Moved FTA into Lib and cleaned it up a little.
1 (* Title: Univ_Poly.thy
5 header {* Univariate Polynomials *}
11 text{* Application of polynomial as a function. *}
13 primrec (in semiring_0) poly :: "'a list => 'a => 'a" where
14 poly_Nil: "poly [] x = 0"
15 | poly_Cons: "poly (h#t) x = h + x * poly t x"
18 subsection{*Arithmetic Operations on Polynomials*}
22 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "+++" 65)
24 padd_Nil: "[] +++ l2 = l2"
25 | padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t
26 else (h + hd l2)#(t +++ tl l2))"
28 text{*Multiplication by a constant*}
29 primrec (in semiring_0) cmult :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "%*" 70) where
30 cmult_Nil: "c %* [] = []"
31 | cmult_Cons: "c %* (h#t) = (c * h)#(c %* t)"
33 text{*Multiplication by a polynomial*}
34 primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "***" 70)
36 pmult_Nil: "[] *** l2 = []"
37 | pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2
38 else (h %* l2) +++ ((0) # (t *** l2)))"
40 text{*Repeated multiplication by a polynomial*}
41 primrec (in semiring_0) mulexp :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where
42 mulexp_zero: "mulexp 0 p q = q"
43 | mulexp_Suc: "mulexp (Suc n) p q = p *** mulexp n p q"
46 primrec (in semiring_1) pexp :: "'a list \<Rightarrow> nat \<Rightarrow> 'a list" (infixl "%^" 80) where
47 pexp_0: "p %^ 0 = [1]"
48 | pexp_Suc: "p %^ (Suc n) = p *** (p %^ n)"
50 text{*Quotient related value of dividing a polynomial by x + a*}
51 (* Useful for divisor properties in inductive proofs *)
52 primrec (in field) "pquot" :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a list" where
53 pquot_Nil: "pquot [] a= []"
54 | pquot_Cons: "pquot (h#t) a = (if t = [] then [h]
55 else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))"
57 text{*normalization of polynomials (remove extra 0 coeff)*}
58 primrec (in semiring_0) pnormalize :: "'a list \<Rightarrow> 'a list" where
59 pnormalize_Nil: "pnormalize [] = []"
60 | pnormalize_Cons: "pnormalize (h#p) = (if ( (pnormalize p) = [])
61 then (if (h = 0) then [] else [h])
62 else (h#(pnormalize p)))"
64 definition (in semiring_0) "pnormal p = ((pnormalize p = p) \<and> p \<noteq> [])"
65 definition (in semiring_0) "nonconstant p = (pnormal p \<and> (\<forall>x. p \<noteq> [x]))"
66 text{*Other definitions*}
68 definition (in ring_1)
69 poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where
72 definition (in semiring_0)
73 divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "divides" 70) where
74 [code del]: "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))"
76 --{*order of a polynomial*}
77 definition (in ring_1) order :: "'a => 'a list => nat" where
78 "order a p = (SOME n. ([-a, 1] %^ n) divides p &
79 ~ (([-a, 1] %^ (Suc n)) divides p))"
81 --{*degree of a polynomial*}
82 definition (in semiring_0) degree :: "'a list => nat" where
83 "degree p = length (pnormalize p) - 1"
85 --{*squarefree polynomials --- NB with respect to real roots only.*}
86 definition (in ring_1)
87 rsquarefree :: "'a list => bool" where
88 "rsquarefree p = (poly p \<noteq> poly [] &
89 (\<forall>a. (order a p = 0) | (order a p = 1)))"
94 lemma padd_Nil2[simp]: "p +++ [] = p"
97 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)"
100 lemma pminus_Nil[simp]: "-- [] = []"
101 by (simp add: poly_minus_def)
103 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp
106 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct "t", auto)
108 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)"
111 text{*Handy general properties*}
113 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b"
114 proof(induct b arbitrary: a)
115 case Nil thus ?case by auto
117 case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute)
120 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)"
121 apply (induct a arbitrary: b c)
122 apply (simp, clarify)
123 apply (case_tac b, simp_all add: add_ac)
126 lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)"
127 apply (induct p arbitrary: q,simp)
128 apply (case_tac q, simp_all add: right_distrib)
131 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)"
132 apply (induct "t", simp)
133 apply (auto simp add: mult_zero_left poly_ident_mult padd_commut)
134 apply (case_tac t, auto)
137 text{*properties of evaluation of polynomials.*}
139 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x"
140 proof(induct p1 arbitrary: p2)
141 case Nil thus ?case by simp
143 case (Cons a as p2) thus ?case
144 by (cases p2, simp_all add: add_ac right_distrib)
147 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x"
149 apply (case_tac [2] "x=zero")
150 apply (auto simp add: right_distrib mult_ac)
153 lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x"
154 by (induct p, auto simp add: right_distrib mult_ac)
156 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)"
157 apply (simp add: poly_minus_def)
158 apply (auto simp add: poly_cmult minus_mult_left[symmetric])
161 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x"
162 proof(induct p1 arbitrary: p2)
163 case Nil thus ?case by simp
166 thus ?case by (cases as,
167 simp_all add: poly_cmult poly_add left_distrib right_distrib mult_ac)
170 class recpower_semiring = semiring + recpower
171 class recpower_semiring_1 = semiring_1 + recpower
172 class recpower_semiring_0 = semiring_0 + recpower
173 class recpower_ring = ring + recpower
174 class recpower_ring_1 = ring_1 + recpower
175 subclass (in recpower_ring_1) recpower_ring ..
