src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author blanchet Fri Apr 04 14:44:51 2014 +0200 (2014-04-04) changeset 56403 ae4f904c98b0 parent 55735 81ba62493610 child 56544 b60d5d119489 permissions -rw-r--r--
tuned spaces
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" where
12 "csqrt z = (if Im z = 0 then
13             if 0 \<le> Re z then Complex (sqrt(Re z)) 0
14             else Complex 0 (sqrt(- Re z))
15            else Complex (sqrt((cmod z + Re z) /2))
16                         ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
18 lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
19 proof-
20   obtain x y where xy: "z = Complex x y" by (cases z)
21   {assume y0: "y = 0"
22     {assume x0: "x \<ge> 0"
23       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
24         by (simp add: csqrt_def power2_eq_square)}
25     moreover
26     {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
27       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
28         by (simp add: csqrt_def power2_eq_square) }
29     ultimately have ?thesis by blast}
30   moreover
31   {assume y0: "y\<noteq>0"
32     {fix x y
33       let ?z = "Complex x y"
34       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
35       hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
36       hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }
37     note th = this
38     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
39       by (simp add: power2_eq_square)
40     from th[of x y]
41     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
42       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
43       unfolding sq4 by simp_all
44     then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) - sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
45       unfolding power2_eq_square by simp
46     have "sqrt 4 = sqrt (2\<^sup>2)" by simp
47     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
48     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
49       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
50       unfolding power2_eq_square
51       by (simp add: algebra_simps real_sqrt_divide sqrt4)
52      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
53        apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y] real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square] real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square] real_sqrt_mult[symmetric])
54       using th1 th2  ..}
55   ultimately show ?thesis by blast
56 qed
58 lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
59   by (simp add: csqrt_def)
61 lemma csqrt_0 [simp]: "csqrt 0 = 0"
62   by (simp add: csqrt_def)
64 lemma csqrt_1 [simp]: "csqrt 1 = 1"
65   by (simp add: csqrt_def)
67 lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
68 proof (cases z)
69   case (Complex x y)
70   then show ?thesis
71     using real_sqrt_sum_squares_ge1 [of "x" y]
72           real_sqrt_sum_squares_ge1 [of "-x" y]
73           real_sqrt_sum_squares_eq_cancel [of x y]
74     apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
76     by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
77 qed
79 lemma Re_csqrt: "0 \<le> Re(csqrt z)"
80   by (metis csqrt_principal le_less)
82 lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
83   using csqrt [of "z^2"] csqrt_principal [of "z^2"]
84   by (cases z) (auto simp: power2_eq_iff)
86 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
87   by auto (metis csqrt power_eq_0_iff)
89 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
90   by auto (metis csqrt power2_eq_1_iff)
92 subsection{* More lemmas about module of complex numbers *}
94 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
95   by (rule of_real_power [symmetric])
97 text{* The triangle inequality for cmod *}
98 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
99   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
101 subsection{* Basic lemmas about polynomials *}
103 lemma poly_bound_exists:
104   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
105   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
106 proof(induct p)
107   case 0 thus ?case by (rule exI[where x=1], simp)
108 next
109   case (pCons c cs)
110   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
111     by blast
112   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
113   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
114   {fix z :: 'a
115     assume H: "norm z \<le> r"
116     from m H have th: "norm (poly cs z) \<le> m" by blast
117     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
118     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
119       using norm_triangle_ineq[of c "z* poly cs z"] by simp
120     also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
121       by (simp add: norm_mult)
122     also have "\<dots> \<le> ?k" by simp
123     finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
124   with kp show ?case by blast
125 qed
128 text{* Offsetting the variable in a polynomial gives another of same degree *}
130 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
131 where
132   "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
134 lemma offset_poly_0: "offset_poly 0 h = 0"
135   by (simp add: offset_poly_def)
137 lemma offset_poly_pCons:
138   "offset_poly (pCons a p) h =
139     smult h (offset_poly p h) + pCons a (offset_poly p h)"
140   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
142 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
143 by (simp add: offset_poly_pCons offset_poly_0)
145 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
146 apply (induct p)
147 apply (simp add: offset_poly_0)
148 apply (simp add: offset_poly_pCons algebra_simps)
149 done
151 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
152 by (induct p arbitrary: a, simp, force)
154 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
155 apply (safe intro!: offset_poly_0)
156 apply (induct p, simp)
