1 (* Title: Univ_Poly.thy |
|
2 Author: Amine Chaieb |
|
3 *) |
|
4 |
|
5 header {* Univariate Polynomials *} |
|
6 |
|
7 theory Univ_Poly |
|
8 imports Plain List |
|
9 begin |
|
10 |
|
11 text{* Application of polynomial as a function. *} |
|
12 |
|
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" |
|
16 |
|
17 |
|
18 subsection{*Arithmetic Operations on Polynomials*} |
|
19 |
|
20 text{*addition*} |
|
21 |
|
22 primrec (in semiring_0) padd :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infixl "+++" 65) |
|
23 where |
|
24 padd_Nil: "[] +++ l2 = l2" |
|
25 | padd_Cons: "(h#t) +++ l2 = (if l2 = [] then h#t |
|
26 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" 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))" |
|
56 |
|
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)))" |
|
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) |
|
69 poly_minus :: "'a list => 'a list" ("-- _" [80] 80) where |
|
70 "-- p = (- 1) %* p" |
|
71 |
|
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))" |
|
75 |
|
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))" |
|
80 |
|
81 --{*degree of a polynomial*} |
|
82 definition (in semiring_0) degree :: "'a list => nat" where |
|
83 "degree p = length (pnormalize p) - 1" |
|
84 |
|
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)))" |
|
90 |
|
91 context semiring_0 |
|
92 begin |
|
93 |
|
94 lemma padd_Nil2[simp]: "p +++ [] = p" |
|
95 by (induct p) auto |
|
96 |
|
97 lemma padd_Cons_Cons: "(h1 # p1) +++ (h2 # p2) = (h1 + h2) # (p1 +++ p2)" |
|
98 by auto |
|
99 |
|
100 lemma pminus_Nil[simp]: "-- [] = []" |
|
101 by (simp add: poly_minus_def) |
|
102 |
|
103 lemma pmult_singleton: "[h1] *** p1 = h1 %* p1" by simp |
|
104 end |
|
105 |
|
106 lemma (in semiring_1) poly_ident_mult[simp]: "1 %* t = t" by (induct "t", auto) |
|
107 |
|
108 lemma (in semiring_0) poly_simple_add_Cons[simp]: "[a] +++ ((0)#t) = (a#t)" |
|
109 by simp |
|
110 |
|
111 text{*Handy general properties*} |
|
112 |
|
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 |
|
116 next |
|
117 case (Cons b bs a) thus ?case by (cases a, simp_all add: add_commute) |
|
118 qed |
|
119 |
|
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) |
|
124 done |
|
125 |
|
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) |
|
129 done |
|
130 |
|
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) |
|
135 done |
|
136 |
|
137 text{*properties of evaluation of polynomials.*} |
|
138 |
|
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 |
|
142 next |
|
143 case (Cons a as p2) thus ?case |
|
144 by (cases p2, simp_all add: add_ac right_distrib) |
|
145 qed |
|
146 |
|
147 lemma (in comm_semiring_0) poly_cmult: "poly (c %* p) x = c * poly p x" |
|
148 apply (induct "p") |
|
149 apply (case_tac [2] "x=zero") |
|
150 apply (auto simp add: right_distrib mult_ac) |
|
151 done |
|
152 |
|
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) |
|
155 |
|
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]) |
|
159 done |
|
160 |
|
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 |
|
164 next |
|
165 case (Cons a as p2) |
|
166 thus ?case by (cases as, |
|
167 simp_all add: poly_cmult poly_add left_distrib right_distrib mult_ac) |
|
168 qed |
|
169 |
|
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 .. |
|
184 |
|
185 lemma (in recpower_comm_ring_1) poly_exp: "poly (p %^ n) x = (poly p x) ^ n" |
|
186 apply (induct "n") |
|
187 apply (auto simp add: poly_cmult poly_mult power_Suc) |
|
188 done |
|
189 |
|
190 text{*More Polynomial Evaluation Lemmas*} |
|
191 |
|
192 lemma (in semiring_0) poly_add_rzero[simp]: "poly (a +++ []) x = poly a x" |
|
193 by simp |
|
194 |
|
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) |
|
197 |
|
198 lemma (in semiring_0) poly_mult_Nil2[simp]: "poly (p *** []) x = 0" |
|
199 by (induct "p", auto) |
|
200 |
|
201 lemma (in comm_semiring_1) poly_exp_add: "poly (p %^ (n + d)) x = poly( p %^ n *** p %^ d) x" |
|
202 apply (induct "n") |
|
203 apply (auto simp add: poly_mult mult_assoc) |
|
204 done |
|
205 |
|
206 subsection{*Key Property: if @{term "f(a) = 0"} then @{term "(x - a)"} divides |
|
207 @{term "p(x)"} *} |
|
208 |
|
209 lemma (in comm_ring_1) lemma_poly_linear_rem: "\<forall>h. \<exists>q r. h#t = [r] +++ [-a, 1] *** q" |
|
210 proof(induct t) |
|
211 case Nil |
|
212 {fix h have "[h] = [h] +++ [- a, 1] *** []" by simp} |
|
213 thus ?case by blast |
|
214 next |
|
215 case (Cons x xs) |
|
216 {fix h |
|
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: ring_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} |
|
223 thus ?case by blast |
|
224 qed |
|
225 |
|
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) |
|
228 |
|
229 |
|
230 lemma (in comm_ring_1) poly_linear_divides: "(poly p a = 0) = ((p = []) | (\<exists>q. p = [-a, 1] *** q))" |
|
231 proof- |
|
232 {assume p: "p = []" hence ?thesis by simp} |
|
233 moreover |
|
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])} |
|
237 moreover |
|
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) |
|
245 qed |
|
246 |
|
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) |
|
249 |
|
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) |
|
252 |
|
253 lemma (in ring_1) poly_length_mult[simp]: "length([-a,1] *** q) = Suc (length q)" |
|
254 by auto |
|
255 |
|
256 subsection{*Polynomial length*} |
|
257 |
|
258 lemma (in semiring_0) poly_cmult_length[simp]: "length (a %* p) = length p" |
|
259 by (induct "p", auto) |
|
260 |
|
261 lemma (in semiring_0) poly_add_length: "length (p1 +++ p2) = max (length p1) (length p2)" |
|
262 apply (induct p1 arbitrary: p2, simp_all) |
|
263 apply arith |
|
264 done |
|
265 |
|
266 lemma (in semiring_0) poly_root_mult_length[simp]: "length([a,b] *** p) = Suc (length p)" |
|
267 by (simp add: poly_add_length) |
|
268 |
|
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) |
|
272 |
|
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) |
|
275 |
|
276 text{*Normalisation Properties*} |
|
277 |
|
278 lemma (in semiring_0) poly_normalized_nil: "(pnormalize p = []) --> (poly p x = 0)" |
|
279 by (induct "p", auto) |
|
280 |
|
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)" |
|
285 using p n |
|
286 proof(induct n arbitrary: p x) |
|
287 case 0 thus ?case by simp |
|
288 next |
|
289 case (Suc n p x) |
|
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" |
|
304 by blast |
|
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: ring_simps) |
|
307 with i[rule_format, of y] y(1) y(2) have False apply auto |
|
308 apply (erule_tac x="m" in allE) |
|
309 apply auto |
|
310 done} |
|
311 thus ?case by blast |
|
312 qed |
|
313 |
|
314 |
|
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) |
|
318 |
|
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) |
|
325 done |
|
326 |
|
327 |
|
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}" |
|
331 proof- |
|
332 let ?M = "{x. P x}" |
|
333 let ?N = "set j" |
|
334 have "?M \<subseteq> ?N" using P by auto |
|
335 thus ?thesis using finite_subset by auto |
|
336 qed |
|
337 |
|
338 |
|
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) |
|
346 |
|
347 lemma UNIV_nat_infinite: "\<not> finite (UNIV :: nat set)" |
|
348 unfolding finite_conv_nat_seg_image |
|
349 proof(auto simp add: expand_set_eq image_iff) |
|
350 fix n::nat and f:: "nat \<Rightarrow> nat" |
|
351 let ?N = "{i. i < n}" |
|
352 let ?fN = "f ` ?N" |
|
353 let ?y = "Max ?fN + 1" |
|
354 from nat_seg_image_imp_finite[of "?fN" "f" n] |
|
355 have thfN: "finite ?fN" by simp |
|
356 {assume "n =0" hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto} |
|
357 moreover |
|
358 {assume nz: "n \<noteq> 0" |
|
359 hence thne: "?fN \<noteq> {}" by (auto simp add: neq0_conv) |
|
360 have "\<forall>x\<in> ?fN. Max ?fN \<ge> x" using nz Max_ge_iff[OF thfN thne] by auto |
|
361 hence "\<forall>x\<in> ?fN. ?y > x" by auto |
|
362 hence "?y \<notin> ?fN" by auto |
|
363 hence "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by auto } |
|
364 ultimately show "\<exists>x. \<forall>xa<n. x \<noteq> f xa" by blast |
|
365 qed |
|
366 |
|
367 lemma (in ring_char_0) UNIV_ring_char_0_infinte: |
|
368 "\<not> (finite (UNIV:: 'a set))" |
|
369 proof |
|
370 assume F: "finite (UNIV :: 'a set)" |
|
371 have "finite (UNIV :: nat set)" |
|
372 proof (rule finite_imageD) |
|
373 have "of_nat ` UNIV \<subseteq> UNIV" by simp |
|
374 then show "finite (of_nat ` UNIV :: 'a set)" using F by (rule finite_subset) |
|
375 show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: inj_on_def) |
|
376 qed |
|
377 with UNIV_nat_infinite show False .. |
|
378 qed |
|
379 |
|
380 lemma (in idom_char_0) poly_roots_finite: "(poly p \<noteq> poly []) = |
|
381 finite {x. poly p x = 0}" |
|
382 proof |
|
383 assume H: "poly p \<noteq> poly []" |
|
384 show "finite {x. poly p x = (0::'a)}" |
|
385 using H |
|
386 apply - |
|
387 apply (erule contrapos_np, rule ext) |
|
388 apply (rule ccontr) |
|
389 apply (clarify dest!: poly_roots_finite_lemma2) |
|
390 using finite_subset |
|
391 proof- |
|
392 fix x i |
|
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 |
|
398 qed |
|
399 next |
|
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 |
|
402 qed |
|
403 |
|
404 text{*Entirety and Cancellation for polynomials*} |
|
405 |
|
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 []" |
|
409 proof- |
|
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 |
|
413 qed |
|
414 |
|
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)+ |
|
419 |
|
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) |
|
422 |
|
423 lemma fun_eq: " (f = g) = (\<forall>x. f x = g x)" |
|
424 by (auto intro!: ext) |
|
425 |
|
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]) |
|
428 |
|
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]) |
|
431 |
|
432 subclass (in idom_char_0) comm_ring_1 .. |
|
433 lemma (in idom_char_0) poly_mult_left_cancel: "(poly (p *** q) = poly (p *** r)) = (poly p = poly [] | poly q = poly r)" |
|
434 proof- |
|
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 . |
|
439 qed |
|
440 |
|
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]) |
|
445 apply (rule ext) |
|
446 apply (induct n) |
|
447 apply (auto simp add: poly_exp poly_mult) |
|
448 done |
|
449 |
|
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) |
|
457 apply simp |
|
458 done |
|
459 |
|
460 lemma (in recpower_idom) poly_exp_prime_eq_zero: "(poly ([a, 1] %^ n) \<noteq> poly [])" |
|
461 by auto |
|
462 |
|
463 text{*A more constructive notion of polynomials being trivial*} |
|
464 |
|
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) |
|
470 proof- |
|
471 fix x |
|
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 |
|
479 qed |
|
480 |
|
481 lemma (in idom_char_0) poly_zero: "(poly p = poly []) = list_all (%c. c = 0) p" |
|
482 apply (induct "p", simp) |
|
483 apply (rule iffI) |
|
484 apply (drule poly_zero_lemma', auto) |
|
485 done |
|
486 |
|
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 |
|
489 |
|
490 |
|
491 |
|
492 text{*Basics of divisibility.*} |
|
493 |
|
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="[]"]) |
|
500 apply simp |
|
501 apply (cases "q = []") |
|
502 apply (erule allE[where x="[]"], simp) |
|
503 |
|
504 apply clarsimp |
|
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]) |
|
509 apply clarsimp |
|
510 |
|
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) |
|
515 done |
|
516 |
|
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) |
|
521 done |
|
522 |
|
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) |
|
527 done |
|
528 |
|
529 |
|
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) |
|
532 apply (induct_tac k) |
|
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) |
|
537 done |
|
538 |
|
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) |
|
541 |
|
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) |
|
547 done |
|
548 |
|
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) |
|
554 done |
|
555 |
|
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) |
|
559 done |
|
560 |
|
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) |
|
565 done |
|
566 |
|
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) |
|
571 done |
|
572 |
|
573 text{*At last, we can consider the order of a root.*} |
|
574 |
|
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" |
|
578 using lp p |
|
579 proof(induct d arbitrary: p) |
|
580 case 0 thus ?case by simp |
|
581 next |
|
582 case (Suc n p) |
|
583 {assume p0: "poly p a = 0" |
|
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 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 []" |
|
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 minus_mult_left[symmetric]) |
|
594 done |
|
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} |
|
599 moreover |
|
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 |
|
603 qed |
|
604 |
|
605 |
|
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) |
|
608 |
|
609 lemma (in comm_semiring_1) divides_left_mult: |
|
610 assumes d:"(p***q) divides r" shows "p divides r \<and> q divides r" |
|
611 proof- |
|
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 |
|
617 qed |
|
618 |
|
619 |
|
620 |
|
621 (* FIXME: Tidy up *) |
|
622 |
|
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) |
|
626 |
|
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)" |
|
630 proof- |
|
631 let ?poly = poly |
|
632 let ?mulexp = mulexp |
|
633 let ?pexp = pexp |
|
634 from lp p0 |
|
635 show ?thesis |
|
636 apply - |
|
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) |
|
643 apply safe |
|
644 apply (subgoal_tac "?poly (?mulexp n [uminus a, one] q) \<noteq> ?poly (pmult (?pexp [uminus a, one] (Suc n)) qa)") |
|
645 apply simp |
|
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]) |
|
651 apply (rule ccontr) |
|
652 apply (simp add: poly_mult_left_cancel poly_mult_assoc del: pmult_Cons pexp_Suc) |
|
653 done |
|
654 qed |
|
655 |
|
656 |
|
657 lemma (in semiring_1) poly_one_divides[simp]: "[1] divides p" |
|
658 by (simp add: divides_def, auto) |
|
659 |
|
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) |
|
668 done |
|
669 |
|
670 text{*Order*} |
|
671 |
|
672 lemma some1_equalityD: "[| n = (@n. P n); EX! n. P n |] ==> P n" |
|
673 by (blast intro: someI2) |
|
674 |
|
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) |
|
680 apply (rule iffI) |
|
681 apply (blast dest: poly_divides_zero intro!: some1_equality [symmetric] poly_order) |
|
682 apply (blast intro!: poly_order [THEN [2] some1_equalityD]) |
|
683 done |
|
684 |
|
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) |
|
689 |
|
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) |
|
694 |
|
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)) |
|
697 ==> (n = order a p)" |
|
698 by (blast intro: order_unique) |
|
699 |
|
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) |
|
702 |
|
703 lemma (in semiring_1) pexp_one[simp]: "p %^ (Suc 0) = p" |
|
704 apply (induct "p") |
|
705 apply (auto simp add: numeral_1_eq_1) |
|
706 done |
|
707 |
|
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) |
|
713 done |
|
714 |
|
715 lemma (in recpower_idom_char_0) order_root: "(poly p a = 0) = ((poly p = poly []) | order a p \<noteq> 0)" |
|
716 proof- |
|
717 let ?poly = poly |
|
718 show ?thesis |
|
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) |
|
722 apply (rule ccontr) |
|
723 apply (simp add: divides_def poly_mult fun_eq del: pmult_Cons, blast) |
|
724 using neq0_conv |
|
725 apply (blast intro: lemma_order_root) |
|
726 done |
|
727 qed |
|
728 |
|
729 lemma (in recpower_idom_char_0) order_divides: "(([-a, 1] %^ n) divides p) = ((poly p = poly []) | n \<le> order a p)" |
|
730 proof- |
|
731 let ?poly = poly |
|
732 show ?thesis |
|
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) |
|
738 done |
|
739 qed |
|
740 |
|
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) |
|
751 done |
|
752 |
|
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)+ |
|
759 apply safe |
|
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)+ |
|
764 apply safe |
|
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) |
|
774 done |
|
775 |
|
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" |
|
779 proof- |
|
780 let ?order = order |
|
781 let ?divides = "op divides" |
|
782 let ?poly = poly |
|
783 from pq0 |
|
784 show ?thesis |
|
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)+ |
|
788 apply safe |
|
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)+ |
|
793 apply safe |
|
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) |
|
803 done |
|
804 qed |
|
805 |
|
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) |
|
808 |
|
809 lemma (in semiring_1) pmult_one[simp]: "[1] *** p = p" by auto |
|
810 |
|
811 lemma (in semiring_0) poly_Nil_zero: "poly [] = poly [0]" |
|
812 by (simp add: fun_eq) |
|
813 |
|
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) |
|
828 done |
|
829 |
|
830 |
|
831 text{*Normalization of a polynomial.*} |
|
832 |
|
833 lemma (in semiring_0) poly_normalize[simp]: "poly (pnormalize p) = poly p" |
|
834 apply (induct "p") |
|
835 apply (auto simp add: fun_eq) |
|
836 done |
|
837 |
|
838 text{*The degree of a polynomial.*} |
|
839 |
|
840 lemma (in semiring_0) lemma_degree_zero: |
|
841 "list_all (%c. c = 0) p \<longleftrightarrow> pnormalize p = []" |
|
842 by (induct "p", auto) |
|
843 |
|
844 lemma (in idom_char_0) degree_zero: |
|
845 assumes pN: "poly p = poly []" shows"degree p = 0" |
|
846 proof- |
|
847 let ?pn = pnormalize |
|
848 from pN |
|
849 show ?thesis |
|
850 apply (simp add: degree_def) |
|
851 apply (case_tac "?pn p = []") |
|
852 apply (auto simp add: poly_zero lemma_degree_zero ) |
|
853 done |
|
854 qed |
|
855 |
|
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) |
|
864 |
|
865 |
|
866 lemma (in semiring_0) pnormal_last_nonzero: "pnormal p ==> last p \<noteq> 0" |
|
867 proof(induct p) |
|
868 case Nil thus ?case by (simp add: pnormal_def) |
|
869 next |
|
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) |
|
876 done |
|
877 qed |
|
878 |
|
879 lemma (in semiring_0) pnormal_length: "pnormal p \<Longrightarrow> 0 < length p" |
|
880 unfolding pnormal_def length_greater_0_conv by blast |
|
881 |
|
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) |
|
887 |
|
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 |
|
890 |
|
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") |
|
892 proof |
|
893 assume eq: ?lhs |
|
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 |
|
902 next |
|
903 assume ?rhs then show ?lhs by - (rule ext,simp) |
|
904 qed |
|
905 |
|
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 |
|
909 next |
|
910 case (Cons c cs p) |
|
911 thus ?case |
|
912 proof(induct p) |
|
913 case Nil |
|
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 |
|
917 next |
|
918 case (Cons d ds) |
|
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 |
|
927 qed |
|
928 qed |
|
929 |
|
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 |
|
933 |
|
934 lemma (in semiring_0) pnormalize_length: "length (pnormalize p) \<le> length p" by (induct p, auto) |
|
935 |
|
936 lemma (in semiring_0) last_linear_mul_lemma: |
|
937 "last ((a %* p) +++ (x#(b %* p))) = (if p=[] then x else b*last p)" |
|
938 |
|
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) |
|
942 done |
|
943 |
|
944 lemma (in semiring_1) last_linear_mul: assumes p:"p\<noteq>[]" shows "last ([a,1] *** p) = last p" |
|
945 proof- |
|
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 |
|
951 qed |
|
952 |
|
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)+ |
|
956 done |
|
957 |
|
958 lemma (in semiring_0) last_pnormalize: "pnormalize p \<noteq> [] \<Longrightarrow> last (pnormalize p) \<noteq> 0" |
|
959 by (induct p, auto) |
|
960 |
|
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 |
|
963 |
|
964 lemma (in semiring_0) poly_Nil_ext: "poly [] = (\<lambda>x. 0)" by (rule ext) simp |
|
965 |
|
966 lemma (in idom_char_0) linear_mul_degree: assumes p: "poly p \<noteq> poly []" |
|
967 shows "degree ([a,1] *** p) = degree p + 1" |
|
968 proof- |
|
969 from p have pnz: "pnormalize p \<noteq> []" |
|
970 unfolding poly_zero lemma_degree_zero . |
|
971 |
|
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 |
|
976 |
|
977 |
|
978 have th: "degree ([a,1] *** pnormalize p) = degree (pnormalize p) + 1" |
|
979 by (auto simp add: poly_length_mult) |
|
980 |
|
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]) |
|
985 qed |
|
986 |
|
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) |
|
990 case (0 a p) |
|
991 {assume p: "poly p = poly []" |
|
992 hence ?case using degree_unique[OF p] by (simp add: degree_def)} |
|
993 moreover |
|
994 {assume p: "poly p \<noteq> poly []" hence ?case by (auto simp add: poly_Nil_ext) } |
|
995 ultimately show ?case by blast |
|
996 next |
|
997 case (Suc n a p) |
|
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)} |
|
1006 moreover |
|
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') |
|
1015 by simp |
|
1016 |
|
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 |
|
1020 qed |
|
1021 |
|
1022 lemma (in recpower_idom_char_0) order_degree: |
|
1023 assumes p0: "poly p \<noteq> poly []" |
|
1024 shows "order a p \<le> degree p" |
|
1025 proof- |
|
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 |
|
1032 qed |
|
1033 |
|
1034 text{*Tidier versions of finiteness of roots.*} |
|
1035 |
|
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 . |
|
1038 |
|
1039 text{*bound for polynomial.*} |
|
1040 |
|
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) |
|
1046 done |
|
1047 |
|
1048 lemma (in semiring_0) poly_Sing: "poly [c] x = c" by simp |
|
1049 |
|
1050 end |
|