src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author paulson Tue Feb 25 16:17:20 2014 +0000 (2014-02-25) changeset 55734 3f5b2745d659 parent 55358 85d81bc281d0 child 55735 81ba62493610 permissions -rw-r--r--
More complex-related lemmas
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 complex polynomials *}
103 lemma poly_bound_exists:
104   shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
105 proof(induct p)
106   case 0 thus ?case by (rule exI[where x=1], simp)
107 next
108   case (pCons c cs)
109   from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
110     by blast
111   let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
112   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
113   {fix z
114     assume H: "cmod z \<le> r"
115     from m H have th: "cmod (poly cs z) \<le> m" by blast
116     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
117     have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
118       using norm_triangle_ineq[of c "z* poly cs z"] by simp
119     also have "\<dots> \<le> cmod c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult)
120     also have "\<dots> \<le> ?k" by simp
121     finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
122   with kp show ?case by blast
123 qed
126 text{* Offsetting the variable in a polynomial gives another of same degree *}
128 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
129 where
130   "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
132 lemma offset_poly_0: "offset_poly 0 h = 0"
133   by (simp add: offset_poly_def)
135 lemma offset_poly_pCons:
136   "offset_poly (pCons a p) h =
137     smult h (offset_poly p h) + pCons a (offset_poly p h)"
138   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
140 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
141 by (simp add: offset_poly_pCons offset_poly_0)
143 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
144 apply (induct p)
145 apply (simp add: offset_poly_0)
146 apply (simp add: offset_poly_pCons algebra_simps)
147 done
149 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
150 by (induct p arbitrary: a, simp, force)
152 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
153 apply (safe intro!: offset_poly_0)
154 apply (induct p, simp)
155 apply (simp add: offset_poly_pCons)
156 apply (frule offset_poly_eq_0_lemma, simp)
157 done
159 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
160 apply (induct p)
161 apply (simp add: offset_poly_0)
162 apply (case_tac "p = 0")
163 apply (simp add: offset_poly_0 offset_poly_pCons)
164 apply (simp add: offset_poly_pCons)
165 apply (subst degree_add_eq_right)
166 apply (rule le_less_trans [OF degree_smult_le])
167 apply (simp add: offset_poly_eq_0_iff)
168 apply (simp add: offset_poly_eq_0_iff)
169 done
171 definition
172   "psize p = (if p = 0 then 0 else Suc (degree p))"
174 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
175   unfolding psize_def by simp
177 lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
178 proof (intro exI conjI)
179   show "psize (offset_poly p a) = psize p"
180     unfolding psize_def
181     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
182   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
183     by (simp add: poly_offset_poly)
184 qed
186 text{* An alternative useful formulation of completeness of the reals *}
187 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
188   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
189 proof
190   from bz have "bdd_above (Collect P)"
191     by (force intro: less_imp_le)
192   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
193     using ex bz by (subst less_cSup_iff) auto
194 qed
196 subsection {* Fundamental theorem of algebra *}
197 lemma  unimodular_reduce_norm:
198   assumes md: "cmod z = 1"
199   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
200 proof-
201   obtain x y where z: "z = Complex x y " by (cases z, auto)
202   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
203   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
204     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
205       by (simp_all add: cmod_def power2_eq_square algebra_simps)
206     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
207     hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
208       by - (rule power_mono, simp, simp)+
209     hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
210       by (simp_all add: power_mult_distrib)
211     from add_mono[OF th0] xy have False by simp }
212   thus ?thesis unfolding linorder_not_le[symmetric] by blast
213 qed
215 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
216 lemma reduce_poly_simple:
217  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
218   shows "\<exists>z. cmod (1 + b * z^n) < 1"
219 using n
220 proof(induct n rule: nat_less_induct)
221   fix n
222   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
223   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
224   {assume e: "even n"
225     hence "\<exists>m. n = 2*m" by presburger
226     then obtain m where m: "n = 2*m" by blast
227     from n m have "m\<noteq>0" "m < n" by presburger+
228     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
229     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
230     hence "\<exists>z. ?P z n" ..}
231   moreover
232   {assume o: "odd n"
233     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
234       using b by (simp add: norm_divide)
235     from o have "\<exists>m. n = Suc (2*m)" by presburger+
236     then obtain m where m: "n = Suc (2*m)" by blast
237     from unimodular_reduce_norm[OF th0] o
238     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
239       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
240       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp)
241       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
242       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
243       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
244       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
245       apply (auto simp add: m power_mult)
246       apply (rule_tac x="ii" in exI)
247       apply (auto simp add: m power_mult)
248       done
249     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
250     let ?w = "v / complex_of_real (root n (cmod b))"
251     from odd_real_root_pow[OF o, of "cmod b"]
252     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
253       by (simp add: power_divide complex_of_real_power)
254     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
255     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
256     have th4: "cmod (complex_of_real (cmod b) / b) *
257    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
258    < cmod (complex_of_real (cmod b) / b) * 1"
259       apply (simp only: norm_mult[symmetric] distrib_left)
260       using b v by (simp add: th2)
262     from mult_less_imp_less_left[OF th4 th3]
263     have "?P ?w n" unfolding th1 .
264     hence "\<exists>z. ?P z n" .. }
265   ultimately show "\<exists>z. ?P z n" by blast
266 qed
268 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
270 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
271   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
272   unfolding cmod_def by simp
274 lemma bolzano_weierstrass_complex_disc:
275   assumes r: "\<forall>n. cmod (s n) \<le> r"
276   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
277 proof-
278   from seq_monosub[of "Re o s"]
279   obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
280     unfolding o_def by blast
281   from seq_monosub[of "Im o s o f"]
282   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
283   let ?h = "f o g"
284   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
285   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
286   proof
287     fix n
288     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
289   qed
290   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
291     apply (rule Bseq_monoseq_convergent)
292     apply (simp add: Bseq_def)
293     apply (metis gt_ex le_less_linear less_trans order.trans th)
294     using f(2) .
295   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
296   proof
297     fix n
298     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
299   qed
301   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
302     apply (rule Bseq_monoseq_convergent)
303     apply (simp add: Bseq_def)
304     apply (metis gt_ex le_less_linear less_trans order.trans th)
305     using g(2) .
307   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
308     by blast
309   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
310     unfolding LIMSEQ_iff real_norm_def .
312   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
313     by blast
314   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
315     unfolding LIMSEQ_iff real_norm_def .
316   let ?w = "Complex x y"
317   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
318   {fix e assume ep: "e > (0::real)"
319     hence e2: "e/2 > 0" by simp
320     from x[rule_format, OF e2] y[rule_format, OF e2]
321     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
322     {fix n assume nN12: "n \<ge> N1 + N2"
323       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
324       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
325       have "cmod (s (?h n) - ?w) < e"
326         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
327     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
328   with hs show ?thesis  by blast
329 qed
331 text{* Polynomial is continuous. *}
333 lemma poly_cont:
334   assumes ep: "e > 0"
335   shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
336 proof-
337   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
338   proof
339     show "degree (offset_poly p z) = degree p"
340       by (rule degree_offset_poly)
341     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
342       by (rule poly_offset_poly)
343   qed
344   {fix w
345     note q(2)[of "w - z", simplified]}
346   note th = this
347   show ?thesis unfolding th[symmetric]
348   proof(induct q)
349     case 0 thus ?case  using ep by auto
350   next
351     case (pCons c cs)
352     from poly_bound_exists[of 1 "cs"]
353     obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
354     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
355     have one0: "1 > (0::real)"  by arith
356     from real_lbound_gt_zero[OF one0 em0]
357     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
358     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
359       by (simp_all add: field_simps mult_pos_pos)
360     show ?case
361       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
362         fix d w
363         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
364         hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
365         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
366         from H have th: "cmod (w-z) \<le> d" by simp
367         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
368         show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
369       qed
370     qed
371 qed
373 text{* Hence a polynomial attains minimum on a closed disc
374   in the complex plane. *}
375 lemma  poly_minimum_modulus_disc:
376   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
377 proof-
378   {assume "\<not> r \<ge> 0" hence ?thesis
379     by (metis norm_ge_zero order.trans)}
380   moreover
381   {assume rp: "r \<ge> 0"
382     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
383     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
384     {fix x z
385       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
386       hence "- x < 0 " by arith
387       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
388     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
389     from real_sup_exists[OF mth1 mth2] obtain s where
390       s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow>(y < s)" by blast
391     let ?m = "-s"
392     {fix y
393       from s[rule_format, of "-y"] have
394     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
395         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
396     note s1 = this[unfolded minus_minus]
397     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
398       by auto
399     {fix n::nat
400       from s1[rule_format, of "?m + 1/real (Suc n)"]
401       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
402         by simp}
403     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
404     from choice[OF th] obtain g where
405       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
406       by blast
407     from bolzano_weierstrass_complex_disc[OF g(1)]
408     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
409       by blast
410     {fix w
411       assume wr: "cmod w \<le> r"
412       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
413       {assume e: "?e > 0"
414         hence e2: "?e/2 > 0" by simp
415         from poly_cont[OF e2, of z p] obtain d where
416           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
417         {fix w assume w: "cmod (w - z) < d"
418           have "cmod(poly p w - poly p z) < ?e / 2"
419             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
420         note th1 = this
422         from fz(2)[rule_format, OF d(1)] obtain N1 where
423           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
424         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
425           N2: "2/?e < real N2" by blast
426         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
427           using N1[rule_format, of "N1 + N2"] th1 by simp
428         {fix a b e2 m :: real
429         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
430           ==> False" by arith}
431       note th0 = this
432       have ath:
433         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
434       from s1m[OF g(1)[rule_format]]
435       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
436       from seq_suble[OF fz(1), of "N1+N2"]
437       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
438       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
439         using N2 by auto
440       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
441       from g(2)[rule_format, of "f (N1 + N2)"]
442       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
443       from order_less_le_trans[OF th01 th00]
444       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
445       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
446       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
447       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
448       with ath[OF th31 th32]
449       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
450       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
451         by arith
452       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
453 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
454         by (simp add: norm_triangle_ineq3)
455       from ath2[OF th22, of ?m]
456       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
457       from th0[OF th2 thc1 thc2] have False .}
458       hence "?e = 0" by auto
459       then have "cmod (poly p z) = ?m" by simp
460       with s1m[OF wr]
461       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
462     hence ?thesis by blast}
463   ultimately show ?thesis by blast
464 qed
466 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
467   unfolding power2_eq_square
468   apply (simp add: rcis_mult)
469   apply (simp add: power2_eq_square[symmetric])
470   done
472 lemma cispi: "cis pi = -1"
473   unfolding cis_def
474   by simp
476 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
477   unfolding power2_eq_square
479   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
480   done
482 text {* Nonzero polynomial in z goes to infinity as z does. *}
484 lemma poly_infinity:
485   assumes ex: "p \<noteq> 0"
486   shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
487 using ex
488 proof(induct p arbitrary: a d)
489   case (pCons c cs a d)
490   {assume H: "cs \<noteq> 0"
491     with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast
492     let ?r = "1 + \<bar>r\<bar>"
493     {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
494       have r0: "r \<le> cmod z" using h by arith
495       from r[rule_format, OF r0]
496       have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
497       from h have z1: "cmod z \<ge> 1" by arith
498       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
499       have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
500         unfolding norm_mult by (simp add: algebra_simps)
501       from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
502       have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"
503         by (simp add: algebra_simps)
504       from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
505     hence ?case by blast}
506   moreover
507   {assume cs0: "\<not> (cs \<noteq> 0)"
508     with pCons.prems have c0: "c \<noteq> 0" by simp
509     from cs0 have cs0': "cs = 0" by simp
510     {fix z
511       assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
512       from c0 have "cmod c > 0" by simp
513       from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
514         by (simp add: field_simps norm_mult)
515       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
516       from complex_mod_triangle_sub[of "z*c" a ]
517       have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
518         by (simp add: algebra_simps)
519       from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"
520         using cs0' by simp}
521     then have ?case  by blast}
522   ultimately show ?case by blast
523 qed simp
525 text {* Hence polynomial's modulus attains its minimum somewhere. *}
526 lemma poly_minimum_modulus:
527   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
528 proof(induct p)
529   case (pCons c cs)
530   {assume cs0: "cs \<noteq> 0"
531     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
532     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
533     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
534     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
535     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
536     {fix z assume z: "r \<le> cmod z"
537       from v[of 0] r[OF z]
538       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
539         by simp }
540     note v0 = this
541     from v0 v ath[of r] have ?case by blast}
542   moreover
543   {assume cs0: "\<not> (cs \<noteq> 0)"
544     hence th:"cs = 0" by simp
545     from th pCons.hyps have ?case by simp}
546   ultimately show ?case by blast
547 qed simp
549 text{* Constant function (non-syntactic characterization). *}
550 definition "constant f = (\<forall>x y. f x = f y)"
552 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
553   unfolding constant_def psize_def
554   apply (induct p, auto)
555   done
557 lemma poly_replicate_append:
558   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
559   by (simp add: poly_monom)
561 text {* Decomposition of polynomial, skipping zero coefficients
562   after the first.  *}
564 lemma poly_decompose_lemma:
565  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
566   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
567                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
568 unfolding psize_def
569 using nz
570 proof(induct p)
571   case 0 thus ?case by simp
572 next
573   case (pCons c cs)
574   {assume c0: "c = 0"
575     from pCons.hyps pCons.prems c0 have ?case
576       apply (auto)
577       apply (rule_tac x="k+1" in exI)
578       apply (rule_tac x="a" in exI, clarsimp)
579       apply (rule_tac x="q" in exI)
580       by (auto)}
581   moreover
582   {assume c0: "c\<noteq>0"
583     have ?case
584       apply (rule exI[where x=0])
585       apply (rule exI[where x=c], auto simp add: c0)
586       done}
587   ultimately show ?case by blast
588 qed
590 lemma poly_decompose:
591   assumes nc: "~constant(poly p)"
592   shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
593                psize q + k + 1 = psize p \<and>
594               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
595 using nc
596 proof(induct p)
597   case 0 thus ?case by (simp add: constant_def)
598 next
599   case (pCons c cs)
600   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
601     {fix x y
602       from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
603     with pCons.prems have False by (auto simp add: constant_def)}
604   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
605   from poly_decompose_lemma[OF th]
606   show ?case
607     apply clarsimp
608     apply (rule_tac x="k+1" in exI)
609     apply (rule_tac x="a" in exI)
610     apply simp
611     apply (rule_tac x="q" in exI)
612     apply (auto simp add: psize_def split: if_splits)
613     done
614 qed
616 text{* Fundamental theorem of algebra *}
618 lemma fundamental_theorem_of_algebra:
619   assumes nc: "~constant(poly p)"
620   shows "\<exists>z::complex. poly p z = 0"
621 using nc
622 proof(induct "psize p" arbitrary: p rule: less_induct)
623   case less
624   let ?p = "poly p"
625   let ?ths = "\<exists>z. ?p z = 0"
627   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
628   from poly_minimum_modulus obtain c where
629     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
630   {assume pc: "?p c = 0" hence ?ths by blast}
631   moreover
632   {assume pc0: "?p c \<noteq> 0"
633     from poly_offset[of p c] obtain q where
634       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
635     {assume h: "constant (poly q)"
636       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
637       {fix x y
638         from th have "?p x = poly q (x - c)" by auto
639         also have "\<dots> = poly q (y - c)"
640           using h unfolding constant_def by blast
641         also have "\<dots> = ?p y" using th by auto
642         finally have "?p x = ?p y" .}
643       with less(2) have False unfolding constant_def by blast }
644     hence qnc: "\<not> constant (poly q)" by blast
645     from q(2) have pqc0: "?p c = poly q 0" by simp
646     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
647     let ?a0 = "poly q 0"
648     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
649     from a00
650     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
651       by simp
652     let ?r = "smult (inverse ?a0) q"
653     have lgqr: "psize q = psize ?r"
654       using a00 unfolding psize_def degree_def
655       by (simp add: poly_eq_iff)
656     {assume h: "\<And>x y. poly ?r x = poly ?r y"
657       {fix x y
658         from qr[rule_format, of x]
659         have "poly q x = poly ?r x * ?a0" by auto
660         also have "\<dots> = poly ?r y * ?a0" using h by simp
661         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
662         finally have "poly q x = poly q y" .}
663       with qnc have False unfolding constant_def by blast}
664     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
665     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
666     {fix w
667       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
668         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
669       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
670         using a00 unfolding norm_divide by (simp add: field_simps)
671       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
672     note mrmq_eq = this
673     from poly_decompose[OF rnc] obtain k a s where
674       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
675       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
676     {assume "psize p = k + 1"
677       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
678       {fix w
679         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
680           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
681       note hth = this [symmetric]
682         from reduce_poly_simple[OF kas(1,2)]
683       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
684     moreover
685     {assume kn: "psize p \<noteq> k+1"
686       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
687       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
688         unfolding constant_def poly_pCons poly_monom
689         using kas(1) apply simp
690         by (rule exI[where x=0], rule exI[where x=1], simp)
691       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
692         by (simp add: psize_def degree_monom_eq)
693       from less(1) [OF k1n [simplified th02] th01]
694       obtain w where w: "1 + w^k * a = 0"
695         unfolding poly_pCons poly_monom
696         using kas(2) by (cases k, auto simp add: algebra_simps)
697       from poly_bound_exists[of "cmod w" s] obtain m where
698         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
699       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
700       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
701       then have wm1: "w^k * a = - 1" by simp
702       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
703         using norm_ge_zero[of w] w0 m(1)
704           by (simp add: inverse_eq_divide zero_less_mult_iff)
705       with real_lbound_gt_zero[OF zero_less_one] obtain t where
706         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
707       let ?ct = "complex_of_real t"
708       let ?w = "?ct * w"
709       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)
710       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
711         unfolding wm1 by (simp)
712       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
713         by metis
714       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
715       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
716       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
717       have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
718       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
719       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
720         by (simp add: inverse_eq_divide field_simps)
721       with zero_less_power[OF t(1), of k]
722       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
723         by (metis comm_mult_strict_left_mono)
724       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
725         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
726       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
727         using t(1,2) m(2)[rule_format, OF tw] w0
728         by auto
729       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
730       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
731         by auto
732       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
733       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
734       from th11 th12
735       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
736       then have "cmod (poly ?r ?w) < 1"
737         unfolding kas(4)[rule_format, of ?w] r01 by simp
738       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
739     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
740     from cr0_contr cq0 q(2)
741     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
742   ultimately show ?ths by blast
743 qed
745 text {* Alternative version with a syntactic notion of constant polynomial. *}
747 lemma fundamental_theorem_of_algebra_alt:
748   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
749   shows "\<exists>z. poly p z = (0::complex)"
750 using nc
751 proof(induct p)
752   case (pCons c cs)
753   {assume "c=0" hence ?case by auto}
754   moreover
755   {assume c0: "c\<noteq>0"
756     {assume nc: "constant (poly (pCons c cs))"
757       from nc[unfolded constant_def, rule_format, of 0]
758       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
759       hence "cs = 0"
760         proof(induct cs)
761           case (pCons d ds)
762           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
763           moreover
764           {assume d0: "d\<noteq>0"
765             from poly_bound_exists[of 1 ds] obtain m where
766               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
767             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
768             from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
769               x: "x > 0" "x < cmod d / m" "x < 1" by blast
770             let ?x = "complex_of_real x"
771             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
772             from pCons.prems[rule_format, OF cx(1)]
773             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
774             from m(2)[rule_format, OF cx(2)] x(1)
775             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
776               by (simp add: norm_mult)
777             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
778             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
779             with cth  have ?case by blast}
780           ultimately show ?case by blast
781         qed simp}
782       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
783         by blast
784       from fundamental_theorem_of_algebra[OF nc] have ?case .}
785   ultimately show ?case by blast
786 qed simp
789 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
791 lemma nullstellensatz_lemma:
792   fixes p :: "complex poly"
793   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
794   and "degree p = n" and "n \<noteq> 0"
795   shows "p dvd (q ^ n)"
796 using assms
797 proof(induct n arbitrary: p q rule: nat_less_induct)
798   fix n::nat fix p q :: "complex poly"
799   assume IH: "\<forall>m<n. \<forall>p q.
800                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
801                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
802     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
803     and dpn: "degree p = n" and n0: "n \<noteq> 0"
804   from dpn n0 have pne: "p \<noteq> 0" by auto
805   let ?ths = "p dvd (q ^ n)"
806   {fix a assume a: "poly p a = 0"
807     {assume oa: "order a p \<noteq> 0"
808       let ?op = "order a p"
809       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
810         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
811       note oop = order_degree[OF pne, unfolded dpn]
812       {assume q0: "q = 0"
813         hence ?ths using n0
814           by (simp add: power_0_left)}
815       moreover
816       {assume q0: "q \<noteq> 0"
817         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
818         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
819         from ap(1) obtain s where
820           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
821         have sne: "s \<noteq> 0"
822           using s pne by auto
823         {assume ds0: "degree s = 0"
824           from ds0 obtain k where kpn: "s = [:k:]"
825             by (cases s) (auto split: if_splits)
826           from sne kpn have k: "k \<noteq> 0" by simp
827           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
828           have "q ^ n = p * ?w"
829             apply (subst r, subst s, subst kpn)
830             using k oop [of a]
831             apply (subst power_mult_distrib, simp)
832             apply (subst power_add [symmetric], simp)
833             done
834           hence ?ths unfolding dvd_def by blast}
835         moreover
836         {assume ds0: "degree s \<noteq> 0"
837           from ds0 sne dpn s oa
838             have dsn: "degree s < n" apply auto
839               apply (erule ssubst)
840               apply (simp add: degree_mult_eq degree_linear_power)
841               done
842             {fix x assume h: "poly s x = 0"
843               {assume xa: "x = a"
844                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
845                   u: "s = [:- a, 1:] * u" by (rule dvdE)
846                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
847                   by (subst s, subst u, simp only: power_Suc mult_ac)
848                 with ap(2)[unfolded dvd_def] have False by blast}
849               note xa = this
850               from h have "poly p x = 0" by (subst s, simp)
851               with pq0 have "poly q x = 0" by blast
852               with r xa have "poly r x = 0"
854             note impth = this
855             from IH[rule_format, OF dsn, of s r] impth ds0
856             have "s dvd (r ^ (degree s))" by blast
857             then obtain u where u: "r ^ (degree s) = s * u" ..
858             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
859               by (simp only: poly_mult[symmetric] poly_power[symmetric])
860             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
861             from oop[of a] dsn have "q ^ n = p * ?w"
862               apply -
863               apply (subst s, subst r)
864               apply (simp only: power_mult_distrib)
865               apply (subst mult_assoc [where b=s])
866               apply (subst mult_assoc [where a=u])
867               apply (subst mult_assoc [where b=u, symmetric])
868               apply (subst u [symmetric])
869               apply (simp add: mult_ac power_add [symmetric])
870               done
871             hence ?ths unfolding dvd_def by blast}
872       ultimately have ?ths by blast }
873       ultimately have ?ths by blast}
874     then have ?ths using a order_root pne by blast}
875   moreover
876   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
877     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
878       ccs: "c\<noteq>0" "p = pCons c 0" by blast
880     then have pp: "\<And>x. poly p x =  c" by simp
881     let ?w = "[:1/c:] * (q ^ n)"
882     from ccs have "(q ^ n) = (p * ?w)" by simp
883     hence ?ths unfolding dvd_def by blast}
884   ultimately show ?ths by blast
885 qed
887 lemma nullstellensatz_univariate:
888   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
889     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
890 proof-
891   {assume pe: "p = 0"
892     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
893       by (auto simp add: poly_all_0_iff_0)
894     {assume "p dvd (q ^ (degree p))"
895       then obtain r where r: "q ^ (degree p) = p * r" ..
896       from r pe have False by simp}
897     with eq pe have ?thesis by blast}
898   moreover
899   {assume pe: "p \<noteq> 0"
900     {assume dp: "degree p = 0"
901       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
902         by (cases p) (simp split: if_splits)
903       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
904       from k dp have "q ^ (degree p) = p * [:1/k:]"
905         by (simp add: one_poly_def)
906       hence th2: "p dvd (q ^ (degree p))" ..
907       from th1 th2 pe have ?thesis by blast}
908     moreover
909     {assume dp: "degree p \<noteq> 0"
910       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
911       {assume "p dvd (q ^ (Suc n))"
912         then obtain u where u: "q ^ (Suc n) = p * u" ..
913         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
914           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
915           hence False using u h(1) by (simp only: poly_mult) simp}}
916         with n nullstellensatz_lemma[of p q "degree p"] dp
917         have ?thesis by auto}
918     ultimately have ?thesis by blast}
919   ultimately show ?thesis by blast
920 qed
922 text{* Useful lemma *}
924 lemma constant_degree:
925   fixes p :: "'a::{idom,ring_char_0} poly"
926   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
927 proof
928   assume l: ?lhs
929   from l[unfolded constant_def, rule_format, of _ "0"]
930   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
931   then have "p = [:poly p 0:]" by (simp add: poly_eq_poly_eq_iff)
932   then have "degree p = degree [:poly p 0:]" by simp
933   then show ?rhs by simp
934 next
935   assume r: ?rhs
936   then obtain k where "p = [:k:]"
937     by (cases p) (simp split: if_splits)
938   then show ?lhs unfolding constant_def by auto
939 qed
941 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
942   shows "degree p \<le> degree q \<or> q = 0"
943 by (metis dvd_imp_degree_le pq)
945 (* Arithmetic operations on multivariate polynomials.                        *)
947 lemma mpoly_base_conv:
948   "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
950 lemma mpoly_norm_conv:
951   "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
953 lemma mpoly_sub_conv:
954   "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
955   by simp
957 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
959 lemma poly_cancel_eq_conv: "p = (0::complex) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto
961 lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
964   fixes p q :: "complex poly"
965   assumes pq: "p dvd q"
966   shows "p dvd (pCons (0::complex) q)"
967 proof-
968   have "pCons 0 q = q * [:0,1:]" by simp
969   then have "q dvd (pCons 0 q)" ..
970   with pq show ?thesis by (rule dvd_trans)
971 qed
974   fixes p q :: "complex poly"
975   assumes pq: "p dvd q"
976   shows "p dvd (smult a q)"
977 proof-
978   have "smult a q = q * [:a:]" by simp
979   then have "q dvd smult a q" ..
980   with pq show ?thesis by (rule dvd_trans)
981 qed
984 lemma poly_divides_conv0:
985   fixes p :: "complex 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   assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
1007   and qrp': "smult a q - p' \<equiv> r"
1008   shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
1009 proof-
1010   {
1011   from pp' obtain t where t: "p' = p * t" ..
1012   {assume l: ?lhs
1013     then obtain u where u: "q = p * u" ..
1014      have "r = p * (smult a u - t)"
1015        using u qrp' [symmetric] t by (simp add: algebra_simps)
1016      then have ?rhs ..}
1017   moreover
1018   {assume r: ?rhs
1019     then obtain u where u: "r = p * u" ..
1020     from u [symmetric] t qrp' [symmetric] a0
1021     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1022     hence ?lhs ..}
1023   ultimately have "?lhs = ?rhs" by blast }
1024 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
1025 qed
1027 lemma basic_cqe_conv1:
1028   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1029   "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
1030   "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c\<noteq>0"
1031   "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
1032   "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0" by simp_all
1034 lemma basic_cqe_conv2:
1035   assumes l:"p \<noteq> 0"
1036   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex))"
1037 proof-
1038   {fix h t
1039     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
1040     with l have False by simp}
1041   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
1042     by blast
1043   from fundamental_theorem_of_algebra_alt[OF th]
1044   show ?thesis by auto
1045 qed
1047 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> (p \<noteq> 0)"
1048 by (metis poly_all_0_iff_0)
1050 lemma basic_cqe_conv3:
1051   fixes p q :: "complex poly"
1052   assumes l: "p \<noteq> 0"
1053   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> ((pCons a p) dvd (q ^ (psize p)))"
1054 proof -
1055   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
1056   from nullstellensatz_univariate[of "pCons a p" q] l
1057   show ?thesis
1058     by (metis dp pCons_eq_0_iff)
1059 qed
1061 lemma basic_cqe_conv4:
1062   fixes p q :: "complex poly"
1063   assumes h: "\<And>x. poly (q ^ n) x = poly r x"
1064   shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
1065 proof-
1066   from h have "poly (q ^ n) = poly r" by auto
1067   then have "(q ^ n) = r" by (simp add: poly_eq_poly_eq_iff)
1068   thus "p dvd (q ^ n) \<longleftrightarrow> p dvd r" by simp
1069 qed
1071 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
1073 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
1075 end