ==>
427 (a \<odot>\<^bsub>P\<^esub> p) \<otimes>\<^bsub>P\<^esub> q = a \<odot>\<^bsub>P\<^esub> (p \<otimes>\<^bsub>P\<^esub> q)"
428 by (rule up_eqI) (simp_all add: R.finsum_rdistr R.m_assoc Pi_def)
431 Interpretation of lemmas from @{term algebra}.
434 lemma (in cring) cring:
436 by (fast intro: cring.intro prems)
438 lemma (in UP_cring) UP_algebra:
440 by (auto intro: algebraI R.cring UP_cring UP_smult_l_distr UP_smult_r_distr
441 UP_smult_assoc1 UP_smult_assoc2)
443 interpretation UP_cring < algebra R P
445 (rule module.axioms algebra.axioms UP_algebra)+
448 subsection {* Further lemmas involving monomials *}
450 lemma (in UP_cring) monom_zero [simp]:
451 "monom P \<zero> n = \<zero>\<^bsub>P\<^esub>"
452 by (simp add: UP_def P_def)
454 lemma (in UP_cring) monom_mult_is_smult:
455 assumes R: "a \<in> carrier R" "p \<in> carrier P"
456 shows "monom P a 0 \<otimes>\<^bsub>P\<^esub> p = a \<odot>\<^bsub>P\<^esub> p"
459 have "coeff P (p \<otimes>\<^bsub>P\<^esub> monom P a 0) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
461 case 0 with R show ?thesis by (simp add: R.m_comm)
463 case Suc with R show ?thesis
464 by (simp cong: R.finsum_cong add: R.r_null Pi_def)
467 with R show "coeff P (monom P a 0 \<otimes>\<^bsub>P\<^esub> p) n = coeff P (a \<odot>\<^bsub>P\<^esub> p) n"
468 by (simp add: UP_m_comm)
469 qed (simp_all add: R)
471 lemma (in UP_cring) monom_add [simp]:
472 "[| a \<in> carrier R; b \<in> carrier R |] ==>
473 monom P (a \<oplus> b) n = monom P a n \<oplus>\<^bsub>P\<^esub> monom P b n"
474 by (rule up_eqI) simp_all
476 lemma (in UP_cring) monom_one_Suc:
477 "monom P \<one> (Suc n) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1"
480 show "coeff P (monom P \<one> (Suc n)) k = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
481 proof (cases "k = Suc n")
482 case True show ?thesis
484 from True have less_add_diff:
485 "!!i. [| n < i; i <= n + m |] ==> n + m - i < m" by arith
486 from True have "coeff P (monom P \<one> (Suc n)) k = \<one>" by simp
488 have "... = (\<Oplus>i \<in> {..<n} \<union> {n}. coeff P (monom P \<one> n) i \<otimes>
489 coeff P (monom P \<one> 1) (k - i))"
490 by (simp cong: R.finsum_cong add: Pi_def)
491 also have "... = (\<Oplus>i \<in> {..n}. coeff P (monom P \<one> n) i \<otimes>
492 coeff P (monom P \<one> 1) (k - i))"
493 by (simp only: ivl_disj_un_singleton)
495 have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. coeff P (monom P \<one> n) i \<otimes>
496 coeff P (monom P \<one> 1) (k - i))"
497 by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
498 order_less_imp_not_eq Pi_def)
499 also from True have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k"
500 by (simp add: ivl_disj_un_one)
501 finally show ?thesis .
507 "\<lambda>i. (if n = i then \<one> else \<zero>) \<otimes> (if Suc 0 = k - i then \<one> else \<zero>)"
508 from neq have "coeff P (monom P \<one> (Suc n)) k = \<zero>" by simp
509 also have "... = (\<Oplus>i \<in> {..k}. ?s i)"
511 have f1: "(\<Oplus>i \<in> {..<n}. ?s i) = \<zero>"
512 by (simp cong: R.finsum_cong add: Pi_def)
513 from neq have f2: "(\<Oplus>i \<in> {n}. ?s i) = \<zero>"
514 by (simp cong: R.finsum_cong add: Pi_def) arith
515 have f3: "n < k ==> (\<Oplus>i \<in> {n<..k}. ?s i) = \<zero>"
516 by (simp cong: R.finsum_cong add: order_less_imp_not_eq Pi_def)
518 proof (cases "k < n")
519 case True then show ?thesis by (simp cong: R.finsum_cong add: Pi_def)
521 case False then have n_le_k: "n <= k" by arith
523 proof (cases "n = k")
525 then have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
526 by (simp cong: R.finsum_cong add: ivl_disj_int_singleton Pi_def)
527 also from True have "... = (\<Oplus>i \<in> {..k}. ?s i)"
528 by (simp only: ivl_disj_un_singleton)
529 finally show ?thesis .
531 case False with n_le_k have n_less_k: "n < k" by arith
532 with neq have "\<zero> = (\<Oplus>i \<in> {..<n} \<union> {n}. ?s i)"
533 by (simp add: R.finsum_Un_disjoint f1 f2
534 ivl_disj_int_singleton Pi_def del: Un_insert_right)
535 also have "... = (\<Oplus>i \<in> {..n}. ?s i)"
536 by (simp only: ivl_disj_un_singleton)
537 also from n_less_k neq have "... = (\<Oplus>i \<in> {..n} \<union> {n<..k}. ?s i)"
538 by (simp add: R.finsum_Un_disjoint f3 ivl_disj_int_one Pi_def)
539 also from n_less_k have "... = (\<Oplus>i \<in> {..k}. ?s i)"
540 by (simp only: ivl_disj_un_one)
541 finally show ?thesis .
545 also have "... = coeff P (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> 1) k" by simp
546 finally show ?thesis .
550 lemma (in UP_cring) monom_mult_smult:
551 "[| a \<in> carrier R; b \<in> carrier R |] ==> monom P (a \<otimes> b) n = a \<odot>\<^bsub>P\<^esub> monom P b n"
552 by (rule up_eqI) simp_all
554 lemma (in UP_cring) monom_one [simp]:
555 "monom P \<one> 0 = \<one>\<^bsub>P\<^esub>"
556 by (rule up_eqI) simp_all
558 lemma (in UP_cring) monom_one_mult:
559 "monom P \<one> (n + m) = monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m"
561 case 0 show ?case by simp
563 case Suc then show ?case
564 by (simp only: add_Suc monom_one_Suc) (simp add: P.m_ac)
567 lemma (in UP_cring) monom_mult [simp]:
568 assumes R: "a \<in> carrier R" "b \<in> carrier R"
569 shows "monom P (a \<otimes> b) (n + m) = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m"
571 from R have "monom P (a \<otimes> b) (n + m) = monom P (a \<otimes> b \<otimes> \<one>) (n + m)" by simp
572 also from R have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> monom P \<one> (n + m)"
573 by (simp add: monom_mult_smult del: R.r_one)
574 also have "... = a \<otimes> b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m)"
575 by (simp only: monom_one_mult)
576 also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> monom P \<one> m))"
577 by (simp add: UP_smult_assoc1)
578 also from R have "... = a \<odot>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> (monom P \<one> m \<otimes>\<^bsub>P\<^esub> monom P \<one> n))"
579 by (simp add: P.m_comm)
580 also from R have "... = a \<odot>\<^bsub>P\<^esub> ((b \<odot>\<^bsub>P\<^esub> monom P \<one> m) \<otimes>\<^bsub>P\<^esub> monom P \<one> n)"
581 by (simp add: UP_smult_assoc2)
582 also from R have "... = a \<odot>\<^bsub>P\<^esub> (monom P \<one> n \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m))"
583 by (simp add: P.m_comm)
584 also from R have "... = (a \<odot>\<^bsub>P\<^esub> monom P \<one> n) \<otimes>\<^bsub>P\<^esub> (b \<odot>\<^bsub>P\<^esub> monom P \<one> m)"
585 by (simp add: UP_smult_assoc2)
586 also from R have "... = monom P (a \<otimes> \<one>) n \<otimes>\<^bsub>P\<^esub> monom P (b \<otimes> \<one>) m"
587 by (simp add: monom_mult_smult del: R.r_one)
588 also from R have "... = monom P a n \<otimes>\<^bsub>P\<^esub> monom P b m" by simp
589 finally show ?thesis .
592 lemma (in UP_cring) monom_a_inv [simp]:
593 "a \<in> carrier R ==> monom P (\<ominus> a) n = \<ominus>\<^bsub>P\<^esub> monom P a n"
594 by (rule up_eqI) simp_all
596 lemma (in UP_cring) monom_inj:
597 "inj_on (%a. monom P a n) (carrier R)"
600 assume R: "x \<in> carrier R" "y \<in> carrier R" and eq: "monom P x n = monom P y n"
601 then have "coeff P (monom P x n) n = coeff P (monom P y n) n" by simp
602 with R show "x = y" by simp
606 subsection {* The degree function *}
608 constdefs (structure R)
609 deg :: "[('a, 'm) ring_scheme, nat => 'a] => nat"
610 "deg R p == LEAST n. bound \<zero> n (coeff (UP R) p)"
612 lemma (in UP_cring) deg_aboveI:
613 "[| (!!m. n < m ==> coeff P p m = \<zero>); p \<in> carrier P |] ==> deg R p <= n"
614 by (unfold deg_def P_def) (fast intro: Least_le)
617 lemma coeff_bound_ex: "EX n. bound n (coeff p)"
619 have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
620 then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
624 lemma bound_coeff_obtain:
625 assumes prem: "(!!n. bound n (coeff p) ==> P)" shows "P"
627 have "(%n. coeff p n) : UP" by (simp add: coeff_def Rep_UP)
628 then obtain n where "bound n (coeff p)" by (unfold UP_def) fast
633 lemma (in UP_cring) deg_aboveD:
634 "[| deg R p < m; p \<in> carrier P |] ==> coeff P p m = \<zero>"
636 assume R: "p \<in> carrier P" and "deg R p < m"
637 from R obtain n where "bound \<zero> n (coeff P p)"
638 by (auto simp add: UP_def P_def)
639 then have "bound \<zero> (deg R p) (coeff P p)"
640 by (auto simp: deg_def P_def dest: LeastI)
644 lemma (in UP_cring) deg_belowI:
645 assumes non_zero: "n ~= 0 ==> coeff P p n ~= \<zero>"
646 and R: "p \<in> carrier P"
648 -- {* Logically, this is a slightly stronger version of
649 @{thm [source] deg_aboveD} *}
651 case True then show ?thesis by simp
653 case False then have "coeff P p n ~= \<zero>" by (rule non_zero)
654 then have "~ deg R p < n" by (fast dest: deg_aboveD intro: R)
655 then show ?thesis by arith
658 lemma (in UP_cring) lcoeff_nonzero_deg:
659 assumes deg: "deg R p ~= 0" and R: "p \<in> carrier P"
660 shows "coeff P p (deg R p) ~= \<zero>"
662 from R obtain m where "deg R p <= m" and m_coeff: "coeff P p m ~= \<zero>"
664 have minus: "!!(n::nat) m. n ~= 0 ==> (n - Suc 0 < m) = (n <= m)"
666 (* TODO: why does simplification below not work with "1" *)
667 from deg have "deg R p - 1 < (LEAST n. bound \<zero> n (coeff P p))"
668 by (unfold deg_def P_def) arith
669 then have "~ bound \<zero> (deg R p - 1) (coeff P p)" by (rule not_less_Least)
670 then have "EX m. deg R p - 1 < m & coeff P p m ~= \<zero>"
671 by (unfold bound_def) fast
672 then have "EX m. deg R p <= m & coeff P p m ~= \<zero>" by (simp add: deg minus)
673 then show ?thesis by auto
675 with deg_belowI R have "deg R p = m" by fastsimp
676 with m_coeff show ?thesis by simp
679 lemma (in UP_cring) lcoeff_nonzero_nonzero:
680 assumes deg: "deg R p = 0" and nonzero: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
681 shows "coeff P p 0 ~= \<zero>"
683 have "EX m. coeff P p m ~= \<zero>"
684 proof (rule classical)
686 with R have "p = \<zero>\<^bsub>P\<^esub>" by (auto intro: up_eqI)
687 with nonzero show ?thesis by contradiction
689 then obtain m where coeff: "coeff P p m ~= \<zero>" ..
690 then have "m <= deg R p" by (rule deg_belowI)
691 then have "m = 0" by (simp add: deg)
692 with coeff show ?thesis by simp
695 lemma (in UP_cring) lcoeff_nonzero:
696 assumes neq: "p ~= \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P"
697 shows "coeff P p (deg R p) ~= \<zero>"
698 proof (cases "deg R p = 0")
699 case True with neq R show ?thesis by (simp add: lcoeff_nonzero_nonzero)
701 case False with neq R show ?thesis by (simp add: lcoeff_nonzero_deg)
704 lemma (in UP_cring) deg_eqI:
705 "[| !!m. n < m ==> coeff P p m = \<zero>;
706 !!n. n ~= 0 ==> coeff P p n ~= \<zero>; p \<in> carrier P |] ==> deg R p = n"
707 by (fast intro: le_anti_sym deg_aboveI deg_belowI)
709 text {* Degree and polynomial operations *}
711 lemma (in UP_cring) deg_add [simp]:
712 assumes R: "p \<in> carrier P" "q \<in> carrier P"
713 shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) <= max (deg R p) (deg R q)"
714 proof (cases "deg R p <= deg R q")
715 case True show ?thesis
716 by (rule deg_aboveI) (simp_all add: True R deg_aboveD)
718 case False show ?thesis
719 by (rule deg_aboveI) (simp_all add: False R deg_aboveD)
722 lemma (in UP_cring) deg_monom_le:
723 "a \<in> carrier R ==> deg R (monom P a n) <= n"
724 by (intro deg_aboveI) simp_all
726 lemma (in UP_cring) deg_monom [simp]:
727 "[| a ~= \<zero>; a \<in> carrier R |] ==> deg R (monom P a n) = n"
728 by (fastsimp intro: le_anti_sym deg_aboveI deg_belowI)
730 lemma (in UP_cring) deg_const [simp]:
731 assumes R: "a \<in> carrier R" shows "deg R (monom P a 0) = 0"
732 proof (rule le_anti_sym)
733 show "deg R (monom P a 0) <= 0" by (rule deg_aboveI) (simp_all add: R)
735 show "0 <= deg R (monom P a 0)" by (rule deg_belowI) (simp_all add: R)
738 lemma (in UP_cring) deg_zero [simp]:
739 "deg R \<zero>\<^bsub>P\<^esub> = 0"
740 proof (rule le_anti_sym)
741 show "deg R \<zero>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
743 show "0 <= deg R \<zero>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
746 lemma (in UP_cring) deg_one [simp]:
747 "deg R \<one>\<^bsub>P\<^esub> = 0"
748 proof (rule le_anti_sym)
749 show "deg R \<one>\<^bsub>P\<^esub> <= 0" by (rule deg_aboveI) simp_all
751 show "0 <= deg R \<one>\<^bsub>P\<^esub>" by (rule deg_belowI) simp_all
754 lemma (in UP_cring) deg_uminus [simp]:
755 assumes R: "p \<in> carrier P" shows "deg R (\<ominus>\<^bsub>P\<^esub> p) = deg R p"
756 proof (rule le_anti_sym)
757 show "deg R (\<ominus>\<^bsub>P\<^esub> p) <= deg R p" by (simp add: deg_aboveI deg_aboveD R)
759 show "deg R p <= deg R (\<ominus>\<^bsub>P\<^esub> p)"
760 by (simp add: deg_belowI lcoeff_nonzero_deg
761 inj_on_iff [OF R.a_inv_inj, of _ "\<zero>", simplified] R)
764 lemma (in UP_domain) deg_smult_ring:
765 "[| a \<in> carrier R; p \<in> carrier P |] ==>
766 deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
767 by (cases "a = \<zero>") (simp add: deg_aboveI deg_aboveD)+
769 lemma (in UP_domain) deg_smult [simp]:
770 assumes R: "a \<in> carrier R" "p \<in> carrier P"
771 shows "deg R (a \<odot>\<^bsub>P\<^esub> p) = (if a = \<zero> then 0 else deg R p)"
772 proof (rule le_anti_sym)
773 show "deg R (a \<odot>\<^bsub>P\<^esub> p) <= (if a = \<zero> then 0 else deg R p)"
774 by (rule deg_smult_ring)
776 show "(if a = \<zero> then 0 else deg R p) <= deg R (a \<odot>\<^bsub>P\<^esub> p)"
777 proof (cases "a = \<zero>")
778 qed (simp, simp add: deg_belowI lcoeff_nonzero_deg integral_iff R)
781 lemma (in UP_cring) deg_mult_cring:
782 assumes R: "p \<in> carrier P" "q \<in> carrier P"
783 shows "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q"
784 proof (rule deg_aboveI)
786 assume boundm: "deg R p + deg R q < m"
789 assume boundk: "deg R p + deg R q < k"
790 then have "coeff P p i \<otimes> coeff P q (k - i) = \<zero>"
791 proof (cases "deg R p < i")
792 case True then show ?thesis by (simp add: deg_aboveD R)
794 case False with boundk have "deg R q < k - i" by arith
795 then show ?thesis by (simp add: deg_aboveD R)
798 with boundm R show "coeff P (p \<otimes>\<^bsub>P\<^esub> q) m = \<zero>" by simp
801 lemma (in UP_domain) deg_mult [simp]:
802 "[| p ~= \<zero>\<^bsub>P\<^esub>; q ~= \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==>
803 deg R (p \<otimes>\<^bsub>P\<^esub> q) = deg R p + deg R q"
804 proof (rule le_anti_sym)
805 assume "p \<in> carrier P" " q \<in> carrier P"
806 show "deg R (p \<otimes>\<^bsub>P\<^esub> q) <= deg R p + deg R q" by (rule deg_mult_cring)
808 let ?s = "(%i. coeff P p i \<otimes> coeff P q (deg R p + deg R q - i))"
809 assume R: "p \<in> carrier P" "q \<in> carrier P" and nz: "p ~= \<zero>\<^bsub>P\<^esub>" "q ~= \<zero>\<^bsub>P\<^esub>"
810 have less_add_diff: "!!(k::nat) n m. k < n ==> m < n + m - k" by arith
811 show "deg R p + deg R q <= deg R (p \<otimes>\<^bsub>P\<^esub> q)"
812 proof (rule deg_belowI, simp add: R)
813 have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
814 = (\<Oplus>i \<in> {..< deg R p} \<union> {deg R p .. deg R p + deg R q}. ?s i)"
815 by (simp only: ivl_disj_un_one)
816 also have "... = (\<Oplus>i \<in> {deg R p .. deg R p + deg R q}. ?s i)"
817 by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint ivl_disj_int_one
818 deg_aboveD less_add_diff R Pi_def)
819 also have "...= (\<Oplus>i \<in> {deg R p} \<union> {deg R p <.. deg R p + deg R q}. ?s i)"
820 by (simp only: ivl_disj_un_singleton)
821 also have "... = coeff P p (deg R p) \<otimes> coeff P q (deg R q)"
822 by (simp cong: R.finsum_cong
823 add: ivl_disj_int_singleton deg_aboveD R Pi_def)
824 finally have "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i)
825 = coeff P p (deg R p) \<otimes> coeff P q (deg R q)" .
826 with nz show "(\<Oplus>i \<in> {.. deg R p + deg R q}. ?s i) ~= \<zero>"
827 by (simp add: integral_iff lcoeff_nonzero R)
831 lemma (in UP_cring) coeff_finsum:
832 assumes fin: "finite A"
833 shows "p \<in> A -> carrier P ==>
834 coeff P (finsum P p A) k = (\<Oplus>i \<in> A. coeff P (p i) k)"
835 using fin by induct (auto simp: Pi_def)
837 lemma (in UP_cring) up_repr:
838 assumes R: "p \<in> carrier P"
839 shows "(\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. monom P (coeff P p i) i) = p"
841 let ?s = "(%i. monom P (coeff P p i) i)"
843 from R have RR: "!!i. (if i = k then coeff P p i else \<zero>) \<in> carrier R"
845 show "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k = coeff P p k"
846 proof (cases "k <= deg R p")
848 hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
849 coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k} \<union> {k<..deg R p}. ?s i) k"
850 by (simp only: ivl_disj_un_one)
852 have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..k}. ?s i) k"
853 by (simp cong: R.finsum_cong add: R.finsum_Un_disjoint
854 ivl_disj_int_one order_less_imp_not_eq2 coeff_finsum R RR Pi_def)
856 have "... = coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<k} \<union> {k}. ?s i) k"
857 by (simp only: ivl_disj_un_singleton)
858 also have "... = coeff P p k"
859 by (simp cong: R.finsum_cong
860 add: ivl_disj_int_singleton coeff_finsum deg_aboveD R RR Pi_def)
861 finally show ?thesis .
864 hence "coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..deg R p}. ?s i) k =
865 coeff P (\<Oplus>\<^bsub>P\<^esub> i \<in> {..<deg R p} \<union> {deg R p}. ?s i) k"
866 by (simp only: ivl_disj_un_singleton)
867 also from False have "... = coeff P p k"
868 by (simp cong: R.finsum_cong
869 add: ivl_disj_int_singleton coeff_finsum deg_aboveD R Pi_def)
870 finally show ?thesis .
872 qed (simp_all add: R Pi_def)
874 lemma (in UP_cring) up_repr_le:
875 "[| deg R p <= n; p \<in> carrier P |] ==>
876 (\<Oplus>\<^bsub>P\<^esub> i \<in> {..n}. monom P (coeff P p i) i) = p"
878 let ?s = "(%i. monom P (coeff P p i) i)"
879 assume R: "p \<in> carrier P" and "deg R p <= n"
880 then have "finsum P ?s {..n} = finsum P ?s ({..deg R p} \<union> {deg R p<..n})"
881 by (simp only: ivl_disj_un_one)
882 also have "... = finsum P ?s {..deg R p}"
883 by (simp cong: P.finsum_cong add: P.finsum_Un_disjoint ivl_disj_int_one
885 also have "... = p" by (rule up_repr)
886 finally show ?thesis .
890 subsection {* Polynomials over an integral domain form an integral domain *}
893 assumes cring: "cring R"
894 and one_not_zero: "one R ~= zero R"
895 and integral: "!!a b. [| mult R a b = zero R; a \<in> carrier R;
896 b \<in> carrier R |] ==> a = zero R | b = zero R"
898 by (auto intro!: domain.intro domain_axioms.intro cring.axioms prems
901 lemma (in UP_domain) UP_one_not_zero:
902 "\<one>\<^bsub>P\<^esub> ~= \<zero>\<^bsub>P\<^esub>"
904 assume "\<one>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
905 hence "coeff P \<one>\<^bsub>P\<^esub> 0 = (coeff P \<zero>\<^bsub>P\<^esub> 0)" by simp
906 hence "\<one> = \<zero>" by simp
907 with one_not_zero show "False" by contradiction
910 lemma (in UP_domain) UP_integral:
911 "[| p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>; p \<in> carrier P; q \<in> carrier P |] ==> p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
914 assume pq: "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>" and R: "p \<in> carrier P" "q \<in> carrier P"
915 show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>"
916 proof (rule classical)
917 assume c: "~ (p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>)"
918 with R have "deg R p + deg R q = deg R (p \<otimes>\<^bsub>P\<^esub> q)" by simp
919 also from pq have "... = 0" by simp
920 finally have "deg R p + deg R q = 0" .
921 then have f1: "deg R p = 0 & deg R q = 0" by simp
922 from f1 R have "p = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P p i) i)"
923 by (simp only: up_repr_le)
924 also from R have "... = monom P (coeff P p 0) 0" by simp
925 finally have p: "p = monom P (coeff P p 0) 0" .
926 from f1 R have "q = (\<Oplus>\<^bsub>P\<^esub> i \<in> {..0}. monom P (coeff P q i) i)"
927 by (simp only: up_repr_le)
928 also from R have "... = monom P (coeff P q 0) 0" by simp
929 finally have q: "q = monom P (coeff P q 0) 0" .
930 from R have "coeff P p 0 \<otimes> coeff P q 0 = coeff P (p \<otimes>\<^bsub>P\<^esub> q) 0" by simp
931 also from pq have "... = \<zero>" by simp
932 finally have "coeff P p 0 \<otimes> coeff P q 0 = \<zero>" .
933 with R have "coeff P p 0 = \<zero> | coeff P q 0 = \<zero>"
934 by (simp add: R.integral_iff)
935 with p q show "p = \<zero>\<^bsub>P\<^esub> | q = \<zero>\<^bsub>P\<^esub>" by fastsimp
939 theorem (in UP_domain) UP_domain:
941 by (auto intro!: domainI UP_cring UP_one_not_zero UP_integral del: disjCI)
944 Interpretation of theorems from @{term domain}.
947 interpretation UP_domain < "domain" P
948 by intro_locales (rule domain.axioms UP_domain)+
951 subsection {* Evaluation Homomorphism and Universal Property*}
953 (* alternative congruence rule (possibly more efficient)
954 lemma (in abelian_monoid) finsum_cong2:
955 "[| !!i. i \<in> A ==> f i \<in> carrier G = True; A = B;
956 !!i. i \<in> B ==> f i = g i |] ==> finsum G f A = finsum G g B"
959 theorem (in cring) diagonal_sum:
960 "[| f \<in> {..n + m::nat} -> carrier R; g \<in> {..n + m} -> carrier R |] ==>
961 (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
962 (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
964 assume Rf: "f \<in> {..n + m} -> carrier R" and Rg: "g \<in> {..n + m} -> carrier R"
968 (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
969 (\<Oplus>k \<in> {..j}. \<Oplus>i \<in> {..j - k}. f k \<otimes> g i)"
971 case 0 from Rf Rg show ?case by (simp add: Pi_def)
974 have R6: "!!i k. [| k <= j; i <= Suc j - k |] ==> g i \<in> carrier R"
975 using Suc by (auto intro!: funcset_mem [OF Rg])
976 have R8: "!!i k. [| k <= Suc j; i <= k |] ==> g (k - i) \<in> carrier R"
977 using Suc by (auto intro!: funcset_mem [OF Rg])
978 have R9: "!!i k. [| k <= Suc j |] ==> f k \<in> carrier R"
979 using Suc by (auto intro!: funcset_mem [OF Rf])
980 have R10: "!!i k. [| k <= Suc j; i <= Suc j - k |] ==> g i \<in> carrier R"
981 using Suc by (auto intro!: funcset_mem [OF Rg])
982 have R11: "g 0 \<in> carrier R"
983 using Suc by (auto intro!: funcset_mem [OF Rg])
985 by (simp cong: finsum_cong add: Suc_diff_le a_ac
986 Pi_def R6 R8 R9 R10 R11)
989 then show ?thesis by fast
992 lemma (in abelian_monoid) boundD_carrier:
993 "[| bound \<zero> n f; n < m |] ==> f m \<in> carrier G"
996 theorem (in cring) cauchy_product:
997 assumes bf: "bound \<zero> n f" and bg: "bound \<zero> m g"
998 and Rf: "f \<in> {..n} -> carrier R" and Rg: "g \<in> {..m} -> carrier R"
999 shows "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1000 (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)" (* State reverse direction? *)
1002 have f: "!!x. f x \<in> carrier R"
1005 show "f x \<in> carrier R"
1006 using Rf bf boundD_carrier by (cases "x <= n") (auto simp: Pi_def)
1008 have g: "!!x. g x \<in> carrier R"
1011 show "g x \<in> carrier R"
1012 using Rg bg boundD_carrier by (cases "x <= m") (auto simp: Pi_def)
1014 from f g have "(\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..k}. f i \<otimes> g (k - i)) =
1015 (\<Oplus>k \<in> {..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1016 by (simp add: diagonal_sum Pi_def)
1017 also have "... = (\<Oplus>k \<in> {..n} \<union> {n<..n + m}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1018 by (simp only: ivl_disj_un_one)
1019 also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..n + m - k}. f k \<otimes> g i)"
1020 by (simp cong: finsum_cong
1021 add: bound.bound [OF bf] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1023 have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m} \<union> {m<..n + m - k}. f k \<otimes> g i)"
1024 by (simp cong: finsum_cong add: ivl_disj_un_one le_add_diff Pi_def)
1025 also from f g have "... = (\<Oplus>k \<in> {..n}. \<Oplus>i \<in> {..m}. f k \<otimes> g i)"
1026 by (simp cong: finsum_cong
1027 add: bound.bound [OF bg] finsum_Un_disjoint ivl_disj_int_one Pi_def)
1028 also from f g have "... = (\<Oplus>i \<in> {..n}. f i) \<otimes> (\<Oplus>i \<in> {..m}. g i)"
1029 by (simp add: finsum_ldistr diagonal_sum Pi_def,
1030 simp cong: finsum_cong add: finsum_rdistr Pi_def)
1031 finally show ?thesis .
1034 lemma (in UP_cring) const_ring_hom:
1035 "(%a. monom P a 0) \<in> ring_hom R P"
1036 by (auto intro!: ring_hom_memI intro: up_eqI simp: monom_mult_is_smult)
1038 constdefs (structure S)
1039 eval :: "[('a, 'm) ring_scheme, ('b, 'n) ring_scheme,
1040 'a => 'b, 'b, nat => 'a] => 'b"
1041 "eval R S phi s == \<lambda>p \<in> carrier (UP R).
1042 \<Oplus>i \<in> {..deg R p}. phi (coeff (UP R) p i) \<otimes> s (^) i"
1045 lemma (in UP) eval_on_carrier:
1047 shows "p \<in> carrier P ==>
1048 eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1049 by (unfold eval_def, fold P_def) simp
1051 lemma (in UP) eval_extensional:
1052 "eval R S phi p \<in> extensional (carrier P)"
1053 by (unfold eval_def, fold P_def) simp
1056 text {* The universal property of the polynomial ring *}
1058 locale UP_pre_univ_prop = ring_hom_cring R S h + UP_cring R P
1060 locale UP_univ_prop = UP_pre_univ_prop +
1062 assumes indet_img_carrier [simp, intro]: "s \<in> carrier S"
1063 defines Eval_def: "Eval == eval R S h s"
1065 theorem (in UP_pre_univ_prop) eval_ring_hom:
1066 assumes S: "s \<in> carrier S"
1067 shows "eval R S h s \<in> ring_hom P S"
1068 proof (rule ring_hom_memI)
1070 assume R: "p \<in> carrier P"
1071 then show "eval R S h s p \<in> carrier S"
1072 by (simp only: eval_on_carrier) (simp add: S Pi_def)
1075 assume R: "p \<in> carrier P" "q \<in> carrier P"
1076 then show "eval R S h s (p \<otimes>\<^bsub>P\<^esub> q) = eval R S h s p \<otimes>\<^bsub>S\<^esub> eval R S h s q"
1077 proof (simp only: eval_on_carrier UP_mult_closed)
1079 "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
1080 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<otimes>\<^bsub>P\<^esub> q)<..deg R p + deg R q}.
1081 h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1082 by (simp cong: S.finsum_cong
1083 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
1085 also from R have "... =
1086 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1087 by (simp only: ivl_disj_un_one deg_mult_cring)
1088 also from R S have "... =
1089 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p + deg R q}.
1090 \<Oplus>\<^bsub>S\<^esub> k \<in> {..i}.
1091 h (coeff P p k) \<otimes>\<^bsub>S\<^esub> h (coeff P q (i - k)) \<otimes>\<^bsub>S\<^esub>
1092 (s (^)\<^bsub>S\<^esub> k \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> (i - k)))"
1093 by (simp cong: S.finsum_cong add: S.nat_pow_mult Pi_def
1094 S.m_ac S.finsum_rdistr)
1095 also from R S have "... =
1096 (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
1097 (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1098 by (simp add: S.cauchy_product [THEN sym] bound.intro deg_aboveD S.m_ac
1101 "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (p \<otimes>\<^bsub>P\<^esub> q)}. h (coeff P (p \<otimes>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
1102 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<otimes>\<^bsub>S\<^esub>
1103 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
1107 assume R: "p \<in> carrier P" "q \<in> carrier P"
1108 then show "eval R S h s (p \<oplus>\<^bsub>P\<^esub> q) = eval R S h s p \<oplus>\<^bsub>S\<^esub> eval R S h s q"
1109 proof (simp only: eval_on_carrier P.a_closed)
1111 "(\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
1112 (\<Oplus>\<^bsub>S \<^esub>i\<in>{..deg R (p \<oplus>\<^bsub>P\<^esub> q)} \<union> {deg R (p \<oplus>\<^bsub>P\<^esub> q)<..max (deg R p) (deg R q)}.
1113 h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1114 by (simp cong: S.finsum_cong
1115 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def
1117 also from R have "... =
1118 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..max (deg R p) (deg R q)}.
1119 h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1120 by (simp add: ivl_disj_un_one)
1121 also from R S have "... =
1122 (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
1123 (\<Oplus>\<^bsub>S\<^esub>i\<in>{..max (deg R p) (deg R q)}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1124 by (simp cong: S.finsum_cong
1125 add: S.l_distr deg_aboveD ivl_disj_int_one Pi_def)
1127 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p} \<union> {deg R p<..max (deg R p) (deg R q)}.
1128 h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
1129 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q} \<union> {deg R q<..max (deg R p) (deg R q)}.
1130 h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1131 by (simp only: ivl_disj_un_one le_maxI1 le_maxI2)
1132 also from R S have "... =
1133 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
1134 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1135 by (simp cong: S.finsum_cong
1136 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
1138 "(\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R (p \<oplus>\<^bsub>P\<^esub> q)}. h (coeff P (p \<oplus>\<^bsub>P\<^esub> q) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
1139 (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R p}. h (coeff P p i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) \<oplus>\<^bsub>S\<^esub>
1140 (\<Oplus>\<^bsub>S\<^esub>i \<in> {..deg R q}. h (coeff P q i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)" .
1143 show "eval R S h s \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>S\<^esub>"
1144 by (simp only: eval_on_carrier UP_one_closed) simp
1147 text {* Interpretation of ring homomorphism lemmas. *}
1149 interpretation UP_univ_prop < ring_hom_cring P S Eval
1150 apply (unfold Eval_def)
1152 apply (rule ring_hom_cring.axioms)
1153 apply (rule ring_hom_cring.intro)
1154 apply unfold_locales
1155 apply (rule eval_ring_hom)
1160 text {* Further properties of the evaluation homomorphism. *}
1162 (* The following lemma could be proved in UP\_cring with the additional
1163 assumption that h is closed. *)
1165 lemma (in UP_pre_univ_prop) eval_const:
1166 "[| s \<in> carrier S; r \<in> carrier R |] ==> eval R S h s (monom P r 0) = h r"
1167 by (simp only: eval_on_carrier monom_closed) simp
1169 text {* The following proof is complicated by the fact that in arbitrary
1170 rings one might have @{term "one R = zero R"}. *}
1172 (* TODO: simplify by cases "one R = zero R" *)
1174 lemma (in UP_pre_univ_prop) eval_monom1:
1175 assumes S: "s \<in> carrier S"
1176 shows "eval R S h s (monom P \<one> 1) = s"
1177 proof (simp only: eval_on_carrier monom_closed R.one_closed)
1179 "(\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) =
1180 (\<Oplus>\<^bsub>S\<^esub> i\<in>{..deg R (monom P \<one> 1)} \<union> {deg R (monom P \<one> 1)<..1}.
1181 h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1182 by (simp cong: S.finsum_cong del: coeff_monom
1183 add: deg_aboveD S.finsum_Un_disjoint ivl_disj_int_one Pi_def)
1185 (\<Oplus>\<^bsub>S\<^esub> i \<in> {..1}. h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i)"
1186 by (simp only: ivl_disj_un_one deg_monom_le R.one_closed)
1188 proof (cases "s = \<zero>\<^bsub>S\<^esub>")
1189 case True then show ?thesis by (simp add: Pi_def)
1191 case False then show ?thesis by (simp add: S Pi_def)
1193 finally show "(\<Oplus>\<^bsub>S\<^esub> i \<in> {..deg R (monom P \<one> 1)}.
1194 h (coeff P (monom P \<one> 1) i) \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> i) = s" .
1197 lemma (in UP_cring) monom_pow:
1198 assumes R: "a \<in> carrier R"
1199 shows "(monom P a n) (^)\<^bsub>P\<^esub> m = monom P (a (^) m) (n * m)"
1201 case 0 from R show ?case by simp
1203 case Suc with R show ?case
1204 by (simp del: monom_mult add: monom_mult [THEN sym] add_commute)
1207 lemma (in ring_hom_cring) hom_pow [simp]:
1208 "x \<in> carrier R ==> h (x (^) n) = h x (^)\<^bsub>S\<^esub> (n::nat)"
1209 by (induct n) simp_all
1211 lemma (in UP_univ_prop) Eval_monom:
1212 "r \<in> carrier R ==> Eval (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
1214 assume R: "r \<in> carrier R"
1215 from R have "Eval (monom P r n) = Eval (monom P r 0 \<otimes>\<^bsub>P\<^esub> (monom P \<one> 1) (^)\<^bsub>P\<^esub> n)"
1216 by (simp del: monom_mult add: monom_mult [THEN sym] monom_pow)
1218 from R eval_monom1 [where s = s, folded Eval_def]
1219 have "... = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
1220 by (simp add: eval_const [where s = s, folded Eval_def])
1221 finally show ?thesis .
1224 lemma (in UP_pre_univ_prop) eval_monom:
1225 assumes R: "r \<in> carrier R" and S: "s \<in> carrier S"
1226 shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s (^)\<^bsub>S\<^esub> n"
1228 interpret UP_univ_prop [R S h P s _]
1229 by (auto! intro: UP_univ_prop.intro UP_univ_prop_axioms.intro)
1231 show ?thesis by (rule Eval_monom)
1234 lemma (in UP_univ_prop) Eval_smult:
1235 "[| r \<in> carrier R; p \<in> carrier P |] ==> Eval (r \<odot>\<^bsub>P\<^esub> p) = h r \<otimes>\<^bsub>S\<^esub> Eval p"
1237 assume R: "r \<in> carrier R" and P: "p \<in> carrier P"
1239 by (simp add: monom_mult_is_smult [THEN sym]
1240 eval_const [where s = s, folded Eval_def])
1243 lemma ring_hom_cringI:
1246 and "h \<in> ring_hom R S"
1247 shows "ring_hom_cring R S h"
1248 by (fast intro: ring_hom_cring.intro ring_hom_cring_axioms.intro
1251 lemma (in UP_pre_univ_prop) UP_hom_unique:
1252 includes ring_hom_cring P S Phi
1253 assumes Phi: "Phi (monom P \<one> (Suc 0)) = s"
1254 "!!r. r \<in> carrier R ==> Phi (monom P r 0) = h r"
1255 includes ring_hom_cring P S Psi
1256 assumes Psi: "Psi (monom P \<one> (Suc 0)) = s"
1257 "!!r. r \<in> carrier R ==> Psi (monom P r 0) = h r"
1258 and P: "p \<in> carrier P" and S: "s \<in> carrier S"
1259 shows "Phi p = Psi p"
1262 Phi (\<Oplus>\<^bsub>P \<^esub>i \<in> {..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
1263 by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
1266 Psi (\<Oplus>\<^bsub>P \<^esub>i\<in>{..deg R p}. monom P (coeff P p i) 0 \<otimes>\<^bsub>P\<^esub> monom P \<one> 1 (^)\<^bsub>P\<^esub> i)"
1267 by (simp add: Phi Psi P Pi_def comp_def)
1268 also have "... = Psi p"
1269 by (simp add: up_repr P monom_mult [THEN sym] monom_pow del: monom_mult)
1270 finally show ?thesis .
1273 lemma (in UP_pre_univ_prop) ring_homD:
1274 assumes Phi: "Phi \<in> ring_hom P S"
1275 shows "ring_hom_cring P S Phi"
1276 proof (rule ring_hom_cring.intro)
1277 show "ring_hom_cring_axioms P S Phi"
1278 by (rule ring_hom_cring_axioms.intro) (rule Phi)
1281 theorem (in UP_pre_univ_prop) UP_universal_property:
1282 assumes S: "s \<in> carrier S"
1283 shows "EX! Phi. Phi \<in> ring_hom P S \<inter> extensional (carrier P) &
1284 Phi (monom P \<one> 1) = s &
1285 (ALL r : carrier R. Phi (monom P r 0) = h r)"
1287 apply (auto intro: eval_ring_hom eval_const eval_extensional)
1288 apply (rule extensionalityI)
1289 apply (auto intro: UP_hom_unique ring_homD)
1293 subsection {* Sample application of evaluation homomorphism *}
1295 lemma UP_pre_univ_propI:
1298 and "h \<in> ring_hom R S"
1299 shows "UP_pre_univ_prop R S h"
1300 by (auto intro!: UP_pre_univ_prop.intro ring_hom_cring.intro
1301 ring_hom_cring_axioms.intro UP_cring.intro)
1305 "INTEG == (| carrier = UNIV, mult = op *, one = 1, zero = 0, add = op + |)"
1309 by (unfold INTEG_def) (auto intro!: cringI abelian_groupI comm_monoidI
1310 zadd_zminus_inverse2 zadd_zmult_distrib)
1312 lemma INTEG_id_eval:
1313 "UP_pre_univ_prop INTEG INTEG id"
1314 by (fast intro: UP_pre_univ_propI INTEG_cring id_ring_hom)
1317 Interpretation now enables to import all theorems and lemmas
1318 valid in the context of homomorphisms between @{term INTEG} and @{term
1319 "UP INTEG"} globally.
1322 interpretation INTEG: UP_pre_univ_prop [INTEG INTEG id]
1328 lemma INTEG_closed [intro, simp]:
1329 "z \<in> carrier INTEG"
1330 by (unfold INTEG_def) simp
1332 lemma INTEG_mult [simp]:
1333 "mult INTEG z w = z * w"
1334 by (unfold INTEG_def) simp
1336 lemma INTEG_pow [simp]:
1337 "pow INTEG z n = z ^ n"
1338 by (induct n) (simp_all add: INTEG_def nat_pow_def)
1340 lemma "eval INTEG INTEG id 10 (monom (UP INTEG) 5 2) = 500"
1341 by (simp add: INTEG.eval_monom)