move Lubs from HOL to HOL-Library (replaced by conditionally complete lattices)
1 (* Author: Amine Chaieb, TU Muenchen *)
3 header{*Fundamental Theorem of Algebra*}
5 theory Fundamental_Theorem_Algebra
6 imports Polynomial Complex_Main
9 subsection {* Square root of complex numbers *}
10 definition csqrt :: "complex \<Rightarrow> complex" where
11 "csqrt z = (if Im z = 0 then
12 if 0 \<le> Re z then Complex (sqrt(Re z)) 0
13 else Complex 0 (sqrt(- Re z))
14 else Complex (sqrt((cmod z + Re z) /2))
15 ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"
17 lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
19 obtain x y where xy: "z = Complex x y" by (cases z)
21 {assume x0: "x \<ge> 0"
22 then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
23 by (simp add: csqrt_def power2_eq_square)}
25 {assume "\<not> x \<ge> 0" hence x0: "- x \<ge> 0" by arith
26 then have ?thesis using y0 xy real_sqrt_pow2[OF x0]
27 by (simp add: csqrt_def power2_eq_square) }
28 ultimately have ?thesis by blast}
30 {assume y0: "y\<noteq>0"
32 let ?z = "Complex x y"
33 from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto
34 hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+
35 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 have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
38 by (simp add: power2_eq_square)
40 have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
41 "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
42 unfolding sq4 by simp_all
43 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"
44 unfolding power2_eq_square by simp
45 have "sqrt 4 = sqrt (2\<^sup>2)" by simp
46 hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
47 have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
48 using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
49 unfolding power2_eq_square
50 by (simp add: algebra_simps real_sqrt_divide sqrt4)
51 from y0 xy have ?thesis apply (simp add: csqrt_def power2_eq_square)
52 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 ultimately show ?thesis by blast
58 subsection{* More lemmas about module of complex numbers *}
60 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
61 by (rule of_real_power [symmetric])
63 lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
64 apply (rule exI[where x = "min d1 d2 / 2"])
65 by (simp add: field_simps min_def)
67 text{* The triangle inequality for cmod *}
68 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
69 using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
71 subsection{* Basic lemmas about complex polynomials *}
73 lemma poly_bound_exists:
74 shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
76 case 0 thus ?case by (rule exI[where x=1], simp)
79 from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
81 let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
82 have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
84 assume H: "cmod z \<le> r"
85 from m H have th: "cmod (poly cs z) \<le> m" by blast
86 from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
87 have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
88 using norm_triangle_ineq[of c "z* poly cs z"] by simp
89 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)
90 also have "\<dots> \<le> ?k" by simp
91 finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
92 with kp show ?case by blast
96 text{* Offsetting the variable in a polynomial gives another of same degree *}
98 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
100 "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
102 lemma offset_poly_0: "offset_poly 0 h = 0"
103 by (simp add: offset_poly_def)
105 lemma offset_poly_pCons:
106 "offset_poly (pCons a p) h =
107 smult h (offset_poly p h) + pCons a (offset_poly p h)"
108 by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
110 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
111 by (simp add: offset_poly_pCons offset_poly_0)
113 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
115 apply (simp add: offset_poly_0)
116 apply (simp add: offset_poly_pCons algebra_simps)
119 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
120 by (induct p arbitrary: a, simp, force)
122 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
123 apply (safe intro!: offset_poly_0)
124 apply (induct p, simp)
125 apply (simp add: offset_poly_pCons)
126 apply (frule offset_poly_eq_0_lemma, simp)
129 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
131 apply (simp add: offset_poly_0)
132 apply (case_tac "p = 0")
133 apply (simp add: offset_poly_0 offset_poly_pCons)
134 apply (simp add: offset_poly_pCons)
135 apply (subst degree_add_eq_right)
136 apply (rule le_less_trans [OF degree_smult_le])
137 apply (simp add: offset_poly_eq_0_iff)
138 apply (simp add: offset_poly_eq_0_iff)
142 "psize p = (if p = 0 then 0 else Suc (degree p))"
144 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
145 unfolding psize_def by simp
147 lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
148 proof (intro exI conjI)
149 show "psize (offset_poly p a) = psize p"
151 by (simp add: offset_poly_eq_0_iff degree_offset_poly)
152 show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
153 by (simp add: poly_offset_poly)
