src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author wenzelm Wed Jun 10 23:34:23 2015 +0200 (2015-06-10) changeset 60424 c96fff9dcdbc parent 59557 ebd8ecacfba6 child 60449 229bad93377e permissions -rw-r--r--
misc tuning;
1 (* Author: Amine Chaieb, TU Muenchen *)
3 section \<open>Fundamental Theorem of Algebra\<close>
5 theory Fundamental_Theorem_Algebra
6 imports Polynomial Complex_Main
7 begin
9 subsection \<open>More lemmas about module of complex numbers\<close>
11 text \<open>The triangle inequality for cmod\<close>
13 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
14   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
17 subsection \<open>Basic lemmas about polynomials\<close>
19 lemma poly_bound_exists:
20   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
21   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
22 proof (induct p)
23   case 0
24   then show ?case by (rule exI[where x=1]) simp
25 next
26   case (pCons c cs)
27   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
28     by blast
29   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
30   have kp: "?k > 0"
31     using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
32   have "norm (poly (pCons c cs) z) \<le> ?k" if H: "norm z \<le> r" for z
33   proof -
34     from m H have th: "norm (poly cs z) \<le> m"
35       by blast
36     from H have rp: "r \<ge> 0"
37       using norm_ge_zero[of z] by arith
38     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)"
39       using norm_triangle_ineq[of c "z* poly cs z"] by simp
40     also have "\<dots> \<le> norm c + r * m"
41       using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
42       by (simp add: norm_mult)
43     also have "\<dots> \<le> ?k"
44       by simp
45     finally show ?thesis .
46   qed
47   with kp show ?case by blast
48 qed
51 text \<open>Offsetting the variable in a polynomial gives another of same degree\<close>
53 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
54   where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
56 lemma offset_poly_0: "offset_poly 0 h = 0"
57   by (simp add: offset_poly_def)
59 lemma offset_poly_pCons:
60   "offset_poly (pCons a p) h =
61     smult h (offset_poly p h) + pCons a (offset_poly p h)"
62   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
64 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
65   by (simp add: offset_poly_pCons offset_poly_0)
67 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
68   apply (induct p)
69   apply (simp add: offset_poly_0)
70   apply (simp add: offset_poly_pCons algebra_simps)
71   done
73 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
74   by (induct p arbitrary: a) (simp, force)
76 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
77   apply (safe intro!: offset_poly_0)
78   apply (induct p)
79   apply simp
80   apply (simp add: offset_poly_pCons)
81   apply (frule offset_poly_eq_0_lemma, simp)
82   done
84 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
85   apply (induct p)
86   apply (simp add: offset_poly_0)
87   apply (case_tac "p = 0")
88   apply (simp add: offset_poly_0 offset_poly_pCons)
89   apply (simp add: offset_poly_pCons)
90   apply (subst degree_add_eq_right)
91   apply (rule le_less_trans [OF degree_smult_le])
92   apply (simp add: offset_poly_eq_0_iff)
93   apply (simp add: offset_poly_eq_0_iff)
94   done
96 definition "psize p = (if p = 0 then 0 else Suc (degree p))"
98 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
99   unfolding psize_def by simp
101 lemma poly_offset:
102   fixes p :: "'a::comm_ring_1 poly"
103   shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
104 proof (intro exI conjI)
105   show "psize (offset_poly p a) = psize p"
106     unfolding psize_def
107     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
108   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
109     by (simp add: poly_offset_poly)
110 qed
112 text \<open>An alternative useful formulation of completeness of the reals\<close>
113 lemma real_sup_exists:
114   assumes ex: "\<exists>x. P x"
115     and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
116   shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
117 proof
118   from bz have "bdd_above (Collect P)"
119     by (force intro: less_imp_le)
120   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
121     using ex bz by (subst less_cSup_iff) auto
122 qed
125 subsection \<open>Fundamental theorem of algebra\<close>
127 lemma unimodular_reduce_norm:
128   assumes md: "cmod z = 1"
129   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
130 proof -
131   obtain x y where z: "z = Complex x y "
132     by (cases z) auto
133   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
134     by (simp add: cmod_def)
135   {
136     assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
137     from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
138       by (simp_all add: cmod_def power2_eq_square algebra_simps)
139     then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
140       by simp_all
141     then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
142       by - (rule power_mono, simp, simp)+
143     then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
144       by (simp_all add: power_mult_distrib)
145     from add_mono[OF th0] xy have False by simp
146   }
147   then show ?thesis
148     unfolding linorder_not_le[symmetric] by blast
149 qed
151 text \<open>Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero\<close>
152 lemma reduce_poly_simple:
153   assumes b: "b \<noteq> 0"
154     and n: "n \<noteq> 0"
155   shows "\<exists>z. cmod (1 + b * z^n) < 1"
156   using n
157 proof (induct n rule: nat_less_induct)
158   fix n
159   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
160   assume n: "n \<noteq> 0"
161   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
162   {
163     assume e: "even n"
164     then have "\<exists>m. n = 2 * m"
165       by presburger
166     then obtain m where m: "n = 2 * m"
167       by blast
168     from n m have "m \<noteq> 0" "m < n"
169       by presburger+
170     with IH[rule_format, of m] obtain z where z: "?P z m"
171       by blast
172     from z have "?P (csqrt z) n"
173       by (simp add: m power_mult power2_csqrt)
174     then have "\<exists>z. ?P z n" ..
175   }
176   moreover
177   {
178     assume o: "odd n"
179     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
180       using b by (simp add: norm_divide)
181     from o have "\<exists>m. n = Suc (2 * m)"
182       by presburger+
183     then obtain m where m: "n = Suc (2 * m)"
184       by blast
185     from unimodular_reduce_norm[OF th0] o
186     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
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 - 1) < 1")
191       apply (rule_tac x="-1" in exI)
192       apply simp
193       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
194       apply (cases "even m")
195       apply (rule_tac x="ii" in exI)
196       apply (simp add: m power_mult)
197       apply (rule_tac x="- ii" in exI)
198       apply (simp add: m power_mult)
199       apply (cases "even m")
200       apply (rule_tac x="- ii" in exI)
201       apply (simp add: m power_mult)
202       apply (auto simp add: m power_mult)
203       apply (rule_tac x="ii" in exI)
204       apply (auto simp add: m power_mult)
205       done
206     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
207       by blast
208     let ?w = "v / complex_of_real (root n (cmod b))"
209     from odd_real_root_pow[OF o, of "cmod b"]
210     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
211       by (simp add: power_divide of_real_power[symmetric])
212     have th2:"cmod (complex_of_real (cmod b) / b) = 1"
213       using b by (simp add: norm_divide)
214     then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
215       by simp
216     have th4: "cmod (complex_of_real (cmod b) / b) *
217         cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
218         cmod (complex_of_real (cmod b) / b) * 1"
219       apply (simp only: norm_mult[symmetric] distrib_left)
220       using b v
221       apply (simp add: th2)
222       done
223     from mult_left_less_imp_less[OF th4 th3]
224     have "?P ?w n" unfolding th1 .
225     then have "\<exists>z. ?P z n" ..
226   }
227   ultimately show "\<exists>z. ?P z n" by blast
228 qed
230 text \<open>Bolzano-Weierstrass type property for closed disc in complex plane.\<close>
232 lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
233   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]
234   unfolding cmod_def by simp
236 lemma bolzano_weierstrass_complex_disc:
237   assumes r: "\<forall>n. cmod (s n) \<le> r"
238   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
239 proof -
240   from seq_monosub[of "Re \<circ> s"]
241   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
242     unfolding o_def by blast
243   from seq_monosub[of "Im \<circ> s \<circ> f"]
244   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
245     unfolding o_def by blast
246   let ?h = "f \<circ> g"
247   from r[rule_format, of 0] have rp: "r \<ge> 0"
248     using norm_ge_zero[of "s 0"] by arith
249   have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
250   proof
251     fix n
252     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
253     show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
254   qed
255   have conv1: "convergent (\<lambda>n. Re (s (f n)))"
256     apply (rule Bseq_monoseq_convergent)
257     apply (simp add: Bseq_def)
258     apply (metis gt_ex le_less_linear less_trans order.trans th)
259     apply (rule f(2))
260     done
261   have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
262   proof
263     fix n
264     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
265     show "\<bar>Im (s n)\<bar> \<le> r + 1"
266       by arith
267   qed
269   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
270     apply (rule Bseq_monoseq_convergent)
271     apply (simp add: Bseq_def)
272     apply (metis gt_ex le_less_linear less_trans order.trans th)
273     apply (rule g(2))
274     done
276   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
277     by blast
278   then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
279     unfolding LIMSEQ_iff real_norm_def .
281   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
282     by blast
283   then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
284     unfolding LIMSEQ_iff real_norm_def .
285   let ?w = "Complex x y"
286   from f(1) g(1) have hs: "subseq ?h"
287     unfolding subseq_def by auto
288   {
289     fix e :: real
290     assume ep: "e > 0"
291     then have e2: "e/2 > 0"
292       by simp
293     from x[rule_format, OF e2] y[rule_format, OF e2]
294     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
295       and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
296       by blast
297     {
298       fix n
299       assume nN12: "n \<ge> N1 + N2"
300       then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
301         using seq_suble[OF g(1), of n] by arith+
302       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
303       have "cmod (s (?h n) - ?w) < e"
304         using metric_bound_lemma[of "s (f (g n))" ?w] by simp
305     }
306     then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e"
307       by blast
308   }
309   with hs show ?thesis by blast
310 qed
312 text \<open>Polynomial is continuous.\<close>
314 lemma poly_cont:
315   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
316   assumes ep: "e > 0"
317   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
318 proof -
319   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
320   proof
321     show "degree (offset_poly p z) = degree p"
322       by (rule degree_offset_poly)
323     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
324       by (rule poly_offset_poly)
325   qed
326   have th: "\<And>w. poly q (w - z) = poly p w"
327     using q(2)[of "w - z" for w] by simp
328   show ?thesis unfolding th[symmetric]
329   proof (induct q)
330     case 0
331     then show ?case
332       using ep by auto
333   next
334     case (pCons c cs)
335     from poly_bound_exists[of 1 "cs"]
336     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
337       by blast
338     from ep m(1) have em0: "e/m > 0"
339       by (simp add: field_simps)
340     have one0: "1 > (0::real)"
341       by arith
342     from real_lbound_gt_zero[OF one0 em0]
343     obtain d where d: "d > 0" "d < 1" "d < e / m"
344       by blast
345     from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
346       by (simp_all add: field_simps)
347     show ?case
348     proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
349       fix d w
350       assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
351       then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
352         by simp_all
353       from H(3) m(1) have dme: "d*m < e"
354         by (simp add: field_simps)
355       from H have th: "norm (w - z) \<le> d"
356         by simp
357       from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
358       show "norm (w - z) * norm (poly cs (w - z)) < e"
359         by simp
360     qed
361   qed
362 qed
364 text \<open>Hence a polynomial attains minimum on a closed disc
365   in the complex plane.\<close>
366 lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
367 proof -
368   show ?thesis
369   proof (cases "r \<ge> 0")
370     case False
371     then show ?thesis
372       by (metis norm_ge_zero order.trans)
373   next
374     case True
375     then have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
376       by simp
377     then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
378       by blast
379     {
380       fix x z
381       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
382       then have "- x < 0 "
383         by arith
384       with H(2) norm_ge_zero[of "poly p z"] have False
385         by simp
386     }
387     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
388       by blast
389     from real_sup_exists[OF mth1 mth2] obtain s where
390       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
391     let ?m = "- s"
392     {
393       fix y
394       from s[rule_format, of "-y"]
395       have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
396         unfolding minus_less_iff[of y ] equation_minus_iff by blast
397     }
398     note s1 = this[unfolded minus_minus]
399     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
400       by auto
401     {
402       fix n :: nat
403       from s1[rule_format, of "?m + 1/real (Suc n)"]
404       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
405         by simp
406     }
407     then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
408     from choice[OF th] obtain g where
409         g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
410       by blast
411     from bolzano_weierstrass_complex_disc[OF g(1)]
412     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
413       by blast
414     {
415       fix w
416       assume wr: "cmod w \<le> r"
417       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
418       {
419         assume e: "?e > 0"
420         then have e2: "?e/2 > 0"
421           by simp
422         from poly_cont[OF e2, of z p] obtain d where
423             d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
424           by blast
425         {
426           fix w
427           assume w: "cmod (w - z) < d"
428           have "cmod(poly p w - poly p z) < ?e / 2"
429             using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
430         }
431         note th1 = this
433         from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
434           by blast
435         from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
436           by blast
437         have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
438           using N1[rule_format, of "N1 + N2"] th1 by simp
439         have th0: "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
440           for a b e2 m :: real
441           by arith
442         have ath: "m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e" for m x e :: real
443           by arith
444         from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
445         from seq_suble[OF fz(1), of "N1 + N2"]
446         have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
447           by simp
448         have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
449           using N2 by auto
450         from frac_le[OF th000 th00]
451         have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
452           by simp
453         from g(2)[rule_format, of "f (N1 + N2)"]
454         have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
455         from order_less_le_trans[OF th01 th00]
456         have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
457         from N2 have "2/?e < real (Suc (N1 + N2))"
458           by arith
459         with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
460         have "?e/2 > 1/ real (Suc (N1 + N2))"
461           by (simp add: inverse_eq_divide)
462         with ath[OF th31 th32] have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
463           by arith
464         have ath2: "\<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c" for a b c m :: real
465           by arith
466         have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
467             cmod (poly p (g (f (N1 + N2))) - poly p z)"
468           by (simp add: norm_triangle_ineq3)
469         from ath2[OF th22, of ?m]
470         have thc2: "2 * (?e/2) \<le>
471             \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
472           by simp
473         from th0[OF th2 thc1 thc2] have False .
474       }
475       then have "?e = 0"
476         by auto
477       then have "cmod (poly p z) = ?m"
478         by simp
479       with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
480         by simp
481     }
482     then show ?thesis by blast
483   qed
484 qed
486 text \<open>Nonzero polynomial in z goes to infinity as z does.\<close>
488 lemma poly_infinity:
489   fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
490   assumes ex: "p \<noteq> 0"
491   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
492   using ex
493 proof (induct p arbitrary: a d)
494   case 0
495   then show ?case by simp
496 next
497   case (pCons c cs a d)
498   show ?case
499   proof (cases "cs = 0")
500     case False
501     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
502       by blast
503     let ?r = "1 + \<bar>r\<bar>"
504     {
505       fix z :: 'a
506       assume h: "1 + \<bar>r\<bar> \<le> norm z"
507       have r0: "r \<le> norm z"
508         using h by arith
509       from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
510         by arith
511       from h have z1: "norm z \<ge> 1"
512         by arith
513       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
514       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
515         unfolding norm_mult by (simp add: algebra_simps)
516       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
517       have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
518         by (simp add: algebra_simps)
519       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
520         by arith
521     }
522     then show ?thesis by blast
523   next
524     case True
525     with pCons.prems have c0: "c \<noteq> 0"
526       by simp
527     have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
528       if h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z" for z :: 'a
529     proof -
530       from c0 have "norm c > 0"
531         by simp
532       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
533         by (simp add: field_simps norm_mult)
534       have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
535         by arith
536       from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
537         by (simp add: algebra_simps)
538       from ath[OF th1 th0] show ?thesis
539         using True by simp
540     qed
541     then show ?thesis by blast
542   qed
543 qed
545 text \<open>Hence polynomial's modulus attains its minimum somewhere.\<close>
546 lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
547 proof (induct p)
548   case 0
549   then show ?case by simp
550 next
551   case (pCons c cs)
552   show ?case
553   proof (cases "cs = 0")
554     case False
555     from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
556     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
557       by blast
558     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
559       by arith
560     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
561     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)"
562       by blast
563     have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" if z: "r \<le> cmod z" for z
564       using v[of 0] r[OF z] by simp
565     with v ath[of r] show ?thesis
566       by blast
567   next
568     case True
569     with pCons.hyps show ?thesis
570       by simp
571   qed
572 qed
574 text \<open>Constant function (non-syntactic characterization).\<close>
575 definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"
577 lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
578   by (induct p) (auto simp: constant_def psize_def)
580 lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
581   by (simp add: poly_monom)
583 text \<open>Decomposition of polynomial, skipping zero coefficients after the first.\<close>
585 lemma poly_decompose_lemma:
586   assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
587   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)"
588   unfolding psize_def
589   using nz
590 proof (induct p)
591   case 0
592   then show ?case by simp
593 next
594   case (pCons c cs)
595   show ?case
596   proof (cases "c = 0")
597     case True
598     from pCons.hyps pCons.prems True show ?thesis
599       apply auto
600       apply (rule_tac x="k+1" in exI)
601       apply (rule_tac x="a" in exI, clarsimp)
602       apply (rule_tac x="q" in exI)
603       apply auto
604       done
605   next
606     case False
607     show ?thesis
608       apply (rule exI[where x=0])
609       apply (rule exI[where x=c])
610       apply (auto simp: False)
611       done
612   qed
613 qed
615 lemma poly_decompose:
616   assumes nc: "\<not> constant (poly p)"
617   shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
618                psize q + k + 1 = psize p \<and>
619               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
620   using nc
621 proof (induct p)
622   case 0
623   then show ?case
624     by (simp add: constant_def)
625 next
626   case (pCons c cs)
627   {
628     assume "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
629     then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y
630       by (cases "x = 0") auto
631     with pCons.prems have False
632       by (auto simp add: constant_def)
633   }
634   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
635   from poly_decompose_lemma[OF th]
636   show ?case
637     apply clarsimp
638     apply (rule_tac x="k+1" in exI)
639     apply (rule_tac x="a" in exI)
640     apply simp
641     apply (rule_tac x="q" in exI)
642     apply (auto simp add: psize_def split: if_splits)
643     done
644 qed
646 text \<open>Fundamental theorem of algebra\<close>
648 lemma fundamental_theorem_of_algebra:
649   assumes nc: "\<not> constant (poly p)"
650   shows "\<exists>z::complex. poly p z = 0"
651   using nc
652 proof (induct "psize p" arbitrary: p rule: less_induct)
653   case less
654   let ?p = "poly p"
655   let ?ths = "\<exists>z. ?p z = 0"
657   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
658   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
659     by blast
661   show ?ths
662   proof (cases "?p c = 0")
663     case True
664     then show ?thesis by blast
665   next
666     case False
667     from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
668       by blast
669     have False if h: "constant (poly q)"
670     proof -
671       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
672         by auto
673       have "?p x = ?p y" for x y
674       proof -
675         from th have "?p x = poly q (x - c)"
676           by auto
677         also have "\<dots> = poly q (y - c)"
678           using h unfolding constant_def by blast
679         also have "\<dots> = ?p y"
680           using th by auto
681         finally show ?thesis .
682       qed
683       with less(2) show ?thesis
684         unfolding constant_def by blast
685     qed
686     then have qnc: "\<not> constant (poly q)"
687       by blast
688     from q(2) have pqc0: "?p c = poly q 0"
689       by simp
690     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
691       by simp
692     let ?a0 = "poly q 0"
693     from False pqc0 have a00: "?a0 \<noteq> 0"
694       by simp
695     from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
696       by simp
697     let ?r = "smult (inverse ?a0) q"
698     have lgqr: "psize q = psize ?r"
699       using a00
700       unfolding psize_def degree_def
701       by (simp add: poly_eq_iff)
702     have False if h: "\<And>x y. poly ?r x = poly ?r y"
703     proof -
704       {
705         fix x y
706         from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
707           by auto
708         also have "\<dots> = poly ?r y * ?a0"
709           using h by simp
710         also have "\<dots> = poly q y"
711           using qr[rule_format, of y] by simp
712         finally have "poly q x = poly q y" .
713       }
714       with qnc show ?thesis
715         unfolding constant_def by blast
716     qed
717     then have rnc: "\<not> constant (poly ?r)"
718       unfolding constant_def by blast
719     from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
720       by auto
721     have mrmq_eq: "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" for w
722     proof -
723       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
724         using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)
725       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
726         using a00 unfolding norm_divide by (simp add: field_simps)
727       finally show ?thesis .
728     qed
729     from poly_decompose[OF rnc] obtain k a s where
730       kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
731         "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
732     have "\<exists>w. cmod (poly ?r w) < 1"
733     proof (cases "psize p = k + 1")
734       case True
735       with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
736         by auto
737       have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w
738         using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
739       from reduce_poly_simple[OF kas(1,2)] show ?thesis
740         unfolding hth by blast
741     next
742       case False note kn = this
743       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
744         by simp
745       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
746         unfolding constant_def poly_pCons poly_monom
747         using kas(1)
748         apply simp
749         apply (rule exI[where x=0])
750         apply (rule exI[where x=1])
751         apply simp
752         done
753       from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
754         by (simp add: psize_def degree_monom_eq)
755       from less(1) [OF k1n [simplified th02] th01]
756       obtain w where w: "1 + w^k * a = 0"
757         unfolding poly_pCons poly_monom
758         using kas(2) by (cases k) (auto simp add: algebra_simps)
759       from poly_bound_exists[of "cmod w" s] obtain m where
760         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
761       have w0: "w \<noteq> 0"
762         using kas(2) w by (auto simp add: power_0_left)
763       from w have "(1 + w ^ k * a) - 1 = 0 - 1"
764         by simp
765       then have wm1: "w^k * a = - 1"
766         by simp
767       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
768         using norm_ge_zero[of w] w0 m(1)
769         by (simp add: inverse_eq_divide zero_less_mult_iff)
770       with real_lbound_gt_zero[OF zero_less_one] obtain t where
771         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
772       let ?ct = "complex_of_real t"
773       let ?w = "?ct * w"
774       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
775         using kas(1) by (simp add: algebra_simps power_mult_distrib)
776       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
777         unfolding wm1 by simp
778       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
779         cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
780         by metis
781       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
782       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
783         unfolding norm_of_real by simp
784       have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
785         by arith
786       have "t * cmod w \<le> 1 * cmod w"
787         apply (rule mult_mono)
788         using t(1,2)
789         apply auto
790         done
791       then have tw: "cmod ?w \<le> cmod w"
792         using t(1) by (simp add: norm_mult)
793       from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
794         by (simp add: field_simps)
795       with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
796         by simp
797       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
798         using w0 t(1)
799         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
800       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
801         using t(1,2) m(2)[rule_format, OF tw] w0
802         by auto
803       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
804         by simp
805       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
806         by auto
807       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
808       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
809       from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
810         by arith
811       then have "cmod (poly ?r ?w) < 1"
812         unfolding kas(4)[rule_format, of ?w] r01 by simp
813       then show ?thesis
814         by blast
815     qed
816     with cq0 q(2) show ?thesis
817       unfolding mrmq_eq not_less[symmetric] by auto
818   qed
819 qed
821 text \<open>Alternative version with a syntactic notion of constant polynomial.\<close>
823 lemma fundamental_theorem_of_algebra_alt:
824   assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
825   shows "\<exists>z. poly p z = (0::complex)"
826   using nc
827 proof (induct p)
828   case 0
829   then show ?case by simp
830 next
831   case (pCons c cs)
832   show ?case
833   proof (cases "c = 0")
834     case True
835     then show ?thesis by auto
836   next
837     case False
838     {
839       assume nc: "constant (poly (pCons c cs))"
840       from nc[unfolded constant_def, rule_format, of 0]
841       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
842       then have "cs = 0"
843       proof (induct cs)
844         case 0
845         then show ?case by simp
846       next
847         case (pCons d ds)
848         show ?case
849         proof (cases "d = 0")
850           case True
851           then show ?thesis
852             using pCons.prems pCons.hyps by simp
853         next
854           case False
855           from poly_bound_exists[of 1 ds] obtain m where
856             m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
857           have dm: "cmod d / m > 0"
858             using False m(1) by (simp add: field_simps)
859           from real_lbound_gt_zero[OF dm zero_less_one]
860           obtain x where x: "x > 0" "x < cmod d / m" "x < 1"
861             by blast
862           let ?x = "complex_of_real x"
863           from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1"
864             by simp_all
865           from pCons.prems[rule_format, OF cx(1)]
866           have cth: "cmod (?x*poly ds ?x) = cmod d"
867             by (simp add: eq_diff_eq[symmetric])
868           from m(2)[rule_format, OF cx(2)] x(1)
869           have th0: "cmod (?x*poly ds ?x) \<le> x*m"
870             by (simp add: norm_mult)
871           from x(2) m(1) have "x * m < cmod d"
872             by (simp add: field_simps)
873           with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
874             by auto
875           with cth show ?thesis
876             by blast
877         qed
878       qed
879     }
880     then have nc: "\<not> constant (poly (pCons c cs))"
881       using pCons.prems False by blast
882     from fundamental_theorem_of_algebra[OF nc] show ?thesis .
883   qed
884 qed
887 subsection \<open>Nullstellensatz, degrees and divisibility of polynomials\<close>
889 lemma nullstellensatz_lemma:
890   fixes p :: "complex poly"
891   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
892     and "degree p = n"
893     and "n \<noteq> 0"
894   shows "p dvd (q ^ n)"
895   using assms
896 proof (induct n arbitrary: p q rule: nat_less_induct)
897   fix n :: nat
898   fix p q :: "complex poly"
899   assume IH: "\<forall>m<n. \<forall>p q.
900                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
901                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
902     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
903     and dpn: "degree p = n"
904     and n0: "n \<noteq> 0"
905   from dpn n0 have pne: "p \<noteq> 0" by auto
906   let ?ths = "p dvd (q ^ n)"
907   {
908     fix a
909     assume a: "poly p a = 0"
910     have ?ths if oa: "order a p \<noteq> 0"
911     proof -
912       let ?op = "order a p"
913       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
914         using order by blast+
915       note oop = order_degree[OF pne, unfolded dpn]
916       show ?thesis
917       proof (cases "q = 0")
918         case True
919         with n0 show ?thesis by (simp add: power_0_left)
920       next
921         case False
922         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
923         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
924         from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
925           by (rule dvdE)
926         have sne: "s \<noteq> 0"
927           using s pne by auto
928         show ?thesis
929         proof (cases "degree s = 0")
930           case True
931           then obtain k where kpn: "s = [:k:]"
932             by (cases s) (auto split: if_splits)
933           from sne kpn have k: "k \<noteq> 0" by simp
934           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
935           have "q ^ n = p * ?w"
936             apply (subst r)
937             apply (subst s)
938             apply (subst kpn)
939             using k oop [of a]
940             apply (subst power_mult_distrib)
941             apply simp
942             apply (subst power_add [symmetric])
943             apply simp
944             done
945           then show ?thesis
946             unfolding dvd_def by blast
947         next
948           case False
949           with sne dpn s oa have dsn: "degree s < n"
950               apply auto
951               apply (erule ssubst)
952               apply (simp add: degree_mult_eq degree_linear_power)
953               done
954             {
955               fix x assume h: "poly s x = 0"
956               {
957                 assume xa: "x = a"
958                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
959                   by (rule dvdE)
960                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
961                   apply (subst s)
962                   apply (subst u)
963                   apply (simp only: power_Suc ac_simps)
964                   done
965                 with ap(2)[unfolded dvd_def] have False
966                   by blast
967               }
968               note xa = this
969               from h have "poly p x = 0"
970                 by (subst s) simp
971               with pq0 have "poly q x = 0"
972                 by blast
973               with r xa have "poly r x = 0"
974                 by auto
975             }
976             note impth = this
977             from IH[rule_format, OF dsn, of s r] impth False
978             have "s dvd (r ^ (degree s))"
979               by blast
980             then obtain u where u: "r ^ (degree s) = s * u" ..
981             then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
982               by (simp only: poly_mult[symmetric] poly_power[symmetric])
983             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
984             from oop[of a] dsn have "q ^ n = p * ?w"
985               apply -
986               apply (subst s)
987               apply (subst r)
988               apply (simp only: power_mult_distrib)
989               apply (subst mult.assoc [where b=s])
990               apply (subst mult.assoc [where a=u])
991               apply (subst mult.assoc [where b=u, symmetric])
992               apply (subst u [symmetric])
993               apply (simp add: ac_simps power_add [symmetric])
994               done
995             then show ?thesis
996               unfolding dvd_def by blast
997         qed
998       qed
999     qed
1000     then have ?ths using a order_root pne by blast
1001   }
1002   moreover
1003   {
1004     assume exa: "\<not> (\<exists>a. poly p a = 0)"
1005     from fundamental_theorem_of_algebra_alt[of p] exa
1006     obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
1007       by blast
1008     then have pp: "\<And>x. poly p x = c"
1009       by simp
1010     let ?w = "[:1/c:] * (q ^ n)"
1011     from ccs have "(q ^ n) = (p * ?w)"
1012       by simp
1013     then have ?ths
1014       unfolding dvd_def by blast
1015   }
1016   ultimately show ?ths by blast
1017 qed
1019 lemma nullstellensatz_univariate:
1020   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
1021     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
1022 proof -
1023   show ?thesis
1024   proof (cases "p = 0")
1025     case True
1026     then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
1027       by (auto simp add: poly_all_0_iff_0)
1028     {
1029       assume "p dvd (q ^ (degree p))"
1030       then obtain r where r: "q ^ (degree p) = p * r" ..
1031       from r True have False by simp
1032     }
1033     with eq True show ?thesis by blast
1034   next
1035     case False
1036     show ?thesis
1037     proof (cases "degree p = 0")
1038       case True
1039       with \<open>p \<noteq> 0\<close> obtain k where k: "p = [:k:]" "k \<noteq> 0"
1040         by (cases p) (simp split: if_splits)
1041       then have th1: "\<forall>x. poly p x \<noteq> 0"
1042         by simp
1043       from k True have "q ^ (degree p) = p * [:1/k:]"
1044         by (simp add: one_poly_def)
1045       then have th2: "p dvd (q ^ (degree p))" ..
1046       from False th1 th2 show ?thesis
1047         by blast
1048     next
1049       case False
1050       assume dp: "degree p \<noteq> 0"
1051       then obtain n where n: "degree p = Suc n "
1052         by (cases "degree p") auto
1053       {
1054         assume "p dvd (q ^ (Suc n))"
1055         then obtain u where u: "q ^ (Suc n) = p * u" ..
1056         {
1057           fix x
1058           assume h: "poly p x = 0" "poly q x \<noteq> 0"
1059           then have "poly (q ^ (Suc n)) x \<noteq> 0"
1060             by simp
1061           then have False using u h(1)
1062             by (simp only: poly_mult) simp
1063         }
1064       }
1065       with n nullstellensatz_lemma[of p q "degree p"] dp
1066       show ?thesis by auto
1067     qed
1068   qed
1069 qed
1071 text \<open>Useful lemma\<close>
1073 lemma constant_degree:
1074   fixes p :: "'a::{idom,ring_char_0} poly"
1075   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
1076 proof
1077   assume l: ?lhs
1078   from l[unfolded constant_def, rule_format, of _ "0"]
1079   have th: "poly p = poly [:poly p 0:]"
1080     by auto
1081   then have "p = [:poly p 0:]"
1082     by (simp add: poly_eq_poly_eq_iff)
1083   then have "degree p = degree [:poly p 0:]"
1084     by simp
1085   then show ?rhs
1086     by simp
1087 next
1088   assume r: ?rhs
1089   then obtain k where "p = [:k:]"
1090     by (cases p) (simp split: if_splits)
1091   then show ?lhs
1092     unfolding constant_def by auto
1093 qed
1095 lemma divides_degree:
1096   assumes pq: "p dvd (q:: complex poly)"
1097   shows "degree p \<le> degree q \<or> q = 0"
1098   by (metis dvd_imp_degree_le pq)
1100 text \<open>Arithmetic operations on multivariate polynomials.\<close>
1102 lemma mpoly_base_conv:
1103   fixes x :: "'a::comm_ring_1"
1104   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
1105   by simp_all
1107 lemma mpoly_norm_conv:
1108   fixes x :: "'a::comm_ring_1"
1109   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
1110   by simp_all
1112 lemma mpoly_sub_conv:
1113   fixes x :: "'a::comm_ring_1"
1114   shows "poly p x - poly q x = poly p x + -1 * poly q x"
1115   by simp
1117 lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
1118   by simp
1120 lemma poly_cancel_eq_conv:
1121   fixes x :: "'a::field"
1122   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
1123   by auto
1126   fixes p:: "('a::comm_ring_1) poly"
1127   assumes pq: "p dvd q"
1128   shows "p dvd (pCons 0 q)"
1129 proof -
1130   have "pCons 0 q = q * [:0,1:]" by simp
1131   then have "q dvd (pCons 0 q)" ..
1132   with pq show ?thesis by (rule dvd_trans)
1133 qed
1135 lemma poly_divides_conv0:
1136   fixes p:: "'a::field poly"
1137   assumes lgpq: "degree q < degree p"
1138     and lq: "p \<noteq> 0"
1139   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
1140 proof
1141   assume r: ?rhs
1142   then have "q = p * 0" by simp
1143   then show ?lhs ..
1144 next
1145   assume l: ?lhs
1146   show ?rhs
1147   proof (cases "q = 0")
1148     case True
1149     then show ?thesis by simp
1150   next
1151     assume q0: "q \<noteq> 0"
1152     from l q0 have "degree p \<le> degree q"
1153       by (rule dvd_imp_degree_le)
1154     with lgpq show ?thesis by simp
1155   qed
1156 qed
1158 lemma poly_divides_conv1:
1159   fixes p :: "'a::field poly"
1160   assumes a0: "a \<noteq> 0"
1161     and pp': "p dvd p'"
1162     and qrp': "smult a q - p' = r"
1163   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
1164 proof
1165   from pp' obtain t where t: "p' = p * t" ..
1166   {
1167     assume l: ?lhs
1168     then obtain u where u: "q = p * u" ..
1169     have "r = p * (smult a u - t)"
1170       using u qrp' [symmetric] t by (simp add: algebra_simps)
1171     then show ?rhs ..
1172   next
1173     assume r: ?rhs
1174     then obtain u where u: "r = p * u" ..
1175     from u [symmetric] t qrp' [symmetric] a0
1176     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1177     then show ?lhs ..
1178   }
1179 qed
1181 lemma basic_cqe_conv1:
1182   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1183   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1184   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
1185   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
1186   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
1187   by simp_all
1189 lemma basic_cqe_conv2:
1190   assumes l: "p \<noteq> 0"
1191   shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
1192 proof -
1193   have False if "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t
1194     using l prems by simp
1195   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
1196     by blast
1197   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
1198     by auto
1199 qed
1201 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
1202   by (metis poly_all_0_iff_0)
1204 lemma basic_cqe_conv3:
1205   fixes p q :: "complex poly"
1206   assumes l: "p \<noteq> 0"
1207   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)"
1208 proof -
1209   from l have dp: "degree (pCons a p) = psize p"
1210     by (simp add: psize_def)
1211   from nullstellensatz_univariate[of "pCons a p" q] l
1212   show ?thesis
1213     by (metis dp pCons_eq_0_iff)
1214 qed
1216 lemma basic_cqe_conv4:
1217   fixes p q :: "complex poly"
1218   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
1219   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1220 proof -
1221   from h have "poly (q ^ n) = poly r"
1222     by auto
1223   then have "(q ^ n) = r"
1224     by (simp add: poly_eq_poly_eq_iff)
1225   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1226     by simp
1227 qed
1229 lemma poly_const_conv:
1230   fixes x :: "'a::comm_ring_1"
1231   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
1232   by simp
1234 end