157 apply (simp add: offset_poly_pCons)
158 apply (frule offset_poly_eq_0_lemma, simp)
159 done
161 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
162 apply (induct p)
163 apply (simp add: offset_poly_0)
164 apply (case_tac "p = 0")
165 apply (simp add: offset_poly_0 offset_poly_pCons)
166 apply (simp add: offset_poly_pCons)
167 apply (subst degree_add_eq_right)
168 apply (rule le_less_trans [OF degree_smult_le])
169 apply (simp add: offset_poly_eq_0_iff)
170 apply (simp add: offset_poly_eq_0_iff)
171 done
173 definition
174   "psize p = (if p = 0 then 0 else Suc (degree p))"
176 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
177   unfolding psize_def by simp
179 lemma poly_offset:
180   fixes p:: "('a::comm_ring_1) poly"
181   shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
182 proof (intro exI conjI)
183   show "psize (offset_poly p a) = psize p"
184     unfolding psize_def
185     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
186   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
187     by (simp add: poly_offset_poly)
188 qed
190 text{* An alternative useful formulation of completeness of the reals *}
191 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
192   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
193 proof
194   from bz have "bdd_above (Collect P)"
195     by (force intro: less_imp_le)
196   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
197     using ex bz by (subst less_cSup_iff) auto
198 qed
200 subsection {* Fundamental theorem of algebra *}
201 lemma  unimodular_reduce_norm:
202   assumes md: "cmod z = 1"
203   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
204 proof-
205   obtain x y where z: "z = Complex x y " by (cases z, auto)
206   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
207   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
208     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
209       by (simp_all add: cmod_def power2_eq_square algebra_simps)
210     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
211     hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
212       by - (rule power_mono, simp, simp)+
213     hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
214       by (simp_all add: power_mult_distrib)
215     from add_mono[OF th0] xy have False by simp }
216   thus ?thesis unfolding linorder_not_le[symmetric] by blast
217 qed
219 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
220 lemma reduce_poly_simple:
221  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
222   shows "\<exists>z. cmod (1 + b * z^n) < 1"
223 using n
224 proof(induct n rule: nat_less_induct)
225   fix n
226   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
227   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
228   {assume e: "even n"
229     hence "\<exists>m. n = 2*m" by presburger
230     then obtain m where m: "n = 2*m" by blast
231     from n m have "m\<noteq>0" "m < n" by presburger+
232     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
233     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
234     hence "\<exists>z. ?P z n" ..}
235   moreover
236   {assume o: "odd n"
237     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
238       using b by (simp add: norm_divide)
239     from o have "\<exists>m. n = Suc (2*m)" by presburger+
240     then obtain m where m: "n = Suc (2*m)" by blast
241     from unimodular_reduce_norm[OF th0] o
242     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
243       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
244       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
245       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
246       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
247       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
248       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
249       apply (auto simp add: m power_mult)
250       apply (rule_tac x="ii" in exI)
251       apply (auto simp add: m power_mult)
252       done
253     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
254     let ?w = "v / complex_of_real (root n (cmod b))"
255     from odd_real_root_pow[OF o, of "cmod b"]
256     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
257       by (simp add: power_divide complex_of_real_power)
258     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
259     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
260     have th4: "cmod (complex_of_real (cmod b) / b) *
261    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
262    < cmod (complex_of_real (cmod b) / b) * 1"
263       apply (simp only: norm_mult[symmetric] distrib_left)
264       using b v by (simp add: th2)
266     from mult_less_imp_less_left[OF th4 th3]
267     have "?P ?w n" unfolding th1 .
268     hence "\<exists>z. ?P z n" .. }
269   ultimately show "\<exists>z. ?P z n" by blast
270 qed
272 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
274 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
275   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
276   unfolding cmod_def by simp
278 lemma bolzano_weierstrass_complex_disc:
279   assumes r: "\<forall>n. cmod (s n) \<le> r"
280   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
281 proof-
282   from seq_monosub[of "Re o s"]
283   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
284     unfolding o_def by blast
285   from seq_monosub[of "Im o s o f"]
286   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
287   let ?h = "f o g"
288   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
289   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
290   proof
291     fix n
292     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
293   qed
294   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
295     apply (rule Bseq_monoseq_convergent)
296     apply (simp add: Bseq_def)
297     apply (metis gt_ex le_less_linear less_trans order.trans th)
298     using f(2) .
299   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
300   proof
301     fix n
302     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
303   qed
305   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
306     apply (rule Bseq_monoseq_convergent)
307     apply (simp add: Bseq_def)
308     apply (metis gt_ex le_less_linear less_trans order.trans th)
309     using g(2) .
311   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
312     by blast
313   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
314     unfolding LIMSEQ_iff real_norm_def .
316   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
317     by blast
318   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
319     unfolding LIMSEQ_iff real_norm_def .
320   let ?w = "Complex x y"
321   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
322   {fix e assume ep: "e > (0::real)"
323     hence e2: "e/2 > 0" by simp
324     from x[rule_format, OF e2] y[rule_format, OF e2]
325     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2" and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2" by blast
326     {fix n assume nN12: "n \<ge> N1 + N2"
327       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
328       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
329       have "cmod (s (?h n) - ?w) < e"
330         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
331     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
332   with hs show ?thesis  by blast
333 qed
335 text{* Polynomial is continuous. *}
337 lemma poly_cont:
338   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
339   assumes ep: "e > 0"
340   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
341 proof-
342   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
343   proof
344     show "degree (offset_poly p z) = degree p"
345       by (rule degree_offset_poly)
346     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
347       by (rule poly_offset_poly)
348   qed
349   {fix w
350     note q(2)[of "w - z", simplified]}
351   note th = this
352   show ?thesis unfolding th[symmetric]
353   proof(induct q)
354     case 0 thus ?case  using ep by auto
355   next
356     case (pCons c cs)
357     from poly_bound_exists[of 1 "cs"]
358     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
359     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
360     have one0: "1 > (0::real)"  by arith
361     from real_lbound_gt_zero[OF one0 em0]
362     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
363     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
364       by (simp_all add: field_simps mult_pos_pos)
365     show ?case
366       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
367         fix d w
368         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
369         hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
370         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
371         from H have th: "norm (w-z) \<le> d" by simp
372         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
373         show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
374       qed
375     qed
376 qed
378 text{* Hence a polynomial attains minimum on a closed disc
379   in the complex plane. *}
380 lemma  poly_minimum_modulus_disc:
381   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
382 proof-
383   {assume "\<not> r \<ge> 0" hence ?thesis
384     by (metis norm_ge_zero order.trans)}
385   moreover
386   {assume rp: "r \<ge> 0"
387     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
388     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
389     {fix x z
390       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
391       hence "- x < 0 " by arith
392       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
393     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
394     from real_sup_exists[OF mth1 mth2] obtain s where
395       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
396     let ?m = "-s"
397     {fix y
398       from s[rule_format, of "-y"] have
399     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
400         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
401     note s1 = this[unfolded minus_minus]
402     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
403       by auto
404     {fix n::nat
405       from s1[rule_format, of "?m + 1/real (Suc n)"]
406       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
407         by simp}
408     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
409     from choice[OF th] obtain g where
410       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
411       by blast
412     from bolzano_weierstrass_complex_disc[OF g(1)]
413     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
414       by blast
415     {fix w
416       assume wr: "cmod w \<le> r"
417       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
418       {assume e: "?e > 0"
419         hence e2: "?e/2 > 0" by simp
420         from poly_cont[OF e2, of z p] obtain d where
421           d: "d>0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2" by blast
422         {fix w assume w: "cmod (w - z) < d"
423           have "cmod(poly p w - poly p z) < ?e / 2"
424             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
425         note th1 = this
427         from fz(2)[rule_format, OF d(1)] obtain N1 where
428           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
429         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
430           N2: "2/?e < real N2" by blast
431         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
432           using N1[rule_format, of "N1 + N2"] th1 by simp
433         {fix a b e2 m :: real
434         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
435           ==> False" by arith}
436       note th0 = this
437       have ath:
438         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
439       from s1m[OF g(1)[rule_format]]
440       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
441       from seq_suble[OF fz(1), of "N1+N2"]
442       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
443       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
444         using N2 by auto
445       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
446       from g(2)[rule_format, of "f (N1 + N2)"]
447       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
448       from order_less_le_trans[OF th01 th00]
449       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
450       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
451       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
452       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
453       with ath[OF th31 th32]
454       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
455       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
456         by arith
457       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
458 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
459         by (simp add: norm_triangle_ineq3)
460       from ath2[OF th22, of ?m]
461       have thc2: "2*(?e/2) \<le> \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)" by simp
462       from th0[OF th2 thc1 thc2] have False .}
463       hence "?e = 0" by auto
464       then have "cmod (poly p z) = ?m" by simp
465       with s1m[OF wr]
466       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
467     hence ?thesis by blast}
468   ultimately show ?thesis by blast
469 qed
471 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
472   unfolding power2_eq_square
473   apply (simp add: rcis_mult)
474   apply (simp add: power2_eq_square[symmetric])
475   done
477 lemma cispi: "cis pi = -1"
478   unfolding cis_def
479   by simp
481 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
482   unfolding power2_eq_square
484   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
485   done
487 text {* Nonzero polynomial in z goes to infinity as z does. *}
489 lemma poly_infinity:
490   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
491   assumes ex: "p \<noteq> 0"
492   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
493 using ex
494 proof(induct p arbitrary: a d)
495   case (pCons c cs a d)
496   {assume H: "cs \<noteq> 0"
497     with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)" by blast
498     let ?r = "1 + \<bar>r\<bar>"
499     {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
500       have r0: "r \<le> norm z" using h by arith
501       from r[rule_format, OF r0]
502       have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
503       from h have z1: "norm z \<ge> 1" by arith
504       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
505       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
506         unfolding norm_mult by (simp add: algebra_simps)
507       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
508       have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
509         by (simp add: algebra_simps)
510       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"  by arith}
511     hence ?case by blast}
512   moreover
513   {assume cs0: "\<not> (cs \<noteq> 0)"
514     with pCons.prems have c0: "c \<noteq> 0" by simp
515     from cs0 have cs0': "cs = 0" by simp
516     {fix z::'a
517       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
518       from c0 have "norm c > 0" by simp
519       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
520         by (simp add: field_simps norm_mult)
521       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
522       from norm_diff_ineq[of "z * c" a ]
523       have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
524         by (simp add: algebra_simps)
525       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
526         using cs0' by simp}
527     then have ?case  by blast}
528   ultimately show ?case by blast
529 qed simp
531 text {* Hence polynomial's modulus attains its minimum somewhere. *}
532 lemma poly_minimum_modulus:
533   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
534 proof(induct p)
535   case (pCons c cs)
536   {assume cs0: "cs \<noteq> 0"
537     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
538     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)" by blast
539     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
540     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
541     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)" by blast
542     {fix z assume z: "r \<le> cmod z"
543       from v[of 0] r[OF z]
544       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
545         by simp }
546     note v0 = this
547     from v0 v ath[of r] have ?case by blast}
548   moreover
549   {assume cs0: "\<not> (cs \<noteq> 0)"
550     hence th:"cs = 0" by simp
551     from th pCons.hyps have ?case by simp}
552   ultimately show ?case by blast
553 qed simp
555 text{* Constant function (non-syntactic characterization). *}
556 definition "constant f = (\<forall>x y. f x = f y)"
558 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
559   unfolding constant_def psize_def
560   apply (induct p, auto)
561   done
563 lemma poly_replicate_append:
564   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
565   by (simp add: poly_monom)
567 text {* Decomposition of polynomial, skipping zero coefficients
568   after the first.  *}
570 lemma poly_decompose_lemma:
571  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
572   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
573                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
574 unfolding psize_def
575 using nz
576 proof(induct p)
577   case 0 thus ?case by simp
578 next
579   case (pCons c cs)
580   {assume c0: "c = 0"
581     from pCons.hyps pCons.prems c0 have ?case
582       apply (auto)
583       apply (rule_tac x="k+1" in exI)
584       apply (rule_tac x="a" in exI, clarsimp)
585       apply (rule_tac x="q" in exI)
586       by (auto)}
587   moreover
588   {assume c0: "c\<noteq>0"
589     have ?case
590       apply (rule exI[where x=0])
591       apply (rule exI[where x=c], auto simp add: c0)
592       done}
593   ultimately show ?case by blast
594 qed
596 lemma poly_decompose:
597   assumes nc: "~constant(poly p)"
598   shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
599                psize q + k + 1 = psize p \<and>
600               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
601 using nc
602 proof(induct p)
603   case 0 thus ?case by (simp add: constant_def)
604 next
605   case (pCons c cs)
606   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
607     {fix x y
608       from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
609     with pCons.prems have False by (auto simp add: constant_def)}
610   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
611   from poly_decompose_lemma[OF th]
612   show ?case
613     apply clarsimp
614     apply (rule_tac x="k+1" in exI)
615     apply (rule_tac x="a" in exI)
616     apply simp
617     apply (rule_tac x="q" in exI)
618     apply (auto simp add: psize_def split: if_splits)
619     done
620 qed
622 text{* Fundamental theorem of algebra *}
624 lemma fundamental_theorem_of_algebra:
625   assumes nc: "~constant(poly p)"
626   shows "\<exists>z::complex. poly p z = 0"
627 using nc
628 proof(induct "psize p" arbitrary: p rule: less_induct)
629   case less
630   let ?p = "poly p"
631   let ?ths = "\<exists>z. ?p z = 0"
633   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
634   from poly_minimum_modulus obtain c where
635     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
636   {assume pc: "?p c = 0" hence ?ths by blast}
637   moreover
638   {assume pc0: "?p c \<noteq> 0"
639     from poly_offset[of p c] obtain q where
640       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
641     {assume h: "constant (poly q)"
642       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
643       {fix x y
644         from th have "?p x = poly q (x - c)" by auto
645         also have "\<dots> = poly q (y - c)"
646           using h unfolding constant_def by blast
647         also have "\<dots> = ?p y" using th by auto
648         finally have "?p x = ?p y" .}
649       with less(2) have False unfolding constant_def by blast }
650     hence qnc: "\<not> constant (poly q)" by blast
651     from q(2) have pqc0: "?p c = poly q 0" by simp
652     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
653     let ?a0 = "poly q 0"
654     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
655     from a00
656     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
657       by simp
658     let ?r = "smult (inverse ?a0) q"
659     have lgqr: "psize q = psize ?r"
660       using a00 unfolding psize_def degree_def
661       by (simp add: poly_eq_iff)
662     {assume h: "\<And>x y. poly ?r x = poly ?r y"
663       {fix x y
664         from qr[rule_format, of x]
665         have "poly q x = poly ?r x * ?a0" by auto
666         also have "\<dots> = poly ?r y * ?a0" using h by simp
667         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
668         finally have "poly q x = poly q y" .}
669       with qnc have False unfolding constant_def by blast}
670     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
671     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
672     {fix w
673       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
674         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
675       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
676         using a00 unfolding norm_divide by (simp add: field_simps)
677       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
678     note mrmq_eq = this
679     from poly_decompose[OF rnc] obtain k a s where
680       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
681       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
682     {assume "psize p = k + 1"
683       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
684       {fix w
685         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
686           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
687       note hth = this [symmetric]
688         from reduce_poly_simple[OF kas(1,2)]
689       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
690     moreover
691     {assume kn: "psize p \<noteq> k+1"
692       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
693       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
694         unfolding constant_def poly_pCons poly_monom
695         using kas(1) apply simp
696         by (rule exI[where x=0], rule exI[where x=1], simp)
697       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
698         by (simp add: psize_def degree_monom_eq)
699       from less(1) [OF k1n [simplified th02] th01]
700       obtain w where w: "1 + w^k * a = 0"
701         unfolding poly_pCons poly_monom
702         using kas(2) by (cases k, auto simp add: algebra_simps)
703       from poly_bound_exists[of "cmod w" s] obtain m where
704         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
705       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
706       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
707       then have wm1: "w^k * a = - 1" by simp
708       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
709         using norm_ge_zero[of w] w0 m(1)
710           by (simp add: inverse_eq_divide zero_less_mult_iff)
711       with real_lbound_gt_zero[OF zero_less_one] obtain t where
712         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
713       let ?ct = "complex_of_real t"
714       let ?w = "?ct * w"
715       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w" using kas(1) by (simp add: algebra_simps power_mult_distrib)
716       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
717         unfolding wm1 by (simp)
718       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
719         by metis
720       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
721       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)" unfolding norm_of_real by simp
722       have ath: "\<And>x (t::real). 0\<le> x \<Longrightarrow> x < t \<Longrightarrow> t\<le>1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1" by arith
723       have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
724       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
725       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
726         by (simp add: inverse_eq_divide field_simps)
727       with zero_less_power[OF t(1), of k]
728       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
729         by (metis comm_mult_strict_left_mono)
730       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
731         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
732       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
733         using t(1,2) m(2)[rule_format, OF tw] w0
734         by auto
735       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
736       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
737         by auto
738       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
739       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
740       from th11 th12
741       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
742       then have "cmod (poly ?r ?w) < 1"
743         unfolding kas(4)[rule_format, of ?w] r01 by simp
744       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
745     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
746     from cr0_contr cq0 q(2)
747     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
748   ultimately show ?ths by blast
749 qed
751 text {* Alternative version with a syntactic notion of constant polynomial. *}
753 lemma fundamental_theorem_of_algebra_alt:
754   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
755   shows "\<exists>z. poly p z = (0::complex)"
756 using nc
757 proof(induct p)
758   case (pCons c cs)
759   {assume "c=0" hence ?case by auto}
760   moreover
761   {assume c0: "c\<noteq>0"
762     {assume nc: "constant (poly (pCons c cs))"
763       from nc[unfolded constant_def, rule_format, of 0]
764       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
765       hence "cs = 0"
766         proof(induct cs)
767           case (pCons d ds)
768           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
769           moreover
770           {assume d0: "d\<noteq>0"
771             from poly_bound_exists[of 1 ds] obtain m where
772               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
773             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
774             from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
775               x: "x > 0" "x < cmod d / m" "x < 1" by blast
776             let ?x = "complex_of_real x"
777             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
778             from pCons.prems[rule_format, OF cx(1)]
779             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
780             from m(2)[rule_format, OF cx(2)] x(1)
781             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
782               by (simp add: norm_mult)
783             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
784             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
785             with cth  have ?case by blast}
786           ultimately show ?case by blast
787         qed simp}
788       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
789         by blast
790       from fundamental_theorem_of_algebra[OF nc] have ?case .}
791   ultimately show ?case by blast
792 qed simp
795 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
797 lemma nullstellensatz_lemma:
798   fixes p :: "complex poly"
799   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
800   and "degree p = n" and "n \<noteq> 0"
801   shows "p dvd (q ^ n)"
802 using assms
803 proof(induct n arbitrary: p q rule: nat_less_induct)
804   fix n::nat fix p q :: "complex poly"
805   assume IH: "\<forall>m<n. \<forall>p q.
806                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
807                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
808     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
809     and dpn: "degree p = n" and n0: "n \<noteq> 0"
810   from dpn n0 have pne: "p \<noteq> 0" by auto
811   let ?ths = "p dvd (q ^ n)"
812   {fix a assume a: "poly p a = 0"
813     {assume oa: "order a p \<noteq> 0"
814       let ?op = "order a p"
815       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
816         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
817       note oop = order_degree[OF pne, unfolded dpn]
818       {assume q0: "q = 0"
819         hence ?ths using n0
820           by (simp add: power_0_left)}
821       moreover
822       {assume q0: "q \<noteq> 0"
823         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
824         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
825         from ap(1) obtain s where
826           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
827         have sne: "s \<noteq> 0"
828           using s pne by auto
829         {assume ds0: "degree s = 0"
830           from ds0 obtain k where kpn: "s = [:k:]"
831             by (cases s) (auto split: if_splits)
832           from sne kpn have k: "k \<noteq> 0" by simp
833           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
834           have "q ^ n = p * ?w"
835             apply (subst r, subst s, subst kpn)
836             using k oop [of a]
837             apply (subst power_mult_distrib, simp)
838             apply (subst power_add [symmetric], simp)
839             done
840           hence ?ths unfolding dvd_def by blast}
841         moreover
842         {assume ds0: "degree s \<noteq> 0"
843           from ds0 sne dpn s oa
844             have dsn: "degree s < n" apply auto
845               apply (erule ssubst)
846               apply (simp add: degree_mult_eq degree_linear_power)
847               done
848             {fix x assume h: "poly s x = 0"
849               {assume xa: "x = a"
850                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
851                   u: "s = [:- a, 1:] * u" by (rule dvdE)
852                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
853                   by (subst s, subst u, simp only: power_Suc mult_ac)
854                 with ap(2)[unfolded dvd_def] have False by blast}
855               note xa = this
856               from h have "poly p x = 0" by (subst s, simp)
857               with pq0 have "poly q x = 0" by blast
858               with r xa have "poly r x = 0"
859                 by auto}
860             note impth = this
861             from IH[rule_format, OF dsn, of s r] impth ds0
862             have "s dvd (r ^ (degree s))" by blast
863             then obtain u where u: "r ^ (degree s) = s * u" ..
864             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
865               by (simp only: poly_mult[symmetric] poly_power[symmetric])
866             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
867             from oop[of a] dsn have "q ^ n = p * ?w"
868               apply -
869               apply (subst s, subst r)
870               apply (simp only: power_mult_distrib)
871               apply (subst mult_assoc [where b=s])
872               apply (subst mult_assoc [where a=u])
873               apply (subst mult_assoc [where b=u, symmetric])
874               apply (subst u [symmetric])
875               apply (simp add: mult_ac power_add [symmetric])
876               done
877             hence ?ths unfolding dvd_def by blast}
878       ultimately have ?ths by blast }
879       ultimately have ?ths by blast}
880     then have ?ths using a order_root pne by blast}
881   moreover
882   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
883     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
884       ccs: "c\<noteq>0" "p = pCons c 0" by blast
886     then have pp: "\<And>x. poly p x =  c" by simp
887     let ?w = "[:1/c:] * (q ^ n)"
888     from ccs have "(q ^ n) = (p * ?w)" by simp
889     hence ?ths unfolding dvd_def by blast}
890   ultimately show ?ths by blast
891 qed
893 lemma nullstellensatz_univariate:
894   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
895     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
896 proof-
897   {assume pe: "p = 0"
898     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
899       by (auto simp add: poly_all_0_iff_0)
900     {assume "p dvd (q ^ (degree p))"
901       then obtain r where r: "q ^ (degree p) = p * r" ..
902       from r pe have False by simp}
903     with eq pe have ?thesis by blast}
904   moreover
905   {assume pe: "p \<noteq> 0"
906     {assume dp: "degree p = 0"
907       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
908         by (cases p) (simp split: if_splits)
909       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
910       from k dp have "q ^ (degree p) = p * [:1/k:]"
911         by (simp add: one_poly_def)
912       hence th2: "p dvd (q ^ (degree p))" ..
913       from th1 th2 pe have ?thesis by blast}
914     moreover
915     {assume dp: "degree p \<noteq> 0"
916       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
917       {assume "p dvd (q ^ (Suc n))"
918         then obtain u where u: "q ^ (Suc n) = p * u" ..
919         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
920           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
921           hence False using u h(1) by (simp only: poly_mult) simp}}
922         with n nullstellensatz_lemma[of p q "degree p"] dp
923         have ?thesis by auto}
924     ultimately have ?thesis by blast}
925   ultimately show ?thesis by blast
926 qed
928 text{* Useful lemma *}
930 lemma constant_degree:
931   fixes p :: "'a::{idom,ring_char_0} poly"
932   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
933 proof
934   assume l: ?lhs
935   from l[unfolded constant_def, rule_format, of _ "0"]
936   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
937   then have "p = [:poly p 0:]" by (simp add: poly_eq_poly_eq_iff)
938   then have "degree p = degree [:poly p 0:]" by simp
939   then show ?rhs by simp
940 next
941   assume r: ?rhs
942   then obtain k where "p = [:k:]"
943     by (cases p) (simp split: if_splits)
944   then show ?lhs unfolding constant_def by auto
945 qed
947 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
948   shows "degree p \<le> degree q \<or> q = 0"
949 by (metis dvd_imp_degree_le pq)
951 (* Arithmetic operations on multivariate polynomials.                        *)
953 lemma mpoly_base_conv:
954   fixes x :: "'a::comm_ring_1"
955   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
956   by simp_all
958 lemma mpoly_norm_conv:
959   fixes x :: "'a::comm_ring_1"
960   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x" by simp_all
962 lemma mpoly_sub_conv:
963   fixes x :: "'a::comm_ring_1"
964   shows "poly p x - poly q x = poly p x + -1 * poly q x"
965   by simp
967 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
969 lemma poly_cancel_eq_conv:
970   fixes x :: "'a::field"
971   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
972   by auto
975   fixes p:: "('a::comm_ring_1) poly"
976   assumes pq: "p dvd q"
977 shows "p dvd (pCons 0 q)"
978 proof-
979   have "pCons 0 q = q * [:0,1:]" by simp
980   then have "q dvd (pCons 0 q)" ..
981   with pq show ?thesis by (rule dvd_trans)
982 qed
984 lemma poly_divides_conv0:
985   fixes p:: "('a::field) poly"
986   assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
987   shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
988 proof-
989   {assume r: ?rhs
990     hence "q = p * 0" by simp
991     hence ?lhs ..}
992   moreover
993   {assume l: ?lhs
994     {assume q0: "q = 0"
995       hence ?rhs by simp}
996     moreover
997     {assume q0: "q \<noteq> 0"
998       from l q0 have "degree p \<le> degree q"
999         by (rule dvd_imp_degree_le)
1000       with lgpq have ?rhs by simp }
1001     ultimately have ?rhs by blast }
1002   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
1003 qed
1005 lemma poly_divides_conv1:
1006   fixes p:: "('a::field) poly"
1007   assumes a0: "a\<noteq> 0" and pp': "p dvd p'"
1008   and qrp': "smult a q - p' \<equiv> r"
1009   shows "p dvd q \<equiv> p dvd r" (is "?lhs \<equiv> ?rhs")
1010 proof-
1011   {
1012   from pp' obtain t where t: "p' = p * t" ..
1013   {assume l: ?lhs
1014     then obtain u where u: "q = p * u" ..
1015      have "r = p * (smult a u - t)"
1016        using u qrp' [symmetric] t by (simp add: algebra_simps)
1017      then have ?rhs ..}
1018   moreover
1019   {assume r: ?rhs
1020     then obtain u where u: "r = p * u" ..
1021     from u [symmetric] t qrp' [symmetric] a0
1022     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1023     hence ?lhs ..}
1024   ultimately have "?lhs = ?rhs" by blast }
1025 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
1026 qed
1028 lemma basic_cqe_conv1:
1029   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1030   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1031   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c\<noteq>0"
1032   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
1033   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0" by simp_all
1035 lemma basic_cqe_conv2:
1036   assumes l:"p \<noteq> 0"
1037   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
1038 proof-
1039   {fix h t
1040     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
1041     with l have False by simp}
1042   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
1043     by blast
1044   from fundamental_theorem_of_algebra_alt[OF th]
1045   show ?thesis by auto
1046 qed
1048 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> (p \<noteq> 0)"
1049 by (metis poly_all_0_iff_0)
1051 lemma basic_cqe_conv3:
1052   fixes p q :: "complex poly"
1053   assumes l: "p \<noteq> 0"
1054   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ (psize p)))"
1055 proof -
1056   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
1057   from nullstellensatz_univariate[of "pCons a p" q] l
1058   show ?thesis
1059     by (metis dp pCons_eq_0_iff)
1060 qed
1062 lemma basic_cqe_conv4:
1063   fixes p q :: "complex poly"
1064   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
1065   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1066 proof-
1067   from h have "poly (q ^ n) = poly r" by auto
1068   then have "(q ^ n) = r" by (simp add: poly_eq_poly_eq_iff)
1069   thus "p dvd (q ^ n) \<longleftrightarrow> p dvd r" by simp
1070 qed
1072 lemma poly_const_conv:
1073   fixes x :: "'a::comm_ring_1"
1074   shows "poly [:c:] x = y \<longleftrightarrow> c = y" by simp
1076 end