src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author wenzelm Mon Apr 28 17:48:59 2014 +0200 (2014-04-28) changeset 56776 309e1a61ee7c parent 56544 b60d5d119489 child 56778 cb0929421ca6 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   {assume y0: "y = 0"
23     {assume x0: "x \<ge> 0"
24       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
25         by (simp add: csqrt_def power2_eq_square)}
26     moreover
27     {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
28       then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
29         by (simp add: csqrt_def power2_eq_square) }
30     ultimately have ?thesis by blast}
31   moreover
32   {assume y0: "y\<noteq>0"
33     {fix x y
34       let ?z = "Complex x y"
35       from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
36       hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
37       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) }
38     note th = this
39     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
40       by (simp add: power2_eq_square)
41     from th[of x y]
42     have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
43       "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
44       unfolding sq4 by simp_all
45     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"
46       unfolding power2_eq_square by simp
47     have "sqrt 4 = sqrt (2\<^sup>2)" by simp
48     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
49     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
50       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
51       unfolding power2_eq_square
52       by (simp add: algebra_simps real_sqrt_divide sqrt4)
53      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
54        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])
55       using th1 th2  ..}
56   ultimately show ?thesis by blast
57 qed
59 lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
60   by (simp add: csqrt_def)
62 lemma csqrt_0 [simp]: "csqrt 0 = 0"
63   by (simp add: csqrt_def)
65 lemma csqrt_1 [simp]: "csqrt 1 = 1"
66   by (simp add: csqrt_def)
68 lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
69 proof (cases z)
70   case (Complex x y)
71   then show ?thesis
72     using real_sqrt_sum_squares_ge1 [of "x" y]
73           real_sqrt_sum_squares_ge1 [of "-x" y]
74           real_sqrt_sum_squares_eq_cancel [of x y]
75     apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
77     by (metis add_commute less_eq_real_def power_minus_Bit0 real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
78 qed
80 lemma Re_csqrt: "0 \<le> Re(csqrt z)"
81   by (metis csqrt_principal le_less)
83 lemma csqrt_square: "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> csqrt (z^2) = z"
84   using csqrt [of "z^2"] csqrt_principal [of "z^2"]
85   by (cases z) (auto simp: power2_eq_iff)
87 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
88   by auto (metis csqrt power_eq_0_iff)
90 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
91   by auto (metis csqrt power2_eq_1_iff)
93 subsection{* More lemmas about module of complex numbers *}
95 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
96   by (rule of_real_power [symmetric])
98 text{* The triangle inequality for cmod *}
99 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
100   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
102 subsection{* Basic lemmas about polynomials *}
104 lemma poly_bound_exists:
105   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
106   shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z <= r \<longrightarrow> norm (poly p z) \<le> m)"
107 proof(induct p)
108   case 0 thus ?case by (rule exI[where x=1], simp)
109 next
110   case (pCons c cs)
111   from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
112     by blast
113   let ?k = " 1 + norm c + \<bar>r * m\<bar>"
114   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
115   {fix z :: 'a
116     assume H: "norm z \<le> r"
117     from m H have th: "norm (poly cs z) \<le> m" by blast
118     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
119     have "norm (poly (pCons c cs) z) \<le> norm c + norm (z* poly cs z)"
120       using norm_triangle_ineq[of c "z* poly cs z"] by simp
121     also have "\<dots> \<le> norm c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
122       by (simp add: norm_mult)
123     also have "\<dots> \<le> ?k" by simp
124     finally have "norm (poly (pCons c cs) z) \<le> ?k" .}
125   with kp show ?case by blast
126 qed
129 text{* Offsetting the variable in a polynomial gives another of same degree *}
131 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
132 where
133   "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
135 lemma offset_poly_0: "offset_poly 0 h = 0"
136   by (simp add: offset_poly_def)
138 lemma offset_poly_pCons:
139   "offset_poly (pCons a p) h =
140     smult h (offset_poly p h) + pCons a (offset_poly p h)"
141   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
143 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
144 by (simp add: offset_poly_pCons offset_poly_0)
146 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
147 apply (induct p)
148 apply (simp add: offset_poly_0)
149 apply (simp add: offset_poly_pCons algebra_simps)
150 done
152 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
153 by (induct p arbitrary: a, simp, force)
155 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
156 apply (safe intro!: offset_poly_0)
157 apply (induct p, simp)
158 apply (simp add: offset_poly_pCons)
159 apply (frule offset_poly_eq_0_lemma, simp)
160 done
162 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
163 apply (induct p)
164 apply (simp add: offset_poly_0)
165 apply (case_tac "p = 0")
166 apply (simp add: offset_poly_0 offset_poly_pCons)
167 apply (simp add: offset_poly_pCons)
168 apply (subst degree_add_eq_right)
169 apply (rule le_less_trans [OF degree_smult_le])
170 apply (simp add: offset_poly_eq_0_iff)
171 apply (simp add: offset_poly_eq_0_iff)
172 done
174 definition
175   "psize p = (if p = 0 then 0 else Suc (degree p))"
177 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
178   unfolding psize_def by simp
180 lemma poly_offset:
181   fixes p:: "('a::comm_ring_1) poly"
182   shows "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
183 proof (intro exI conjI)
184   show "psize (offset_poly p a) = psize p"
185     unfolding psize_def
186     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
187   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
188     by (simp add: poly_offset_poly)
189 qed
191 text{* An alternative useful formulation of completeness of the reals *}
192 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
193   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
194 proof
195   from bz have "bdd_above (Collect P)"
196     by (force intro: less_imp_le)
197   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
198     using ex bz by (subst less_cSup_iff) auto
199 qed
201 subsection {* Fundamental theorem of algebra *}
202 lemma  unimodular_reduce_norm:
203   assumes md: "cmod z = 1"
204   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
205 proof-
206   obtain x y where z: "z = Complex x y " by (cases z, auto)
207   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
208   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
209     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
210       by (simp_all add: cmod_def power2_eq_square algebra_simps)
211     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
212     hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
213       by - (rule power_mono, simp, simp)+
214     hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
215       by (simp_all add: power_mult_distrib)
216     from add_mono[OF th0] xy have False by simp }
217   thus ?thesis unfolding linorder_not_le[symmetric] by blast
218 qed
220 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
221 lemma reduce_poly_simple:
222  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
223   shows "\<exists>z. cmod (1 + b * z^n) < 1"
224 using n
225 proof(induct n rule: nat_less_induct)
226   fix n
227   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
228   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
229   {assume e: "even n"
230     hence "\<exists>m. n = 2*m" by presburger
231     then obtain m where m: "n = 2*m" by blast
232     from n m have "m\<noteq>0" "m < n" by presburger+
233     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
234     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
235     hence "\<exists>z. ?P z n" ..}
236   moreover
237   {assume o: "odd n"
238     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
239       using b by (simp add: norm_divide)
240     from o have "\<exists>m. n = Suc (2*m)" by presburger+
241     then obtain m where m: "n = Suc (2*m)" by blast
242     from unimodular_reduce_norm[OF th0] o
243     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
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 - 1) < 1", rule_tac x="-1" in exI, simp)
246       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
247       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
248       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
249       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
250       apply (auto simp add: m power_mult)
251       apply (rule_tac x="ii" in exI)
252       apply (auto simp add: m power_mult)
253       done
254     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
255     let ?w = "v / complex_of_real (root n (cmod b))"
256     from odd_real_root_pow[OF o, of "cmod b"]
257     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
258       by (simp add: power_divide complex_of_real_power)
259     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
260     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
261     have th4: "cmod (complex_of_real (cmod b) / b) *
262    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
263    < cmod (complex_of_real (cmod b) / b) * 1"
264       apply (simp only: norm_mult[symmetric] distrib_left)
265       using b v by (simp add: th2)
267     from mult_less_imp_less_left[OF th4 th3]
268     have "?P ?w n" unfolding th1 .
269     hence "\<exists>z. ?P z n" .. }
270   ultimately show "\<exists>z. ?P z n" by blast
271 qed
273 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
275 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
276   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
277   unfolding cmod_def by simp
279 lemma bolzano_weierstrass_complex_disc:
280   assumes r: "\<forall>n. cmod (s n) \<le> r"
281   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
282 proof-
283   from seq_monosub[of "Re o s"]
284   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
285     unfolding o_def by blast
286   from seq_monosub[of "Im o s o f"]
287   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
288   let ?h = "f o g"
289   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
290   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
291   proof
292     fix n
293     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
294   qed
295   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
296     apply (rule Bseq_monoseq_convergent)
297     apply (simp add: Bseq_def)
298     apply (metis gt_ex le_less_linear less_trans order.trans th)
299     using f(2) .
300   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
301   proof
302     fix n
303     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
304   qed
306   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
307     apply (rule Bseq_monoseq_convergent)
308     apply (simp add: Bseq_def)
309     apply (metis gt_ex le_less_linear less_trans order.trans th)
310     using g(2) .
312   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
313     by blast
314   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
315     unfolding LIMSEQ_iff real_norm_def .
317   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
318     by blast
319   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
320     unfolding LIMSEQ_iff real_norm_def .
321   let ?w = "Complex x y"
322   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
323   {fix e assume ep: "e > (0::real)"
324     hence e2: "e/2 > 0" by simp
325     from x[rule_format, OF e2] y[rule_format, OF e2]
326     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
327     {fix n assume nN12: "n \<ge> N1 + N2"
328       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
329       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
330       have "cmod (s (?h n) - ?w) < e"
331         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
332     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
333   with hs show ?thesis  by blast
334 qed
336 text{* Polynomial is continuous. *}
338 lemma poly_cont:
339   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
340   assumes ep: "e > 0"
341   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
342 proof-
343   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
344   proof
345     show "degree (offset_poly p z) = degree p"
346       by (rule degree_offset_poly)
347     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
348       by (rule poly_offset_poly)
349   qed
350   {fix w
351     note q(2)[of "w - z", simplified]}
352   note th = this
353   show ?thesis unfolding th[symmetric]
354   proof(induct q)
355     case 0 thus ?case  using ep by auto
356   next
357     case (pCons c cs)
358     from poly_bound_exists[of 1 "cs"]
359     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m" by blast
360     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
361     have one0: "1 > (0::real)"  by arith
362     from real_lbound_gt_zero[OF one0 em0]
363     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
364     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
365       by (simp_all add: field_simps)
366     show ?case
367       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
368         fix d w
369         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "norm (w-z) < d"
370         hence d1: "norm (w-z) \<le> 1" "d \<ge> 0" by simp_all
371         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
372         from H have th: "norm (w-z) \<le> d" by simp
373         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
374         show "norm (w - z) * norm (poly cs (w - z)) < e" by simp
375       qed
376     qed
377 qed
379 text{* Hence a polynomial attains minimum on a closed disc
380   in the complex plane. *}
381 lemma  poly_minimum_modulus_disc:
382   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
383 proof-
384   {assume "\<not> r \<ge> 0" hence ?thesis
385     by (metis norm_ge_zero order.trans)}
386   moreover
387   {assume rp: "r \<ge> 0"
388     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
389     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
390     {fix x z
391       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
392       hence "- x < 0 " by arith
393       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
394     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
395     from real_sup_exists[OF mth1 mth2] obtain s where
396       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
397     let ?m = "-s"
398     {fix y
399       from s[rule_format, of "-y"] have
400     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
401         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
402     note s1 = this[unfolded minus_minus]
403     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
404       by auto
405     {fix n::nat
406       from s1[rule_format, of "?m + 1/real (Suc n)"]
407       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
408         by simp}
409     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
410     from choice[OF th] obtain g where
411       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
412       by blast
413     from bolzano_weierstrass_complex_disc[OF g(1)]
414     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
415       by blast
416     {fix w
417       assume wr: "cmod w \<le> r"
418       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
419       {assume e: "?e > 0"
420         hence e2: "?e/2 > 0" by simp
421         from poly_cont[OF e2, of z p] obtain d where
422           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
423         {fix w assume w: "cmod (w - z) < d"
424           have "cmod(poly p w - poly p z) < ?e / 2"
425             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
426         note th1 = this
428         from fz(2)[rule_format, OF d(1)] obtain N1 where
429           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
430         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
431           N2: "2/?e < real N2" by blast
432         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
433           using N1[rule_format, of "N1 + N2"] th1 by simp
434         {fix a b e2 m :: real
435         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
436           ==> False" by arith}
437       note th0 = this
438       have ath:
439         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
440       from s1m[OF g(1)[rule_format]]
441       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
442       from seq_suble[OF fz(1), of "N1+N2"]
443       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
444       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
445         using N2 by auto
446       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
447       from g(2)[rule_format, of "f (N1 + N2)"]
448       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
449       from order_less_le_trans[OF th01 th00]
450       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
451       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
452       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
453       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
454       with ath[OF th31 th32]
455       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
456       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
457         by arith
458       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
459 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
460         by (simp add: norm_triangle_ineq3)
461       from ath2[OF th22, of ?m]
462       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
463       from th0[OF th2 thc1 thc2] have False .}
464       hence "?e = 0" by auto
465       then have "cmod (poly p z) = ?m" by simp
466       with s1m[OF wr]
467       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
468     hence ?thesis by blast}
469   ultimately show ?thesis by blast
470 qed
472 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
473   unfolding power2_eq_square
474   apply (simp add: rcis_mult)
475   apply (simp add: power2_eq_square[symmetric])
476   done
478 lemma cispi: "cis pi = -1"
479   unfolding cis_def
480   by simp
482 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
483   unfolding power2_eq_square
485   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
486   done
488 text {* Nonzero polynomial in z goes to infinity as z does. *}
490 lemma poly_infinity:
491   fixes p:: "('a::{comm_semiring_0,real_normed_div_algebra}) poly"
492   assumes ex: "p \<noteq> 0"
493   shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
494 using ex
495 proof(induct p arbitrary: a d)
496   case (pCons c cs a d)
497   {assume H: "cs \<noteq> 0"
498     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
499     let ?r = "1 + \<bar>r\<bar>"
500     {fix z::'a assume h: "1 + \<bar>r\<bar> \<le> norm z"
501       have r0: "r \<le> norm z" using h by arith
502       from r[rule_format, OF r0]
503       have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)" by arith
504       from h have z1: "norm z \<ge> 1" by arith
505       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
506       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
507         unfolding norm_mult by (simp add: algebra_simps)
508       from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
509       have th2: "norm(z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
510         by (simp add: algebra_simps)
511       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"  by arith}
512     hence ?case by blast}
513   moreover
514   {assume cs0: "\<not> (cs \<noteq> 0)"
515     with pCons.prems have c0: "c \<noteq> 0" by simp
516     from cs0 have cs0': "cs = 0" by simp
517     {fix z::'a
518       assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
519       from c0 have "norm c > 0" by simp
520       from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
521         by (simp add: field_simps norm_mult)
522       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
523       from norm_diff_ineq[of "z * c" a ]
524       have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
525         by (simp add: algebra_simps)
526       from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
527         using cs0' by simp}
528     then have ?case  by blast}
529   ultimately show ?case by blast
530 qed simp
532 text {* Hence polynomial's modulus attains its minimum somewhere. *}
533 lemma poly_minimum_modulus:
534   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
535 proof(induct p)
536   case (pCons c cs)
537   {assume cs0: "cs \<noteq> 0"
538     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
539     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
540     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
541     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
542     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
543     {fix z assume z: "r \<le> cmod z"
544       from v[of 0] r[OF z]
545       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
546         by simp }
547     note v0 = this
548     from v0 v ath[of r] have ?case by blast}
549   moreover
550   {assume cs0: "\<not> (cs \<noteq> 0)"
551     hence th:"cs = 0" by simp
552     from th pCons.hyps have ?case by simp}
553   ultimately show ?case by blast
554 qed simp
556 text{* Constant function (non-syntactic characterization). *}
557 definition "constant f = (\<forall>x y. f x = f y)"
559 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
560   unfolding constant_def psize_def
561   apply (induct p, auto)
562   done
564 lemma poly_replicate_append:
565   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
566   by (simp add: poly_monom)
568 text {* Decomposition of polynomial, skipping zero coefficients
569   after the first.  *}
571 lemma poly_decompose_lemma:
572  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
573   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
574                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
575 unfolding psize_def
576 using nz
577 proof(induct p)
578   case 0 thus ?case by simp
579 next
580   case (pCons c cs)
581   {assume c0: "c = 0"
582     from pCons.hyps pCons.prems c0 have ?case
583       apply (auto)
584       apply (rule_tac x="k+1" in exI)
585       apply (rule_tac x="a" in exI, clarsimp)
586       apply (rule_tac x="q" in exI)
587       by (auto)}
588   moreover
589   {assume c0: "c\<noteq>0"
590     have ?case
591       apply (rule exI[where x=0])
592       apply (rule exI[where x=c], auto simp add: c0)
593       done}
594   ultimately show ?case by blast
595 qed
597 lemma poly_decompose:
598   assumes nc: "\<not> constant (poly p)"
599   shows "\<exists>k a q. a \<noteq> (0::'a::{idom}) \<and> k \<noteq> 0 \<and>
600                psize q + k + 1 = psize p \<and>
601               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
602   using nc
603 proof (induct p)
604   case 0
605   then show ?case
606     by (simp add: constant_def)
607 next
608   case (pCons c cs)
609   {
610     assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
611     {
612       fix x y
613       from C have "poly (pCons c cs) x = poly (pCons c cs) y"
614         by (cases "x = 0") auto
615     }
616     with pCons.prems have False by (auto simp add: constant_def)
617   }
618   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
619   from poly_decompose_lemma[OF th]
620   show ?case
621     apply clarsimp
622     apply (rule_tac x="k+1" in exI)
623     apply (rule_tac x="a" in exI)
624     apply simp
625     apply (rule_tac x="q" in exI)
626     apply (auto simp add: psize_def split: if_splits)
627     done
628 qed
630 text{* Fundamental theorem of algebra *}
632 lemma fundamental_theorem_of_algebra:
633   assumes nc: "\<not> constant (poly p)"
634   shows "\<exists>z::complex. poly p z = 0"
635   using nc
636 proof (induct "psize p" arbitrary: p rule: less_induct)
637   case less
638   let ?p = "poly p"
639   let ?ths = "\<exists>z. ?p z = 0"
641   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
642   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
643     by blast
644   {assume pc: "?p c = 0" hence ?ths by blast}
645   moreover
646   {assume pc0: "?p c \<noteq> 0"
647     from poly_offset[of p c] obtain q where
648       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
649     {assume h: "constant (poly q)"
650       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
651       {fix x y
652         from th have "?p x = poly q (x - c)" by auto
653         also have "\<dots> = poly q (y - c)"
654           using h unfolding constant_def by blast
655         also have "\<dots> = ?p y" using th by auto
656         finally have "?p x = ?p y" .}
657       with less(2) have False unfolding constant_def by blast }
658     hence qnc: "\<not> constant (poly q)" by blast
659     from q(2) have pqc0: "?p c = poly q 0" by simp
660     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
661     let ?a0 = "poly q 0"
662     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
663     from a00
664     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
665       by simp
666     let ?r = "smult (inverse ?a0) q"
667     have lgqr: "psize q = psize ?r"
668       using a00 unfolding psize_def degree_def
669       by (simp add: poly_eq_iff)
670     {assume h: "\<And>x y. poly ?r x = poly ?r y"
671       {fix x y
672         from qr[rule_format, of x]
673         have "poly q x = poly ?r x * ?a0" by auto
674         also have "\<dots> = poly ?r y * ?a0" using h by simp
675         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
676         finally have "poly q x = poly q y" .}
677       with qnc have False unfolding constant_def by blast}
678     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
679     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
680     {fix w
681       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
682         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
683       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
684         using a00 unfolding norm_divide by (simp add: field_simps)
685       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
686     note mrmq_eq = this
687     from poly_decompose[OF rnc] obtain k a s where
688       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
689       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
690     {assume "psize p = k + 1"
691       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
692       {fix w
693         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
694           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
695       note hth = this [symmetric]
696         from reduce_poly_simple[OF kas(1,2)]
697       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
698     moreover
699     {assume kn: "psize p \<noteq> k+1"
700       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
701       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
702         unfolding constant_def poly_pCons poly_monom
703         using kas(1) apply simp
704         by (rule exI[where x=0], rule exI[where x=1], simp)
705       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
706         by (simp add: psize_def degree_monom_eq)
707       from less(1) [OF k1n [simplified th02] th01]
708       obtain w where w: "1 + w^k * a = 0"
709         unfolding poly_pCons poly_monom
710         using kas(2) by (cases k, auto simp add: algebra_simps)
711       from poly_bound_exists[of "cmod w" s] obtain m where
712         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
713       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
714       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
715       then have wm1: "w^k * a = - 1" by simp
716       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
717         using norm_ge_zero[of w] w0 m(1)
718           by (simp add: inverse_eq_divide zero_less_mult_iff)
719       with real_lbound_gt_zero[OF zero_less_one] obtain t where
720         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
721       let ?ct = "complex_of_real t"
722       let ?w = "?ct * w"
723       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)
724       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
725         unfolding wm1 by (simp)
726       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
727         by metis
728       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
729       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
730       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
731       have "t * cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
732       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
733       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
734         by (simp add: inverse_eq_divide field_simps)
735       with zero_less_power[OF t(1), of k]
736       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
737         by (metis comm_mult_strict_left_mono)
738       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
739         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
740       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
741         using t(1,2) m(2)[rule_format, OF tw] w0
742         by auto
743       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
744       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
745         by auto
746       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
747       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
748       from th11 th12
749       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
750       then have "cmod (poly ?r ?w) < 1"
751         unfolding kas(4)[rule_format, of ?w] r01 by simp
752       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
753     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
754     from cr0_contr cq0 q(2)
755     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
756   ultimately show ?ths by blast
757 qed
759 text {* Alternative version with a syntactic notion of constant polynomial. *}
761 lemma fundamental_theorem_of_algebra_alt:
762   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
763   shows "\<exists>z. poly p z = (0::complex)"
764 using nc
765 proof(induct p)
766   case (pCons c cs)
767   {assume "c=0" hence ?case by auto}
768   moreover
769   {assume c0: "c\<noteq>0"
770     {assume nc: "constant (poly (pCons c cs))"
771       from nc[unfolded constant_def, rule_format, of 0]
772       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
773       hence "cs = 0"
774         proof(induct cs)
775           case (pCons d ds)
776           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
777           moreover
778           {assume d0: "d\<noteq>0"
779             from poly_bound_exists[of 1 ds] obtain m where
780               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
781             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
782             from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
783               x: "x > 0" "x < cmod d / m" "x < 1" by blast
784             let ?x = "complex_of_real x"
785             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
786             from pCons.prems[rule_format, OF cx(1)]
787             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
788             from m(2)[rule_format, OF cx(2)] x(1)
789             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
790               by (simp add: norm_mult)
791             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
792             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
793             with cth  have ?case by blast}
794           ultimately show ?case by blast
795         qed simp}
796       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
797         by blast
798       from fundamental_theorem_of_algebra[OF nc] have ?case .}
799   ultimately show ?case by blast
800 qed simp
803 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
805 lemma nullstellensatz_lemma:
806   fixes p :: "complex poly"
807   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
808     and "degree p = n"
809     and "n \<noteq> 0"
810   shows "p dvd (q ^ n)"
811   using assms
812 proof (induct n arbitrary: p q rule: nat_less_induct)
813   fix n :: nat
814   fix p q :: "complex poly"
815   assume IH: "\<forall>m<n. \<forall>p q.
816                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
817                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
818     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
819     and dpn: "degree p = n" and n0: "n \<noteq> 0"
820   from dpn n0 have pne: "p \<noteq> 0" by auto
821   let ?ths = "p dvd (q ^ n)"
822   {fix a assume a: "poly p a = 0"
823     {assume oa: "order a p \<noteq> 0"
824       let ?op = "order a p"
825       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
826         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
827       note oop = order_degree[OF pne, unfolded dpn]
828       {assume q0: "q = 0"
829         hence ?ths using n0
830           by (simp add: power_0_left)}
831       moreover
832       {assume q0: "q \<noteq> 0"
833         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
834         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
835         from ap(1) obtain s where
836           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
837         have sne: "s \<noteq> 0"
838           using s pne by auto
839         {assume ds0: "degree s = 0"
840           from ds0 obtain k where kpn: "s = [:k:]"
841             by (cases s) (auto split: if_splits)
842           from sne kpn have k: "k \<noteq> 0" by simp
843           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
844           have "q ^ n = p * ?w"
845             apply (subst r, subst s, subst kpn)
846             using k oop [of a]
847             apply (subst power_mult_distrib, simp)
848             apply (subst power_add [symmetric], simp)
849             done
850           hence ?ths unfolding dvd_def by blast}
851         moreover
852         {assume ds0: "degree s \<noteq> 0"
853           from ds0 sne dpn s oa
854             have dsn: "degree s < n" apply auto
855               apply (erule ssubst)
856               apply (simp add: degree_mult_eq degree_linear_power)
857               done
858             {fix x assume h: "poly s x = 0"
859               {assume xa: "x = a"
860                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
861                   u: "s = [:- a, 1:] * u" by (rule dvdE)
862                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
863                   by (subst s, subst u, simp only: power_Suc mult_ac)
864                 with ap(2)[unfolded dvd_def] have False by blast}
865               note xa = this
866               from h have "poly p x = 0" by (subst s, simp)
867               with pq0 have "poly q x = 0" by blast
868               with r xa have "poly r x = 0"
869                 by auto}
870             note impth = this
871             from IH[rule_format, OF dsn, of s r] impth ds0
872             have "s dvd (r ^ (degree s))" by blast
873             then obtain u where u: "r ^ (degree s) = s * u" ..
874             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
875               by (simp only: poly_mult[symmetric] poly_power[symmetric])
876             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
877             from oop[of a] dsn have "q ^ n = p * ?w"
878               apply -
879               apply (subst s, subst r)
880               apply (simp only: power_mult_distrib)
881               apply (subst mult_assoc [where b=s])
882               apply (subst mult_assoc [where a=u])
883               apply (subst mult_assoc [where b=u, symmetric])
884               apply (subst u [symmetric])
885               apply (simp add: mult_ac power_add [symmetric])
886               done
887             hence ?ths unfolding dvd_def by blast}
888       ultimately have ?ths by blast }
889       ultimately have ?ths by blast}
890     then have ?ths using a order_root pne by blast}
891   moreover
892   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
893     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
894       ccs: "c\<noteq>0" "p = pCons c 0" by blast
896     then have pp: "\<And>x. poly p x =  c" by simp
897     let ?w = "[:1/c:] * (q ^ n)"
898     from ccs have "(q ^ n) = (p * ?w)" by simp
899     hence ?ths unfolding dvd_def by blast}
900   ultimately show ?ths by blast
901 qed
903 lemma nullstellensatz_univariate:
904   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
905     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
906 proof -
907   {assume pe: "p = 0"
908     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
909       by (auto simp add: poly_all_0_iff_0)
910     {assume "p dvd (q ^ (degree p))"
911       then obtain r where r: "q ^ (degree p) = p * r" ..
912       from r pe have False by simp}
913     with eq pe have ?thesis by blast}
914   moreover
915   {assume pe: "p \<noteq> 0"
916     {assume dp: "degree p = 0"
917       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
918         by (cases p) (simp split: if_splits)
919       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
920       from k dp have "q ^ (degree p) = p * [:1/k:]"
921         by (simp add: one_poly_def)
922       hence th2: "p dvd (q ^ (degree p))" ..
923       from th1 th2 pe have ?thesis by blast}
924     moreover
925     {assume dp: "degree p \<noteq> 0"
926       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
927       {assume "p dvd (q ^ (Suc n))"
928         then obtain u where u: "q ^ (Suc n) = p * u" ..
929         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
930           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
931           hence False using u h(1) by (simp only: poly_mult) simp}}
932         with n nullstellensatz_lemma[of p q "degree p"] dp
933         have ?thesis by auto}
934     ultimately have ?thesis by blast}
935   ultimately show ?thesis by blast
936 qed
938 text{* Useful lemma *}
940 lemma constant_degree:
941   fixes p :: "'a::{idom,ring_char_0} poly"
942   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
943 proof
944   assume l: ?lhs
945   from l[unfolded constant_def, rule_format, of _ "0"]
946   have th: "poly p = poly [:poly p 0:]"
947     by auto
948   then have "p = [:poly p 0:]"
949     by (simp add: poly_eq_poly_eq_iff)
950   then have "degree p = degree [:poly p 0:]"
951     by simp
952   then show ?rhs
953     by simp
954 next
955   assume r: ?rhs
956   then obtain k where "p = [:k:]"
957     by (cases p) (simp split: if_splits)
958   then show ?lhs
959     unfolding constant_def by auto
960 qed
962 lemma divides_degree:
963   assumes pq: "p dvd (q:: complex poly)"
964   shows "degree p \<le> degree q \<or> q = 0"
965   by (metis dvd_imp_degree_le pq)
967 (* Arithmetic operations on multivariate polynomials.                        *)
969 lemma mpoly_base_conv:
970   fixes x :: "'a::comm_ring_1"
971   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
972   by simp_all
974 lemma mpoly_norm_conv:
975   fixes x :: "'a::comm_ring_1"
976   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
977   by simp_all
979 lemma mpoly_sub_conv:
980   fixes x :: "'a::comm_ring_1"
981   shows "poly p x - poly q x = poly p x + -1 * poly q x"
982   by simp
984 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = 0" by simp
986 lemma poly_cancel_eq_conv:
987   fixes x :: "'a::field"
988   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (y = 0) = (a * y - b * x = 0)"
989   by auto
992   fixes p:: "('a::comm_ring_1) poly"
993   assumes pq: "p dvd q"
994 shows "p dvd (pCons 0 q)"
995 proof-
996   have "pCons 0 q = q * [:0,1:]" by simp
997   then have "q dvd (pCons 0 q)" ..
998   with pq show ?thesis by (rule dvd_trans)
999 qed
1001 lemma poly_divides_conv0:
1002   fixes p:: "'a::field poly"
1003   assumes lgpq: "degree q < degree p"
1004     and lq: "p \<noteq> 0"
1005   shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
1006 proof
1007   assume r: ?rhs
1008   then have "q = p * 0" by simp
1009   then show ?lhs ..
1010 next
1011   assume l: ?lhs
1012   {
1013     assume q0: "q = 0"
1014     then have ?rhs by simp
1015   }
1016   moreover
1017   {
1018     assume q0: "q \<noteq> 0"
1019     from l q0 have "degree p \<le> degree q"
1020       by (rule dvd_imp_degree_le)
1021     with lgpq have ?rhs by simp
1022   }
1023   ultimately show ?rhs by blast
1024 qed
1026 lemma poly_divides_conv1:
1027   fixes p :: "('a::field) poly"
1028   assumes a0: "a \<noteq> 0"
1029     and pp': "p dvd p'"
1030     and qrp': "smult a q - p' = r"
1031   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
1032 proof
1033   from pp' obtain t where t: "p' = p * t" ..
1034   {
1035     assume l: ?lhs
1036     then obtain u where u: "q = p * u" ..
1037     have "r = p * (smult a u - t)"
1038       using u qrp' [symmetric] t by (simp add: algebra_simps)
1039     then show ?rhs ..
1040   next
1041     assume r: ?rhs
1042     then obtain u where u: "r = p * u" ..
1043     from u [symmetric] t qrp' [symmetric] a0
1044     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1045     then show ?lhs ..
1046   }
1047 qed
1049 lemma basic_cqe_conv1:
1050   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1051   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1052   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
1053   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
1054   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
1055   by simp_all
1057 lemma basic_cqe_conv2:
1058   assumes l:"p \<noteq> 0"
1059   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
1060 proof -
1061   {
1062     fix h t
1063     assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
1064     with l have False by simp
1065   }
1066   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
1067     by blast
1068   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
1069     by auto
1070 qed
1072 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
1073   by (metis poly_all_0_iff_0)
1075 lemma basic_cqe_conv3:
1076   fixes p q :: "complex poly"
1077   assumes l: "p \<noteq> 0"
1078   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ psize p))"
1079 proof -
1080   from l have dp: "degree (pCons a p) = psize p"
1081     by (simp add: psize_def)
1082   from nullstellensatz_univariate[of "pCons a p" q] l
1083   show ?thesis
1084     by (metis dp pCons_eq_0_iff)
1085 qed
1087 lemma basic_cqe_conv4:
1088   fixes p q :: "complex poly"
1089   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
1090   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1091 proof -
1092   from h have "poly (q ^ n) = poly r"
1093     by auto
1094   then have "(q ^ n) = r"
1095     by (simp add: poly_eq_poly_eq_iff)
1096   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1097     by simp
1098 qed
1100 lemma poly_const_conv:
1101   fixes x :: "'a::comm_ring_1"
1102   shows "poly [:c:] x = y \<longleftrightarrow> c = y"
1103   by simp
1105 end