176 class recpower_comm_semiring_1 = recpower + comm_semiring_1
177 class recpower_comm_ring_1 = recpower + comm_ring_1
178 subclass (in recpower_comm_ring_1) recpower_comm_semiring_1 ..
179 class recpower_idom = recpower + idom
180 subclass (in recpower_idom) recpower_comm_ring_1 ..
181 class idom_char_0 = idom + ring_char_0
182 class recpower_idom_char_0 = recpower + idom_char_0
183 subclass (in recpower_idom_char_0) recpower_idom ..
185 lemma (in recpower_comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n"
187 apply (auto simp add: poly_cmult poly_mult power_Suc)
190 text{*More Polynomial Evaluation Lemmas*}
192 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x"
195 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x"
196 by (simp add: poly_mult mult_assoc)
198 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0"
199 by (induct "p", auto)
201 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x"
203 apply (auto simp add: poly_mult mult_assoc)
206 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides
209 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q"
212 {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp}
217 from Cons.hyps[rule_format, of x]
218 obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
219 have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)"
220 using qr by(cases q, simp_all add: algebra_simps diff_def[symmetric]
221 minus_mult_left[symmetric] right_minus)
222 hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast}
226 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q"
227 by (cut_tac t = t and a = a in lemma_poly_linear_rem, auto)
230 lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))"
232 {assume p: "p = []" hence ?thesis by simp}
234 {fix x xs assume p: "p = x#xs"
235 {fix q assume "p = [-a, 1] *** q" hence "poly p a = 0"
236 by (simp add: poly_add poly_cmult minus_mult_left[symmetric])}
238 {assume p0: "poly p a = 0"
239 from poly_linear_rem[of x xs a] obtain q r
240 where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast
241 have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp
242 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}
243 ultimately have ?thesis using p by blast}
244 ultimately show ?thesis by (cases p, auto)
247 lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)"
248 by (induct "p", auto)
250 lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++ (h # p)) = Suc (length p)"
251 by (induct "p", auto)
253 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)"
256 subsection{*Polynomial length*}
258 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p"
259 by (induct "p", auto)
261 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)"
262 apply (induct p1 arbitrary: p2, simp_all)
266 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)"
267 by (simp add: poly_add_length)
269 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]:
270 "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x"
271 by (auto simp add: poly_mult)
273 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0"
274 by (auto simp add: poly_mult)
276 text{*Normalisation Properties*}
278 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)"
279 by (induct "p", auto)
281 text{*A nontrivial polynomial of degree n has no more than n roots*}
282 lemma (in idom) poly_roots_index_lemma:
283 assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n"
284 shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)"
286 proof(induct n arbitrary: p x)
287 case 0 thus ?case by simp
290 {assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)"
291 from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto
292 from p0(1)[unfolded poly_linear_divides[of p x]]
293 have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast
294 from C obtain a where a: "poly p a = 0" by blast
295 from a[unfolded poly_linear_divides[of p a]] p0(2)
296 obtain q where q: "p = [-a, 1] *** q" by blast
297 have lg: "length q = n" using q Suc.prems(2) by simp
298 from q p0 have qx: "poly q x \<noteq> poly [] x"
299 by (auto simp add: poly_mult poly_add poly_cmult)
300 from Suc.hyps[OF qx lg] obtain i where
301 i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast
302 let ?i = "\<lambda>m. if m = Suc n then a else i m"
303 from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m"
305 from y have "y = a \<or> poly q y = 0"
306 by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps)
307 with i[rule_format, of y] y(1) y(2) have False apply auto
308 apply (erule_tac x="m" in allE)
315 lemma (in idom) poly_roots_index_length: "poly p x \<noteq> poly [] x ==>
316 \<exists>i. \<forall>x. (poly p x = 0) --> (\<exists>n. n \<le> length p & x = i n)"
317 by (blast intro: poly_roots_index_lemma)
319 lemma (in idom) poly_roots_finite_lemma1: "poly p x \<noteq> poly [] x ==>
320 \<exists>N i. \<forall>x. (poly p x = 0) --> (\<exists>n. (n::nat) < N & x = i n)"
321 apply (drule poly_roots_index_length, safe)
322 apply (rule_tac x = "Suc (length p)" in exI)
323 apply (rule_tac x = i in exI)
324 apply (simp add: less_Suc_eq_le)
328 lemma (in idom) idom_finite_lemma:
329 assumes P: "\<forall>x. P x --> (\<exists>n. n < length j & x = j!n)"
330 shows "finite {x. P x}"
334 have "?M \<subseteq> ?N" using P by auto
335 thus ?thesis using finite_subset by auto
339 lemma (in idom) poly_roots_finite_lemma2: "poly p x \<noteq> poly [] x ==>
340 \<exists>i. \<forall>x. (poly p x = 0) --> x \<in> set i"
341 apply (drule poly_roots_index_length, safe)
342 apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI)
343 apply (auto simp add: image_iff)
344 apply (erule_tac x="x" in allE, clarsimp)
345 by (case_tac "n=length p", auto simp add: order_le_less)
347 lemma (in ring_char_0) UNIV_ring_char_0_infinte:
348 "\<not> (finite (UNIV:: 'a set))"
350 assume F: "finite (UNIV :: 'a set)"
351 have "finite (UNIV :: nat set)"
352 proof (rule finite_imageD)
353 have "of_nat ` UNIV \<subseteq> UNIV" by simp
354 then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset)
355 show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def)
357 with infinite_UNIV_nat show False ..
360 lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) =
361 finite {x. poly p x = 0}"
363 assume H: "poly p \<noteq> poly []"
364 show "finite {x. poly p x = (0::'a)}"
367 apply (erule contrapos_np, rule ext)
369 apply (clarify dest!: poly_roots_finite_lemma2)
373 assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}"
374 and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i"
375 let ?M= "{x. poly p x = (0\<Colon>'a)}"
376 from P have "?M \<subseteq> set i" by auto
377 with finite_subset F show False by auto
380 assume F: "finite {x. poly p x = (0\<Colon>'a)}"
381 show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto
384 text{*Entirety and Cancellation for polynomials*}
386 lemma (in idom_char_0) poly_entire_lemma2:
387 assumes p0: "poly p \<noteq> poly []" and q0: "poly q \<noteq> poly []"
388 shows "poly (p***q) \<noteq> poly []"
390 let ?S = "\<lambda>p. {x. poly p x = 0}"
391 have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult)
392 with p0 q0 show ?thesis unfolding poly_roots_finite by auto
395 lemma (in idom_char_0) poly_entire:
396 "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []"
397 using poly_entire_lemma2[of p q]
398 by (auto simp add: expand_fun_eq poly_mult)
400 lemma (in idom_char_0) poly_entire_neg: "(poly (p *** q) \<noteq> poly []) = ((poly p \<noteq> poly []) & (poly q \<noteq> poly []))"
401 by (simp add: poly_entire)
403 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)"
404 by (auto intro!: ext)
406 lemma (in comm_ring_1) poly_add_minus_zero_iff: "(poly (p +++ -- q) = poly []) = (poly p = poly q)"
407 by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult minus_mult_left[symmetric])
409 lemma (in comm_ring_1) poly_add_minus_mult_eq: "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))"
410 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])
412 subclass (in idom_char_0) comm_ring_1 ..
413 lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)"
415 have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" by (simp only: poly_add_minus_zero_iff)
416 also have "\<dots> \<longleftrightarrow> poly p = poly [] | poly q = poly r"
417 by (auto intro: ext simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff)
418 finally show ?thesis .
421 lemma (in recpower_idom) poly_exp_eq_zero[simp]:
422 "(poly (p %^ n) = poly []) = (poly p = poly [] & n \<noteq> 0)"
423 apply (simp only: fun_eq add: all_simps [symmetric])
424 apply (rule arg_cong [where f = All])
427 apply (auto simp add: poly_exp poly_mult)
430 lemma (in semiring_1) one_neq_zero[simp]: "1 \<noteq> 0" using zero_neq_one by blast
431 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []"
432 apply (simp add: fun_eq)
433 apply (rule_tac x = "minus one a" in exI)
434 apply (unfold diff_minus)
435 apply (subst add_commute)
436 apply (subst add_assoc)
440 lemma (in recpower_idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])"
443 text{*A more constructive notion of polynomials being trivial*}
445 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] ==> h = 0 & poly t = poly []"
446 apply(simp add: fun_eq)
447 apply (case_tac "h = zero")
448 apply (drule_tac [2] x = zero in spec, auto)
449 apply (cases "poly t = poly []", simp)
452 assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)" and pnz: "poly t \<noteq> poly []"
453 let ?S = "{x. poly t x = 0}"
454 from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast
455 hence th: "?S \<supseteq> UNIV - {0}" by auto
456 from poly_roots_finite pnz have th': "finite ?S" by blast
457 from finite_subset[OF th th'] UNIV_ring_char_0_infinte
458 show "poly t x = (0\<Colon>'a)" by simp
461 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p"
462 apply (induct "p", simp)
464 apply (drule poly_zero_lemma', auto)
467 lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0"
468 unfolding poly_zero[symmetric] by simp
472 text{*Basics of divisibility.*}
474 lemma (in idom) poly_primes: "([a, 1] divides (p *** q)) = ([a, 1] divides p | [a, 1] divides q)"
475 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult left_distrib [symmetric])
476 apply (drule_tac x = "uminus a" in spec)
477 apply (simp add: poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
478 apply (cases "p = []")
479 apply (rule exI[where x="[]"])
481 apply (cases "q = []")
482 apply (erule allE[where x="[]"], simp)
485 apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)")
486 apply (clarsimp simp add: poly_add poly_cmult)
487 apply (rule_tac x="qa" in exI)
488 apply (simp add: left_distrib [symmetric])
491 apply (auto simp add: right_minus poly_linear_divides poly_add poly_cmult left_distrib [symmetric])
492 apply (rule_tac x = "pmult qa q" in exI)
493 apply (rule_tac [2] x = "pmult p qa" in exI)
494 apply (auto simp add: poly_add poly_mult poly_cmult mult_ac)
497 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p"
498 apply (simp add: divides_def)
499 apply (rule_tac x = "[one]" in exI)
500 apply (auto simp add: poly_mult fun_eq)
503 lemma (in comm_semiring_1) poly_divides_trans: "[| p divides q; q divides r |] ==> p divides r"
504 apply (simp add: divides_def, safe)
505 apply (rule_tac x = "pmult qa qaa" in exI)
506 apply (auto simp add: poly_mult fun_eq mult_assoc)
510 lemma (in recpower_comm_semiring_1) poly_divides_exp: "m \<le> n ==> (p %^ m) divides (p %^ n)"
511 apply (auto simp add: le_iff_add)
513 apply (rule_tac [2] poly_divides_trans)
514 apply (auto simp add: divides_def)
515 apply (rule_tac x = p in exI)
516 apply (auto simp add: poly_mult fun_eq mult_ac)
519 lemma (in recpower_comm_semiring_1) poly_exp_divides: "[| (p %^ n) divides q; m\<le>n |] ==> (p %^ m) divides q"
520 by (blast intro: poly_divides_exp poly_divides_trans)
522 lemma (in comm_semiring_0) poly_divides_add:
523 "[| p divides q; p divides r |] ==> p divides (q +++ r)"
524 apply (simp add: divides_def, auto)
525 apply (rule_tac x = "padd qa qaa" in exI)
526 apply (auto simp add: poly_add fun_eq poly_mult right_distrib)
529 lemma (in comm_ring_1) poly_divides_diff:
530 "[| p divides q; p divides (q +++ r) |] ==> p divides r"
531 apply (simp add: divides_def, auto)
532 apply (rule_tac x = "padd qaa (poly_minus qa)" in exI)
533 apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps)
536 lemma (in comm_ring_1) poly_divides_diff2: "[| p divides r; p divides (q +++ r) |] ==> p divides q"
537 apply (erule poly_divides_diff)
538 apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac)
541 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] ==> q divides p"
542 apply (simp add: divides_def)
543 apply (rule exI[where x="[]"])
544 apply (auto simp add: fun_eq poly_mult)
547 lemma (in semiring_0) poly_divides_zero2[simp]: "q divides []"
548 apply (simp add: divides_def)
549 apply (rule_tac x = "[]" in exI)
550 apply (auto simp add: fun_eq)
553 text{*At last, we can consider the order of a root.*}
555 lemma (in idom_char_0) poly_order_exists_lemma:
556 assumes lp: "length p = d" and p: "poly p \<noteq> poly []"
557 shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0"
559 proof(induct d arbitrary: p)
560 case 0 thus ?case by simp
563 {assume p0: "poly p a = 0"
564 from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto
565 hence pN: "p \<noteq> []" by auto
566 from p0[unfolded poly_linear_divides] pN obtain q where
567 q: "p = [-a, 1] *** q" by blast
568 from q h p0 have qh: "length q = n" "poly q \<noteq> poly []"
571 apply (simp only: fun_eq)
573 apply (simp add: fun_eq poly_add poly_cmult minus_mult_left[symmetric])
575 from Suc.hyps[OF qh] obtain m r where
576 mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" by blast
577 from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp
578 hence ?case by blast}
580 {assume p0: "poly p a \<noteq> 0"
581 hence ?case using Suc.prems apply simp by (rule exI[where x="0::nat"], simp)}
582 ultimately show ?case by blast
586 lemma (in recpower_comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x"
587 by(induct n, auto simp add: poly_mult power_Suc mult_ac)
589 lemma (in comm_semiring_1) divides_left_mult:
590 assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r"
592 from d obtain t where r:"poly r = poly (p***q *** t)"
593 unfolding divides_def by blast
594 hence "poly r = poly (p *** (q *** t))"
595 "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac)
596 thus ?thesis unfolding divides_def by blast
603 lemma (in recpower_semiring_1)
604 zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)"
605 by (induct n, simp_all add: power_Suc)
607 lemma (in recpower_idom_char_0) poly_order_exists:
608 assumes lp: "length p = d" and p0: "poly p \<noteq> poly []"
609 shows "\<exists>n. ([-a, 1] %^ n) divides p & ~(([-a, 1] %^ (Suc n)) divides p)"
617 apply (drule poly_order_exists_lemma [where a=a], assumption, clarify)
618 apply (rule_tac x = n in exI, safe)
619 apply (unfold divides_def)
620 apply (rule_tac x = q in exI)
621 apply (induct_tac "n", simp)
622 apply (simp (no_asm_simp) add: poly_add poly_cmult poly_mult right_distrib mult_ac)
624 apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)")
626 apply (induct_tac "n")
627 apply (simp del: pmult_Cons pexp_Suc)
628 apply (erule_tac Q = "?poly q a = zero" in contrapos_np)
629 apply (simp add: poly_add poly_cmult minus_mult_left[symmetric])
630 apply (rule pexp_Suc [THEN ssubst])
632 apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc)
637 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p"
638 by (simp add: divides_def, auto)
640 lemma (in recpower_idom_char_0) poly_order: "poly p \<noteq> poly []
641 ==> EX! n. ([-a, 1] %^ n) divides p &
642 ~(([-a, 1] %^ (Suc n)) divides p)"
643 apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc)
644 apply (cut_tac x = y and y = n in less_linear)
645 apply (drule_tac m = n in poly_exp_divides)
646 apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides]
647 simp del: pmult_Cons pexp_Suc)
652 lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n"
653 by (blast intro: someI2)
655 lemma (in recpower_idom_char_0) order:
656 "(([-a, 1] %^ n) divides p &
657 ~(([-a, 1] %^ (Suc n)) divides p)) =
658 ((n = order a p) & ~(poly p = poly []))"
659 apply (unfold order_def)
661 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order)
662 apply (blast intro!: poly_order [THEN [2] some1_equalityD])
665 lemma (in recpower_idom_char_0) order2: "[| poly p \<noteq> poly [] |]
666 ==> ([-a, 1] %^ (order a p)) divides p &
667 ~(([-a, 1] %^ (Suc(order a p))) divides p)"
668 by (simp add: order del: pexp_Suc)
670 lemma (in recpower_idom_char_0) order_unique: "[| poly p \<noteq> poly []; ([-a, 1] %^ n) divides p;
671 ~(([-a, 1] %^ (Suc n)) divides p)
672 |] ==> (n = order a p)"
673 by (insert order [of a n p], auto)
675 lemma (in recpower_idom_char_0) order_unique_lemma: "(poly p \<noteq> poly [] & ([-a, 1] %^ n) divides p &
676 ~(([-a, 1] %^ (Suc n)) divides p))
678 by (blast intro: order_unique)
680 lemma (in ring_1) order_poly: "poly p = poly q ==> order a p = order a q"
681 by (auto simp add: fun_eq divides_def poly_mult order_def)
683 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p"
685 apply (auto simp add: numeral_1_eq_1)
688 lemma (in comm_ring_1) lemma_order_root:
689 " 0 < n & [- a, 1] %^ n divides p & ~ [- a, 1] %^ (Suc n) divides p
690 \<Longrightarrow> poly p a = 0"
691 apply (induct n arbitrary: a p, blast)
692 apply (auto simp add: divides_def poly_mult simp del: pmult_Cons)
695 lemma (in recpower_idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)"
699 apply (case_tac "?poly p = ?poly []", auto)
700 apply (simp add: poly_linear_divides del: pmult_Cons, safe)
701 apply (drule_tac [!] a = a in order2)
703 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast)
705 apply (blast intro: lemma_order_root)
709 lemma (in recpower_idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)"
713 apply (case_tac "?poly p = ?poly []", auto)
714 apply (simp add: divides_def fun_eq poly_mult)
715 apply (rule_tac x = "[]" in exI)
716 apply (auto dest!: order2 [where a=a]
717 intro: poly_exp_divides simp del: pexp_Suc)
721 lemma (in recpower_idom_char_0) order_decomp:
722 "poly p \<noteq> poly []
723 ==> \<exists>q. (poly p = poly (([-a, 1] %^ (order a p)) *** q)) &
724 ~([-a, 1] divides q)"
725 apply (unfold divides_def)
726 apply (drule order2 [where a = a])
727 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
728 apply (rule_tac x = q in exI, safe)
729 apply (drule_tac x = qa in spec)
730 apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons)
733 text{*Important composition properties of orders.*}
734 lemma order_mult: "poly (p *** q) \<noteq> poly []
735 ==> order a (p *** q) = order a p + order (a::'a::{recpower_idom_char_0}) q"
736 apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order)
737 apply (auto simp add: poly_entire simp del: pmult_Cons)
738 apply (drule_tac a = a in order2)+
740 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
741 apply (rule_tac x = "qa *** qaa" in exI)
742 apply (simp add: poly_mult mult_ac del: pmult_Cons)
743 apply (drule_tac a = a in order_decomp)+
745 apply (subgoal_tac "[-a,1] divides (qa *** qaa) ")
746 apply (simp add: poly_primes del: pmult_Cons)
747 apply (auto simp add: divides_def simp del: pmult_Cons)
748 apply (rule_tac x = qb in exI)
749 apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))")
750 apply (drule poly_mult_left_cancel [THEN iffD1], force)
751 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))) ")
752 apply (drule poly_mult_left_cancel [THEN iffD1], force)
753 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
756 lemma (in recpower_idom_char_0) order_mult:
757 assumes pq0: "poly (p *** q) \<noteq> poly []"
758 shows "order a (p *** q) = order a p + order a q"
761 let ?divides = "op divides"
765 apply (cut_tac a = a and p = "pmult p q" and n = "?order a p + ?order a q" in order)
766 apply (auto simp add: poly_entire simp del: pmult_Cons)
767 apply (drule_tac a = a in order2)+
769 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe)
770 apply (rule_tac x = "pmult qa qaa" in exI)
771 apply (simp add: poly_mult mult_ac del: pmult_Cons)
772 apply (drule_tac a = a in order_decomp)+
774 apply (subgoal_tac "?divides [uminus a,one ] (pmult qa qaa) ")
775 apply (simp add: poly_primes del: pmult_Cons)
776 apply (auto simp add: divides_def simp del: pmult_Cons)
777 apply (rule_tac x = qb in exI)
778 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))")
779 apply (drule poly_mult_left_cancel [THEN iffD1], force)
780 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))) ")
781 apply (drule poly_mult_left_cancel [THEN iffD1], force)
782 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons)
786 lemma (in recpower_idom_char_0) order_root2: "poly p \<noteq> poly [] ==> (poly p a = 0) = (order a p \<noteq> 0)"
787 by (rule order_root [THEN ssubst], auto)
789 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto
791 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]"
792 by (simp add: fun_eq)
794 lemma (in recpower_idom_char_0) rsquarefree_decomp:
795 "[| rsquarefree p; poly p a = 0 |]
796 ==> \<exists>q. (poly p = poly ([-a, 1] *** q)) & poly q a \<noteq> 0"
797 apply (simp add: rsquarefree_def, safe)
798 apply (frule_tac a = a in order_decomp)
799 apply (drule_tac x = a in spec)
800 apply (drule_tac a = a in order_root2 [symmetric])
801 apply (auto simp del: pmult_Cons)
802 apply (rule_tac x = q in exI, safe)
803 apply (simp add: poly_mult fun_eq)
804 apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1])
805 apply (simp add: divides_def del: pmult_Cons, safe)
806 apply (drule_tac x = "[]" in spec)
807 apply (auto simp add: fun_eq)
811 text{*Normalization of a polynomial.*}
813 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p"
815 apply (auto simp add: fun_eq)
818 text{*The degree of a polynomial.*}
820 lemma (in semiring_0) lemma_degree_zero:
821 "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []"
822 by (induct "p", auto)
824 lemma (in idom_char_0) degree_zero:
825 assumes pN: "poly p = poly []" shows"degree p = 0"
830 apply (simp add: degree_def)
831 apply (case_tac "?pn p = []")
832 apply (auto simp add: poly_zero lemma_degree_zero )
836 lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" by simp
837 lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" by simp
838 lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)"
839 unfolding pnormal_def by simp
840 lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p"
841 unfolding pnormal_def
842 apply (cases "pnormalize p = []", auto)
843 by (cases "c = 0", auto)
846 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0"
848 case Nil thus ?case by (simp add: pnormal_def)
850 case (Cons a as) thus ?case
851 apply (simp add: pnormal_def)
852 apply (cases "pnormalize as = []", simp_all)
853 apply (cases "as = []", simp_all)
854 apply (cases "a=0", simp_all)
855 apply (cases "a=0", simp_all)
859 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p"
860 unfolding pnormal_def length_greater_0_conv by blast
862 lemma (in semiring_0) pnormal_last_length: "\<lbrakk>0 < length p ; last p \<noteq> 0\<rbrakk> \<Longrightarrow> pnormal p"
863 apply (induct p, auto)
864 apply (case_tac "p = []", auto)
865 apply (simp add: pnormal_def)
866 by (rule pnormal_cons, auto)
868 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> (0 < length p \<and> last p \<noteq> 0)"
869 using pnormal_last_length pnormal_length pnormal_last_nonzero by blast
871 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")
874 hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0"
875 by (simp only: poly_minus poly_add algebra_simps) simp
876 hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add:expand_fun_eq)
877 hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))"
878 unfolding poly_zero by (simp add: poly_minus_def algebra_simps)
879 hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)"
880 unfolding poly_zero[symmetric] by simp
881 thus ?rhs by (simp add: poly_minus poly_add algebra_simps expand_fun_eq)
883 assume ?rhs then show ?lhs by(simp add:expand_fun_eq)
886 lemma (in idom_char_0) pnormalize_unique: "poly p = poly q \<Longrightarrow> pnormalize p = pnormalize q"
887 proof(induct q arbitrary: p)
888 case Nil thus ?case by (simp only: poly_zero lemma_degree_zero) simp
894 hence "poly [] = poly (c#cs)" by blast
895 then have "poly (c#cs) = poly [] " by simp
896 thus ?case by (simp only: poly_zero lemma_degree_zero) simp
899 hence eq: "poly (d # ds) = poly (c # cs)" by blast
900 hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp
901 hence "poly (d # ds) 0 = poly (c # cs) 0" by blast
902 hence dc: "d = c" by auto
903 with eq have "poly ds = poly cs"
904 unfolding poly_Cons_eq by simp
905 with Cons.prems have "pnormalize ds = pnormalize cs" by blast
906 with dc show ?case by simp
910 lemma (in idom_char_0) degree_unique: assumes pq: "poly p = poly q"
911 shows "degree p = degree q"
