1 (* Title: Univ_Poly.thy
6 header{*Univariate Polynomials*}
9 imports Plain "~~/src/HOL/List"
12 text{* Application of polynomial as a function. *}
14 primrec (in semiring_0) poly :: "'a list => 'a => 'a" where
15 poly_Nil: "poly [] x = 0"
16 | poly_Cons: "poly (h#t) x = h + x * poly t x"
19 subsection{*Arithmetic Operations on Polynomials*}
23 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "+++" 65)
25 padd_Nil: "[] +++ l2 = l2"
26 | padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
27 else (h + hd l2)#(t +++ tl l2))"
29 text{*Multiplication by a constant*}
30 primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "%*" 70) where
31 cmult_Nil: "c %* [] = []"
32 | cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
34 text{*Multiplication by a polynomial*}
35 primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "***" 70)
37 pmult_Nil: "[] *** l2 = []"
38 | pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
39 else (h %* l2) +++ ((0) # (t *** l2)))"
41 text{*Repeated multiplication by a polynomial*}
42 primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
43 mulexp_zero: "mulexp 0 p q = q"
44 | mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
47 primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list" (infixl "%^" 80) where
48 pexp_0: "p %^ 0 = [1]"
49 | pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
51 text{*Quotient related value of dividing a polynomial by x + a*}
52 (* Useful for divisor properties in inductive proofs *)
53 primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
54 pquot_Nil: "pquot [] a= []"
55 | pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
56 else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
58 text{*normalization of polynomials (remove extra 0 coeff)*}
59 primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
60 pnormalize_Nil: "pnormalize [] = []"
61 | pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
62 then (if (h = 0) then [] else [h])
63 else (h#(pnormalize p)))"
65 definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
66 definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
67 text{*Other definitions*}
69 definition (in ring_1)
70 poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where
73 definition (in semiring_0)
74 divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "divides" 70) where
75 [code del]: "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
77 --{*order of a polynomial*}
78 definition (in ring_1) order :: "'a => 'a list => nat" where
79 "order a p = (SOME n. ([-a, 1] %^ n) divides p &
80 ~ (([-a, 1] %^ (Suc n)) divides p))"
82 --{*degree of a polynomial*}
83 definition (in semiring_0) degree :: "'a list => nat" where
84 "degree p = length (pnormalize p) - 1"
86 --{*squarefree polynomials --- NB with respect to real roots only.*}
87 definition (in ring_1)
88 rsquarefree :: "'a list => bool" where
89 "rsquarefree p = (poly p \<noteq> poly [] &
90 (\<forall>a. (order a p = 0) | (order a p = 1)))"
95 lemma padd_Nil2[simp]: "p +++ [] = p"
98 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
101 lemma pminus_Nil[simp]: "-- [] = []"
102 by (simp add: poly_minus_def)
104 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
107 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct "t", auto)
109 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
112 text{*Handy general properties*}
114 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
115 proof(induct b arbitrary: a)
116 case Nil thus ?case by auto
118 case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute)
121 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
122 apply (induct a arbitrary: b c)
123 apply (simp, clarify)
124 apply (case_tac b, simp_all add: add_ac)
127 lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
128 apply (induct p arbitrary: q,simp)
129 apply (case_tac q, simp_all add: right_distrib)
132 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
133 apply (induct "t", simp)
134 apply (auto simp add: mult_zero_left poly_ident_mult padd_commut)
135 apply (case_tac t, auto)
138 text{*properties of evaluation of polynomials.*}
140 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
141 proof(induct p1 arbitrary: p2)
142 case Nil thus ?case by simp
144 case (Cons a as p2) thus ?case
145 by (cases p2, simp_all add: add_ac right_distrib)
148 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
150 apply (case_tac [2] "x=zero")
151 apply (auto simp add: right_distrib mult_ac)
154 lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
155 by (induct p, auto simp add: right_distrib mult_ac)
157 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
158 apply (simp add: poly_minus_def)
