src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author wenzelm Mon Apr 28 23:43:13 2014 +0200 (2014-04-28) changeset 56778 cb0929421ca6 parent 56776 309e1a61ee7c child 56795 e8cce2bd23e5 permissions -rw-r--r--
tuned proofs;
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 {* Square root of complex numbers *}
11 definition csqrt :: "complex \<Rightarrow> complex"
12 where
13   "csqrt z =
14     (if Im z = 0 then
15        if 0 \<le> Re z then Complex (sqrt(Re z)) 0
16        else Complex 0 (sqrt(- Re z))
17      else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
19 lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
20 proof -
21   obtain x y where xy: "z = Complex x y" by (cases z)
22   {
23     assume y0: "y = 0"
24     {
25       assume x0: "x \<ge> 0"
26       then have ?thesis
27         using y0 xy real_sqrt_pow2[OF x0]
28         by (simp add: csqrt_def power2_eq_square)
29     }
30     moreover
31     {
32       assume "\<not> x \<ge> 0"
33       then have x0: "- x \<ge> 0" by arith
34       then have ?thesis
35         using y0 xy real_sqrt_pow2[OF x0]
36         by (simp add: csqrt_def power2_eq_square)
37     }
38     ultimately have ?thesis by blast
39   }
40   moreover
41   {
42     assume y0: "y \<noteq> 0"
43     {
44       fix x y
45       let ?z = "Complex x y"
46       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z"
47         by auto
48       then have "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0"
49         by arith+
50       then have "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0"
51         by (simp_all add: power2_eq_square)
52     }
53     note th = this
54     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
55       by (simp add: power2_eq_square)
56     from th[of x y]
57     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
58       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
59       unfolding sq4 by simp_all
60     then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) -
61         sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
62       unfolding power2_eq_square by simp
63     have "sqrt 4 = sqrt (2\<^sup>2)"
64       by simp
65     then have sqrt4: "sqrt 4 = 2"
66       by (simp only: real_sqrt_abs)
67     have th2: "2 * (y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
68       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
69       unfolding power2_eq_square
70       by (simp add: algebra_simps real_sqrt_divide sqrt4)
71     from y0 xy have ?thesis
72       apply (simp add: csqrt_def power2_eq_square)
73       apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y]
74         real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square]
75         real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square]
76         real_sqrt_mult[symmetric])
77       using th1 th2  ..
78   }
79   ultimately show ?thesis by blast
80 qed
82 lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
83   by (simp add: csqrt_def)
85 lemma csqrt_0 [simp]: "csqrt 0 = 0"
86   by (simp add: csqrt_def)
88 lemma csqrt_1 [simp]: "csqrt 1 = 1"
89   by (simp add: csqrt_def)
91 lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
92 proof (cases z)
93   case (Complex x y)
94   then show ?thesis
95     using real_sqrt_sum_squares_ge1 [of "x" y]
96           real_sqrt_sum_squares_ge1 [of "-x" y]
97           real_sqrt_sum_squares_eq_cancel [of x y]
98     apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
100     apply (metis add_commute less_eq_real_def power_minus_Bit0
102     done
103 qed
105 lemma Re_csqrt: "0 \<le> Re(csqrt z)"
106   by (metis csqrt_principal le_less)
108 lemma csqrt_square: "0 < Re z \<or> Re z = 0 \<and> 0 \<le> Im z \<Longrightarrow> csqrt (z\<^sup>2) = z"
109   using csqrt [of "z\<^sup>2"] csqrt_principal [of "z\<^sup>2"]
110   by (cases z) (auto simp: power2_eq_iff)
112 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
113   by auto (metis csqrt power_eq_0_iff)
115 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
116   by auto (metis csqrt power2_eq_1_iff)
119 subsection {* More lemmas about module of complex numbers *}
121 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
122   by (rule of_real_power [symmetric])
124 text{* The triangle inequality for cmod *}
125 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
126   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
129 subsection {* Basic lemmas about polynomials *}
131 lemma poly_bound_exists:
132   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
133   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
134 proof (induct p)
135   case 0
136   then show ?case by (rule exI[where x=1]) simp
137 next
138   case (pCons c cs)
139   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
140     by blast
141   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
142   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
143   {
144     fix z :: 'a
145     assume H: "norm z \<le> r"
146     from m H have th: "norm (poly cs z) \<le> m"
147       by blast
148     from H have rp: "r \<ge> 0" using norm_ge_zero[of z]
149       by arith
150     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
151       using norm_triangle_ineq[of c "z* poly cs z"] by simp
152     also have "\<dots> \<le> norm c + r * m"
153       using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
154       by (simp add: norm_mult)
155     also have "\<dots> \<le> ?k"
156       by simp
157     finally have "norm (poly (pCons c cs) z) \<le> ?k" .
158   }
159   with kp show ?case by blast
160 qed
163 text{* Offsetting the variable in a polynomial gives another of same degree *}
165 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
166   where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
168 lemma offset_poly_0: "offset_poly 0 h = 0"
169   by (simp add: offset_poly_def)
171 lemma offset_poly_pCons:
172   "offset_poly (pCons a p) h =
173     smult h (offset_poly p h) + pCons a (offset_poly p h)"
174   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
176 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
177   by (simp add: offset_poly_pCons offset_poly_0)
179 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
180   apply (induct p)
181   apply (simp add: offset_poly_0)
182   apply (simp add: offset_poly_pCons algebra_simps)
183   done
185 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
186   by (induct p arbitrary: a) (simp, force)
188 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
189   apply (safe intro!: offset_poly_0)
190   apply (induct p, simp)
191   apply (simp add: offset_poly_pCons)
192   apply (frule offset_poly_eq_0_lemma, simp)
193   done
195 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
196   apply (induct p)
197   apply (simp add: offset_poly_0)
198   apply (case_tac "p = 0")
199   apply (simp add: offset_poly_0 offset_poly_pCons)
200   apply (simp add: offset_poly_pCons)
201   apply (subst degree_add_eq_right)
202   apply (rule le_less_trans [OF degree_smult_le])
203   apply (simp add: offset_poly_eq_0_iff)
204   apply (simp add: offset_poly_eq_0_iff)
205   done
207 definition "psize p = (if p = 0 then 0 else Suc (degree p))"
209 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
210   unfolding psize_def by simp
212 lemma poly_offset:
213   fixes p :: "'a::comm_ring_1 poly"
214   shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
215 proof (intro exI conjI)
216   show "psize (offset_poly p a) = psize p"
217     unfolding psize_def
218     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
219   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
220     by (simp add: poly_offset_poly)
221 qed
223 text{* An alternative useful formulation of completeness of the reals *}
224 lemma real_sup_exists:
225   assumes ex: "\<exists>x. P x"
226     and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
227   shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
228 proof
229   from bz have "bdd_above (Collect P)"
230     by (force intro: less_imp_le)
231   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
232     using ex bz by (subst less_cSup_iff) auto
233 qed
235 subsection {* Fundamental theorem of algebra *}
236 lemma  unimodular_reduce_norm:
237   assumes md: "cmod z = 1"
238   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
239 proof -
240   obtain x y where z: "z = Complex x y "
241     by (cases z) auto
242   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
243     by (simp add: cmod_def)
244   {
245     assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
246     from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
247       by (simp_all add: cmod_def power2_eq_square algebra_simps)
248     then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
249       by simp_all
250     then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
251       by - (rule power_mono, simp, simp)+
252     then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
253       by (simp_all add: power_mult_distrib)
254     from add_mono[OF th0] xy have False by simp
255   }
256   then show ?thesis
257     unfolding linorder_not_le[symmetric] by blast
258 qed
260 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
261 lemma reduce_poly_simple:
262   assumes b: "b \<noteq> 0"
263     and n: "n \<noteq> 0"
264   shows "\<exists>z. cmod (1 + b * z^n) < 1"
265   using n
266 proof (induct n rule: nat_less_induct)
267   fix n
268   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
269   assume n: "n \<noteq> 0"
270   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
271   {
272     assume e: "even n"
273     then have "\<exists>m. n = 2 * m"
274       by presburger
275     then obtain m where m: "n = 2 * m"
276       by blast
277     from n m have "m \<noteq> 0" "m < n"
278       by presburger+
279     with IH[rule_format, of m] obtain z where z: "?P z m"
280       by blast
281     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
282     then have "\<exists>z. ?P z n" ..
283   }
284   moreover
285   {
286     assume o: "odd n"
287     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
288       using b by (simp add: norm_divide)
289     from o have "\<exists>m. n = Suc (2 * m)"
290       by presburger+
291     then obtain m where m: "n = Suc (2*m)"
292       by blast
293     from unimodular_reduce_norm[OF th0] o
294     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
295       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
296       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
297       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
298       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
299       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
300       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
301       apply (auto simp add: m power_mult)
302       apply (rule_tac x="ii" in exI)
303       apply (auto simp add: m power_mult)
304       done
305     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
306       by blast
307     let ?w = "v / complex_of_real (root n (cmod b))"
308     from odd_real_root_pow[OF o, of "cmod b"]
309     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
310       by (simp add: power_divide complex_of_real_power)
311     have th2:"cmod (complex_of_real (cmod b) / b) = 1"
312       using b by (simp add: norm_divide)
313     then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
314       by simp
315     have th4: "cmod (complex_of_real (cmod b) / b) *
316         cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
317         cmod (complex_of_real (cmod b) / b) * 1"
318       apply (simp only: norm_mult[symmetric] distrib_left)
319       using b v
320       apply (simp add: th2)
321       done
322     from mult_less_imp_less_left[OF th4 th3]
323     have "?P ?w n" unfolding th1 .
324     then have "\<exists>z. ?P z n" ..
325   }
326   ultimately show "\<exists>z. ?P z n" by blast
327 qed
329 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
331 lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
332   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
333   unfolding cmod_def by simp
335 lemma bolzano_weierstrass_complex_disc:
336   assumes r: "\<forall>n. cmod (s n) \<le> r"
337   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
338 proof-
339   from seq_monosub[of "Re \<circ> s"]
340   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
341     unfolding o_def by blast
342   from seq_monosub[of "Im \<circ> s \<circ> f"]
343   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
344     unfolding o_def by blast
345   let ?h = "f \<circ> g"
346   from r[rule_format, of 0] have rp: "r \<ge> 0"
347     using norm_ge_zero[of "s 0"] by arith
348   have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
349   proof
350     fix n
351     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
352     show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
353   qed
354   have conv1: "convergent (\<lambda>n. Re (s (f n)))"
355     apply (rule Bseq_monoseq_convergent)
356     apply (simp add: Bseq_def)
357     apply (metis gt_ex le_less_linear less_trans order.trans th)
358     apply (rule f(2))
359     done
360   have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
361   proof
362     fix n
363     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
364     show "\<bar>Im (s n)\<bar> \<le> r + 1"
365       by arith
366   qed
368   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
369     apply (rule Bseq_monoseq_convergent)
370     apply (simp add: Bseq_def)
371     apply (metis gt_ex le_less_linear less_trans order.trans th)
372     apply (rule g(2))
373     done
375   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
376     by blast
377   then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
378     unfolding LIMSEQ_iff real_norm_def .
380   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
381     by blast
382   then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
383     unfolding LIMSEQ_iff real_norm_def .
384   let ?w = "Complex x y"
385   from f(1) g(1) have hs: "subseq ?h"
386     unfolding subseq_def by auto
387   {
388     fix e :: real
389     assume ep: "e > 0"
390     then have e2: "e/2 > 0" by simp
391     from x[rule_format, OF e2] y[rule_format, OF e2]
392     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
393       and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
394     {
395       fix n
396       assume nN12: "n \<ge> N1 + N2"
397       then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
398         using seq_suble[OF g(1), of n] by arith+
399       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
400       have "cmod (s (?h n) - ?w) < e"
401         using metric_bound_lemma[of "s (f (g n))" ?w] by simp
402     }
403     then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast
404   }
405   with hs show ?thesis by blast
406 qed
408 text{* Polynomial is continuous. *}
410 lemma poly_cont:
411   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
412   assumes ep: "e > 0"
413   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
414 proof -
415   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
416   proof
417     show "degree (offset_poly p z) = degree p"
418       by (rule degree_offset_poly)
419     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
420       by (rule poly_offset_poly)
421   qed
422   have th: "\<And>w. poly q (w - z) = poly p w"
423     using q(2)[of "w - z" for w] by simp
424   show ?thesis unfolding th[symmetric]
425   proof (induct q)
426     case 0
427     then show ?case
428       using ep by auto
429   next
430     case (pCons c cs)
431     from poly_bound_exists[of 1 "cs"]
432     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
433       by blast
434     from ep m(1) have em0: "e/m > 0"
435       by (simp add: field_simps)
436     have one0: "1 > (0::real)"
437       by arith
438     from real_lbound_gt_zero[OF one0 em0]
439     obtain d where d: "d > 0" "d < 1" "d < e / m"
440       by blast
441     from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
442       by (simp_all add: field_simps)
443     show ?case
444     proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
445       fix d w
446       assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
447       then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
448         by simp_all
449       from H(3) m(1) have dme: "d*m < e"
450         by (simp add: field_simps)
451       from H have th: "norm (w - z) \<le> d"
452         by simp
453       from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
454       show "norm (w - z) * norm (poly cs (w - z)) < e"
455         by simp
456     qed
457   qed
458 qed
460 text{* Hence a polynomial attains minimum on a closed disc
461   in the complex plane. *}
462 lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
463 proof -
464   {
465     assume "\<not> r \<ge> 0"
466     then have ?thesis
467       by (metis norm_ge_zero order.trans)
468   }
469   moreover
470   {
471     assume rp: "r \<ge> 0"
472     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
473       by simp
474     then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
475       by blast
476     {
477       fix x z
478       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
479       then have "- x < 0 "
480         by arith
481       with H(2) norm_ge_zero[of "poly p z"] have False
482         by simp
483     }
484     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
485       by blast
486     from real_sup_exists[OF mth1 mth2] obtain s where
487       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
488     let ?m = "- s"
489     {
490       fix y
491       from s[rule_format, of "-y"]
492       have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
493         unfolding minus_less_iff[of y ] equation_minus_iff by blast
494     }
495     note s1 = this[unfolded minus_minus]
496     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
497       by auto
498     {
499       fix n :: nat
500       from s1[rule_format, of "?m + 1/real (Suc n)"]
501       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
502         by simp
503     }
504     then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
505     from choice[OF th] obtain g where
506         g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
507       by blast
508     from bolzano_weierstrass_complex_disc[OF g(1)]
509     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
510       by blast
511     {
512       fix w
513       assume wr: "cmod w \<le> r"
514       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
515       {
516         assume e: "?e > 0"
517         then have e2: "?e/2 > 0" by simp
518         from poly_cont[OF e2, of z p] obtain d where
519             d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
520           by blast
521         {
522           fix w
523           assume w: "cmod (w - z) < d"
524           have "cmod(poly p w - poly p z) < ?e / 2"
525             using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
526         }
527         note th1 = this
529         from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
530           by blast
531         from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
532           by blast
533         have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
534           using N1[rule_format, of "N1 + N2"] th1 by simp
535         {
536           fix a b e2 m :: real
537           have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
538             by arith
539         }
540         note th0 = this
541         have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"
542           by arith
543         from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
544         from seq_suble[OF fz(1), of "N1+N2"]
545         have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
546           by simp
547         have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
548           using N2 by auto
549         from frac_le[OF th000 th00]
550         have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
551           by simp
552         from g(2)[rule_format, of "f (N1 + N2)"]
553         have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
554         from order_less_le_trans[OF th01 th00]
555         have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
556         from N2 have "2/?e < real (Suc (N1 + N2))"
557           by arith
558         with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
559         have "?e/2 > 1/ real (Suc (N1 + N2))"
560           by (simp add: inverse_eq_divide)
561         with ath[OF th31 th32]
562         have thc1: "\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
563           by arith
564         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"
565           by arith
566         have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
567             cmod (poly p (g (f (N1 + N2))) - poly p z)"
568           by (simp add: norm_triangle_ineq3)
569         from ath2[OF th22, of ?m]
570         have thc2: "2 * (?e/2) \<le>
571             \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
572           by simp
573         from th0[OF th2 thc1 thc2] have False .
574       }
575       then have "?e = 0"
576         by auto
577       then have "cmod (poly p z) = ?m"
578         by simp
579       with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
580         by simp
581     }
582     then have ?thesis by blast
583   }
584   ultimately show ?thesis by blast
585 qed
587 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
588   unfolding power2_eq_square
589   apply (simp add: rcis_mult)
590   apply (simp add: power2_eq_square[symmetric])
591   done
593 lemma cispi: "cis pi = -1"
594   by (simp add: cis_def)
596 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
597   unfolding power2_eq_square
599   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
600   done
602 text {* Nonzero polynomial in z goes to infinity as z does. *}
604 lemma poly_infinity:
605   fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
606   assumes ex: "p \<noteq> 0"
607   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
608   using ex
609 proof (induct p arbitrary: a d)
610   case (pCons c cs a d)
611   {
612     assume H: "cs \<noteq> 0"
613     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
614       by blast
615     let ?r = "1 + \<bar>r\<bar>"
616     {
617       fix z::'a
618       assume h: "1 + \<bar>r\<bar> \<le> norm z"
619       have r0: "r \<le> norm z" using h by arith
620       from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
621         by arith
622       from h have z1: "norm z \<ge> 1"
623         by arith
624       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
625       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
626         unfolding norm_mult by (simp add: algebra_simps)
627       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
628       have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
629         by (simp add: algebra_simps)
630       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)" by arith
631     }
632     then have ?case by blast
633   }
634   moreover
635   {
636     assume cs0: "\<not> (cs \<noteq> 0)"
637     with pCons.prems have c0: "c \<noteq> 0"
638       by simp
639     from cs0 have cs0': "cs = 0"
640       by simp
641     {
642       fix z::'a
643       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
644       from c0 have "norm c > 0"
645         by simp
646       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
647         by (simp add: field_simps norm_mult)
648       have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
649         by arith
650       from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
651         by (simp add: algebra_simps)
652       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
653         using cs0' by simp
654     }
655     then have ?case  by blast
656   }
657   ultimately show ?case by blast
658 qed simp
660 text {* Hence polynomial's modulus attains its minimum somewhere. *}
661 lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
662 proof (induct p)
663   case 0
664   then show ?case by simp
665 next
666   case (pCons c cs)
667   show ?case
668   proof (cases "cs = 0")
669     case False
670     from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
671     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
672       by blast
673     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
674       by arith
675     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
676     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)"
677       by blast
678     {
679       fix z
680       assume z: "r \<le> cmod z"
681       from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
682         by simp
683     }
684     note v0 = this
685     from v0 v ath[of r] show ?thesis
686       by blast
687   next
688     case True
689     with pCons.hyps show ?thesis by simp
690   qed
691 qed
693 text{* Constant function (non-syntactic characterization). *}
694 definition "constant f = (\<forall>x y. f x = f y)"
696 lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
697   by (induct p) (auto simp: constant_def psize_def)
699 lemma poly_replicate_append:
700   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
701   by (simp add: poly_monom)
703 text {* Decomposition of polynomial, skipping zero coefficients
704   after the first.  *}
706 lemma poly_decompose_lemma:
707   assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
708   shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and>
709     (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
710   unfolding psize_def
711   using nz
712 proof (induct p)
713   case 0
714   then show ?case by simp
715 next
716   case (pCons c cs)
717   show ?case
718   proof (cases "c = 0")
719     case True
720     from pCons.hyps pCons.prems True show ?thesis
721       apply (auto)
722       apply (rule_tac x="k+1" in exI)
723       apply (rule_tac x="a" in exI, clarsimp)
724       apply (rule_tac x="q" in exI)
725       apply auto
726       done
727   next
728     case False
729     show ?thesis
730       apply (rule exI[where x=0])
731       apply (rule exI[where x=c], auto simp add: False)
732       done
733   qed
734 qed
736 lemma poly_decompose:
737   assumes nc: "\<not> constant (poly p)"
738   shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
739                psize q + k + 1 = psize p \<and>
740               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
741   using nc
742 proof (induct p)
743   case 0
744   then show ?case
745     by (simp add: constant_def)
746 next
747   case (pCons c cs)
748   {
749     assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
750     {
751       fix x y
752       from C have "poly (pCons c cs) x = poly (pCons c cs) y"
753         by (cases "x = 0") auto
754     }
755     with pCons.prems have False
756       by (auto simp add: constant_def)
757   }
758   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
759   from poly_decompose_lemma[OF th]
760   show ?case
761     apply clarsimp
762     apply (rule_tac x="k+1" in exI)
763     apply (rule_tac x="a" in exI)
764     apply simp
765     apply (rule_tac x="q" in exI)
766     apply (auto simp add: psize_def split: if_splits)
767     done
768 qed
770 text{* Fundamental theorem of algebra *}
772 lemma fundamental_theorem_of_algebra:
773   assumes nc: "\<not> constant (poly p)"
774   shows "\<exists>z::complex. poly p z = 0"
775   using nc
776 proof (induct "psize p" arbitrary: p rule: less_induct)
777   case less
778   let ?p = "poly p"
779   let ?ths = "\<exists>z. ?p z = 0"
781   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
782   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
783     by blast
785   show ?ths
786   proof (cases "?p c = 0")
787     case True
788     then show ?thesis by blast
789   next
790     case False
791     note pc0 = this
792     from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
793       by blast
794     {
795       assume h: "constant (poly q)"
796       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
797       {
798         fix x y
799         from th have "?p x = poly q (x - c)" by auto
800         also have "\<dots> = poly q (y - c)"
801           using h unfolding constant_def by blast
802         also have "\<dots> = ?p y" using th by auto
803         finally have "?p x = ?p y" .
804       }
805       with less(2) have False
806         unfolding constant_def by blast
807     }
808     then have qnc: "\<not> constant (poly q)"
809       by blast
810     from q(2) have pqc0: "?p c = poly q 0"
811       by simp
812     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
813       by simp
814     let ?a0 = "poly q 0"
815     from pc0 pqc0 have a00: "?a0 \<noteq> 0"
816       by simp
817     from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
818       by simp
819     let ?r = "smult (inverse ?a0) q"
820     have lgqr: "psize q = psize ?r"
821       using a00
822       unfolding psize_def degree_def
823       by (simp add: poly_eq_iff)
824     {
825       assume h: "\<And>x y. poly ?r x = poly ?r y"
826       {
827         fix x y
828         from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
829           by auto
830         also have "\<dots> = poly ?r y * ?a0"
831           using h by simp
832         also have "\<dots> = poly q y"
833           using qr[rule_format, of y] by simp
834         finally have "poly q x = poly q y" .
835       }
836       with qnc have False unfolding constant_def by blast
837     }
838     then have rnc: "\<not> constant (poly ?r)"
839       unfolding constant_def by blast
840     from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
841       by auto
842     {
843       fix w
844       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
845         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
846       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
847         using a00 unfolding norm_divide by (simp add: field_simps)
848       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .
849     }
850     note mrmq_eq = this
851     from poly_decompose[OF rnc] obtain k a s where
852       kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
853         "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
854     {
855       assume "psize p = k + 1"
856       with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
857         by auto
858       {
859         fix w
860         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
861           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
862       }
863       note hth = this [symmetric]
864       from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"
865         unfolding hth by blast
866     }
867     moreover
868     {
869       assume kn: "psize p \<noteq> k + 1"
870       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
871         by simp
872       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
873         unfolding constant_def poly_pCons poly_monom
874         using kas(1) apply simp
875         apply (rule exI[where x=0])
876         apply (rule exI[where x=1])
877         apply simp
878         done
879       from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
880         by (simp add: psize_def degree_monom_eq)
881       from less(1) [OF k1n [simplified th02] th01]
882       obtain w where w: "1 + w^k * a = 0"
883         unfolding poly_pCons poly_monom
884         using kas(2) by (cases k) (auto simp add: algebra_simps)
885       from poly_bound_exists[of "cmod w" s] obtain m where
886         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
887       have w0: "w \<noteq> 0" using kas(2) w
888         by (auto simp add: power_0_left)
889       from w have "(1 + w ^ k * a) - 1 = 0 - 1"
890         by simp
891       then have wm1: "w^k * a = - 1"
892         by simp
893       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
894         using norm_ge_zero[of w] w0 m(1)
895         by (simp add: inverse_eq_divide zero_less_mult_iff)
896       with real_lbound_gt_zero[OF zero_less_one] obtain t where
897         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
898       let ?ct = "complex_of_real t"
899       let ?w = "?ct * w"
900       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
901         using kas(1) by (simp add: algebra_simps power_mult_distrib)
902       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
903         unfolding wm1 by simp
904       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
905         cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
906         by metis
907       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
908       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
909         unfolding norm_of_real by simp
910       have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
911         by arith
912       have "t * cmod w \<le> 1 * cmod w"
913         apply (rule mult_mono)
914         using t(1,2)
915         apply auto
916         done
917       then have tw: "cmod ?w \<le> cmod w"
918         using t(1) by (simp add: norm_mult)
919       from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
920         by (simp add: inverse_eq_divide field_simps)
921       with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
922         by (metis comm_mult_strict_left_mono)
923       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
924         using w0 t(1)
925         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
926       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
927         using t(1,2) m(2)[rule_format, OF tw] w0
928         by auto
929       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
930         by simp
931       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
932         by auto
933       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
934       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
935       from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
936         by arith
937       then have "cmod (poly ?r ?w) < 1"
938         unfolding kas(4)[rule_format, of ?w] r01 by simp
939       then have "\<exists>w. cmod (poly ?r w) < 1"
940         by blast
941     }
942     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"
943       by blast
944     from cr0_contr cq0 q(2) show ?thesis
945       unfolding mrmq_eq not_less[symmetric] by auto
946   qed
947 qed
949 text {* Alternative version with a syntactic notion of constant polynomial. *}
951 lemma fundamental_theorem_of_algebra_alt:
952   assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
953   shows "\<exists>z. poly p z = (0::complex)"
954   using nc
955 proof (induct p)
956   case 0
957   then show ?case by simp
958 next
959   case (pCons c cs)
960   show ?case
961   proof (cases "c = 0")
962     case True
963     then show ?thesis by auto
964   next
965     case False
966     {
967       assume nc: "constant (poly (pCons c cs))"
968       from nc[unfolded constant_def, rule_format, of 0]
969       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
970       then have "cs = 0"
971       proof (induct cs)
972         case 0
973         then show ?case by simp
974       next
975         case (pCons d ds)
976         show ?case
977         proof (cases "d = 0")
978           case True
979           then show ?thesis using pCons.prems pCons.hyps by simp
980         next
981           case False
982           from poly_bound_exists[of 1 ds] obtain m where
983             m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
984           have dm: "cmod d / m > 0" using False m(1) by (simp add: field_simps)
985           from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
986             x: "x > 0" "x < cmod d / m" "x < 1" by blast
987           let ?x = "complex_of_real x"
988           from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
989           from pCons.prems[rule_format, OF cx(1)]
990           have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
991           from m(2)[rule_format, OF cx(2)] x(1)
992           have th0: "cmod (?x*poly ds ?x) \<le> x*m"
993             by (simp add: norm_mult)
994           from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
995           with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
996           with cth show ?thesis by blast
997         qed
998       qed
999     }
1000     then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems False
1001       by blast
1002     from fundamental_theorem_of_algebra[OF nc] show ?thesis .
1003   qed
1004 qed
1007 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
1009 lemma nullstellensatz_lemma:
1010   fixes p :: "complex poly"
1011   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
1012     and "degree p = n"
1013     and "n \<noteq> 0"
1014   shows "p dvd (q ^ n)"
1015   using assms
1016 proof (induct n arbitrary: p q rule: nat_less_induct)
1017   fix n :: nat
1018   fix p q :: "complex poly"
1019   assume IH: "\<forall>m<n. \<forall>p q.
1020                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
1021                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
1022     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
1023     and dpn: "degree p = n"
1024     and n0: "n \<noteq> 0"
1025   from dpn n0 have pne: "p \<noteq> 0" by auto
1026   let ?ths = "p dvd (q ^ n)"
1027   {
1028     fix a
1029     assume a: "poly p a = 0"
1030     {
1031       assume oa: "order a p \<noteq> 0"
1032       let ?op = "order a p"
1033       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
1034         using order by blast+
1035       note oop = order_degree[OF pne, unfolded dpn]
1036       {
1037         assume q0: "q = 0"
1038         then have ?ths using n0
1039           by (simp add: power_0_left)
1040       }
1041       moreover
1042       {
1043         assume q0: "q \<noteq> 0"
1044         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
1045         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
1046         from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
1047           by (rule dvdE)
1048         have sne: "s \<noteq> 0" using s pne by auto
1049         {
1050           assume ds0: "degree s = 0"
1051           from ds0 obtain k where kpn: "s = [:k:]"
1052             by (cases s) (auto split: if_splits)
1053           from sne kpn have k: "k \<noteq> 0" by simp
1054           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
1055           have "q ^ n = p * ?w"
1056             apply (subst r, subst s, subst kpn)
1057             using k oop [of a]
1058             apply (subst power_mult_distrib, simp)
1059             apply (subst power_add [symmetric], simp)
1060             done
1061           then have ?ths unfolding dvd_def by blast
1062         }
1063         moreover
1064         {
1065           assume ds0: "degree s \<noteq> 0"
1066           from ds0 sne dpn s oa
1067             have dsn: "degree s < n"
1068               apply auto
1069               apply (erule ssubst)
1070               apply (simp add: degree_mult_eq degree_linear_power)
1071               done
1072             {
1073               fix x assume h: "poly s x = 0"
1074               {
1075                 assume xa: "x = a"
1076                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
1077                   by (rule dvdE)
1078                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
1079                   by (subst s, subst u, simp only: power_Suc mult_ac)
1080                 with ap(2)[unfolded dvd_def] have False by blast
1081               }
1082               note xa = this
1083               from h have "poly p x = 0" by (subst s) simp
1084               with pq0 have "poly q x = 0" by blast
1085               with r xa have "poly r x = 0"
1086                 by auto
1087             }
1088             note impth = this
1089             from IH[rule_format, OF dsn, of s r] impth ds0
1090             have "s dvd (r ^ (degree s))" by blast
1091             then obtain u where u: "r ^ (degree s) = s * u" ..
1092             then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
1093               by (simp only: poly_mult[symmetric] poly_power[symmetric])
1094             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
1095             from oop[of a] dsn have "q ^ n = p * ?w"
1096               apply -
1097               apply (subst s, subst r)
1098               apply (simp only: power_mult_distrib)
1099               apply (subst mult_assoc [where b=s])
1100               apply (subst mult_assoc [where a=u])
1101               apply (subst mult_assoc [where b=u, symmetric])
1102               apply (subst u [symmetric])
1103               apply (simp add: mult_ac power_add [symmetric])
1104               done
1105             then have ?ths unfolding dvd_def by blast
1106         }
1107         ultimately have ?ths by blast
1108       }
1109       ultimately have ?ths by blast
1110     }
1111     then have ?ths using a order_root pne by blast
1112   }
1113   moreover
1114   {
1115     assume exa: "\<not> (\<exists>a. poly p a = 0)"
1116     from fundamental_theorem_of_algebra_alt[of p] exa
1117     obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
1118       by blast
1119     then have pp: "\<And>x. poly p x = c"
1120       by simp
1121     let ?w = "[:1/c:] * (q ^ n)"
1122     from ccs have "(q ^ n) = (p * ?w)"
1123       by simp
1124     then have ?ths
1125       unfolding dvd_def by blast
1126   }
1127   ultimately show ?ths by blast
1128 qed
1130 lemma nullstellensatz_univariate:
1131   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
1132     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
1133 proof -
1134   {
1135     assume pe: "p = 0"
1136     then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
1137       by (auto simp add: poly_all_0_iff_0)
1138     {
1139       assume "p dvd (q ^ (degree p))"
1140       then obtain r where r: "q ^ (degree p) = p * r" ..
1141       from r pe have False by simp
1142     }
1143     with eq pe have ?thesis by blast
1144   }
1145   moreover
1146   {
1147     assume pe: "p \<noteq> 0"
1148     {
1149       assume dp: "degree p = 0"
1150       then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe
1151         by (cases p) (simp split: if_splits)
1152       then have th1: "\<forall>x. poly p x \<noteq> 0"
1153         by simp
1154       from k dp have "q ^ (degree p) = p * [:1/k:]"
1155         by (simp add: one_poly_def)
1156       then have th2: "p dvd (q ^ (degree p))" ..
1157       from th1 th2 pe have ?thesis by blast
1158     }
1159     moreover
1160     {
1161       assume dp: "degree p \<noteq> 0"
1162       then obtain n where n: "degree p = Suc n "
1163         by (cases "degree p") auto
1164       {
1165         assume "p dvd (q ^ (Suc n))"
1166         then obtain u where u: "q ^ (Suc n) = p * u" ..
1167         {
1168           fix x
1169           assume h: "poly p x = 0" "poly q x \<noteq> 0"
1170           then have "poly (q ^ (Suc n)) x \<noteq> 0"
1171             by simp
1172           then have False using u h(1)
1173             by (simp only: poly_mult) simp
1174         }
1175       }
1176       with n nullstellensatz_lemma[of p q "degree p"] dp
1177       have ?thesis by auto
1178     }
1179     ultimately have ?thesis by blast
1180   }
1181   ultimately show ?thesis by blast
1182 qed
1184 text{* Useful lemma *}
1186 lemma constant_degree:
1187   fixes p :: "'a::{idom,ring_char_0} poly"
1188   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
1189 proof
1190   assume l: ?lhs
1191   from l[unfolded constant_def, rule_format, of _ "0"]
1192   have th: "poly p = poly [:poly p 0:]"
1193     by auto
1194   then have "p = [:poly p 0:]"
1195     by (simp add: poly_eq_poly_eq_iff)
1196   then have "degree p = degree [:poly p 0:]"
1197     by simp
1198   then show ?rhs
1199     by simp
1200 next
1201   assume r: ?rhs
1202   then obtain k where "p = [:k:]"
1203     by (cases p) (simp split: if_splits)
1204   then show ?lhs
1205     unfolding constant_def by auto
1206 qed
1208 lemma divides_degree:
1209   assumes pq: "p dvd (q:: complex poly)"
1210   shows "degree p \<le> degree q \<or> q = 0"
1211   by (metis dvd_imp_degree_le pq)
1213 (* Arithmetic operations on multivariate polynomials.                        *)
1215 lemma mpoly_base_conv:
1216   fixes x :: "'a::comm_ring_1"
1217   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
1218   by simp_all
1220 lemma mpoly_norm_conv:
1221   fixes x :: "'a::comm_ring_1"
1222   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
1223   by simp_all
1225 lemma mpoly_sub_conv:
1226   fixes x :: "'a::comm_ring_1"
1227   shows "poly p x - poly q x = poly p x + -1 * poly q x"
1228   by simp
1230 lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
1231   by simp
1233 lemma poly_cancel_eq_conv:
1234   fixes x :: "'a::field"
1235   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
1236   by auto
1239   fixes p:: "('a::comm_ring_1) poly"
1240   assumes pq: "p dvd q"
1241   shows "p dvd (pCons 0 q)"
1242 proof -
1243   have "pCons 0 q = q * [:0,1:]" by simp
1244   then have "q dvd (pCons 0 q)" ..
1245   with pq show ?thesis by (rule dvd_trans)
1246 qed
1248 lemma poly_divides_conv0:
1249   fixes p:: "'a::field poly"
1250   assumes lgpq: "degree q < degree p"
1251     and lq: "p \<noteq> 0"
1252   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
1253 proof
1254   assume r: ?rhs
1255   then have "q = p * 0" by simp
1256   then show ?lhs ..
1257 next
1258   assume l: ?lhs
1259   show ?rhs
1260   proof (cases "q = 0")
1261     case True
1262     then show ?thesis by simp
1263   next
1264     assume q0: "q \<noteq> 0"
1265     from l q0 have "degree p \<le> degree q"
1266       by (rule dvd_imp_degree_le)
1267     with lgpq show ?thesis by simp
1268   qed
1269 qed
1271 lemma poly_divides_conv1:
1272   fixes p :: "'a::field poly"
1273   assumes a0: "a \<noteq> 0"
1274     and pp': "p dvd p'"
1275     and qrp': "smult a q - p' = r"
1276   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
1277 proof
1278   from pp' obtain t where t: "p' = p * t" ..
1279   {
1280     assume l: ?lhs
1281     then obtain u where u: "q = p * u" ..
1282     have "r = p * (smult a u - t)"
1283       using u qrp' [symmetric] t by (simp add: algebra_simps)
1284     then show ?rhs ..
1285   next
1286     assume r: ?rhs
1287     then obtain u where u: "r = p * u" ..
1288     from u [symmetric] t qrp' [symmetric] a0
1289     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1290     then show ?lhs ..
1291   }
1292 qed
1294 lemma basic_cqe_conv1:
1295   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1296   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1297   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
1298   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
1299   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
1300   by simp_all
1302 lemma basic_cqe_conv2:
1303   assumes l:"p \<noteq> 0"
1304   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
1305 proof -
1306   {
1307     fix h t
1308     assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
1309     with l have False by simp
1310   }
1311   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
1312     by blast
1313   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
1314     by auto
1315 qed
1317 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
1318   by (metis poly_all_0_iff_0)
1320 lemma basic_cqe_conv3:
1321   fixes p q :: "complex poly"
1322   assumes l: "p \<noteq> 0"
1323   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ psize p))"
1324 proof -
1325   from l have dp: "degree (pCons a p) = psize p"
1326     by (simp add: psize_def)
1327   from nullstellensatz_univariate[of "pCons a p" q] l
1328   show ?thesis
1329     by (metis dp pCons_eq_0_iff)
1330 qed
1332 lemma basic_cqe_conv4:
1333   fixes p q :: "complex poly"
1334   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
1335   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1336 proof -
1337   from h have "poly (q ^ n) = poly r"
1338     by auto
1339   then have "(q ^ n) = r"
1340     by (simp add: poly_eq_poly_eq_iff)
1341   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1342     by simp
1343 qed
1345 lemma poly_const_conv:
1346   fixes x :: "'a::comm_ring_1"
1347   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
1348   by simp
1350 end