912 using pnormalize_unique[OF pq] unfolding degree_def by simp
914 lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto)
916 lemma (in semiring_0) last_linear_mul_lemma:
917 "last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)"
919 apply (induct p arbitrary: a x b, auto)
920 apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []", simp)
921 apply (induct_tac p, auto)
924 lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p"
926 from p obtain c cs where cs: "p = c#cs" by (cases p, auto)
927 from cs have eq:"[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))"
928 by (simp add: poly_cmult_distr)
929 show ?thesis using cs
930 unfolding eq last_linear_mul_lemma by simp
933 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p"
934 apply (induct p, auto)
935 apply (case_tac p, auto)+
938 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0"
941 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1"
942 using pnormalize_eq[of p] unfolding degree_def by simp
944 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp
946 lemma (in idom_char_0) linear_mul_degree: assumes p: "poly p \<noteq> poly []"
947 shows "degree ([a,1] *** p) = degree p + 1"
949 from p have pnz: "pnormalize p \<noteq> []"
950 unfolding poly_zero lemma_degree_zero .
952 from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz]
953 have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp
954 from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a]
955 pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz
958 have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1"
959 by (auto simp add: poly_length_mult)
961 have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)"
962 by (rule ext) (simp add: poly_mult poly_add poly_cmult)
963 from degree_unique[OF eqs] th
964 show ?thesis by (simp add: degree_unique[OF poly_normalize])
967 lemma (in idom_char_0) linear_pow_mul_degree:
968 "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)"
969 proof(induct n arbitrary: a p)
971 {assume p: "poly p = poly []"
972 hence ?case using degree_unique[OF p] by (simp add: degree_def)}
974 {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) }
975 ultimately show ?case by blast
978 have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))"
979 apply (rule ext, simp add: poly_mult poly_add poly_cmult)
980 by (simp add: mult_ac add_ac right_distrib)
981 note deq = degree_unique[OF eq]
982 {assume p: "poly p = poly []"
983 with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []"
984 by - (rule ext,simp add: poly_mult poly_cmult poly_add)
985 from degree_unique[OF eq'] p have ?case by (simp add: degree_def)}
987 {assume p: "poly p \<noteq> poly []"
988 from p have ap: "poly ([a,1] *** p) \<noteq> poly []"
989 using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto
990 have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))"
991 by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps)
992 from ap have ap': "(poly ([a,1] *** p) = poly []) = False" by blast
993 have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n"
994 apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap')
997 from degree_unique[OF eq] ap p th0 linear_mul_degree[OF p, of a]
998 have ?case by (auto simp del: poly.simps)}
999 ultimately show ?case by blast
1002 lemma (in recpower_idom_char_0) order_degree:
1003 assumes p0: "poly p \<noteq> poly []"
1004 shows "order a p \<le> degree p"
1006 from order2[OF p0, unfolded divides_def]
1007 obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast
1008 {assume "poly q = poly []"
1009 with q p0 have False by (simp add: poly_mult poly_entire)}
1010 with degree_unique[OF q, unfolded linear_pow_mul_degree]
1011 show ?thesis by auto
1014 text{*Tidier versions of finiteness of roots.*}
1016 lemma (in idom_char_0) poly_roots_finite_set: "poly p \<noteq> poly [] ==> finite {x. poly p x = 0}"
1017 unfolding poly_roots_finite .
1019 text{*bound for polynomial.*}
1021 lemma poly_mono: "abs(x) \<le> k ==> abs(poly p (x::'a::{ordered_idom})) \<le> poly (map abs p) k"
1022 apply (induct "p", auto)
1023 apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans)
1024 apply (rule abs_triangle_ineq)
1025 apply (auto intro!: mult_mono simp add: abs_mult)
1028 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp