1 (* Title: HOL/Library/Univ_Poly.thy |
|
2 Author: Amine Chaieb |
|
3 *) |
|
4 |
|
5 header {* Univariate Polynomials *} |
|
6 |
|
7 theory Univ_Poly |
|
8 imports Main |
|
9 begin |
|
10 |
|
11 text{* Application of polynomial as a function. *} |
|
12 |
|
13 primrec (in semiring_0) poly :: "'a list \<Rightarrow> 'a \<Rightarrow> 'a" |
|
14 where |
|
15 poly_Nil: "poly [] x = 0" |
|
16 | poly_Cons: "poly (h#t) x = h + x * poly t x" |
|
17 |
|
18 |
|
19 subsection{*Arithmetic Operations on Polynomials*} |
|
20 |
|
21 text{*addition*} |
|
22 |
|
23 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "+++" 65) |
|
24 where |
|
25 padd_Nil: "[] +++ l2 = l2" |
|
26 | padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t else (h + hd l2)#(t +++ tl l2))" |
|
27 |
|
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)" |
|
32 |
|
33 text{*Multiplication by a polynomial*} |
|
34 primrec (in semiring_0) pmult :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "***" 70) |
|
35 where |
|
36 pmult_Nil: "[] *** l2 = []" |
|
37 | pmult_Cons: "(h#t) *** l2 = (if t = [] then h %* l2 |
|
38 else (h %* l2) +++ ((0) # (t *** l2)))" |
|
39 |
|
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" |
|
44 |
|
45 text{*Exponential*} |
|
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)" |
|
49 |
|
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" |
|
53 where |
|
54 pquot_Nil: "pquot [] a= []" |
|
55 | pquot_Cons: "pquot (h#t) a = |
|
56 (if t = [] then [h] else (inverse(a) * (h - hd( pquot t a)))#(pquot t a))" |
|
57 |
|
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) = |
|
62 (if pnormalize p = [] then (if h = 0 then [] else [h]) else h # pnormalize p)" |
|
63 |
|
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*} |
|
67 |
|
68 definition (in ring_1) poly_minus :: "'a list \<Rightarrow> 'a list" ("-- _" [80] 80) |
|
69 where "-- p = (- 1) %* p" |
|
70 |
|
71 definition (in semiring_0) divides :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixl "divides" 70) |
|
72 where "p1 divides p2 = (\<exists>q. poly p2 = poly(p1 *** q))" |
|
73 |
|
74 lemma (in semiring_0) dividesI: |
|
75 "poly p2 = poly (p1 *** q) \<Longrightarrow> p1 divides p2" |
|
76 by (auto simp add: divides_def) |
|
77 |
|
78 lemma (in semiring_0) dividesE: |
|
79 assumes "p1 divides p2" |
|
80 obtains q where "poly p2 = poly (p1 *** q)" |
|
81 using assms by (auto simp add: divides_def) |
|
82 |
|
83 --{*order of a polynomial*} |
|
84 definition (in ring_1) order :: "'a \<Rightarrow> 'a list \<Rightarrow> nat" where |
|
85 "order a p = (SOME n. ([-a, 1] %^ n) divides p \<and> ~ (([-a, 1] %^ (Suc n)) divides p))" |
|
86 |
|
87 --{*degree of a polynomial*} |
|
88 definition (in semiring_0) degree :: "'a list \<Rightarrow> nat" |
|
89 where "degree p = length (pnormalize p) - 1" |
|
90 |
|
91 --{*squarefree polynomials --- NB with respect to real roots only.*} |
|
92 definition (in ring_1) rsquarefree :: "'a list \<Rightarrow> bool" |
|
93 where "rsquarefree p \<longleftrightarrow> poly p \<noteq> poly [] \<and> (\<forall>a. order a p = 0 \<or> order a p = 1)" |
|
94 |
|
95 context semiring_0 |
|
96 begin |
|
97 |
|
98 lemma padd_Nil2[simp]: "p +++ [] = p" |
|
99 by (induct p) auto |
|
100 |
|
101 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)" |
|
102 by auto |
|
103 |
|
104 lemma pminus_Nil: "-- [] = []" |
|
105 by (simp add: poly_minus_def) |
|
106 |
|
107 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp |
|
108 |
|
109 end |
|
110 |
|
111 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct t) auto |
|
112 |
|
113 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)" |
|
114 by simp |
|
115 |
|
116 text{*Handy general properties*} |
|
117 |
|
118 lemma (in comm_semiring_0) padd_commut: "b +++ a = a +++ b" |
|
119 proof (induct b arbitrary: a) |
|
120 case Nil |
|
121 thus ?case by auto |
|
122 next |
|
123 case (Cons b bs a) |
|
124 thus ?case by (cases a) (simp_all add: add_commute) |
|
125 qed |
|
126 |
|
127 lemma (in comm_semiring_0) padd_assoc: "\<forall>b c. (a +++ b) +++ c = a +++ (b +++ c)" |
|
128 apply (induct a) |
|
129 apply (simp, clarify) |
|
130 apply (case_tac b, simp_all add: add_ac) |
|
131 done |
|
132 |
|
133 lemma (in semiring_0) poly_cmult_distr: "a %* ( p +++ q) = (a %* p +++ a %* q)" |
|
134 apply (induct p arbitrary: q) |
|
135 apply simp |
|
136 apply (case_tac q, simp_all add: distrib_left) |
|
137 done |
|
138 |
|
139 lemma (in ring_1) pmult_by_x[simp]: "[0, 1] *** t = ((0)#t)" |
|
140 apply (induct t) |
|
141 apply simp |
|
142 apply (auto simp add: padd_commut) |
|
143 apply (case_tac t, auto) |
|
144 done |
|
145 |
|
146 text{*properties of evaluation of polynomials.*} |
|
147 |
|
148 lemma (in semiring_0) poly_add: "poly (p1 +++ p2) x = poly p1 x + poly p2 x" |
|
149 proof(induct p1 arbitrary: p2) |
|
150 case Nil |
|
151 thus ?case by simp |
|
152 next |
|
153 case (Cons a as p2) |
|
154 thus ?case |
|
155 by (cases p2) (simp_all add: add_ac distrib_left) |
|
156 qed |
|
157 |
|
158 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x" |
|
159 apply (induct p) |
|
160 apply (case_tac [2] "x = zero") |
|
161 apply (auto simp add: distrib_left mult_ac) |
|
162 done |
|
163 |
|
164 lemma (in comm_semiring_0) poly_cmult_map: "poly (map (op * c) p) x = c*poly p x" |
|
165 by (induct p) (auto simp add: distrib_left mult_ac) |
|
166 |
|
167 lemma (in comm_ring_1) poly_minus: "poly (-- p) x = - (poly p x)" |
|
168 apply (simp add: poly_minus_def) |
|
169 apply (auto simp add: poly_cmult) |
|
170 done |
|
171 |
|
172 lemma (in comm_semiring_0) poly_mult: "poly (p1 *** p2) x = poly p1 x * poly p2 x" |
|
173 proof (induct p1 arbitrary: p2) |
|
174 case Nil |
|
175 thus ?case by simp |
|
176 next |
|
177 case (Cons a as p2) |
|
178 thus ?case by (cases as) |
|
179 (simp_all add: poly_cmult poly_add distrib_right distrib_left mult_ac) |
|
180 qed |
|
181 |
|
182 class idom_char_0 = idom + ring_char_0 |
|
183 |
|
184 lemma (in comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n" |
|
185 by (induct n) (auto simp add: poly_cmult poly_mult) |
|
186 |
|
187 text{*More Polynomial Evaluation Lemmas*} |
|
188 |
|
189 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x" |
|
190 by simp |
|
191 |
|
192 lemma (in comm_semiring_0) poly_mult_assoc: "poly ((a *** b) *** c) x = poly (a *** (b *** c)) x" |
|
193 by (simp add: poly_mult mult_assoc) |
|
194 |
|
195 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0" |
|
196 by (induct p) auto |
|
197 |
|
198 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x" |
|
199 by (induct n) (auto simp add: poly_mult mult_assoc) |
|
200 |
|
201 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides |
|
202 @{term "p(x)"} *} |
|
203 |
|
204 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q" |
|
205 proof(induct t) |
|
206 case Nil |
|
207 { fix h have "[h] = [h] +++ [- a, 1] *** []" by simp } |
|
208 thus ?case by blast |
|
209 next |
|
210 case (Cons x xs) |
|
211 { fix h |
|
212 from Cons.hyps[rule_format, of x] |
|
213 obtain q r where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast |
|
214 have "h#x#xs = [a*r + h] +++ [-a, 1] *** (r#q)" |
|
215 using qr by (cases q) (simp_all add: algebra_simps) |
|
216 hence "\<exists>q r. h#x#xs = [r] +++ [-a, 1] *** q" by blast} |
|
217 thus ?case by blast |
|
218 qed |
|
219 |
|
220 lemma (in comm_ring_1) poly_linear_rem: "\<exists>q r. h#t = [r] +++ [-a, 1] *** q" |
|
221 using lemma_poly_linear_rem [where t = t and a = a] by auto |
|
222 |
|
223 |
|
224 lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))" |
|
225 proof - |
|
226 { assume p: "p = []" hence ?thesis by simp } |
|
227 moreover |
|
228 { |
|
229 fix x xs assume p: "p = x#xs" |
|
230 { |
|
231 fix q assume "p = [-a, 1] *** q" |
|
232 hence "poly p a = 0" by (simp add: poly_add poly_cmult) |
|
233 } |
|
234 moreover |
|
235 { assume p0: "poly p a = 0" |
|
236 from poly_linear_rem[of x xs a] obtain q r |
|
237 where qr: "x#xs = [r] +++ [- a, 1] *** q" by blast |
|
238 have "r = 0" using p0 by (simp only: p qr poly_mult poly_add) simp |
|
239 hence "\<exists>q. p = [- a, 1] *** q" |
|
240 using p qr |
|
241 apply - |
|
242 apply (rule exI[where x=q]) |
|
243 apply auto |
|
244 apply (cases q) |
|
245 apply auto |
|
246 done |
|
247 } |
|
248 ultimately have ?thesis using p by blast |
|
249 } |
|
250 ultimately show ?thesis by (cases p) auto |
|
251 qed |
|
252 |
|
253 lemma (in semiring_0) lemma_poly_length_mult[simp]: "\<forall>h k a. length (k %* p +++ (h # (a %* p))) = Suc (length p)" |
|
254 by (induct p) auto |
|
255 |
|
256 lemma (in semiring_0) lemma_poly_length_mult2[simp]: "\<forall>h k. length (k %* p +++ (h # p)) = Suc (length p)" |
|
257 by (induct p) auto |
|
258 |
|
259 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)" |
|
260 by auto |
|
261 |
|
262 subsection{*Polynomial length*} |
|
263 |
|
264 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p" |
|
265 by (induct p) auto |
|
266 |
|
267 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)" |
|
268 by (induct p1 arbitrary: p2) (simp_all, arith) |
|
269 |
|
270 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)" |
|
271 by (simp add: poly_add_length) |
|
272 |
|
273 lemma (in idom) poly_mult_not_eq_poly_Nil[simp]: |
|
274 "poly (p *** q) x \<noteq> poly [] x \<longleftrightarrow> poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x" |
|
275 by (auto simp add: poly_mult) |
|
276 |
|
277 lemma (in idom) poly_mult_eq_zero_disj: "poly (p *** q) x = 0 \<longleftrightarrow> poly p x = 0 \<or> poly q x = 0" |
|
278 by (auto simp add: poly_mult) |
|
279 |
|
280 text{*Normalisation Properties*} |
|
281 |
|
282 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)" |
|
283 by (induct p) auto |
|
284 |
|
285 text{*A nontrivial polynomial of degree n has no more than n roots*} |
|
286 lemma (in idom) poly_roots_index_lemma: |
|
287 assumes p: "poly p x \<noteq> poly [] x" and n: "length p = n" |
|
288 shows "\<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" |
|
289 using p n |
|
290 proof (induct n arbitrary: p x) |
|
291 case 0 |
|
292 thus ?case by simp |
|
293 next |
|
294 case (Suc n p x) |
|
295 { |
|
296 assume C: "\<And>i. \<exists>x. poly p x = 0 \<and> (\<forall>m\<le>Suc n. x \<noteq> i m)" |
|
297 from Suc.prems have p0: "poly p x \<noteq> 0" "p\<noteq> []" by auto |
|
298 from p0(1)[unfolded poly_linear_divides[of p x]] |
|
299 have "\<forall>q. p \<noteq> [- x, 1] *** q" by blast |
|
300 from C obtain a where a: "poly p a = 0" by blast |
|
301 from a[unfolded poly_linear_divides[of p a]] p0(2) |
|
302 obtain q where q: "p = [-a, 1] *** q" by blast |
|
303 have lg: "length q = n" using q Suc.prems(2) by simp |
|
304 from q p0 have qx: "poly q x \<noteq> poly [] x" |
|
305 by (auto simp add: poly_mult poly_add poly_cmult) |
|
306 from Suc.hyps[OF qx lg] obtain i where |
|
307 i: "\<forall>x. poly q x = 0 \<longrightarrow> (\<exists>m\<le>n. x = i m)" by blast |
|
308 let ?i = "\<lambda>m. if m = Suc n then a else i m" |
|
309 from C[of ?i] obtain y where y: "poly p y = 0" "\<forall>m\<le> Suc n. y \<noteq> ?i m" |
|
310 by blast |
|
311 from y have "y = a \<or> poly q y = 0" |
|
312 by (simp only: q poly_mult_eq_zero_disj poly_add) (simp add: algebra_simps) |
|
313 with i[rule_format, of y] y(1) y(2) have False |
|
314 apply auto |
|
315 apply (erule_tac x = "m" in allE) |
|
316 apply auto |
|
317 done |
|
318 } |
|
319 thus ?case by blast |
|
320 qed |
|
321 |
|
322 |
|
323 lemma (in idom) poly_roots_index_length: |
|
324 "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. n \<le> length p \<and> x = i n)" |
|
325 by (blast intro: poly_roots_index_lemma) |
|
326 |
|
327 lemma (in idom) poly_roots_finite_lemma1: |
|
328 "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>N i. \<forall>x. (poly p x = 0) \<longrightarrow> (\<exists>n. (n::nat) < N \<and> x = i n)" |
|
329 apply (drule poly_roots_index_length, safe) |
|
330 apply (rule_tac x = "Suc (length p)" in exI) |
|
331 apply (rule_tac x = i in exI) |
|
332 apply (simp add: less_Suc_eq_le) |
|
333 done |
|
334 |
|
335 lemma (in idom) idom_finite_lemma: |
|
336 assumes P: "\<forall>x. P x --> (\<exists>n. n < length j \<and> x = j!n)" |
|
337 shows "finite {x. P x}" |
|
338 proof - |
|
339 let ?M = "{x. P x}" |
|
340 let ?N = "set j" |
|
341 have "?M \<subseteq> ?N" using P by auto |
|
342 thus ?thesis using finite_subset by auto |
|
343 qed |
|
344 |
|
345 lemma (in idom) poly_roots_finite_lemma2: |
|
346 "poly p x \<noteq> poly [] x \<Longrightarrow> \<exists>i. \<forall>x. poly p x = 0 \<longrightarrow> x \<in> set i" |
|
347 apply (drule poly_roots_index_length, safe) |
|
348 apply (rule_tac x="map (\<lambda>n. i n) [0 ..< Suc (length p)]" in exI) |
|
349 apply (auto simp add: image_iff) |
|
350 apply (erule_tac x="x" in allE, clarsimp) |
|
351 apply (case_tac "n = length p") |
|
352 apply (auto simp add: order_le_less) |
|
353 done |
|
354 |
|
355 lemma (in ring_char_0) UNIV_ring_char_0_infinte: "\<not> (finite (UNIV:: 'a set))" |
|
356 proof |
|
357 assume F: "finite (UNIV :: 'a set)" |
|
358 have "finite (UNIV :: nat set)" |
|
359 proof (rule finite_imageD) |
|
360 have "of_nat ` UNIV \<subseteq> UNIV" by simp |
|
361 then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset) |
|
362 show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def) |
|
363 qed |
|
364 with infinite_UNIV_nat show False .. |
|
365 qed |
|
366 |
|
367 lemma (in idom_char_0) poly_roots_finite: "poly p \<noteq> poly [] \<longleftrightarrow> finite {x. poly p x = 0}" |
|
368 proof |
|
369 assume H: "poly p \<noteq> poly []" |
|
370 show "finite {x. poly p x = (0::'a)}" |
|
371 using H |
|
372 apply - |
|
373 apply (erule contrapos_np, rule ext) |
|
374 apply (rule ccontr) |
|
375 apply (clarify dest!: poly_roots_finite_lemma2) |
|
376 using finite_subset |
|
377 proof - |
|
378 fix x i |
|
379 assume F: "\<not> finite {x. poly p x = (0\<Colon>'a)}" |
|
380 and P: "\<forall>x. poly p x = (0\<Colon>'a) \<longrightarrow> x \<in> set i" |
|
381 let ?M= "{x. poly p x = (0\<Colon>'a)}" |
|
382 from P have "?M \<subseteq> set i" by auto |
|
383 with finite_subset F show False by auto |
|
384 qed |
|
385 next |
|
386 assume F: "finite {x. poly p x = (0\<Colon>'a)}" |
|
387 show "poly p \<noteq> poly []" using F UNIV_ring_char_0_infinte by auto |
|
388 qed |
|
389 |
|
390 text{*Entirety and Cancellation for polynomials*} |
|
391 |
|
392 lemma (in idom_char_0) poly_entire_lemma2: |
|
393 assumes p0: "poly p \<noteq> poly []" |
|
394 and q0: "poly q \<noteq> poly []" |
|
395 shows "poly (p***q) \<noteq> poly []" |
|
396 proof - |
|
397 let ?S = "\<lambda>p. {x. poly p x = 0}" |
|
398 have "?S (p *** q) = ?S p \<union> ?S q" by (auto simp add: poly_mult) |
|
399 with p0 q0 show ?thesis unfolding poly_roots_finite by auto |
|
400 qed |
|
401 |
|
402 lemma (in idom_char_0) poly_entire: |
|
403 "poly (p *** q) = poly [] \<longleftrightarrow> poly p = poly [] \<or> poly q = poly []" |
|
404 using poly_entire_lemma2[of p q] |
|
405 by (auto simp add: fun_eq_iff poly_mult) |
|
406 |
|
407 lemma (in idom_char_0) poly_entire_neg: |
|
408 "poly (p *** q) \<noteq> poly [] \<longleftrightarrow> poly p \<noteq> poly [] \<and> poly q \<noteq> poly []" |
|
409 by (simp add: poly_entire) |
|
410 |
|
411 lemma fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)" |
|
412 by auto |
|
413 |
|
414 lemma (in comm_ring_1) poly_add_minus_zero_iff: |
|
415 "poly (p +++ -- q) = poly [] \<longleftrightarrow> poly p = poly q" |
|
416 by (auto simp add: algebra_simps poly_add poly_minus_def fun_eq poly_cmult) |
|
417 |
|
418 lemma (in comm_ring_1) poly_add_minus_mult_eq: |
|
419 "poly (p *** q +++ --(p *** r)) = poly (p *** (q +++ -- r))" |
|
420 by (auto simp add: poly_add poly_minus_def fun_eq poly_mult poly_cmult distrib_left) |
|
421 |
|
422 subclass (in idom_char_0) comm_ring_1 .. |
|
423 |
|
424 lemma (in idom_char_0) poly_mult_left_cancel: |
|
425 "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r" |
|
426 proof - |
|
427 have "poly (p *** q) = poly (p *** r) \<longleftrightarrow> poly (p *** q +++ -- (p *** r)) = poly []" |
|
428 by (simp only: poly_add_minus_zero_iff) |
|
429 also have "\<dots> \<longleftrightarrow> poly p = poly [] \<or> poly q = poly r" |
|
430 by (auto intro: simp add: poly_add_minus_mult_eq poly_entire poly_add_minus_zero_iff) |
|
431 finally show ?thesis . |
|
432 qed |
|
433 |
|
434 lemma (in idom) poly_exp_eq_zero[simp]: |
|
435 "poly (p %^ n) = poly [] \<longleftrightarrow> poly p = poly [] \<and> n \<noteq> 0" |
|
436 apply (simp only: fun_eq add: HOL.all_simps [symmetric]) |
|
437 apply (rule arg_cong [where f = All]) |
|
438 apply (rule ext) |
|
439 apply (induct n) |
|
440 apply (auto simp add: poly_exp poly_mult) |
|
441 done |
|
442 |
|
443 lemma (in comm_ring_1) poly_prime_eq_zero[simp]: "poly [a,1] \<noteq> poly []" |
|
444 apply (simp add: fun_eq) |
|
445 apply (rule_tac x = "minus one a" in exI) |
|
446 apply (unfold diff_minus) |
|
447 apply (subst add_commute) |
|
448 apply (subst add_assoc) |
|
449 apply simp |
|
450 done |
|
451 |
|
452 lemma (in idom) poly_exp_prime_eq_zero: "poly ([a, 1] %^ n) \<noteq> poly []" |
|
453 by auto |
|
454 |
|
455 text{*A more constructive notion of polynomials being trivial*} |
|
456 |
|
457 lemma (in idom_char_0) poly_zero_lemma': "poly (h # t) = poly [] \<Longrightarrow> h = 0 \<and> poly t = poly []" |
|
458 apply (simp add: fun_eq) |
|
459 apply (case_tac "h = zero") |
|
460 apply (drule_tac [2] x = zero in spec, auto) |
|
461 apply (cases "poly t = poly []", simp) |
|
462 proof - |
|
463 fix x |
|
464 assume H: "\<forall>x. x = (0\<Colon>'a) \<or> poly t x = (0\<Colon>'a)" |
|
465 and pnz: "poly t \<noteq> poly []" |
|
466 let ?S = "{x. poly t x = 0}" |
|
467 from H have "\<forall>x. x \<noteq>0 \<longrightarrow> poly t x = 0" by blast |
|
468 hence th: "?S \<supseteq> UNIV - {0}" by auto |
|
469 from poly_roots_finite pnz have th': "finite ?S" by blast |
|
470 from finite_subset[OF th th'] UNIV_ring_char_0_infinte show "poly t x = (0\<Colon>'a)" |
|
471 by simp |
|
472 qed |
|
473 |
|
474 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p" |
|
475 apply (induct p) |
|
476 apply simp |
|
477 apply (rule iffI) |
|
478 apply (drule poly_zero_lemma', auto) |
|
479 done |
|
480 |
|
481 lemma (in idom_char_0) poly_0: "list_all (\<lambda>c. c = 0) p \<Longrightarrow> poly p x = 0" |
|
482 unfolding poly_zero[symmetric] by simp |
|
483 |
|
484 |
|
485 |
|
486 text{*Basics of divisibility.*} |
|
487 |
|
488 lemma (in idom) poly_primes: |
|
489 "[a, 1] divides (p *** q) \<longleftrightarrow> [a, 1] divides p \<or> [a, 1] divides q" |
|
490 apply (auto simp add: divides_def fun_eq poly_mult poly_add poly_cmult distrib_right [symmetric]) |
|
491 apply (drule_tac x = "uminus a" in spec) |
|
492 apply (simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric]) |
|
493 apply (cases "p = []") |
|
494 apply (rule exI[where x="[]"]) |
|
495 apply simp |
|
496 apply (cases "q = []") |
|
497 apply (erule allE[where x="[]"], simp) |
|
498 |
|
499 apply clarsimp |
|
500 apply (cases "\<exists>q\<Colon>'a list. p = a %* q +++ ((0\<Colon>'a) # q)") |
|
501 apply (clarsimp simp add: poly_add poly_cmult) |
|
502 apply (rule_tac x="qa" in exI) |
|
503 apply (simp add: distrib_right [symmetric]) |
|
504 apply clarsimp |
|
505 |
|
506 apply (auto simp add: poly_linear_divides poly_add poly_cmult distrib_right [symmetric]) |
|
507 apply (rule_tac x = "pmult qa q" in exI) |
|
508 apply (rule_tac [2] x = "pmult p qa" in exI) |
|
509 apply (auto simp add: poly_add poly_mult poly_cmult mult_ac) |
|
510 done |
|
511 |
|
512 lemma (in comm_semiring_1) poly_divides_refl[simp]: "p divides p" |
|
513 apply (simp add: divides_def) |
|
514 apply (rule_tac x = "[one]" in exI) |
|
515 apply (auto simp add: poly_mult fun_eq) |
|
516 done |
|
517 |
|
518 lemma (in comm_semiring_1) poly_divides_trans: "p divides q \<Longrightarrow> q divides r \<Longrightarrow> p divides r" |
|
519 apply (simp add: divides_def, safe) |
|
520 apply (rule_tac x = "pmult qa qaa" in exI) |
|
521 apply (auto simp add: poly_mult fun_eq mult_assoc) |
|
522 done |
|
523 |
|
524 lemma (in comm_semiring_1) poly_divides_exp: "m \<le> n \<Longrightarrow> (p %^ m) divides (p %^ n)" |
|
525 apply (auto simp add: le_iff_add) |
|
526 apply (induct_tac k) |
|
527 apply (rule_tac [2] poly_divides_trans) |
|
528 apply (auto simp add: divides_def) |
|
529 apply (rule_tac x = p in exI) |
|
530 apply (auto simp add: poly_mult fun_eq mult_ac) |
|
531 done |
|
532 |
|
533 lemma (in comm_semiring_1) poly_exp_divides: |
|
534 "(p %^ n) divides q \<Longrightarrow> m \<le> n \<Longrightarrow> (p %^ m) divides q" |
|
535 by (blast intro: poly_divides_exp poly_divides_trans) |
|
536 |
|
537 lemma (in comm_semiring_0) poly_divides_add: |
|
538 "p divides q \<Longrightarrow> p divides r \<Longrightarrow> p divides (q +++ r)" |
|
539 apply (simp add: divides_def, auto) |
|
540 apply (rule_tac x = "padd qa qaa" in exI) |
|
541 apply (auto simp add: poly_add fun_eq poly_mult distrib_left) |
|
542 done |
|
543 |
|
544 lemma (in comm_ring_1) poly_divides_diff: |
|
545 "p divides q \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides r" |
|
546 apply (simp add: divides_def, auto) |
|
547 apply (rule_tac x = "padd qaa (poly_minus qa)" in exI) |
|
548 apply (auto simp add: poly_add fun_eq poly_mult poly_minus algebra_simps) |
|
549 done |
|
550 |
|
551 lemma (in comm_ring_1) poly_divides_diff2: |
|
552 "p divides r \<Longrightarrow> p divides (q +++ r) \<Longrightarrow> p divides q" |
|
553 apply (erule poly_divides_diff) |
|
554 apply (auto simp add: poly_add fun_eq poly_mult divides_def add_ac) |
|
555 done |
|
556 |
|
557 lemma (in semiring_0) poly_divides_zero: "poly p = poly [] \<Longrightarrow> q divides p" |
|
558 apply (simp add: divides_def) |
|
559 apply (rule exI[where x="[]"]) |
|
560 apply (auto simp add: fun_eq poly_mult) |
|
561 done |
|
562 |
|
563 lemma (in semiring_0) poly_divides_zero2 [simp]: "q divides []" |
|
564 apply (simp add: divides_def) |
|
565 apply (rule_tac x = "[]" in exI) |
|
566 apply (auto simp add: fun_eq) |
|
567 done |
|
568 |
|
569 text{*At last, we can consider the order of a root.*} |
|
570 |
|
571 lemma (in idom_char_0) poly_order_exists_lemma: |
|
572 assumes lp: "length p = d" |
|
573 and p: "poly p \<noteq> poly []" |
|
574 shows "\<exists>n q. p = mulexp n [-a, 1] q \<and> poly q a \<noteq> 0" |
|
575 using lp p |
|
576 proof (induct d arbitrary: p) |
|
577 case 0 |
|
578 thus ?case by simp |
|
579 next |
|
580 case (Suc n p) |
|
581 show ?case |
|
582 proof (cases "poly p a = 0") |
|
583 case True |
|
584 from Suc.prems have h: "length p = Suc n" "poly p \<noteq> poly []" by auto |
|
585 hence pN: "p \<noteq> []" by auto |
|
586 from True[unfolded poly_linear_divides] pN obtain q where q: "p = [-a, 1] *** q" |
|
587 by blast |
|
588 from q h True have qh: "length q = n" "poly q \<noteq> poly []" |
|
589 apply - |
|
590 apply simp |
|
591 apply (simp only: fun_eq) |
|
592 apply (rule ccontr) |
|
593 apply (simp add: fun_eq poly_add poly_cmult) |
|
594 done |
|
595 from Suc.hyps[OF qh] obtain m r where mr: "q = mulexp m [-a,1] r" "poly r a \<noteq> 0" |
|
596 by blast |
|
597 from mr q have "p = mulexp (Suc m) [-a,1] r \<and> poly r a \<noteq> 0" by simp |
|
598 then show ?thesis by blast |
|
599 next |
|
600 case False |
|
601 then show ?thesis |
|
602 using Suc.prems |
|
603 apply simp |
|
604 apply (rule exI[where x="0::nat"]) |
|
605 apply simp |
|
606 done |
|
607 qed |
|
608 qed |
|
609 |
|
610 |
|
611 lemma (in comm_semiring_1) poly_mulexp: "poly (mulexp n p q) x = (poly p x) ^ n * poly q x" |
|
612 by (induct n) (auto simp add: poly_mult mult_ac) |
|
613 |
|
614 lemma (in comm_semiring_1) divides_left_mult: |
|
615 assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r" |
|
616 proof- |
|
617 from d obtain t where r:"poly r = poly (p***q *** t)" |
|
618 unfolding divides_def by blast |
|
619 hence "poly r = poly (p *** (q *** t))" |
|
620 "poly r = poly (q *** (p***t))" by(auto simp add: fun_eq poly_mult mult_ac) |
|
621 thus ?thesis unfolding divides_def by blast |
|
622 qed |
|
623 |
|
624 |
|
625 (* FIXME: Tidy up *) |
|
626 |
|
627 lemma (in semiring_1) zero_power_iff: "0 ^ n = (if n = 0 then 1 else 0)" |
|
628 by (induct n) simp_all |
|
629 |
|
630 lemma (in idom_char_0) poly_order_exists: |
|
631 assumes "length p = d" and "poly p \<noteq> poly []" |
|
632 shows "\<exists>n. [- a, 1] %^ n divides p \<and> \<not> [- a, 1] %^ Suc n divides p" |
|
633 proof - |
|
634 from assms have "\<exists>n q. p = mulexp n [- a, 1] q \<and> poly q a \<noteq> 0" |
|
635 by (rule poly_order_exists_lemma) |
|
636 then obtain n q where p: "p = mulexp n [- a, 1] q" and "poly q a \<noteq> 0" by blast |
|
637 have "[- a, 1] %^ n divides mulexp n [- a, 1] q" |
|
638 proof (rule dividesI) |
|
639 show "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ n *** q)" |
|
640 by (induct n) (simp_all add: poly_add poly_cmult poly_mult distrib_left mult_ac) |
|
641 qed |
|
642 moreover have "\<not> [- a, 1] %^ Suc n divides mulexp n [- a, 1] q" |
|
643 proof |
|
644 assume "[- a, 1] %^ Suc n divides mulexp n [- a, 1] q" |
|
645 then obtain m where "poly (mulexp n [- a, 1] q) = poly ([- a, 1] %^ Suc n *** m)" |
|
646 by (rule dividesE) |
|
647 moreover have "poly (mulexp n [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc n *** m)" |
|
648 proof (induct n) |
|
649 case 0 show ?case |
|
650 proof (rule ccontr) |
|
651 assume "\<not> poly (mulexp 0 [- a, 1] q) \<noteq> poly ([- a, 1] %^ Suc 0 *** m)" |
|
652 then have "poly q a = 0" |
|
653 by (simp add: poly_add poly_cmult) |
|
654 with `poly q a \<noteq> 0` show False by simp |
|
655 qed |
|
656 next |
|
657 case (Suc n) show ?case |
|
658 by (rule pexp_Suc [THEN ssubst], rule ccontr) |
|
659 (simp add: poly_mult_left_cancel poly_mult_assoc Suc del: pmult_Cons pexp_Suc) |
|
660 qed |
|
661 ultimately show False by simp |
|
662 qed |
|
663 ultimately show ?thesis by (auto simp add: p) |
|
664 qed |
|
665 |
|
666 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p" |
|
667 by (auto simp add: divides_def) |
|
668 |
|
669 lemma (in idom_char_0) poly_order: |
|
670 "poly p \<noteq> poly [] \<Longrightarrow> \<exists>!n. ([-a, 1] %^ n) divides p \<and> \<not> (([-a, 1] %^ Suc n) divides p)" |
|
671 apply (auto intro: poly_order_exists simp add: less_linear simp del: pmult_Cons pexp_Suc) |
|
672 apply (cut_tac x = y and y = n in less_linear) |
|
673 apply (drule_tac m = n in poly_exp_divides) |
|
674 apply (auto dest: Suc_le_eq [THEN iffD2, THEN [2] poly_exp_divides] |
|
675 simp del: pmult_Cons pexp_Suc) |
|
676 done |
|
677 |
|
678 text{*Order*} |
|
679 |
|
680 lemma some1_equalityD: "n = (SOME n. P n) \<Longrightarrow> \<exists>!n. P n \<Longrightarrow> P n" |
|
681 by (blast intro: someI2) |
|
682 |
|
683 lemma (in idom_char_0) order: |
|
684 "(([-a, 1] %^ n) divides p \<and> |
|
685 ~(([-a, 1] %^ (Suc n)) divides p)) = |
|
686 ((n = order a p) \<and> ~(poly p = poly []))" |
|
687 apply (unfold order_def) |
|
688 apply (rule iffI) |
|
689 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order) |
|
690 apply (blast intro!: poly_order [THEN [2] some1_equalityD]) |
|
691 done |
|
692 |
|
693 lemma (in idom_char_0) order2: |
|
694 "poly p \<noteq> poly [] \<Longrightarrow> |
|
695 ([-a, 1] %^ (order a p)) divides p \<and> \<not> (([-a, 1] %^ (Suc (order a p))) divides p)" |
|
696 by (simp add: order del: pexp_Suc) |
|
697 |
|
698 lemma (in idom_char_0) order_unique: |
|
699 "poly p \<noteq> poly [] \<Longrightarrow> ([-a, 1] %^ n) divides p \<Longrightarrow> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow> |
|
700 n = order a p" |
|
701 using order [of a n p] by auto |
|
702 |
|
703 lemma (in idom_char_0) order_unique_lemma: |
|
704 "poly p \<noteq> poly [] \<and> ([-a, 1] %^ n) divides p \<and> ~(([-a, 1] %^ (Suc n)) divides p) \<Longrightarrow> |
|
705 n = order a p" |
|
706 by (blast intro: order_unique) |
|
707 |
|
708 lemma (in ring_1) order_poly: "poly p = poly q \<Longrightarrow> order a p = order a q" |
|
709 by (auto simp add: fun_eq divides_def poly_mult order_def) |
|
710 |
|
711 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p" |
|
712 by (induct "p") auto |
|
713 |
|
714 lemma (in comm_ring_1) lemma_order_root: |
|
715 "0 < n \<and> [- a, 1] %^ n divides p \<and> ~ [- a, 1] %^ (Suc n) divides p \<Longrightarrow> poly p a = 0" |
|
716 by (induct n arbitrary: a p) (auto simp add: divides_def poly_mult simp del: pmult_Cons) |
|
717 |
|
718 lemma (in idom_char_0) order_root: |
|
719 "poly p a = 0 \<longleftrightarrow> poly p = poly [] \<or> order a p \<noteq> 0" |
|
720 apply (cases "poly p = poly []") |
|
721 apply auto |
|
722 apply (simp add: poly_linear_divides del: pmult_Cons, safe) |
|
723 apply (drule_tac [!] a = a in order2) |
|
724 apply (rule ccontr) |
|
725 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast) |
|
726 using neq0_conv |
|
727 apply (blast intro: lemma_order_root) |
|
728 done |
|
729 |
|
730 lemma (in idom_char_0) order_divides: |
|
731 "([-a, 1] %^ n) divides p \<longleftrightarrow> poly p = poly [] \<or> n \<le> order a p" |
|
732 apply (cases "poly p = poly []") |
|
733 apply 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] intro: poly_exp_divides simp del: pexp_Suc) |
|
737 done |
|
738 |
|
739 lemma (in idom_char_0) order_decomp: |
|
740 "poly p \<noteq> poly [] \<Longrightarrow> \<exists>q. poly p = poly (([-a, 1] %^ (order a p)) *** q) \<and> ~([-a, 1] divides q)" |
|
741 apply (unfold divides_def) |
|
742 apply (drule order2 [where a = a]) |
|
743 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe) |
|
744 apply (rule_tac x = q in exI, safe) |
|
745 apply (drule_tac x = qa in spec) |
|
746 apply (auto simp add: poly_mult fun_eq poly_exp mult_ac simp del: pmult_Cons) |
|
747 done |
|
748 |
|
749 text{*Important composition properties of orders.*} |
|
750 lemma order_mult: |
|
751 "poly (p *** q) \<noteq> poly [] \<Longrightarrow> |
|
752 order a (p *** q) = order a p + order (a::'a::{idom_char_0}) q" |
|
753 apply (cut_tac a = a and p = "p *** q" and n = "order a p + order a q" in order) |
|
754 apply (auto simp add: poly_entire simp del: pmult_Cons) |
|
755 apply (drule_tac a = a in order2)+ |
|
756 apply safe |
|
757 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe) |
|
758 apply (rule_tac x = "qa *** qaa" in exI) |
|
759 apply (simp add: poly_mult mult_ac del: pmult_Cons) |
|
760 apply (drule_tac a = a in order_decomp)+ |
|
761 apply safe |
|
762 apply (subgoal_tac "[-a,1] divides (qa *** qaa) ") |
|
763 apply (simp add: poly_primes del: pmult_Cons) |
|
764 apply (auto simp add: divides_def simp del: pmult_Cons) |
|
765 apply (rule_tac x = qb in exI) |
|
766 apply (subgoal_tac "poly ([-a, 1] %^ (order a p) *** (qa *** qaa)) = poly ([-a, 1] %^ (order a p) *** ([-a, 1] *** qb))") |
|
767 apply (drule poly_mult_left_cancel [THEN iffD1], force) |
|
768 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))) ") |
|
769 apply (drule poly_mult_left_cancel [THEN iffD1], force) |
|
770 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons) |
|
771 done |
|
772 |
|
773 lemma (in idom_char_0) order_mult: |
|
774 assumes "poly (p *** q) \<noteq> poly []" |
|
775 shows "order a (p *** q) = order a p + order a q" |
|
776 using assms |
|
777 apply (cut_tac a = a and p = "pmult p q" and n = "order a p + order a q" in order) |
|
778 apply (auto simp add: poly_entire simp del: pmult_Cons) |
|
779 apply (drule_tac a = a in order2)+ |
|
780 apply safe |
|
781 apply (simp add: divides_def fun_eq poly_exp_add poly_mult del: pmult_Cons, safe) |
|
782 apply (rule_tac x = "pmult qa qaa" in exI) |
|
783 apply (simp add: poly_mult mult_ac del: pmult_Cons) |
|
784 apply (drule_tac a = a in order_decomp)+ |
|
785 apply safe |
|
786 apply (subgoal_tac "[uminus a, one] divides pmult qa qaa") |
|
787 apply (simp add: poly_primes del: pmult_Cons) |
|
788 apply (auto simp add: divides_def simp del: pmult_Cons) |
|
789 apply (rule_tac x = qb in exI) |
|
790 apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa)) = |
|
791 poly (pmult (pexp [uminus a, one] (?order a p)) (pmult [uminus a, one] qb))") |
|
792 apply (drule poly_mult_left_cancel [THEN iffD1], force) |
|
793 apply (subgoal_tac "poly (pmult (pexp [uminus a, one] (order a q)) |
|
794 (pmult (pexp [uminus a, one] (order a p)) (pmult qa qaa))) = |
|
795 poly (pmult (pexp [uminus a, one] (order a q)) |
|
796 (pmult (pexp [uminus a, one] (order a p)) (pmult [uminus a, one] qb)))") |
|
797 apply (drule poly_mult_left_cancel [THEN iffD1], force) |
|
798 apply (simp add: fun_eq poly_exp_add poly_mult mult_ac del: pmult_Cons) |
|
799 done |
|
800 |
|
801 lemma (in idom_char_0) order_root2: "poly p \<noteq> poly [] \<Longrightarrow> poly p a = 0 \<longleftrightarrow> order a p \<noteq> 0" |
|
802 by (rule order_root [THEN ssubst]) auto |
|
803 |
|
804 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto |
|
805 |
|
806 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]" |
|
807 by (simp add: fun_eq) |
|
808 |
|
809 lemma (in idom_char_0) rsquarefree_decomp: |
|
810 "rsquarefree p \<Longrightarrow> poly p a = 0 \<Longrightarrow> |
|
811 \<exists>q. poly p = poly ([-a, 1] *** q) \<and> poly q a \<noteq> 0" |
|
812 apply (simp add: rsquarefree_def, safe) |
|
813 apply (frule_tac a = a in order_decomp) |
|
814 apply (drule_tac x = a in spec) |
|
815 apply (drule_tac a = a in order_root2 [symmetric]) |
|
816 apply (auto simp del: pmult_Cons) |
|
817 apply (rule_tac x = q in exI, safe) |
|
818 apply (simp add: poly_mult fun_eq) |
|
819 apply (drule_tac p1 = q in poly_linear_divides [THEN iffD1]) |
|
820 apply (simp add: divides_def del: pmult_Cons, safe) |
|
821 apply (drule_tac x = "[]" in spec) |
|
822 apply (auto simp add: fun_eq) |
|
823 done |
|
824 |
|
825 |
|
826 text{*Normalization of a polynomial.*} |
|
827 |
|
828 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p" |
|
829 by (induct p) (auto simp add: fun_eq) |
|
830 |
|
831 text{*The degree of a polynomial.*} |
|
832 |
|
833 lemma (in semiring_0) lemma_degree_zero: "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []" |
|
834 by (induct p) auto |
|
835 |
|
836 lemma (in idom_char_0) degree_zero: |
|
837 assumes "poly p = poly []" |
|
838 shows "degree p = 0" |
|
839 using assms |
|
840 by (cases "pnormalize p = []") (auto simp add: degree_def poly_zero lemma_degree_zero) |
|
841 |
|
842 lemma (in semiring_0) pnormalize_sing: "(pnormalize [x] = [x]) \<longleftrightarrow> x \<noteq> 0" |
|
843 by simp |
|
844 |
|
845 lemma (in semiring_0) pnormalize_pair: "y \<noteq> 0 \<longleftrightarrow> (pnormalize [x, y] = [x, y])" |
|
846 by simp |
|
847 |
|
848 lemma (in semiring_0) pnormal_cons: "pnormal p \<Longrightarrow> pnormal (c#p)" |
|
849 unfolding pnormal_def by simp |
|
850 |
|
851 lemma (in semiring_0) pnormal_tail: "p\<noteq>[] \<Longrightarrow> pnormal (c#p) \<Longrightarrow> pnormal p" |
|
852 unfolding pnormal_def by(auto split: split_if_asm) |
|
853 |
|
854 |
|
855 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p \<Longrightarrow> last p \<noteq> 0" |
|
856 by (induct p) (simp_all add: pnormal_def split: split_if_asm) |
|
857 |
|
858 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p" |
|
859 unfolding pnormal_def length_greater_0_conv by blast |
|
860 |
|
861 lemma (in semiring_0) pnormal_last_length: "0 < length p \<Longrightarrow> last p \<noteq> 0 \<Longrightarrow> pnormal p" |
|
862 by (induct p) (auto simp: pnormal_def split: split_if_asm) |
|
863 |
|
864 |
|
865 lemma (in semiring_0) pnormal_id: "pnormal p \<longleftrightarrow> 0 < length p \<and> last p \<noteq> 0" |
|
866 using pnormal_last_length pnormal_length pnormal_last_nonzero by blast |
|
867 |
|
868 lemma (in idom_char_0) poly_Cons_eq: |
|
869 "poly (c # cs) = poly (d # ds) \<longleftrightarrow> c = d \<and> poly cs = poly ds" |
|
870 (is "?lhs \<longleftrightarrow> ?rhs") |
|
871 proof |
|
872 assume eq: ?lhs |
|
873 hence "\<And>x. poly ((c#cs) +++ -- (d#ds)) x = 0" |
|
874 by (simp only: poly_minus poly_add algebra_simps) simp |
|
875 hence "poly ((c#cs) +++ -- (d#ds)) = poly []" by(simp add: fun_eq_iff) |
|
876 hence "c = d \<and> list_all (\<lambda>x. x=0) ((cs +++ -- ds))" |
|
877 unfolding poly_zero by (simp add: poly_minus_def algebra_simps) |
|
878 hence "c = d \<and> (\<forall>x. poly (cs +++ -- ds) x = 0)" |
|
879 unfolding poly_zero[symmetric] by simp |
|
880 then show ?rhs by (simp add: poly_minus poly_add algebra_simps fun_eq_iff) |
|
881 next |
|
882 assume ?rhs |
|
883 then show ?lhs by(simp add:fun_eq_iff) |
|
884 qed |
|
885 |
|
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 |
|
889 thus ?case by (simp only: poly_zero lemma_degree_zero) simp |
|
890 next |
|
891 case (Cons c cs p) |
|
892 thus ?case |
|
893 proof (induct p) |
|
894 case Nil |
|
895 hence "poly [] = poly (c#cs)" by blast |
|
896 then have "poly (c#cs) = poly [] " by simp |
|
897 thus ?case by (simp only: poly_zero lemma_degree_zero) simp |
|
898 next |
|
899 case (Cons d ds) |
|
900 hence eq: "poly (d # ds) = poly (c # cs)" by blast |
|
901 hence eq': "\<And>x. poly (d # ds) x = poly (c # cs) x" by simp |
|
902 hence "poly (d # ds) 0 = poly (c # cs) 0" by blast |
|
903 hence dc: "d = c" by auto |
|
904 with eq have "poly ds = poly cs" |
|
905 unfolding poly_Cons_eq by simp |
|
906 with Cons.prems have "pnormalize ds = pnormalize cs" by blast |
|
907 with dc show ?case by simp |
|
908 qed |
|
909 qed |
|
910 |
|
911 lemma (in idom_char_0) degree_unique: |
|
912 assumes pq: "poly p = poly q" |
|
913 shows "degree p = degree q" |
|
914 using pnormalize_unique[OF pq] unfolding degree_def by simp |
|
915 |
|
916 lemma (in semiring_0) pnormalize_length: |
|
917 "length (pnormalize p) \<le> length p" by (induct p) auto |
|
918 |
|
919 lemma (in semiring_0) last_linear_mul_lemma: |
|
920 "last ((a %* p) +++ (x#(b %* p))) = (if p = [] then x else b * last p)" |
|
921 apply (induct p arbitrary: a x b) |
|
922 apply auto |
|
923 apply (subgoal_tac "padd (cmult aa p) (times b a # cmult b p) \<noteq> []") |
|
924 apply simp |
|
925 apply (induct_tac p) |
|
926 apply auto |
|
927 done |
|
928 |
|
929 lemma (in semiring_1) last_linear_mul: |
|
930 assumes p: "p \<noteq> []" |
|
931 shows "last ([a,1] *** p) = last p" |
|
932 proof - |
|
933 from p obtain c cs where cs: "p = c#cs" by (cases p) auto |
|
934 from cs have eq: "[a,1] *** p = (a %* (c#cs)) +++ (0#(1 %* (c#cs)))" |
|
935 by (simp add: poly_cmult_distr) |
|
936 show ?thesis using cs |
|
937 unfolding eq last_linear_mul_lemma by simp |
|
938 qed |
|
939 |
|
940 lemma (in semiring_0) pnormalize_eq: "last p \<noteq> 0 \<Longrightarrow> pnormalize p = p" |
|
941 by (induct p) (auto split: split_if_asm) |
|
942 |
|
943 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0" |
|
944 by (induct p) auto |
|
945 |
|
946 lemma (in semiring_0) pnormal_degree: "last p \<noteq> 0 \<Longrightarrow> degree p = length p - 1" |
|
947 using pnormalize_eq[of p] unfolding degree_def by simp |
|
948 |
|
949 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" |
|
950 by (rule ext) simp |
|
951 |
|
952 lemma (in idom_char_0) linear_mul_degree: |
|
953 assumes p: "poly p \<noteq> poly []" |
|
954 shows "degree ([a,1] *** p) = degree p + 1" |
|
955 proof - |
|
956 from p have pnz: "pnormalize p \<noteq> []" |
|
957 unfolding poly_zero lemma_degree_zero . |
|
958 |
|
959 from last_linear_mul[OF pnz, of a] last_pnormalize[OF pnz] |
|
960 have l0: "last ([a, 1] *** pnormalize p) \<noteq> 0" by simp |
|
961 from last_pnormalize[OF pnz] last_linear_mul[OF pnz, of a] |
|
962 pnormal_degree[OF l0] pnormal_degree[OF last_pnormalize[OF pnz]] pnz |
|
963 |
|
964 have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1" |
|
965 by simp |
|
966 |
|
967 have eqs: "poly ([a,1] *** pnormalize p) = poly ([a,1] *** p)" |
|
968 by (rule ext) (simp add: poly_mult poly_add poly_cmult) |
|
969 from degree_unique[OF eqs] th |
|
970 show ?thesis by (simp add: degree_unique[OF poly_normalize]) |
|
971 qed |
|
972 |
|
973 lemma (in idom_char_0) linear_pow_mul_degree: |
|
974 "degree([a,1] %^n *** p) = (if poly p = poly [] then 0 else degree p + n)" |
|
975 proof (induct n arbitrary: a p) |
|
976 case (0 a p) |
|
977 show ?case |
|
978 proof (cases "poly p = poly []") |
|
979 case True |
|
980 then show ?thesis |
|
981 using degree_unique[OF True] by (simp add: degree_def) |
|
982 next |
|
983 case False |
|
984 then show ?thesis by (auto simp add: poly_Nil_ext) |
|
985 qed |
|
986 next |
|
987 case (Suc n a p) |
|
988 have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1] %^ n *** ([a,1] *** p))" |
|
989 apply (rule ext) |
|
990 apply (simp add: poly_mult poly_add poly_cmult) |
|
991 apply (simp add: mult_ac add_ac distrib_left) |
|
992 done |
|
993 note deq = degree_unique[OF eq] |
|
994 show ?case |
|
995 proof (cases "poly p = poly []") |
|
996 case True |
|
997 with eq have eq': "poly ([a,1] %^(Suc n) *** p) = poly []" |
|
998 apply - |
|
999 apply (rule ext) |
|
1000 apply (simp add: poly_mult poly_cmult poly_add) |
|
1001 done |
|
1002 from degree_unique[OF eq'] True show ?thesis |
|
1003 by (simp add: degree_def) |
|
1004 next |
|
1005 case False |
|
1006 then have ap: "poly ([a,1] *** p) \<noteq> poly []" |
|
1007 using poly_mult_not_eq_poly_Nil unfolding poly_entire by auto |
|
1008 have eq: "poly ([a,1] %^(Suc n) *** p) = poly ([a,1]%^n *** ([a,1] *** p))" |
|
1009 by (rule ext, simp add: poly_mult poly_add poly_exp poly_cmult algebra_simps) |
|
1010 from ap have ap': "(poly ([a,1] *** p) = poly []) = False" |
|
1011 by blast |
|
1012 have th0: "degree ([a,1]%^n *** ([a,1] *** p)) = degree ([a,1] *** p) + n" |
|
1013 apply (simp only: Suc.hyps[of a "pmult [a,one] p"] ap') |
|
1014 apply simp |
|
1015 done |
|
1016 from degree_unique[OF eq] ap False th0 linear_mul_degree[OF False, of a] |
|
1017 show ?thesis by (auto simp del: poly.simps) |
|
1018 qed |
|
1019 qed |
|
1020 |
|
1021 lemma (in idom_char_0) order_degree: |
|
1022 assumes p0: "poly p \<noteq> poly []" |
|
1023 shows "order a p \<le> degree p" |
|
1024 proof - |
|
1025 from order2[OF p0, unfolded divides_def] |
|
1026 obtain q where q: "poly p = poly ([- a, 1]%^ (order a p) *** q)" by blast |
|
1027 { |
|
1028 assume "poly q = poly []" |
|
1029 with q p0 have False by (simp add: poly_mult poly_entire) |
|
1030 } |
|
1031 with degree_unique[OF q, unfolded linear_pow_mul_degree] show ?thesis |
|
1032 by auto |
|
1033 qed |
|
1034 |
|
1035 text{*Tidier versions of finiteness of roots.*} |
|
1036 |
|
1037 lemma (in idom_char_0) poly_roots_finite_set: |
|
1038 "poly p \<noteq> poly [] \<Longrightarrow> finite {x. poly p x = 0}" |
|
1039 unfolding poly_roots_finite . |
|
1040 |
|
1041 text{*bound for polynomial.*} |
|
1042 |
|
1043 lemma poly_mono: "abs(x) \<le> k \<Longrightarrow> abs(poly p (x::'a::{linordered_idom})) \<le> poly (map abs p) k" |
|
1044 apply (induct p) |
|
1045 apply auto |
|
1046 apply (rule_tac y = "abs a + abs (x * poly p x)" in order_trans) |
|
1047 apply (rule abs_triangle_ineq) |
|
1048 apply (auto intro!: mult_mono simp add: abs_mult) |
|
1049 done |
|
1050 |
|
1051 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp |
|
1052 |
|
1053 end |
|