156 text{* An alternative useful formulation of completeness of the reals *}
157 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
158 shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
160 from bz have "bdd_above (Collect P)"
161 by (force intro: less_imp_le)
162 then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
163 using ex bz by (subst less_cSup_iff) auto
166 subsection {* Fundamental theorem of algebra *}
167 lemma unimodular_reduce_norm:
168 assumes md: "cmod z = 1"
169 shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
171 obtain x y where z: "z = Complex x y " by (cases z, auto)
172 from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1" by (simp add: cmod_def)
173 {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
174 from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
175 by (simp_all add: cmod_def power2_eq_square algebra_simps)
176 hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
177 hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"
178 by - (rule power_mono, simp, simp)+
179 hence th0: "4*x\<^sup>2 \<le> 1" "4*y\<^sup>2 \<le> 1"
180 by (simp_all add: power_mult_distrib)
181 from add_mono[OF th0] xy have False by simp }
182 thus ?thesis unfolding linorder_not_le[symmetric] by blast
185 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
186 lemma reduce_poly_simple:
187 assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
188 shows "\<exists>z. cmod (1 + b * z^n) < 1"
190 proof(induct n rule: nat_less_induct)
192 assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
193 let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
195 hence "\<exists>m. n = 2*m" by presburger
196 then obtain m where m: "n = 2*m" by blast
197 from n m have "m\<noteq>0" "m < n" by presburger+
198 with IH[rule_format, of m] obtain z where z: "?P z m" by blast
199 from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
200 hence "\<exists>z. ?P z n" ..}
203 have th0: "cmod (complex_of_real (cmod b) / b) = 1"
204 using b by (simp add: norm_divide)
205 from o have "\<exists>m. n = Suc (2*m)" by presburger+
206 then obtain m where m: "n = Suc (2*m)" by blast
207 from unimodular_reduce_norm[OF th0] o
208 have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
209 apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
210 apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp del: minus_one add: minus_one [symmetric])
211 apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
212 apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
213 apply (rule_tac x="- ii" in exI, simp add: m power_mult)
214 apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult)
215 apply (rule_tac x="ii" in exI, simp add: m power_mult)
217 then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
218 let ?w = "v / complex_of_real (root n (cmod b))"
219 from odd_real_root_pow[OF o, of "cmod b"]
220 have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
221 by (simp add: power_divide complex_of_real_power)
222 have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
223 hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
224 have th4: "cmod (complex_of_real (cmod b) / b) *
225 cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
226 < cmod (complex_of_real (cmod b) / b) * 1"
227 apply (simp only: norm_mult[symmetric] distrib_left)
228 using b v by (simp add: th2)
230 from mult_less_imp_less_left[OF th4 th3]
231 have "?P ?w n" unfolding th1 .
232 hence "\<exists>z. ?P z n" .. }
233 ultimately show "\<exists>z. ?P z n" by blast
236 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
238 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
239 using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
240 unfolding cmod_def by simp
242 lemma bolzano_weierstrass_complex_disc:
243 assumes r: "\<forall>n. cmod (s n) \<le> r"
244 shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
246 from seq_monosub[of "Re o s"]
247 obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
248 unfolding o_def by blast
249 from seq_monosub[of "Im o s o f"]
250 obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
252 from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
253 have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
256 from abs_Re_le_cmod[of "s n"] r[rule_format, of n] show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
258 have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
259 apply (rule Bseq_monoseq_convergent)
260 apply (simp add: Bseq_def)
261 apply (rule exI[where x= "r + 1"])
262 using th rp apply simp
264 have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
267 from abs_Im_le_cmod[of "s n"] r[rule_format, of n] show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
270 have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
271 apply (rule Bseq_monoseq_convergent)
272 apply (simp add: Bseq_def)
273 apply (rule exI[where x= "r + 1"])
274 using th rp apply simp
277 from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
279 hence x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
280 unfolding LIMSEQ_iff real_norm_def .
282 from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
284 hence y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
285 unfolding LIMSEQ_iff real_norm_def .
286 let ?w = "Complex x y"
287 from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
288 {fix e assume ep: "e > (0::real)"
289 hence e2: "e/2 > 0" by simp
290 from x[rule_format, OF e2] y[rule_format, OF e2]
291 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
292 {fix n assume nN12: "n \<ge> N1 + N2"
293 hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
294 from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
295 have "cmod (s (?h n) - ?w) < e"
296 using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
297 hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
298 with hs show ?thesis by blast
301 text{* Polynomial is continuous. *}
305 shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
307 obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
309 show "degree (offset_poly p z) = degree p"
310 by (rule degree_offset_poly)
311 show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
312 by (rule poly_offset_poly)
315 note q(2)[of "w - z", simplified]}
317 show ?thesis unfolding th[symmetric]
319 case 0 thus ?case using ep by auto
322 from poly_bound_exists[of 1 "cs"]
323 obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
324 from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
325 have one0: "1 > (0::real)" by arith
326 from real_lbound_gt_zero[OF one0 em0]
327 obtain d where d: "d >0" "d < 1" "d < e / m" by blast
328 from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
329 by (simp_all add: field_simps mult_pos_pos)
331 proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
333 assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
334 hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
335 from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
336 from H have th: "cmod (w-z) \<le> d" by simp
337 from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
338 show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
343 text{* Hence a polynomial attains minimum on a closed disc
344 in the complex plane. *}
345 lemma poly_minimum_modulus_disc:
346 "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
348 {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
350 apply (rule exI[where x=0])
352 apply (subgoal_tac "cmod w < 0")
357 {assume rp: "r \<ge> 0"
358 from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
359 hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x" by blast
361 assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
362 hence "- x < 0 " by arith
363 with H(2) norm_ge_zero[of "poly p z"] have False by simp }
364 then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
365 from real_sup_exists[OF mth1 mth2] obtain s where
366 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
369 from s[rule_format, of "-y"] have
370 "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
371 unfolding minus_less_iff[of y ] equation_minus_iff by blast }
372 note s1 = this[unfolded minus_minus]
373 from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
376 from s1[rule_format, of "?m + 1/real (Suc n)"]
377 have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
379 hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
380 from choice[OF th] obtain g where
381 g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
383 from bolzano_weierstrass_complex_disc[OF g(1)]
384 obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
387 assume wr: "cmod w \<le> r"
388 let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
390 hence e2: "?e/2 > 0" by simp
391 from poly_cont[OF e2, of z p] obtain d where
392 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
393 {fix w assume w: "cmod (w - z) < d"
394 have "cmod(poly p w - poly p z) < ?e / 2"
395 using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
398 from fz(2)[rule_format, OF d(1)] obtain N1 where
399 N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
400 from reals_Archimedean2[of "2/?e"] obtain N2::nat where
401 N2: "2/?e < real N2" by blast
402 have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
403 using N1[rule_format, of "N1 + N2"] th1 by simp
404 {fix a b e2 m :: real
405 have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
409 "\<And>m x e. m <= x \<Longrightarrow> x < m + e ==> abs(x - m::real) < e" by arith
410 from s1m[OF g(1)[rule_format]]
411 have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
412 from seq_suble[OF fz(1), of "N1+N2"]
413 have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
414 have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
416 from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
417 from g(2)[rule_format, of "f (N1 + N2)"]
418 have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
419 from order_less_le_trans[OF th01 th00]
420 have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
421 from N2 have "2/?e < real (Suc (N1 + N2))" by arith
422 with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
423 have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
424 with ath[OF th31 th32]
425 have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
426 have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
428 have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
429 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
430 by (simp add: norm_triangle_ineq3)
431 from ath2[OF th22, of ?m]
432 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
433 from th0[OF th2 thc1 thc2] have False .}
434 hence "?e = 0" by auto
435 then have "cmod (poly p z) = ?m" by simp
437 have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
438 hence ?thesis by blast}
439 ultimately show ?thesis by blast
442 lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
443 unfolding power2_eq_square
444 apply (simp add: rcis_mult)
445 apply (simp add: power2_eq_square[symmetric])
448 lemma cispi: "cis pi = -1"
452 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
453 unfolding power2_eq_square
454 apply (simp add: rcis_mult add_divide_distrib)
455 apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
458 text {* Nonzero polynomial in z goes to infinity as z does. *}
461 assumes ex: "p \<noteq> 0"
462 shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
464 proof(induct p arbitrary: a d)
465 case (pCons c cs a d)
466 {assume H: "cs \<noteq> 0"
467 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
468 let ?r = "1 + \<bar>r\<bar>"
469 {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
470 have r0: "r \<le> cmod z" using h by arith
471 from r[rule_format, OF r0]
472 have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
473 from h have z1: "cmod z \<ge> 1" by arith
474 from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
475 have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
476 unfolding norm_mult by (simp add: algebra_simps)
477 from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
478 have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"
479 by (simp add: algebra_simps)
480 from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" by arith}
481 hence ?case by blast}
483 {assume cs0: "\<not> (cs \<noteq> 0)"
484 with pCons.prems have c0: "c \<noteq> 0" by simp
485 from cs0 have cs0': "cs = 0" by simp
487 assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
488 from c0 have "cmod c > 0" by simp
489 from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
490 by (simp add: field_simps norm_mult)
491 have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
492 from complex_mod_triangle_sub[of "z*c" a ]
493 have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
494 by (simp add: algebra_simps)
495 from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"
497 then have ?case by blast}
498 ultimately show ?case by blast
501 text {* Hence polynomial's modulus attains its minimum somewhere. *}
502 lemma poly_minimum_modulus:
503 "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
506 {assume cs0: "cs \<noteq> 0"
507 from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
508 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
509 have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
510 from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
511 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
512 {fix z assume z: "r \<le> cmod z"
514 have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
517 from v0 v ath[of r] have ?case by blast}
519 {assume cs0: "\<not> (cs \<noteq> 0)"
520 hence th:"cs = 0" by simp
521 from th pCons.hyps have ?case by simp}
522 ultimately show ?case by blast
525 text{* Constant function (non-syntactic characterization). *}
526 definition "constant f = (\<forall>x y. f x = f y)"
528 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
529 unfolding constant_def psize_def
530 apply (induct p, auto)
533 lemma poly_replicate_append:
534 "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
535 by (simp add: poly_monom)
537 text {* Decomposition of polynomial, skipping zero coefficients
540 lemma poly_decompose_lemma:
541 assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
542 shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
543 (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
547 case 0 thus ?case by simp
551 from pCons.hyps pCons.prems c0 have ?case
553 apply (rule_tac x="k+1" in exI)
554 apply (rule_tac x="a" in exI, clarsimp)
555 apply (rule_tac x="q" in exI)
558 {assume c0: "c\<noteq>0"
560 apply (rule exI[where x=0])
561 apply (rule exI[where x=c], clarsimp)
562 apply (rule exI[where x=cs])
565 ultimately show ?case by blast
568 lemma poly_decompose:
569 assumes nc: "~constant(poly p)"
570 shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
571 psize q + k + 1 = psize p \<and>
572 (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
575 case 0 thus ?case by (simp add: constant_def)
578 {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
580 from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
581 with pCons.prems have False by (auto simp add: constant_def)}
582 hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
583 from poly_decompose_lemma[OF th]
586 apply (rule_tac x="k+1" in exI)
587 apply (rule_tac x="a" in exI)
589 apply (rule_tac x="q" in exI)
590 apply (auto simp add: psize_def split: if_splits)
594 text{* Fundamental theorem of algebra *}
596 lemma fundamental_theorem_of_algebra:
597 assumes nc: "~constant(poly p)"
598 shows "\<exists>z::complex. poly p z = 0"
600 proof(induct "psize p" arbitrary: p rule: less_induct)
603 let ?ths = "\<exists>z. ?p z = 0"
605 from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
606 from poly_minimum_modulus obtain c where
607 c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
608 {assume pc: "?p c = 0" hence ?ths by blast}
610 {assume pc0: "?p c \<noteq> 0"
611 from poly_offset[of p c] obtain q where
612 q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
613 {assume h: "constant (poly q)"
614 from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
616 from th have "?p x = poly q (x - c)" by auto
617 also have "\<dots> = poly q (y - c)"
618 using h unfolding constant_def by blast
619 also have "\<dots> = ?p y" using th by auto
620 finally have "?p x = ?p y" .}
621 with less(2) have False unfolding constant_def by blast }
622 hence qnc: "\<not> constant (poly q)" by blast
623 from q(2) have pqc0: "?p c = poly q 0" by simp
624 from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
626 from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
628 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
630 let ?r = "smult (inverse ?a0) q"
631 have lgqr: "psize q = psize ?r"
632 using a00 unfolding psize_def degree_def
633 by (simp add: poly_eq_iff)
634 {assume h: "\<And>x y. poly ?r x = poly ?r y"
636 from qr[rule_format, of x]
637 have "poly q x = poly ?r x * ?a0" by auto
638 also have "\<dots> = poly ?r y * ?a0" using h by simp
639 also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
640 finally have "poly q x = poly q y" .}
641 with qnc have False unfolding constant_def by blast}
642 hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
643 from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1" by auto
645 have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
646 using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
647 also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
648 using a00 unfolding norm_divide by (simp add: field_simps)
649 finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
651 from poly_decompose[OF rnc] obtain k a s where
652 kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
653 "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
654 {assume "psize p = k + 1"
655 with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
657 have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
658 using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
659 note hth = this [symmetric]
660 from reduce_poly_simple[OF kas(1,2)]
661 have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
663 {assume kn: "psize p \<noteq> k+1"
664 from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
665 have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
666 unfolding constant_def poly_pCons poly_monom
667 using kas(1) apply simp
668 by (rule exI[where x=0], rule exI[where x=1], simp)
669 from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
670 by (simp add: psize_def degree_monom_eq)
671 from less(1) [OF k1n [simplified th02] th01]
672 obtain w where w: "1 + w^k * a = 0"
673 unfolding poly_pCons poly_monom
674 using kas(2) by (cases k, auto simp add: algebra_simps)
675 from poly_bound_exists[of "cmod w" s] obtain m where
676 m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
677 have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
678 from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
679 then have wm1: "w^k * a = - 1" by simp
680 have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
681 using norm_ge_zero[of w] w0 m(1)
682 by (simp add: inverse_eq_divide zero_less_mult_iff)
683 with real_down2[OF zero_less_one] obtain t where
684 t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
685 let ?ct = "complex_of_real t"
687 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)
688 also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
689 unfolding wm1 by (simp)
690 finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
692 apply (rule cong[OF refl[of cmod]])
695 with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
696 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
697 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
698 have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
699 then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
700 from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
701 by (simp add: inverse_eq_divide field_simps)
702 with zero_less_power[OF t(1), of k]
703 have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
704 apply - apply (rule mult_strict_left_mono) by simp_all
705 have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))" using w0 t(1)
706 by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
707 then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
708 using t(1,2) m(2)[rule_format, OF tw] w0
712 with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
713 from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
715 from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
716 have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
718 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1" by arith
719 then have "cmod (poly ?r ?w) < 1"
720 unfolding kas(4)[rule_format, of ?w] r01 by simp
721 then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
722 ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
723 from cr0_contr cq0 q(2)
724 have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
725 ultimately show ?ths by blast
728 text {* Alternative version with a syntactic notion of constant polynomial. *}
730 lemma fundamental_theorem_of_algebra_alt:
731 assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
732 shows "\<exists>z. poly p z = (0::complex)"
736 {assume "c=0" hence ?case by auto}
738 {assume c0: "c\<noteq>0"
739 {assume nc: "constant (poly (pCons c cs))"
740 from nc[unfolded constant_def, rule_format, of 0]
741 have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
745 {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
747 {assume d0: "d\<noteq>0"
748 from poly_bound_exists[of 1 ds] obtain m where
749 m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
750 have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
751 from real_down2[OF dm zero_less_one] obtain x where
752 x: "x > 0" "x < cmod d / m" "x < 1" by blast
753 let ?x = "complex_of_real x"
754 from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1" by simp_all
755 from pCons.prems[rule_format, OF cx(1)]
756 have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
757 from m(2)[rule_format, OF cx(2)] x(1)
758 have th0: "cmod (?x*poly ds ?x) \<le> x*m"
759 by (simp add: norm_mult)
760 from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
761 with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
762 with cth have ?case by blast}
763 ultimately show ?case by blast
765 then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
767 from fundamental_theorem_of_algebra[OF nc] have ?case .}
768 ultimately show ?case by blast
772 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
774 lemma nullstellensatz_lemma:
775 fixes p :: "complex poly"
776 assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
777 and "degree p = n" and "n \<noteq> 0"
778 shows "p dvd (q ^ n)"
780 proof(induct n arbitrary: p q rule: nat_less_induct)
781 fix n::nat fix p q :: "complex poly"
782 assume IH: "\<forall>m<n. \<forall>p q.
783 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
784 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
785 and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
786 and dpn: "degree p = n" and n0: "n \<noteq> 0"
787 from dpn n0 have pne: "p \<noteq> 0" by auto
788 let ?ths = "p dvd (q ^ n)"
789 {fix a assume a: "poly p a = 0"
790 {assume oa: "order a p \<noteq> 0"
791 let ?op = "order a p"
792 from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
793 "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
794 note oop = order_degree[OF pne, unfolded dpn]
797 by (simp add: power_0_left)}
799 {assume q0: "q \<noteq> 0"
800 from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
801 obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
802 from ap(1) obtain s where
803 s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
804 have sne: "s \<noteq> 0"
806 {assume ds0: "degree s = 0"
807 from ds0 obtain k where kpn: "s = [:k:]"
808 by (cases s) (auto split: if_splits)
809 from sne kpn have k: "k \<noteq> 0" by simp
810 let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
811 from k oop [of a] have "q ^ n = p * ?w"
813 apply (subst r, subst s, subst kpn)
814 apply (subst power_mult_distrib, simp)
815 apply (subst power_add [symmetric], simp)
817 hence ?ths unfolding dvd_def by blast}
819 {assume ds0: "degree s \<noteq> 0"
820 from ds0 sne dpn s oa
821 have dsn: "degree s < n" apply auto
823 apply (simp add: degree_mult_eq degree_linear_power)
825 {fix x assume h: "poly s x = 0"
827 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
828 u: "s = [:- a, 1:] * u" by (rule dvdE)
829 have "p = [:- a, 1:] ^ (Suc ?op) * u"
830 by (subst s, subst u, simp only: power_Suc mult_ac)
831 with ap(2)[unfolded dvd_def] have False by blast}
833 from h have "poly p x = 0" by (subst s, simp)
834 with pq0 have "poly q x = 0" by blast
835 with r xa have "poly r x = 0"
836 by (auto simp add: uminus_add_conv_diff)}
838 from IH[rule_format, OF dsn, of s r] impth ds0
839 have "s dvd (r ^ (degree s))" by blast
840 then obtain u where u: "r ^ (degree s) = s * u" ..
841 hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
842 by (simp only: poly_mult[symmetric] poly_power[symmetric])
843 let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
844 from oop[of a] dsn have "q ^ n = p * ?w"
846 apply (subst s, subst r)
847 apply (simp only: power_mult_distrib)
848 apply (subst mult_assoc [where b=s])
849 apply (subst mult_assoc [where a=u])
850 apply (subst mult_assoc [where b=u, symmetric])
851 apply (subst u [symmetric])
852 apply (simp add: mult_ac power_add [symmetric])
854 hence ?ths unfolding dvd_def by blast}
855 ultimately have ?ths by blast }
856 ultimately have ?ths by blast}
857 then have ?ths using a order_root pne by blast}
859 {assume exa: "\<not> (\<exists>a. poly p a = 0)"
860 from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
861 ccs: "c\<noteq>0" "p = pCons c 0" by blast
863 then have pp: "\<And>x. poly p x = c" by simp
864 let ?w = "[:1/c:] * (q ^ n)"
865 from ccs have "(q ^ n) = (p * ?w)" by simp
866 hence ?ths unfolding dvd_def by blast}
867 ultimately show ?ths by blast
870 lemma nullstellensatz_univariate:
871 "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
872 p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
875 hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
876 by (auto simp add: poly_all_0_iff_0)
877 {assume "p dvd (q ^ (degree p))"
878 then obtain r where r: "q ^ (degree p) = p * r" ..
879 from r pe have False by simp}
880 with eq pe have ?thesis by blast}
882 {assume pe: "p \<noteq> 0"
883 {assume dp: "degree p = 0"
884 then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
885 by (cases p) (simp split: if_splits)
886 hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
887 from k dp have "q ^ (degree p) = p * [:1/k:]"
888 by (simp add: one_poly_def)
889 hence th2: "p dvd (q ^ (degree p))" ..
890 from th1 th2 pe have ?thesis by blast}
892 {assume dp: "degree p \<noteq> 0"
893 then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
894 {assume "p dvd (q ^ (Suc n))"
895 then obtain u where u: "q ^ (Suc n) = p * u" ..
896 {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
897 hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
898 hence False using u h(1) by (simp only: poly_mult) simp}}
899 with n nullstellensatz_lemma[of p q "degree p"] dp
900 have ?thesis by auto}
901 ultimately have ?thesis by blast}
902 ultimately show ?thesis by blast
905 text{* Useful lemma *}
907 lemma constant_degree:
908 fixes p :: "'a::{idom,ring_char_0} poly"
909 shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
912 from l[unfolded constant_def, rule_format, of _ "0"]
913 have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
914 then have "p = [:poly p 0:]" by (simp add: poly_eq_poly_eq_iff)
915 then have "degree p = degree [:poly p 0:]" by simp
916 then show ?rhs by simp
919 then obtain k where "p = [:k:]"
920 by (cases p) (simp split: if_splits)
921 then show ?lhs unfolding constant_def by auto
924 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
925 shows "degree p \<le> degree q \<or> q = 0"
926 apply (cases "q = 0", simp_all)
927 apply (erule dvd_imp_degree_le [OF pq])
930 (* Arithmetic operations on multivariate polynomials. *)
932 lemma mpoly_base_conv:
933 "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
935 lemma mpoly_norm_conv:
936 "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
938 lemma mpoly_sub_conv:
939 "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
942 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
944 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
946 lemma resolve_eq_raw: "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
947 lemma resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
948 \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast
950 lemma poly_divides_pad_rule:
951 fixes p q :: "complex poly"
952 assumes pq: "p dvd q"
953 shows "p dvd (pCons (0::complex) q)"
955 have "pCons 0 q = q * [:0,1:]" by simp
956 then have "q dvd (pCons 0 q)" ..
957 with pq show ?thesis by (rule dvd_trans)
960 lemma poly_divides_pad_const_rule:
961 fixes p q :: "complex poly"
962 assumes pq: "p dvd q"
963 shows "p dvd (smult a q)"
965 have "smult a q = q * [:a:]" by simp
966 then have "q dvd smult a q" ..
967 with pq show ?thesis by (rule dvd_trans)
971 lemma poly_divides_conv0:
972 fixes p :: "complex poly"
973 assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
974 shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
977 hence "q = p * 0" by simp
984 {assume q0: "q \<noteq> 0"
985 from l q0 have "degree p \<le> degree q"
986 by (rule dvd_imp_degree_le)
987 with lgpq have ?rhs by simp }
988 ultimately have ?rhs by blast }
989 ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
992 lemma poly_divides_conv1:
993 assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
994 and qrp': "smult a q - p' \<equiv> r"
995 shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
998 from pp' obtain t where t: "p' = p * t" ..
1000 then obtain u where u: "q = p * u" ..
1001 have "r = p * (smult a u - t)"
1002 using u qrp' [symmetric] t by (simp add: algebra_simps)
1006 then obtain u where u: "r = p * u" ..
1007 from u [symmetric] t qrp' [symmetric] a0
1008 have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
1010 ultimately have "?lhs = ?rhs" by blast }
1011 thus "?lhs \<equiv> ?rhs" by - (atomize(full), blast)
1014 lemma basic_cqe_conv1:
1015 "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
1016 "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
1017 "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
1018 "(\<exists>x. poly 0 x = 0) \<equiv> True"
1019 "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
1021 lemma basic_cqe_conv2:
1022 assumes l:"p \<noteq> 0"
1023 shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
1026 assume h: "h\<noteq>0" "t=0" "pCons a (pCons b p) = pCons h t"
1027 with l have False by simp}
1028 hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
1030 from fundamental_theorem_of_algebra_alt[OF th]
1031 show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
1034 lemma basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
1036 have "p = 0 \<longleftrightarrow> poly p = poly 0"
1037 by (simp add: poly_eq_poly_eq_iff)
1038 also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by auto
1039 finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
1040 by - (atomize (full), blast)
1043 lemma basic_cqe_conv3:
1044 fixes p q :: "complex poly"
1045 assumes l: "p \<noteq> 0"
1046 shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
1048 from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
1049 from nullstellensatz_univariate[of "pCons a p" q] l
1050 show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
1052 by - (atomize (full), auto)
1055 lemma basic_cqe_conv4:
1056 fixes p q :: "complex poly"
1057 assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
1058 shows "p dvd (q ^ n) \<equiv> p dvd r"
1060 from h have "poly (q ^ n) = poly r" by auto
1061 then have "(q ^ n) = r" by (simp add: poly_eq_poly_eq_iff)
1062 thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
1065 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
1068 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
1069 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
1070 lemma negate_negate_rule: "Trueprop P \<equiv> (\<not> P \<equiv> False)" by (atomize (full), auto)
1072 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
1073 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"
1074 by (atomize (full)) simp_all
1075 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True" by simp
1076 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))" (is "?l \<equiv> ?r")
1078 assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
1080 assume "p \<and> q \<equiv> p \<and> r" "p"
1081 thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
1083 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp