src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author hoelzl Tue Nov 05 09:45:02 2013 +0100 (2013-11-05) changeset 54263 c4159fe6fa46 parent 54230 b1d955791529 child 54489 03ff4d1e6784 permissions -rw-r--r--
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
7 begin
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"
18 proof-
19   obtain x y where xy: "z = Complex x y" by (cases z)
20   {assume y0: "y = 0"
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)}
24     moreover
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}
29   moreover
30   {assume y0: "y\<noteq>0"
31     {fix x y
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) }
36     note th = this
37     have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
38       by (simp add: power2_eq_square)
39     from th[of x y]
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])
53       using th1 th2  ..}
54   ultimately show ?thesis by blast
55 qed
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)"
75 proof(induct p)
76   case 0 thus ?case by (rule exI[where x=1], simp)
77 next
78   case (pCons c cs)
79   from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
80     by blast
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
83   {fix z
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
93 qed
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"
99 where
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)"
114 apply (induct p)
115 apply (simp add: offset_poly_0)
116 apply (simp add: offset_poly_pCons algebra_simps)
117 done
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)
127 done
129 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
130 apply (induct 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)
139 done
141 definition
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"
150     unfolding psize_def
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)
154 qed
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"
159 proof
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
164 qed
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"
170 proof-
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
183 qed
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"
189 using n
190 proof(induct n rule: nat_less_induct)
191   fix n
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"
194   {assume e: "even n"
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" ..}
201   moreover
202   {assume o: "odd 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)
216       done
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
234 qed
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)"
245 proof-
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
251   let ?h = "f o g"
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>"
254   proof
255     fix n
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
257   qed
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
263     using f(2) .
264   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
265   proof
266     fix n
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
268   qed
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
275     using g(2) .
277   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
278     by blast
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"
283     by blast
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
299 qed
301 text{* Polynomial is continuous. *}
303 lemma poly_cont:
304   assumes ep: "e > 0"
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"
306 proof-
307   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
308   proof
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)
313   qed
314   {fix w
315     note q(2)[of "w - z", simplified]}
316   note th = this
317   show ?thesis unfolding th[symmetric]
318   proof(induct q)
319     case 0 thus ?case  using ep by auto
320   next
321     case (pCons c cs)
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)
330     show ?case
331       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
332         fix d w
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
339       qed
340     qed
341 qed
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)"
347 proof-
348   {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
349       apply -
350       apply (rule exI[where x=0])
351       apply auto
352       apply (subgoal_tac "cmod w < 0")
353       apply simp
354       apply arith
355       done }
356   moreover
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
360     {fix x z
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
367     let ?m = "-s"
368     {fix y
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"
374       by auto
375     {fix n::nat
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)"
378         by simp}
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)"
382       by blast
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"
385       by blast
386     {fix w
387       assume wr: "cmod w \<le> r"
388       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
389       {assume e: "?e > 0"
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)}
396         note th1 = this
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
406           ==> False" by arith}
407       note th0 = this
408       have ath:
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"
415         using N2 by auto
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"
427         by arith
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
436       with s1m[OF wr]
437       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
438     hence ?thesis by blast}
439   ultimately show ?thesis by blast
440 qed
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])
446   done
448 lemma cispi: "cis pi = -1"
449   unfolding cis_def
450   by simp
452 lemma "(rcis (sqrt (abs r)) ((pi + a)/2))\<^sup>2 = rcis (- abs r) a"
453   unfolding power2_eq_square
455   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
456   done
458 text {* Nonzero polynomial in z goes to infinity as z does. *}
460 lemma poly_infinity:
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)"
463 using ex
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}
482   moreover
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
486     {fix z
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)"
496         using cs0' by simp}
497     then have ?case  by blast}
498   ultimately show ?case by blast
499 qed simp
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)"
504 proof(induct p)
505   case (pCons c cs)
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"
513       from v[of 0] r[OF z]
514       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
515         by simp }
516     note v0 = this
517     from v0 v ath[of r] have ?case by blast}
518   moreover
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
523 qed simp
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)
531   done
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
538   after the first.  *}
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)"
544 unfolding psize_def
545 using nz
546 proof(induct p)
547   case 0 thus ?case by simp
548 next
549   case (pCons c cs)
550   {assume c0: "c = 0"
551     from pCons.hyps pCons.prems c0 have ?case
552       apply (auto)
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)
556       by (auto)}
557   moreover
558   {assume c0: "c\<noteq>0"
559     hence ?case apply-
560       apply (rule exI[where x=0])
561       apply (rule exI[where x=c], clarsimp)
562       apply (rule exI[where x=cs])
563       apply auto
564       done}
565   ultimately show ?case by blast
566 qed
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)"
573 using nc
574 proof(induct p)
575   case 0 thus ?case by (simp add: constant_def)
576 next
577   case (pCons c cs)
578   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
579     {fix x y
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]
584   show ?case
585     apply clarsimp
586     apply (rule_tac x="k+1" in exI)
587     apply (rule_tac x="a" in exI)
588     apply simp
589     apply (rule_tac x="q" in exI)
590     apply (auto simp add: psize_def split: if_splits)
591     done
592 qed
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"
599 using nc
600 proof(induct "psize p" arbitrary: p rule: less_induct)
601   case less
602   let ?p = "poly p"
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}
609   moreover
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
615       {fix x y
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
625     let ?a0 = "poly q 0"
626     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
627     from a00
628     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
629       by simp
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"
635       {fix x 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
644     {fix w
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" .}
650     note mrmq_eq = this
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
656       {fix w
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}
662     moreover
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"
686       let ?w = "?ct * w"
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)"
691         apply -
692         apply (rule cong[OF refl[of cmod]])
693         apply assumption
694         done
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
709         apply (simp only: )
710         apply auto
711         done
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"
714         by auto
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" .
717       from th11 th12
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
726 qed
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)"
733 using nc
734 proof(induct p)
735   case (pCons c cs)
736   {assume "c=0" hence ?case by auto}
737   moreover
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
742       hence "cs = 0"
743         proof(induct cs)
744           case (pCons d ds)
745           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
746           moreover
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
764         qed simp}
765       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
766         by blast
767       from fundamental_theorem_of_algebra[OF nc] have ?case .}
768   ultimately show ?case by blast
769 qed simp
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)"
779 using assms
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]
795       {assume q0: "q = 0"
796         hence ?ths using n0
797           by (simp add: power_0_left)}
798       moreover
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"
805           using s pne by auto
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"
812             apply -
813             apply (subst r, subst s, subst kpn)
814             apply (subst power_mult_distrib, simp)
815             apply (subst power_add [symmetric], simp)
816             done
817           hence ?ths unfolding dvd_def by blast}
818         moreover
819         {assume ds0: "degree s \<noteq> 0"
820           from ds0 sne dpn s oa
821             have dsn: "degree s < n" apply auto
822               apply (erule ssubst)
823               apply (simp add: degree_mult_eq degree_linear_power)
824               done
825             {fix x assume h: "poly s x = 0"
826               {assume xa: "x = a"
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}
832               note xa = this
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"
837             note impth = this
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"
845               apply -
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])
853               done
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}
858   moreover
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
868 qed
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)"
873 proof-
874   {assume pe: "p = 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}
881   moreover
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}
891     moreover
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
903 qed
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")
910 proof
911   assume l: ?lhs
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
917 next
918   assume r: ?rhs
919   then obtain k where "p = [:k:]"
920     by (cases p) (simp split: if_splits)
921   then show ?lhs unfolding constant_def by auto
922 qed
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])
928 done
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"
940   by simp
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
951   fixes p q :: "complex poly"
952   assumes pq: "p dvd q"
953   shows "p dvd (pCons (0::complex) q)"
954 proof-
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)
958 qed
961   fixes p q :: "complex poly"
962   assumes pq: "p dvd q"
963   shows "p dvd (smult a q)"
964 proof-
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)
968 qed
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")
975 proof-
976   {assume r: ?rhs
977     hence "q = p * 0" by simp
978     hence ?lhs ..}
979   moreover
980   {assume l: ?lhs
981     {assume q0: "q = 0"
982       hence ?rhs by simp}
983     moreover
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)
990 qed
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")
996 proof-
997   {
998   from pp' obtain t where t: "p' = p * t" ..
999   {assume l: ?lhs
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)
1003      then have ?rhs ..}
1004   moreover
1005   {assume r: ?rhs
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)
1009     hence ?lhs ..}
1010   ultimately have "?lhs = ?rhs" by blast }
1011 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
1012 qed
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"
1024 proof-
1025   {fix h t
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)"
1029     by blast
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
1032 qed
1034 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
1035 proof-
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)
1041 qed
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)))"
1047 proof-
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)))"
1051     unfolding dp
1052     by - (atomize (full), auto)
1053 qed
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"
1059 proof-
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
1063 qed
1065 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
1066   by simp
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")
1077 proof
1078   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
1079 next
1080   assume "p \<and> q \<equiv> p \<and> r" "p"
1081   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
1082 qed
1083 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
1085 end