159 apply (auto simp add: poly_cmult minus_mult_left[symmetric])
162 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
163 proof(induct p1 arbitrary: p2)
164 case Nil thus ?case by simp
167 thus ?case by (cases as,
168 simp_all add: poly_cmult poly_add left_distrib right_distrib mult_ac)
171 class recpower_semiring = semiring + recpower
172 class recpower_semiring_1 = semiring_1 + recpower
173 class recpower_semiring_0 = semiring_0 + recpower
174 class recpower_ring = ring + recpower
175 class recpower_ring_1 = ring_1 + recpower
176 subclass (in recpower_ring_1) recpower_ring by unfold_locales
177 class recpower_comm_semiring_1 = recpower + comm_semiring_1
178 class recpower_comm_ring_1 = recpower + comm_ring_1
179 subclass (in recpower_comm_ring_1) recpower_comm_semiring_1 by unfold_locales
180 class recpower_idom = recpower + idom
181 subclass (in recpower_idom) recpower_comm_ring_1 by unfold_locales
182 class idom_char_0 = idom + ring_char_0
183 class recpower_idom_char_0 = recpower + idom_char_0
184 subclass (in recpower_idom_char_0) recpower_idom by unfold_locales
186 lemma (in recpower_comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
188 apply (auto simp add: poly_cmult poly_mult power_Suc)
191 text{*More Polynomial Evaluation Lemmas*}
193 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
196 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
197 by (simp add: poly_mult mult_assoc)
199 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
200 by (induct "p", auto)
202 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
204 apply (auto simp add: poly_mult mult_assoc)
207 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
210 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
213 {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp}
218 from Cons.hyps[rule_format, of x]
219 obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
220 have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
221 using qr by(cases q, simp_all add: ring_simps diff_def[symmetric]
222 minus_mult_left[symmetric] right_minus)
223 hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
227 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
228 by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
231 lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
233 {assume p: "p = []" hence ?thesis by simp}
235 {fix x xs assume p: "p = x#xs"
236 {fix q assume "p = [-a, 1] *** q" hence "poly p a = 0"
237 by (simp add: poly_add poly_cmult minus_mult_left[symmetric])}
239 {assume p0: "poly p a = 0"
240 from poly_linear_rem[of x xs a] obtain q r
241 where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
242 have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
243 hence "\<exists>q. p = [- a, 1] *** q" using p qr apply - apply (rule exI[where x=q])apply auto apply (cases q) apply auto done}
244 ultimately have ?thesis using p by blast}
245 ultimately show ?thesis by (cases p, auto)
248 lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)"
249 by (induct "p", auto)
251 lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++ (h # p)) = Suc (length p)"
252 by (induct "p", auto)
254 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
257 subsection{*Polynomial length*}
259 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
260 by (induct "p", auto)
262 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
263 apply (induct p1 arbitrary: p2, simp_all)
267 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
268 by (simp add: poly_add_length)
270 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
271 "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
272 by (auto simp add: poly_mult)
274 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
275 by (auto simp add: poly_mult)
277 text{*Normalisation Properties*}
279 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
280 by (induct "p", auto)
282 text{*A nontrivial polynomial of degree n has no more than n roots*}
283 lemma (in idom) poly_roots_index_lemma:
284 assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
285 shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
287 proof(induct n arbitrary: p x)
288 case 0 thus ?case by simp
291 {assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
292 from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
293 from p0(1)[unfolded poly_linear_divides[of p x]]
294 have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
295 from C obtain a where a: "poly p a = 0" by blast
296 from a[unfolded poly_linear_divides[of p a]] p0(2)
297 obtain q where q: "p = [-a, 1] *** q" by blast
298 have lg: "length q = n" using q Suc.prems(2) by simp
299 from q p0 have qx: "poly q x \<noteq> poly [] x"
300 by (auto simp add: poly_mult poly_add poly_cmult)
301 from Suc.hyps[OF qx lg] obtain i where
302 i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
303 let ?i = "\<lambda>m. if m = Suc n then a else i m"
304 from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
306 from y have "y = a \<or> poly q y = 0"
307 by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: ring_simps)
308 with i[rule_format, of y] y(1) y(2) have False apply auto
309 apply (erule_tac x="m" in allE)
316 lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==>
317 \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
318 by (blast intro: poly_roots_index_lemma)
320 lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>
321 \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
322 apply (drule poly_roots_index_length, safe)
323 apply (rule_tac x = "Suc (length p)" in exI)
324 apply (rule_tac x = i in exI)
325 apply (simp add: less_Suc_eq_le)
329 lemma (in idom) idom_finite_lemma:
330 assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
331 shows "finite {x. P x}"
335 have "?M \<subseteq> ?N" using P by auto
336 thus ?thesis using finite_subset by auto
340 lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==>
341 \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
342 apply (drule poly_roots_index_length, safe)
343 apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
344 apply (auto simp add: image_iff)
345 apply (erule_tac x="x" in allE, clarsimp)
346 by (case_tac "n=length p", auto simp add: order_le_less)
348 lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)"
349 unfolding finite_conv_nat_seg_image
350 proof(auto simp add: expand_set_eq image_iff)
351 fix n::nat and f:: "nat \<Rightarrow> nat"
352 let ?N = "{i. i < n}"
354 let ?y = "Max ?fN + 1"
355 from nat_seg_image_imp_finite[of "?fN" "f" n]
356 have thfN: "finite ?fN" by simp
357 {assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto}
359 {assume nz: "n \<noteq> 0"
360 hence thne: "?fN \<noteq> {}" by (auto simp add: neq0_conv)
361 have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto
362 hence "\<forall>x\<in> ?fN. ?y > x" by auto
363 hence "?y \<notin> ?fN" by auto
364 hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto }
365 ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast
368 lemma (in ring_char_0) UNIV_ring_char_0_infinte:
369 "\<not> (finite (UNIV:: 'a set))"
371 assume F: "finite (UNIV :: 'a set)"
372 have th0: "of_nat ` UNIV \<subseteq> UNIV" by simp
373 from finite_subset[OF th0] have th: "finite (of_nat ` UNIV :: 'a set)" .
374 have th': "inj_on (of_nat::nat \<Rightarrow> 'a) (UNIV)"
375 unfolding inj_on_def by auto
376 from finite_imageD[OF th th'] UNIV_nat_infinite
380 lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) =
381 finite {x. poly p x = 0}"
383 assume H: "poly p \<noteq> poly []"
384 show "finite {x. poly p x = (0::'a)}"
387 apply (erule contrapos_np, rule ext)
389 apply (clarify dest!: poly_roots_finite_lemma2)
393 assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
394 and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
395 let ?M= "{x. poly p x = (0\<Colon>'a)}"
396 from P have "?M \<subseteq> set i" by auto
397 with finite_subset F show False by auto
400 assume F: "finite {x. poly p x = (0\<Colon>'a)}"
401 show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
404 text{*Entirety and Cancellation for polynomials*}
406 lemma (in idom_char_0) poly_entire_lemma2:
407 assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []"
408 shows "poly (p***q) \<noteq> poly []"
410 let ?S = "\<lambda>p. {x. poly p x = 0}"
411 have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
412 with p0 q0 show ?thesis unfolding poly_roots_finite by auto
415 lemma (in idom_char_0) poly_entire:
416 "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
417 using poly_entire_lemma2[of p q]
418 by auto (rule ext, simp add: poly_mult)+
420 lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
421 by (simp add: poly_entire)
423 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
424 by (auto intro!: ext)
426 lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
427 by (auto simp add: ring_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])
429 lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
430 by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult right_distrib minus_mult_left[symmetric] minus_mult_right[symmetric])
432 subclass (in idom_char_0) comm_ring_1 by unfold_locales
433 lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"
435 have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff)
436 also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r"
437 by (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
438 finally show ?thesis .
441 lemma (in recpower_idom) poly_exp_eq_zero[simp]:
442 "(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)"
443 apply (simp only: fun_eq add: all_simps [symmetric])
444 apply (rule arg_cong [where f = All])
447 apply (auto simp add: poly_exp poly_mult)
450 lemma (in semiring_1) one_neq_zero[simp]: "1 \<noteq> 0" using zero_neq_one by blast
451 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
452 apply (simp add: fun_eq)
453 apply (rule_tac x = "minus one a" in exI)
454 apply (unfold diff_minus)
455 apply (subst add_commute)
456 apply (subst add_assoc)
460 lemma (in recpower_idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])"
463 text{*A more constructive notion of polynomials being trivial*}
465 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
466 apply(simp add: fun_eq)
467 apply (case_tac "h = zero")
468 apply (drule_tac [2] x = zero in spec, auto)
469 apply (cases "poly t = poly []", simp)
472 assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)" and pnz: "poly t \<noteq> poly []"
473 let ?S = "{x. poly t x = 0}"
474 from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
475 hence th: "?S \<supseteq> UNIV - {0}" by auto
476 from poly_roots_finite pnz have th': "finite ?S" by blast
477 from finite_subset[OF th th'] UNIV_ring_char_0_infinte
478 show "poly t x = (0\<Colon>'a)" by simp
481 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
482 apply (induct "p", simp)
484 apply (drule poly_zero_lemma', auto)
487 lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
488 unfolding poly_zero[symmetric] by simp
492 text{*Basics of divisibility.*}
494 lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
495 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
496 apply (drule_tac x = "uminus a" in spec)
497 apply (simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
498 apply (cases "p = []")
499 apply (rule exI[where x="[]"])
501 apply (cases "q = []")
502 apply (erule allE[where x="[]"], simp)
505 apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
506 apply (clarsimp simp add: poly_add poly_cmult)
507 apply (rule_tac x="qa" in exI)
508 apply (simp add: left_distrib [symmetric])
511 apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
512 apply (rule_tac x = "pmult qa q" in exI)
513 apply (rule_tac [2] x = "pmult p qa" in exI)
514 apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
517 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
518 apply (simp add: divides_def)
519 apply (rule_tac x = "[one]" in exI)
520 apply (auto simp add: poly_mult fun_eq)
523 lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
524 apply (simp add: divides_def, safe)
525 apply (rule_tac x = "pmult qa qaa" in exI)
526 apply (auto simp add: poly_mult fun_eq mult_assoc)
530 lemma (in recpower_comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
531 apply (auto simp add: le_iff_add)
533 apply (rule_tac [2] poly_divides_trans)
534 apply (auto simp add: divides_def)
535 apply (rule_tac x = p in exI)
536 apply (auto simp add: poly_mult fun_eq mult_ac)
539 lemma (in recpower_comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q; m\<le>n |] ==> (p %^ m) divides q"
540 by (blast intro: poly_divides_exp poly_divides_trans)
542 lemma (in comm_semiring_0) poly_divides_add:
543 "[| p divides q; p divides r |] ==> p divides (q +++ r)"
544 apply (simp add: divides_def, auto)
545 apply (rule_tac x = "padd qa qaa" in exI)
546 apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
549 lemma (in comm_ring_1) poly_divides_diff:
550 "[| p divides q; p divides (q +++ r) |] ==> p divides r"
551 apply (simp add: divides_def, auto)
552 apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
553 apply (auto simp add: poly_add fun_eq poly_mult poly_minus right_diff_distrib compare_rls add_ac)
556 lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
557 apply (erule poly_divides_diff)
558 apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
561 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p"
562 apply (simp add: divides_def)
563 apply (rule exI[where x="[]"])
564 apply (auto simp add: fun_eq poly_mult)
567 lemma (in semiring_0) poly_divides_zero2[simp]: "q divides []"
568 apply (simp add: divides_def)
569 apply (rule_tac x = "[]" in exI)
570 apply (auto simp add: fun_eq)
573 text{*At last, we can consider the order of a root.*}
575 lemma (in idom_char_0) poly_order_exists_lemma:
576 assumes lp: "length p = d" and p: "poly p \<noteq> poly []"
577 shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
579 proof(induct d arbitrary: p)
580 case 0 thus ?case by simp
583 {assume p0: "poly p a = 0"
584 from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by blast
585 hence pN: "p \<noteq> []" by - (rule ccontr, simp)
586 from p0[unfolded poly_linear_divides] pN obtain q where
587 q: "p = [-a, 1] *** q" by blast
588 from q h p0 have qh: "length q = n" "poly q \<noteq> poly []"
591 apply (simp only: fun_eq)
593 apply (simp add: fun_eq poly_add poly_cmult minus_mult_left[symmetric])
595 from Suc.hyps[OF qh] obtain m r where
596 mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" by blast
597 from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
598 hence ?case by blast}
600 {assume p0: "poly p a \<noteq> 0"
601 hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}
602 ultimately show ?case by blast
606 lemma (in recpower_comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
607 by(induct n, auto simp add: poly_mult power_Suc mult_ac)
609 lemma (in comm_semiring_1) divides_left_mult:
610 assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
612 from d obtain t where r:"poly r = poly (p***q *** t)"
613 unfolding divides_def by blast
614 hence "poly r = poly (p *** (q *** t))"
615 "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)
616 thus ?thesis unfolding divides_def by blast
623 lemma (in recpower_semiring_1)
624 zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
625 by (induct n, simp_all add: power_Suc)
627 lemma (in recpower_idom_char_0) poly_order_exists:
628 assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"
629 shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"
637 apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
638 apply (rule_tac x = n in exI, safe)
639 apply (unfold divides_def)
640 apply (rule_tac x = q in exI)
641 apply (induct_tac "n", simp)
642 apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
644 apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")
646 apply (induct_tac "n")
647 apply (simp del: pmult_Cons pexp_Suc)
648 apply (erule_tac Q = "?poly q a = zero" in contrapos_np)
649 apply (simp add: poly_add poly_cmult minus_mult_left[symmetric])
650 apply (rule pexp_Suc [THEN ssubst])
652 apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
657 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
658 by (simp add: divides_def, auto)
660 lemma (in recpower_idom_char_0) poly_order: "poly p \<noteq> poly []
661 ==> EX! n. ([-a, 1] %^ n) divides p &
662 ~(([-a, 1] %^ (Suc n)) divides p)"
663 apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
664 apply (cut_tac x = y and y = n in less_linear)
665 apply (drule_tac m = n in poly_exp_divides)
666 apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
667 simp del: pmult_Cons pexp_Suc)
672 lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
673 by (blast intro: someI2)
675 lemma (in recpower_idom_char_0) order:
676 "(([-a, 1] %^ n) divides p &
677 ~(([-a, 1] %^ (Suc n)) divides p)) =
678 ((n = order a p) & ~(poly p = poly []))"
679 apply (unfold order_def)
681 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
682 apply (blast intro!: poly_order [THEN [2] some1_equalityD])
685 lemma (in recpower_idom_char_0) order2: "[| poly p \<noteq> poly [] |]
686 ==> ([-a, 1] %^ (order a p)) divides p &
687 ~(([-a, 1] %^ (Suc(order a p))) divides p)"
688 by (simp add: order del: pexp_Suc)
690 lemma (in recpower_idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
691 ~(([-a, 1] %^ (Suc n)) divides p)
692 |] ==> (n = order a p)"
693 by (insert order [of a n p], auto)
695 lemma (in recpower_idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
696 ~(([-a, 1] %^ (Suc n)) divides p))
698 by (blast intro: order_unique)
700 lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q"
701 by (auto simp add: fun_eq divides_def poly_mult order_def)
703 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
705 apply (auto simp add: numeral_1_eq_1)
708 lemma (in comm_ring_1) lemma_order_root:
709 " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
710 \<Longrightarrow> poly p a = 0"
711 apply (induct n arbitrary: a p, blast)
712 apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
715 lemma (in recpower_idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)"
719 apply (case_tac "?poly p = ?poly []", auto)
720 apply (simp add: poly_linear_divides del: pmult_Cons, safe)
721 apply (drule_tac [!] a = a in order2)
723 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
725 apply (blast intro: lemma_order_root)
729 lemma (in recpower_idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
733 apply (case_tac "?poly p = ?poly []", auto)
734 apply (simp add: divides_def fun_eq poly_mult)
735 apply (rule_tac x = "[]" in exI)
736 apply (auto dest!: order2 [where a=a]
737 intro: poly_exp_divides simp del: pexp_Suc)
741 lemma (in recpower_idom_char_0) order_decomp:
742 "poly p \<noteq> poly []
743 ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
744 ~([-a, 1] divides q)"
745 apply (unfold divides_def)
746 apply (drule order2 [where a = a])
747 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
748 apply (rule_tac x = q in exI, safe)
749 apply (drule_tac x = qa in spec)
750 apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
753 text{*Important composition properties of orders.*}
754 lemma order_mult: "poly (p *** q) \<noteq> poly []
755 ==> order a (p *** q) = order a p + order (a::'a::{recpower_idom_char_0}) q"
756 apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
757 apply (auto simp add: poly_entire simp del: pmult_Cons)
758 apply (drule_tac a = a in order2)+
760 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
761 apply (rule_tac x = "qa *** qaa" in exI)
762 apply (simp add: poly_mult mult_ac del: pmult_Cons)
763 apply (drule_tac a = a in order_decomp)+
765 apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
766 apply (simp add: poly_primes del: pmult_Cons)
767 apply (auto simp add: divides_def simp del: pmult_Cons)
768 apply (rule_tac x = qb in exI)
769 apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
770 apply (drule poly_mult_left_cancel [THEN iffD1], force)
771 apply (subgoal_tac "poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** (qa *** qaa))) = poly ([-a, 1] %^ (order a q) *** ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))) ")
772 apply (drule poly_mult_left_cancel [THEN iffD1], force)
773 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
776 lemma (in recpower_idom_char_0) order_mult:
777 assumes pq0: "poly (p *** q) \<noteq> poly []"
778 shows "order a (p *** q) = order a p + order a q"
781 let ?divides = "op divides"
785 apply (cut_tac a = a and p = "pmult p q" and n = "?order a p + ?order a q" in order)
786 apply (auto simp add: poly_entire simp del: pmult_Cons)
787 apply (drule_tac a = a in order2)+
789 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
790 apply (rule_tac x = "pmult qa qaa" in exI)
791 apply (simp add: poly_mult mult_ac del: pmult_Cons)
792 apply (drule_tac a = a in order_decomp)+
794 apply (subgoal_tac "?divides [uminus a,one ] (pmult qa qaa) ")
795 apply (simp add: poly_primes del: pmult_Cons)
796 apply (auto simp add: divides_def simp del: pmult_Cons)
797 apply (rule_tac x = qb in exI)
798 apply (subgoal_tac "?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult qa qaa)) = ?poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))")
799 apply (drule poly_mult_left_cancel [THEN iffD1], force)
800 apply (subgoal_tac "?poly (pmult (pexp [uminus a, one ] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) = ?poly (pmult (pexp [uminus a, one] (order a q)) (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb))) ")
801 apply (drule poly_mult_left_cancel [THEN iffD1], force)
802 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
806 lemma (in recpower_idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)"
807 by (rule order_root [THEN ssubst], auto)
809 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
811 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
812 by (simp add: fun_eq)
814 lemma (in recpower_idom_char_0) rsquarefree_decomp:
815 "[| rsquarefree p; poly p a = 0 |]
816 ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
817 apply (simp add: rsquarefree_def, safe)
818 apply (frule_tac a = a in order_decomp)
819 apply (drule_tac x = a in spec)
820 apply (drule_tac a = a in order_root2 [symmetric])
821 apply (auto simp del: pmult_Cons)
822 apply (rule_tac x = q in exI, safe)
823 apply (simp add: poly_mult fun_eq)
824 apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
825 apply (simp add: divides_def del: pmult_Cons, safe)
826 apply (drule_tac x = "[]" in spec)
827 apply (auto simp add: fun_eq)
831 text{*Normalization of a polynomial.*}
833 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
835 apply (auto simp add: fun_eq)
838 text{*The degree of a polynomial.*}
840 lemma (in semiring_0) lemma_degree_zero:
841 "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
842 by (induct "p", auto)
844 lemma (in idom_char_0) degree_zero:
845 assumes pN: "poly p = poly []" shows"degree p = 0"
850 apply (simp add: degree_def)
851 apply (case_tac "?pn p = []")
852 apply (auto simp add: poly_zero lemma_degree_zero )
856 lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" by simp
857 lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
858 lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
859 unfolding pnormal_def by simp
860 lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
861 unfolding pnormal_def
862 apply (cases "pnormalize p = []", auto)
863 by (cases "c = 0", auto)
866 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
868 case Nil thus ?case by (simp add: pnormal_def)
870 case (Cons a as) thus ?case
871 apply (simp add: pnormal_def)
872 apply (cases "pnormalize as = []", simp_all)
873 apply (cases "as = []", simp_all)
874 apply (cases "a=0", simp_all)
875 apply (cases "a=0", simp_all)
879 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
880 unfolding pnormal_def length_greater_0_conv by blast
882 lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
883 apply (induct p, auto)
884 apply (case_tac "p = []", auto)
885 apply (simp add: pnormal_def)
886 by (rule pnormal_cons, auto)
888 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
889 using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
891 lemma (in idom_char_0) poly_Cons_eq: "poly (c#cs) = poly (d#ds) \<longleftrightarrow> c=d \<and> poly cs = poly ds" (is "?lhs \<longleftrightarrow> ?rhs")
894 hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
895 by (simp only: poly_minus poly_add ring_simps) simp
896 hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by - (rule ext, simp)
897 hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
898 unfolding poly_zero by (simp add: poly_minus_def ring_simps minus_mult_left[symmetric])
899 hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
900 unfolding poly_zero[symmetric] by simp
901 thus ?rhs apply (simp add: poly_minus poly_add ring_simps) apply (rule ext, simp) done
903 assume ?rhs then show ?lhs by - (rule ext,simp)
906 lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
907 proof(induct q arbitrary: p)
908 case Nil thus ?case by (simp only: poly_zero lemma_degree_zero) simp
914 hence "poly [] = poly (c#cs)" by blast
915 then have "poly (c#cs) = poly [] " by simp
916 thus ?case by (simp only: poly_zero lemma_degree_zero) simp
919 hence eq: "poly (d # ds) = poly (c # cs)" by blast
920 hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
921 hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
922 hence dc: "d = c" by auto
923 with eq have "poly ds = poly cs"
924 unfolding poly_Cons_eq by simp
925 with Cons.prems have "pnormalize ds = pnormalize cs" by blast
926 with dc show ?case by simp
930 lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q"
931 shows "degree p = degree q"
932 using pnormalize_unique[OF pq] unfolding degree_def by simp
934 lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto)
936 lemma (in semiring_0) last_linear_mul_lemma:
937 "last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)"
939 apply (induct p arbitrary: a x b, auto)
940 apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)
941 apply (induct_tac p, auto)
944 lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p"
946 from p obtain c cs where cs: "p = c#cs" by (cases p, auto)
947 from cs have eq:"[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
948 by (simp add: poly_cmult_distr)
949 show ?thesis using cs
950 unfolding eq last_linear_mul_lemma by simp
953 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
954 apply (induct p, auto)
955 apply (case_tac p, auto)+
958 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
961 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
962 using pnormalize_eq[of p] unfolding degree_def by simp
964 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp
966 lemma (in idom_char_0) linear_mul_degree: assumes p: "poly p \<noteq> poly []"
967 shows "degree ([a,1] *** p) = degree p + 1"
969 from p have pnz: "pnormalize p \<noteq> []"
970 unfolding poly_zero lemma_degree_zero .
972 from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
973 have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
974 from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
975 pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
978 have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
979 by (auto simp add: poly_length_mult)
981 have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
982 by (rule ext) (simp add: poly_mult poly_add poly_cmult)
983 from degree_unique[OF eqs] th
984 show ?thesis by (simp add: degree_unique[OF poly_normalize])
987 lemma (in idom_char_0) linear_pow_mul_degree:
988 "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
989 proof(induct n arbitrary: a p)
991 {assume p: "poly p = poly []"
992 hence ?case using degree_unique[OF p] by (simp add: degree_def)}
994 {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }
995 ultimately show ?case by blast
998 have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
999 apply (rule ext, simp add: poly_mult poly_add poly_cmult)
1000 by (simp add: mult_ac add_ac right_distrib)
1001 note deq = degree_unique[OF eq]
1002 {assume p: "poly p = poly []"
1003 with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
1004 by - (rule ext,simp add: poly_mult poly_cmult poly_add)
1005 from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}
1007 {assume p: "poly p \<noteq> poly []"
1008 from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
1009 using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
1010 have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
1011 by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult mult_ac add_ac right_distrib)
1012 from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast
1013 have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
1014 apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
1017 from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]
1018 have ?case by (auto simp del: poly.simps)}
1019 ultimately show ?case by blast
1022 lemma (in recpower_idom_char_0) order_degree:
1023 assumes p0: "poly p \<noteq> poly []"
1024 shows "order a p \<le> degree p"
1026 from order2[OF p0, unfolded divides_def]
1027 obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
1028 {assume "poly q = poly []"
1029 with q p0 have False by (simp add: poly_mult poly_entire)}
1030 with degree_unique[OF q, unfolded linear_pow_mul_degree]
1031 show ?thesis by auto
1034 text{*Tidier versions of finiteness of roots.*}
1036 lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}"
1037 unfolding poly_roots_finite .
1039 text{*bound for polynomial.*}
1041 lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
1042 apply (induct "p", auto)
1043 apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
1044 apply (rule abs_triangle_ineq)
1045 apply (auto intro!: mult_mono simp add: abs_mult)
1048 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp