src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author hoelzl Wed May 07 12:25:35 2014 +0200 (2014-05-07) changeset 56889 48a745e1bde7 parent 56795 e8cce2bd23e5 child 57512 cc97b347b301 permissions -rw-r--r--
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
1 (* Author: Amine Chaieb, TU Muenchen *)
3 header{*Fundamental Theorem of Algebra*}
5 theory Fundamental_Theorem_Algebra
6 imports Polynomial Complex_Main
7 begin
9 subsection {* More lemmas about module of complex numbers *}
11 text{* The triangle inequality for cmod *}
12 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
13   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
15 subsection {* Basic lemmas about polynomials *}
17 lemma poly_bound_exists:
18   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
19   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
20 proof (induct p)
21   case 0
22   then show ?case by (rule exI[where x=1]) simp
23 next
24   case (pCons c cs)
25   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
26     by blast
27   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
28   have kp: "?k > 0"
29     using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
30   {
31     fix z :: 'a
32     assume H: "norm z \<le> r"
33     from m H have th: "norm (poly cs z) \<le> m"
34       by blast
35     from H have rp: "r \<ge> 0"
36       using norm_ge_zero[of z] by arith
37     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)"
38       using norm_triangle_ineq[of c "z* poly cs z"] by simp
39     also have "\<dots> \<le> norm c + r * m"
40       using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
41       by (simp add: norm_mult)
42     also have "\<dots> \<le> ?k"
43       by simp
44     finally have "norm (poly (pCons c cs) z) \<le> ?k" .
45   }
46   with kp show ?case by blast
47 qed
50 text{* Offsetting the variable in a polynomial gives another of same degree *}
52 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
53   where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
55 lemma offset_poly_0: "offset_poly 0 h = 0"
56   by (simp add: offset_poly_def)
58 lemma offset_poly_pCons:
59   "offset_poly (pCons a p) h =
60     smult h (offset_poly p h) + pCons a (offset_poly p h)"
61   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
63 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
64   by (simp add: offset_poly_pCons offset_poly_0)
66 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
67   apply (induct p)
68   apply (simp add: offset_poly_0)
69   apply (simp add: offset_poly_pCons algebra_simps)
70   done
72 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
73   by (induct p arbitrary: a) (simp, force)
75 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
76   apply (safe intro!: offset_poly_0)
77   apply (induct p)
78   apply simp
79   apply (simp add: offset_poly_pCons)
80   apply (frule offset_poly_eq_0_lemma, simp)
81   done
83 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
84   apply (induct p)
85   apply (simp add: offset_poly_0)
86   apply (case_tac "p = 0")
87   apply (simp add: offset_poly_0 offset_poly_pCons)
88   apply (simp add: offset_poly_pCons)
89   apply (subst degree_add_eq_right)
90   apply (rule le_less_trans [OF degree_smult_le])
91   apply (simp add: offset_poly_eq_0_iff)
92   apply (simp add: offset_poly_eq_0_iff)
93   done
95 definition "psize p = (if p = 0 then 0 else Suc (degree p))"
97 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
98   unfolding psize_def by simp
100 lemma poly_offset:
101   fixes p :: "'a::comm_ring_1 poly"
102   shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
103 proof (intro exI conjI)
104   show "psize (offset_poly p a) = psize p"
105     unfolding psize_def
106     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
107   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
108     by (simp add: poly_offset_poly)
109 qed
111 text{* An alternative useful formulation of completeness of the reals *}
112 lemma real_sup_exists:
113   assumes ex: "\<exists>x. P x"
114     and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
115   shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
116 proof
117   from bz have "bdd_above (Collect P)"
118     by (force intro: less_imp_le)
119   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
120     using ex bz by (subst less_cSup_iff) auto
121 qed
123 subsection {* Fundamental theorem of algebra *}
124 lemma  unimodular_reduce_norm:
125   assumes md: "cmod z = 1"
126   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
127 proof -
128   obtain x y where z: "z = Complex x y "
129     by (cases z) auto
130   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
131     by (simp add: cmod_def)
132   {
133     assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
134     from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
135       by (simp_all add: cmod_def power2_eq_square algebra_simps)
136     then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
137       by simp_all
138     then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
139       by - (rule power_mono, simp, simp)+
140     then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
141       by (simp_all add: power_mult_distrib)
142     from add_mono[OF th0] xy have False by simp
143   }
144   then show ?thesis
145     unfolding linorder_not_le[symmetric] by blast
146 qed
148 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
149 lemma reduce_poly_simple:
150   assumes b: "b \<noteq> 0"
151     and n: "n \<noteq> 0"
152   shows "\<exists>z. cmod (1 + b * z^n) < 1"
153   using n
154 proof (induct n rule: nat_less_induct)
155   fix n
156   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
157   assume n: "n \<noteq> 0"
158   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
159   {
160     assume e: "even n"
161     then have "\<exists>m. n = 2 * m"
162       by presburger
163     then obtain m where m: "n = 2 * m"
164       by blast
165     from n m have "m \<noteq> 0" "m < n"
166       by presburger+
167     with IH[rule_format, of m] obtain z where z: "?P z m"
168       by blast
169     from z have "?P (csqrt z) n"
170       by (simp add: m power_mult power2_csqrt)
171     then have "\<exists>z. ?P z n" ..
172   }
173   moreover
174   {
175     assume o: "odd n"
176     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
177       using b by (simp add: norm_divide)
178     from o have "\<exists>m. n = Suc (2 * m)"
179       by presburger+
180     then obtain m where m: "n = Suc (2 * m)"
181       by blast
182     from unimodular_reduce_norm[OF th0] o
183     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
184       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
185       apply (rule_tac x="1" in exI)
186       apply simp
187       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")
188       apply (rule_tac x="-1" in exI)
189       apply simp
190       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
191       apply (cases "even m")
192       apply (rule_tac x="ii" in exI)
193       apply (simp add: m power_mult)
194       apply (rule_tac x="- ii" in exI)
195       apply (simp add: m power_mult)
196       apply (cases "even m")
197       apply (rule_tac x="- ii" in exI)
198       apply (simp add: m power_mult)
199       apply (auto simp add: m power_mult)
200       apply (rule_tac x="ii" in exI)
201       apply (auto simp add: m power_mult)
202       done
203     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
204       by blast
205     let ?w = "v / complex_of_real (root n (cmod b))"
206     from odd_real_root_pow[OF o, of "cmod b"]
207     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
208       by (simp add: power_divide of_real_power[symmetric])
209     have th2:"cmod (complex_of_real (cmod b) / b) = 1"
210       using b by (simp add: norm_divide)
211     then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
212       by simp
213     have th4: "cmod (complex_of_real (cmod b) / b) *
214         cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
215         cmod (complex_of_real (cmod b) / b) * 1"
216       apply (simp only: norm_mult[symmetric] distrib_left)
217       using b v
218       apply (simp add: th2)
219       done
220     from mult_less_imp_less_left[OF th4 th3]
221     have "?P ?w n" unfolding th1 .
222     then have "\<exists>z. ?P z n" ..
223   }
224   ultimately show "\<exists>z. ?P z n" by blast
225 qed
227 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
229 lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
230   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]
231   unfolding cmod_def by simp
233 lemma bolzano_weierstrass_complex_disc:
234   assumes r: "\<forall>n. cmod (s n) \<le> r"
235   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
236 proof-
237   from seq_monosub[of "Re \<circ> s"]
238   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
239     unfolding o_def by blast
240   from seq_monosub[of "Im \<circ> s \<circ> f"]
241   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
242     unfolding o_def by blast
243   let ?h = "f \<circ> g"
244   from r[rule_format, of 0] have rp: "r \<ge> 0"
245     using norm_ge_zero[of "s 0"] by arith
246   have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
247   proof
248     fix n
249     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
250     show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
251   qed
252   have conv1: "convergent (\<lambda>n. Re (s (f n)))"
253     apply (rule Bseq_monoseq_convergent)
254     apply (simp add: Bseq_def)
255     apply (metis gt_ex le_less_linear less_trans order.trans th)
256     apply (rule f(2))
257     done
258   have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
259   proof
260     fix n
261     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
262     show "\<bar>Im (s n)\<bar> \<le> r + 1"
263       by arith
264   qed
266   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
267     apply (rule Bseq_monoseq_convergent)
268     apply (simp add: Bseq_def)
269     apply (metis gt_ex le_less_linear less_trans order.trans th)
270     apply (rule g(2))
271     done
273   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
274     by blast
275   then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
276     unfolding LIMSEQ_iff real_norm_def .
278   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
279     by blast
280   then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
281     unfolding LIMSEQ_iff real_norm_def .
282   let ?w = "Complex x y"
283   from f(1) g(1) have hs: "subseq ?h"
284     unfolding subseq_def by auto
285   {
286     fix e :: real
287     assume ep: "e > 0"
288     then have e2: "e/2 > 0"
289       by simp
290     from x[rule_format, OF e2] y[rule_format, OF e2]
291     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
292       and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
293       by blast
294     {
295       fix n
296       assume nN12: "n \<ge> N1 + N2"
297       then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
298         using seq_suble[OF g(1), of n] by arith+
299       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
300       have "cmod (s (?h n) - ?w) < e"
301         using metric_bound_lemma[of "s (f (g n))" ?w] by simp
302     }
303     then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e"
304       by blast
305   }
306   with hs show ?thesis by blast
307 qed
309 text{* Polynomial is continuous. *}
311 lemma poly_cont:
312   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
313   assumes ep: "e > 0"
314   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
315 proof -
316   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
317   proof
318     show "degree (offset_poly p z) = degree p"
319       by (rule degree_offset_poly)
320     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
321       by (rule poly_offset_poly)
322   qed
323   have th: "\<And>w. poly q (w - z) = poly p w"
324     using q(2)[of "w - z" for w] by simp
325   show ?thesis unfolding th[symmetric]
326   proof (induct q)
327     case 0
328     then show ?case
329       using ep by auto
330   next
331     case (pCons c cs)
332     from poly_bound_exists[of 1 "cs"]
333     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
334       by blast
335     from ep m(1) have em0: "e/m > 0"
336       by (simp add: field_simps)
337     have one0: "1 > (0::real)"
338       by arith
339     from real_lbound_gt_zero[OF one0 em0]
340     obtain d where d: "d > 0" "d < 1" "d < e / m"
341       by blast
342     from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
343       by (simp_all add: field_simps)
344     show ?case
345     proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
346       fix d w
347       assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
348       then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
349         by simp_all
350       from H(3) m(1) have dme: "d*m < e"
351         by (simp add: field_simps)
352       from H have th: "norm (w - z) \<le> d"
353         by simp
354       from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
355       show "norm (w - z) * norm (poly cs (w - z)) < e"
356         by simp
357     qed
358   qed
359 qed
361 text{* Hence a polynomial attains minimum on a closed disc
362   in the complex plane. *}
363 lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
364 proof -
365   {
366     assume "\<not> r \<ge> 0"
367     then have ?thesis
368       by (metis norm_ge_zero order.trans)
369   }
370   moreover
371   {
372     assume rp: "r \<ge> 0"
373     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
374       by simp
375     then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
376       by blast
377     {
378       fix x z
379       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
380       then have "- x < 0 "
381         by arith
382       with H(2) norm_ge_zero[of "poly p z"] have False
383         by simp
384     }
385     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
386       by blast
387     from real_sup_exists[OF mth1 mth2] obtain s where
388       s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow> y < s" by blast
389     let ?m = "- s"
390     {
391       fix y
392       from s[rule_format, of "-y"]
393       have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
394         unfolding minus_less_iff[of y ] equation_minus_iff by blast
395     }
396     note s1 = this[unfolded minus_minus]
397     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
398       by auto
399     {
400       fix n :: nat
401       from s1[rule_format, of "?m + 1/real (Suc n)"]
402       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
403         by simp
404     }
405     then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
406     from choice[OF th] obtain g where
407         g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
408       by blast
409     from bolzano_weierstrass_complex_disc[OF g(1)]
410     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
411       by blast
412     {
413       fix w
414       assume wr: "cmod w \<le> r"
415       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
416       {
417         assume e: "?e > 0"
418         then have e2: "?e/2 > 0"
419           by simp
420         from poly_cont[OF e2, of z p] obtain d where
421             d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
422           by blast
423         {
424           fix w
425           assume w: "cmod (w - z) < d"
426           have "cmod(poly p w - poly p z) < ?e / 2"
427             using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
428         }
429         note th1 = this
431         from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
432           by blast
433         from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
434           by blast
435         have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
436           using N1[rule_format, of "N1 + N2"] th1 by simp
437         {
438           fix a b e2 m :: real
439           have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
440             by arith
441         }
442         note th0 = this
443         have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"
444           by arith
445         from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
446         from seq_suble[OF fz(1), of "N1 + N2"]
447         have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
448           by simp
449         have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
450           using N2 by auto
451         from frac_le[OF th000 th00]
452         have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
453           by simp
454         from g(2)[rule_format, of "f (N1 + N2)"]
455         have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
456         from order_less_le_trans[OF th01 th00]
457         have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
458         from N2 have "2/?e < real (Suc (N1 + N2))"
459           by arith
460         with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
461         have "?e/2 > 1/ real (Suc (N1 + N2))"
462           by (simp add: inverse_eq_divide)
463         with ath[OF th31 th32]
464         have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
465           by arith
466         have ath2: "\<And>a b c m::real. \<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c"
467           by arith
468         have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
469             cmod (poly p (g (f (N1 + N2))) - poly p z)"
470           by (simp add: norm_triangle_ineq3)
471         from ath2[OF th22, of ?m]
472         have thc2: "2 * (?e/2) \<le>
473             \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
474           by simp
475         from th0[OF th2 thc1 thc2] have False .
476       }
477       then have "?e = 0"
478         by auto
479       then have "cmod (poly p z) = ?m"
480         by simp
481       with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
482         by simp
483     }
484     then have ?thesis by blast
485   }
486   ultimately show ?thesis by blast
487 qed
489 text {* Nonzero polynomial in z goes to infinity as z does. *}
491 lemma poly_infinity:
492   fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
493   assumes ex: "p \<noteq> 0"
494   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
495   using ex
496 proof (induct p arbitrary: a d)
497   case 0
498   then show ?case by simp
499 next
500   case (pCons c cs a d)
501   show ?case
502   proof (cases "cs = 0")
503     case False
504     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
505       by blast
506     let ?r = "1 + \<bar>r\<bar>"
507     {
508       fix z :: 'a
509       assume h: "1 + \<bar>r\<bar> \<le> norm z"
510       have r0: "r \<le> norm z"
511         using h by arith
512       from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
513         by arith
514       from h have z1: "norm z \<ge> 1"
515         by arith
516       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
517       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
518         unfolding norm_mult by (simp add: algebra_simps)
519       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
520       have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
521         by (simp add: algebra_simps)
522       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
523         by arith
524     }
525     then show ?thesis by blast
526   next
527     case True
528     with pCons.prems have c0: "c \<noteq> 0"
529       by simp
530     {
531       fix z :: 'a
532       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
533       from c0 have "norm c > 0"
534         by simp
535       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
536         by (simp add: field_simps norm_mult)
537       have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
538         by arith
539       from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
540         by (simp add: algebra_simps)
541       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
542         using True by simp
543     }
544     then show ?thesis by blast
545   qed
546 qed
548 text {* Hence polynomial's modulus attains its minimum somewhere. *}
549 lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
550 proof (induct p)
551   case 0
552   then show ?case by simp
553 next
554   case (pCons c cs)
555   show ?case
556   proof (cases "cs = 0")
557     case False
558     from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
559     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
560       by blast
561     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
562       by arith
563     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
564     obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)"
565       by blast
566     {
567       fix z
568       assume z: "r \<le> cmod z"
569       from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
570         by simp
571     }
572     note v0 = this
573     from v0 v ath[of r] show ?thesis
574       by blast
575   next
576     case True
577     with pCons.hyps show ?thesis by simp
578   qed
579 qed
581 text{* Constant function (non-syntactic characterization). *}
582 definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"
584 lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
585   by (induct p) (auto simp: constant_def psize_def)
587 lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
588   by (simp add: poly_monom)
590 text {* Decomposition of polynomial, skipping zero coefficients
591   after the first.  *}
593 lemma poly_decompose_lemma:
594   assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
595   shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and> (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
596   unfolding psize_def
597   using nz
598 proof (induct p)
599   case 0
600   then show ?case by simp
601 next
602   case (pCons c cs)
603   show ?case
604   proof (cases "c = 0")
605     case True
606     from pCons.hyps pCons.prems True show ?thesis
607       apply (auto)
608       apply (rule_tac x="k+1" in exI)
609       apply (rule_tac x="a" in exI, clarsimp)
610       apply (rule_tac x="q" in exI)
611       apply auto
612       done
613   next
614     case False
615     show ?thesis
616       apply (rule exI[where x=0])
617       apply (rule exI[where x=c], auto simp add: False)
618       done
619   qed
620 qed
622 lemma poly_decompose:
623   assumes nc: "\<not> constant (poly p)"
624   shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
625                psize q + k + 1 = psize p \<and>
626               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
627   using nc
628 proof (induct p)
629   case 0
630   then show ?case
631     by (simp add: constant_def)
632 next
633   case (pCons c cs)
634   {
635     assume C: "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
636     {
637       fix x y
638       from C have "poly (pCons c cs) x = poly (pCons c cs) y"
639         by (cases "x = 0") auto
640     }
641     with pCons.prems have False
642       by (auto simp add: constant_def)
643   }
644   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
645   from poly_decompose_lemma[OF th]
646   show ?case
647     apply clarsimp
648     apply (rule_tac x="k+1" in exI)
649     apply (rule_tac x="a" in exI)
650     apply simp
651     apply (rule_tac x="q" in exI)
652     apply (auto simp add: psize_def split: if_splits)
653     done
654 qed
656 text{* Fundamental theorem of algebra *}
658 lemma fundamental_theorem_of_algebra:
659   assumes nc: "\<not> constant (poly p)"
660   shows "\<exists>z::complex. poly p z = 0"
661   using nc
662 proof (induct "psize p" arbitrary: p rule: less_induct)
663   case less
664   let ?p = "poly p"
665   let ?ths = "\<exists>z. ?p z = 0"
667   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
668   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
669     by blast
671   show ?ths
672   proof (cases "?p c = 0")
673     case True
674     then show ?thesis by blast
675   next
676     case False
677     note pc0 = this
678     from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
679       by blast
680     {
681       assume h: "constant (poly q)"
682       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
683         by auto
684       {
685         fix x y
686         from th have "?p x = poly q (x - c)"
687           by auto
688         also have "\<dots> = poly q (y - c)"
689           using h unfolding constant_def by blast
690         also have "\<dots> = ?p y"
691           using th by auto
692         finally have "?p x = ?p y" .
693       }
694       with less(2) have False
695         unfolding constant_def by blast
696     }
697     then have qnc: "\<not> constant (poly q)"
698       by blast
699     from q(2) have pqc0: "?p c = poly q 0"
700       by simp
701     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
702       by simp
703     let ?a0 = "poly q 0"
704     from pc0 pqc0 have a00: "?a0 \<noteq> 0"
705       by simp
706     from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
707       by simp
708     let ?r = "smult (inverse ?a0) q"
709     have lgqr: "psize q = psize ?r"
710       using a00
711       unfolding psize_def degree_def
712       by (simp add: poly_eq_iff)
713     {
714       assume h: "\<And>x y. poly ?r x = poly ?r y"
715       {
716         fix x y
717         from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
718           by auto
719         also have "\<dots> = poly ?r y * ?a0"
720           using h by simp
721         also have "\<dots> = poly q y"
722           using qr[rule_format, of y] by simp
723         finally have "poly q x = poly q y" .
724       }
725       with qnc have False
726         unfolding constant_def by blast
727     }
728     then have rnc: "\<not> constant (poly ?r)"
729       unfolding constant_def by blast
730     from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
731       by auto
732     {
733       fix w
734       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
735         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
736       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
737         using a00 unfolding norm_divide by (simp add: field_simps)
738       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .
739     }
740     note mrmq_eq = this
741     from poly_decompose[OF rnc] obtain k a s where
742       kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
743         "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
744     {
745       assume "psize p = k + 1"
746       with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
747         by auto
748       {
749         fix w
750         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
751           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
752       }
753       note hth = this [symmetric]
754       from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"
755         unfolding hth by blast
756     }
757     moreover
758     {
759       assume kn: "psize p \<noteq> k + 1"
760       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
761         by simp
762       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
763         unfolding constant_def poly_pCons poly_monom
764         using kas(1)
765         apply simp
766         apply (rule exI[where x=0])
767         apply (rule exI[where x=1])
768         apply simp
769         done
770       from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
771         by (simp add: psize_def degree_monom_eq)
772       from less(1) [OF k1n [simplified th02] th01]
773       obtain w where w: "1 + w^k * a = 0"
774         unfolding poly_pCons poly_monom
775         using kas(2) by (cases k) (auto simp add: algebra_simps)
776       from poly_bound_exists[of "cmod w" s] obtain m where
777         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
778       have w0: "w \<noteq> 0"
779         using kas(2) w by (auto simp add: power_0_left)
780       from w have "(1 + w ^ k * a) - 1 = 0 - 1"
781         by simp
782       then have wm1: "w^k * a = - 1"
783         by simp
784       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
785         using norm_ge_zero[of w] w0 m(1)
786         by (simp add: inverse_eq_divide zero_less_mult_iff)
787       with real_lbound_gt_zero[OF zero_less_one] obtain t where
788         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
789       let ?ct = "complex_of_real t"
790       let ?w = "?ct * w"
791       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
792         using kas(1) by (simp add: algebra_simps power_mult_distrib)
793       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
794         unfolding wm1 by simp
795       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
796         cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
797         by metis
798       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
799       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
800         unfolding norm_of_real by simp
801       have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
802         by arith
803       have "t * cmod w \<le> 1 * cmod w"
804         apply (rule mult_mono)
805         using t(1,2)
806         apply auto
807         done
808       then have tw: "cmod ?w \<le> cmod w"
809         using t(1) by (simp add: norm_mult)
810       from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
811         by (simp add: inverse_eq_divide field_simps)
812       with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
813         by (metis comm_mult_strict_left_mono)
814       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
815         using w0 t(1)
816         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
817       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
818         using t(1,2) m(2)[rule_format, OF tw] w0
819         by auto
820       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
821         by simp
822       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
823         by auto
824       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
825       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
826       from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
827         by arith
828       then have "cmod (poly ?r ?w) < 1"
829         unfolding kas(4)[rule_format, of ?w] r01 by simp
830       then have "\<exists>w. cmod (poly ?r w) < 1"
831         by blast
832     }
833     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"
834       by blast
835     from cr0_contr cq0 q(2) show ?thesis
836       unfolding mrmq_eq not_less[symmetric] by auto
837   qed
838 qed
840 text {* Alternative version with a syntactic notion of constant polynomial. *}
842 lemma fundamental_theorem_of_algebra_alt:
843   assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
844   shows "\<exists>z. poly p z = (0::complex)"
845   using nc
846 proof (induct p)
847   case 0
848   then show ?case by simp
849 next
850   case (pCons c cs)
851   show ?case
852   proof (cases "c = 0")
853     case True
854     then show ?thesis by auto
855   next
856     case False
857     {
858       assume nc: "constant (poly (pCons c cs))"
859       from nc[unfolded constant_def, rule_format, of 0]
860       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
861       then have "cs = 0"
862       proof (induct cs)
863         case 0
864         then show ?case by simp
865       next
866         case (pCons d ds)
867         show ?case
868         proof (cases "d = 0")
869           case True
870           then show ?thesis using pCons.prems pCons.hyps by simp
871         next
872           case False
873           from poly_bound_exists[of 1 ds] obtain m where
874             m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
875           have dm: "cmod d / m > 0"
876             using False m(1) by (simp add: field_simps)
877           from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
878             x: "x > 0" "x < cmod d / m" "x < 1" by blast
879           let ?x = "complex_of_real x"
880           from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1"
881             by simp_all
882           from pCons.prems[rule_format, OF cx(1)]
883           have cth: "cmod (?x*poly ds ?x) = cmod d"
884             by (simp add: eq_diff_eq[symmetric])
885           from m(2)[rule_format, OF cx(2)] x(1)
886           have th0: "cmod (?x*poly ds ?x) \<le> x*m"
887             by (simp add: norm_mult)
888           from x(2) m(1) have "x * m < cmod d"
889             by (simp add: field_simps)
890           with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
891             by auto
892           with cth show ?thesis
893             by blast
894         qed
895       qed
896     }
897     then have nc: "\<not> constant (poly (pCons c cs))"
898       using pCons.prems False by blast
899     from fundamental_theorem_of_algebra[OF nc] show ?thesis .
900   qed
901 qed
904 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
906 lemma nullstellensatz_lemma:
907   fixes p :: "complex poly"
908   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
909     and "degree p = n"
910     and "n \<noteq> 0"
911   shows "p dvd (q ^ n)"
912   using assms
913 proof (induct n arbitrary: p q rule: nat_less_induct)
914   fix n :: nat
915   fix p q :: "complex poly"
916   assume IH: "\<forall>m<n. \<forall>p q.
917                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
918                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
919     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
920     and dpn: "degree p = n"
921     and n0: "n \<noteq> 0"
922   from dpn n0 have pne: "p \<noteq> 0" by auto
923   let ?ths = "p dvd (q ^ n)"
924   {
925     fix a
926     assume a: "poly p a = 0"
927     {
928       assume oa: "order a p \<noteq> 0"
929       let ?op = "order a p"
930       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
931         using order by blast+
932       note oop = order_degree[OF pne, unfolded dpn]
933       {
934         assume q0: "q = 0"
935         then have ?ths using n0
936           by (simp add: power_0_left)
937       }
938       moreover
939       {
940         assume q0: "q \<noteq> 0"
941         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
942         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
943         from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
944           by (rule dvdE)
945         have sne: "s \<noteq> 0" using s pne by auto
946         {
947           assume ds0: "degree s = 0"
948           from ds0 obtain k where kpn: "s = [:k:]"
949             by (cases s) (auto split: if_splits)
950           from sne kpn have k: "k \<noteq> 0" by simp
951           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
952           have "q ^ n = p * ?w"
953             apply (subst r)
954             apply (subst s)
955             apply (subst kpn)
956             using k oop [of a]
957             apply (subst power_mult_distrib)
958             apply simp
959             apply (subst power_add [symmetric])
960             apply simp
961             done
962           then have ?ths
963             unfolding dvd_def by blast
964         }
965         moreover
966         {
967           assume ds0: "degree s \<noteq> 0"
968           from ds0 sne dpn s oa
969             have dsn: "degree s < n"
970               apply auto
971               apply (erule ssubst)
972               apply (simp add: degree_mult_eq degree_linear_power)
973               done
974             {
975               fix x assume h: "poly s x = 0"
976               {
977                 assume xa: "x = a"
978                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
979                   by (rule dvdE)
980                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
981                   apply (subst s)
982                   apply (subst u)
983                   apply (simp only: power_Suc mult_ac)
984                   done
985                 with ap(2)[unfolded dvd_def] have False
986                   by blast
987               }
988               note xa = this
989               from h have "poly p x = 0"
990                 by (subst s) simp
991               with pq0 have "poly q x = 0"
992                 by blast
993               with r xa have "poly r x = 0"
994                 by auto
995             }
996             note impth = this
997             from IH[rule_format, OF dsn, of s r] impth ds0
998             have "s dvd (r ^ (degree s))"
999               by blast
1000             then obtain u where u: "r ^ (degree s) = s * u" ..
1001             then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
1002               by (simp only: poly_mult[symmetric] poly_power[symmetric])
1003             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
1004             from oop[of a] dsn have "q ^ n = p * ?w"
1005               apply -
1006               apply (subst s)
1007               apply (subst r)
1008               apply (simp only: power_mult_distrib)
1009               apply (subst mult_assoc [where b=s])
1010               apply (subst mult_assoc [where a=u])
1011               apply (subst mult_assoc [where b=u, symmetric])
1012               apply (subst u [symmetric])
1013               apply (simp add: mult_ac power_add [symmetric])
1014               done
1015             then have ?ths
1016               unfolding dvd_def by blast
1017         }
1018         ultimately have ?ths by blast
1019       }
1020       ultimately have ?ths by blast
1021     }
1022     then have ?ths using a order_root pne by blast
1023   }
1024   moreover
1025   {
1026     assume exa: "\<not> (\<exists>a. poly p a = 0)"
1027     from fundamental_theorem_of_algebra_alt[of p] exa
1028     obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
1029       by blast
1030     then have pp: "\<And>x. poly p x = c"
1031       by simp
1032     let ?w = "[:1/c:] * (q ^ n)"
1033     from ccs have "(q ^ n) = (p * ?w)"
1034       by simp
1035     then have ?ths
1036       unfolding dvd_def by blast
1037   }
1038   ultimately show ?ths by blast
1039 qed
1041 lemma nullstellensatz_univariate:
1042   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
1043     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
1044 proof -
1045   {
1046     assume pe: "p = 0"
1047     then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
1048       by (auto simp add: poly_all_0_iff_0)
1049     {
1050       assume "p dvd (q ^ (degree p))"
1051       then obtain r where r: "q ^ (degree p) = p * r" ..
1052       from r pe have False by simp
1053     }
1054     with eq pe have ?thesis by blast
1055   }
1056   moreover
1057   {
1058     assume pe: "p \<noteq> 0"
1059     {
1060       assume dp: "degree p = 0"
1061       then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe
1062         by (cases p) (simp split: if_splits)
1063       then have th1: "\<forall>x. poly p x \<noteq> 0"
1064         by simp
1065       from k dp have "q ^ (degree p) = p * [:1/k:]"
1066         by (simp add: one_poly_def)
1067       then have th2: "p dvd (q ^ (degree p))" ..
1068       from th1 th2 pe have ?thesis
1069         by blast
1070     }
1071     moreover
1072     {
1073       assume dp: "degree p \<noteq> 0"
1074       then obtain n where n: "degree p = Suc n "
1075         by (cases "degree p") auto
1076       {
1077         assume "p dvd (q ^ (Suc n))"
1078         then obtain u where u: "q ^ (Suc n) = p * u" ..
1079         {
1080           fix x
1081           assume h: "poly p x = 0" "poly q x \<noteq> 0"
1082           then have "poly (q ^ (Suc n)) x \<noteq> 0"
1083             by simp
1084           then have False using u h(1)
1085             by (simp only: poly_mult) simp
1086         }
1087       }
1088       with n nullstellensatz_lemma[of p q "degree p"] dp
1089       have ?thesis by auto
1090     }
1091     ultimately have ?thesis by blast
1092   }
1093   ultimately show ?thesis by blast
1094 qed
1096 text {* Useful lemma *}
1098 lemma constant_degree:
1099   fixes p :: "'a::{idom,ring_char_0} poly"
1100   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
1101 proof
1102   assume l: ?lhs
1103   from l[unfolded constant_def, rule_format, of _ "0"]
1104   have th: "poly p = poly [:poly p 0:]"
1105     by auto
1106   then have "p = [:poly p 0:]"
1107     by (simp add: poly_eq_poly_eq_iff)
1108   then have "degree p = degree [:poly p 0:]"
1109     by simp
1110   then show ?rhs
1111     by simp
1112 next
1113   assume r: ?rhs
1114   then obtain k where "p = [:k:]"
1115     by (cases p) (simp split: if_splits)
1116   then show ?lhs
1117     unfolding constant_def by auto
1118 qed
1120 lemma divides_degree:
1121   assumes pq: "p dvd (q:: complex poly)"
1122   shows "degree p \<le> degree q \<or> q = 0"
1123   by (metis dvd_imp_degree_le pq)
1125 text {* Arithmetic operations on multivariate polynomials. *}
1127 lemma mpoly_base_conv:
1128   fixes x :: "'a::comm_ring_1"
1129   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
1130   by simp_all
1132 lemma mpoly_norm_conv:
1133   fixes x :: "'a::comm_ring_1"
1134   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
1135   by simp_all
1137 lemma mpoly_sub_conv:
1138   fixes x :: "'a::comm_ring_1"
1139   shows "poly p x - poly q x = poly p x + -1 * poly q x"
1140   by simp
1142 lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
1143   by simp
1145 lemma poly_cancel_eq_conv:
1146   fixes x :: "'a::field"
1147   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
1148   by auto
1151   fixes p:: "('a::comm_ring_1) poly"
1152   assumes pq: "p dvd q"
1153   shows "p dvd (pCons 0 q)"
1154 proof -
1155   have "pCons 0 q = q * [:0,1:]" by simp
1156   then have "q dvd (pCons 0 q)" ..
1157   with pq show ?thesis by (rule dvd_trans)
1158 qed
1160 lemma poly_divides_conv0:
1161   fixes p:: "'a::field poly"
1162   assumes lgpq: "degree q < degree p"
1163     and lq: "p \<noteq> 0"
1164   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
1165 proof
1166   assume r: ?rhs
1167   then have "q = p * 0" by simp
1168   then show ?lhs ..
1169 next
1170   assume l: ?lhs
1171   show ?rhs
1172   proof (cases "q = 0")
1173     case True
1174     then show ?thesis by simp
1175   next
1176     assume q0: "q \<noteq> 0"
1177     from l q0 have "degree p \<le> degree q"
1178       by (rule dvd_imp_degree_le)
1179     with lgpq show ?thesis by simp
1180   qed
1181 qed
1183 lemma poly_divides_conv1:
1184   fixes p :: "'a::field poly"
1185   assumes a0: "a \<noteq> 0"
1186     and pp': "p dvd p'"
1187     and qrp': "smult a q - p' = r"
1188   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
1189 proof
1190   from pp' obtain t where t: "p' = p * t" ..
1191   {
1192     assume l: ?lhs
1193     then obtain u where u: "q = p * u" ..
1194     have "r = p * (smult a u - t)"
1195       using u qrp' [symmetric] t by (simp add: algebra_simps)
1196     then show ?rhs ..
1197   next
1198     assume r: ?rhs
1199     then obtain u where u: "r = p * u" ..
1200     from u [symmetric] t qrp' [symmetric] a0
1201     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1202     then show ?lhs ..
1203   }
1204 qed
1206 lemma basic_cqe_conv1:
1207   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1208   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1209   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
1210   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
1211   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
1212   by simp_all
1214 lemma basic_cqe_conv2:
1215   assumes l: "p \<noteq> 0"
1216   shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
1217 proof -
1218   {
1219     fix h t
1220     assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
1221     with l have False by simp
1222   }
1223   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
1224     by blast
1225   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
1226     by auto
1227 qed
1229 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
1230   by (metis poly_all_0_iff_0)
1232 lemma basic_cqe_conv3:
1233   fixes p q :: "complex poly"
1234   assumes l: "p \<noteq> 0"
1235   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)"
1236 proof -
1237   from l have dp: "degree (pCons a p) = psize p"
1238     by (simp add: psize_def)
1239   from nullstellensatz_univariate[of "pCons a p" q] l
1240   show ?thesis
1241     by (metis dp pCons_eq_0_iff)
1242 qed
1244 lemma basic_cqe_conv4:
1245   fixes p q :: "complex poly"
1246   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
1247   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1248 proof -
1249   from h have "poly (q ^ n) = poly r"
1250     by auto
1251   then have "(q ^ n) = r"
1252     by (simp add: poly_eq_poly_eq_iff)
1253   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1254     by simp
1255 qed
1257 lemma poly_const_conv:
1258   fixes x :: "'a::comm_ring_1"
1259   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
1260   by simp
1262 end