src/HOL/Library/Fundamental_Theorem_Algebra.thy
 author huffman Tue Jan 17 16:30:54 2012 +0100 (2012-01-17) changeset 46240 933f35c4e126 parent 41529 ba60efa2fd08 child 49962 a8cc904a6820 permissions -rw-r--r--
factor-cancellation simprocs now call the full simplifier to prove that factors are non-zero
1 (* Author: Amine Chaieb, TU Muenchen *)
3 header{*Fundamental Theorem of Algebra*}
5 theory Fundamental_Theorem_Algebra
6 imports Polynomial Complex
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 ^ 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^2 / 4 = (x / 2) ^ 2"
38       by (simp add: power2_eq_square)
39     from th[of x y]
40     have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^2 / 4)) = (sqrt (x * x + y * y) - x) / 2" unfolding sq4 by simp_all
41     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"
42       unfolding power2_eq_square by simp
43     have "sqrt 4 = sqrt (2^2)" by simp
44     hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs)
45     have th2: "2 *(y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
46       using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
47       unfolding power2_eq_square
48       by (simp add: algebra_simps real_sqrt_divide sqrt4)
49      from y0 xy have ?thesis  apply (simp add: csqrt_def power2_eq_square)
50        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])
51       using th1 th2  ..}
52   ultimately show ?thesis by blast
53 qed
56 subsection{* More lemmas about module of complex numbers *}
58 lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
59   by (rule of_real_power [symmetric])
61 lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2"
62   apply (rule exI[where x = "min d1 d2 / 2"])
63   by (simp add: field_simps min_def)
65 text{* The triangle inequality for cmod *}
66 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
67   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto
69 subsection{* Basic lemmas about complex polynomials *}
71 lemma poly_bound_exists:
72   shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)"
73 proof(induct p)
74   case 0 thus ?case by (rule exI[where x=1], simp)
75 next
76   case (pCons c cs)
77   from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"
78     by blast
79   let ?k = " 1 + cmod c + \<bar>r * m\<bar>"
80   have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
81   {fix z
82     assume H: "cmod z \<le> r"
83     from m H have th: "cmod (poly cs z) \<le> m" by blast
84     from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith
85     have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"
86       using norm_triangle_ineq[of c "z* poly cs z"] by simp
87     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)
88     also have "\<dots> \<le> ?k" by simp
89     finally have "cmod (poly (pCons c cs) z) \<le> ?k" .}
90   with kp show ?case by blast
91 qed
94 text{* Offsetting the variable in a polynomial gives another of same degree *}
96 definition
97   "offset_poly p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p"
99 lemma offset_poly_0: "offset_poly 0 h = 0"
100   unfolding offset_poly_def by (simp add: poly_rec_0)
102 lemma offset_poly_pCons:
103   "offset_poly (pCons a p) h =
104     smult h (offset_poly p h) + pCons a (offset_poly p h)"
105   unfolding offset_poly_def by (simp add: poly_rec_pCons)
107 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
108 by (simp add: offset_poly_pCons offset_poly_0)
110 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
111 apply (induct p)
112 apply (simp add: offset_poly_0)
113 apply (simp add: offset_poly_pCons algebra_simps)
114 done
116 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
117 by (induct p arbitrary: a, simp, force)
119 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
120 apply (safe intro!: offset_poly_0)
121 apply (induct p, simp)
122 apply (simp add: offset_poly_pCons)
123 apply (frule offset_poly_eq_0_lemma, simp)
124 done
126 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
127 apply (induct p)
128 apply (simp add: offset_poly_0)
129 apply (case_tac "p = 0")
130 apply (simp add: offset_poly_0 offset_poly_pCons)
131 apply (simp add: offset_poly_pCons)
132 apply (subst degree_add_eq_right)
133 apply (rule le_less_trans [OF degree_smult_le])
134 apply (simp add: offset_poly_eq_0_iff)
135 apply (simp add: offset_poly_eq_0_iff)
136 done
138 definition
139   "psize p = (if p = 0 then 0 else Suc (degree p))"
141 lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
142   unfolding psize_def by simp
144 lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
145 proof (intro exI conjI)
146   show "psize (offset_poly p a) = psize p"
147     unfolding psize_def
148     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
149   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
150     by (simp add: poly_offset_poly)
151 qed
153 text{* An alternative useful formulation of completeness of the reals *}
154 lemma real_sup_exists: assumes ex: "\<exists>x. P x" and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
155   shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
156 proof-
157   from ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast
158   from ex have thx:"\<exists>x. x \<in> Collect P" by blast
159   from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y"
160     by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)
161   from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"
162     by blast
163   from Y[OF x] have xY: "x < Y" .
164   from L have L': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
165   from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y"
166     apply (clarsimp, atomize (full)) by auto
167   from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
168   {fix y
169     {fix z assume z: "P z" "y < z"
170       from L' z have "y < L" by auto }
171     moreover
172     {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"
173       hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto
174       from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)
175       with yL(1) have False  by arith}
176     ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}
177   thus ?thesis by blast
178 qed
180 subsection {* Fundamental theorem of algebra *}
181 lemma  unimodular_reduce_norm:
182   assumes md: "cmod z = 1"
183   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
184 proof-
185   obtain x y where z: "z = Complex x y " by (cases z, auto)
186   from md z have xy: "x^2 + y^2 = 1" by (simp add: cmod_def)
187   {assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
188     from C z xy have "2*x \<le> 1" "2*x \<ge> -1" "2*y \<le> 1" "2*y \<ge> -1"
189       by (simp_all add: cmod_def power2_eq_square algebra_simps)
190     hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all
191     hence "(abs (2 * x))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"
192       by - (rule power_mono, simp, simp)+
193     hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1"
194       by (simp_all  add: power2_abs power_mult_distrib)
195     from add_mono[OF th0] xy have False by simp }
196   thus ?thesis unfolding linorder_not_le[symmetric] by blast
197 qed
199 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
200 lemma reduce_poly_simple:
201  assumes b: "b \<noteq> 0" and n: "n\<noteq>0"
202   shows "\<exists>z. cmod (1 + b * z^n) < 1"
203 using n
204 proof(induct n rule: nat_less_induct)
205   fix n
206   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0"
207   let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
208   {assume e: "even n"
209     hence "\<exists>m. n = 2*m" by presburger
210     then obtain m where m: "n = 2*m" by blast
211     from n m have "m\<noteq>0" "m < n" by presburger+
212     with IH[rule_format, of m] obtain z where z: "?P z m" by blast
213     from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt)
214     hence "\<exists>z. ?P z n" ..}
215   moreover
216   {assume o: "odd n"
217     have th0: "cmod (complex_of_real (cmod b) / b) = 1"
218       using b by (simp add: norm_divide)
219     from o have "\<exists>m. n = Suc (2*m)" by presburger+
220     then obtain m where m: "n = Suc (2*m)" by blast
221     from unimodular_reduce_norm[OF th0] o
222     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
223       apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp)
224       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp add: diff_minus)
225       apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
226       apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult)
227       apply (rule_tac x="- ii" in exI, simp add: m power_mult)
228       apply (cases "even m", rule_tac x="- ii" in exI, simp add: m power_mult diff_minus)
229       apply (rule_tac x="ii" in exI, simp add: m power_mult diff_minus)
230       done
231     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast
232     let ?w = "v / complex_of_real (root n (cmod b))"
233     from odd_real_root_pow[OF o, of "cmod b"]
234     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
235       by (simp add: power_divide complex_of_real_power)
236     have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide)
237     hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp
238     have th4: "cmod (complex_of_real (cmod b) / b) *
239    cmod (1 + b * (v ^ n / complex_of_real (cmod b)))
240    < cmod (complex_of_real (cmod b) / b) * 1"
241       apply (simp only: norm_mult[symmetric] right_distrib)
242       using b v by (simp add: th2)
244     from mult_less_imp_less_left[OF th4 th3]
245     have "?P ?w n" unfolding th1 .
246     hence "\<exists>z. ?P z n" .. }
247   ultimately show "\<exists>z. ?P z n" by blast
248 qed
250 text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}
252 lemma metric_bound_lemma: "cmod (x - y) <= \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
253   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y" ]
254   unfolding cmod_def by simp
256 lemma bolzano_weierstrass_complex_disc:
257   assumes r: "\<forall>n. cmod (s n) \<le> r"
258   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
259 proof-
260   from seq_monosub[of "Re o s"]
261   obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
262     unfolding o_def by blast
263   from seq_monosub[of "Im o s o f"]
264   obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast
265   let ?h = "f o g"
266   from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith
267   have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>"
268   proof
269     fix n
270     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
271   qed
272   have conv1: "convergent (\<lambda>n. Re (s ( f n)))"
273     apply (rule Bseq_monoseq_convergent)
274     apply (simp add: Bseq_def)
275     apply (rule exI[where x= "r + 1"])
276     using th rp apply simp
277     using f(2) .
278   have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>"
279   proof
280     fix n
281     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]  show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith
282   qed
284   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
285     apply (rule Bseq_monoseq_convergent)
286     apply (simp add: Bseq_def)
287     apply (rule exI[where x= "r + 1"])
288     using th rp apply simp
289     using g(2) .
291   from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
292     by blast
293   hence  x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n)) - x \<bar> < r"
294     unfolding LIMSEQ_iff real_norm_def .
296   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
297     by blast
298   hence  y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n))) - y \<bar> < r"
299     unfolding LIMSEQ_iff real_norm_def .
300   let ?w = "Complex x y"
301   from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto
302   {fix e assume ep: "e > (0::real)"
303     hence e2: "e/2 > 0" by simp
304     from x[rule_format, OF e2] y[rule_format, OF e2]
305     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
306     {fix n assume nN12: "n \<ge> N1 + N2"
307       hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+
308       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
309       have "cmod (s (?h n) - ?w) < e"
310         using metric_bound_lemma[of "s (f (g n))" ?w] by simp }
311     hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e" by blast }
312   with hs show ?thesis  by blast
313 qed
315 text{* Polynomial is continuous. *}
317 lemma poly_cont:
318   assumes ep: "e > 0"
319   shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (poly p w - poly p z) < e"
320 proof-
321   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
322   proof
323     show "degree (offset_poly p z) = degree p"
324       by (rule degree_offset_poly)
325     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
326       by (rule poly_offset_poly)
327   qed
328   {fix w
329     note q(2)[of "w - z", simplified]}
330   note th = this
331   show ?thesis unfolding th[symmetric]
332   proof(induct q)
333     case 0 thus ?case  using ep by auto
334   next
335     case (pCons c cs)
336     from poly_bound_exists[of 1 "cs"]
337     obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast
338     from ep m(1) have em0: "e/m > 0" by (simp add: field_simps)
339     have one0: "1 > (0::real)"  by arith
340     from real_lbound_gt_zero[OF one0 em0]
341     obtain d where d: "d >0" "d < 1" "d < e / m" by blast
342     from d(1,3) m(1) have dm: "d*m > 0" "d*m < e"
343       by (simp_all add: field_simps mult_pos_pos)
344     show ?case
345       proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
346         fix d w
347         assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (w-z) < d"
348         hence d1: "cmod (w-z) \<le> 1" "d \<ge> 0" by simp_all
349         from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps)
350         from H have th: "cmod (w-z) \<le> d" by simp
351         from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
352         show "cmod (w - z) * cmod (poly cs (w - z)) < e" by simp
353       qed
354     qed
355 qed
357 text{* Hence a polynomial attains minimum on a closed disc
358   in the complex plane. *}
359 lemma  poly_minimum_modulus_disc:
360   "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
361 proof-
362   {assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le
363       apply -
364       apply (rule exI[where x=0])
365       apply auto
366       apply (subgoal_tac "cmod w < 0")
367       apply simp
368       apply arith
369       done }
370   moreover
371   {assume rp: "r \<ge> 0"
372     from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp
373     hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"  by blast
374     {fix x z
375       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not>x < 1"
376       hence "- x < 0 " by arith
377       with H(2) norm_ge_zero[of "poly p z"]  have False by simp }
378     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z" by blast
379     from real_sup_exists[OF mth1 mth2] obtain s where
380       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
381     let ?m = "-s"
382     {fix y
383       from s[rule_format, of "-y"] have
384     "(\<exists>z x. cmod z \<le> r \<and> -(- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
385         unfolding minus_less_iff[of y ] equation_minus_iff by blast }
386     note s1 = this[unfolded minus_minus]
387     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
388       by auto
389     {fix n::nat
390       from s1[rule_format, of "?m + 1/real (Suc n)"]
391       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
392         by simp}
393     hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
394     from choice[OF th] obtain g where
395       g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)"
396       by blast
397     from bolzano_weierstrass_complex_disc[OF g(1)]
398     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
399       by blast
400     {fix w
401       assume wr: "cmod w \<le> r"
402       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
403       {assume e: "?e > 0"
404         hence e2: "?e/2 > 0" by simp
405         from poly_cont[OF e2, of z p] obtain d where
406           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
407         {fix w assume w: "cmod (w - z) < d"
408           have "cmod(poly p w - poly p z) < ?e / 2"
409             using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)}
410         note th1 = this
412         from fz(2)[rule_format, OF d(1)] obtain N1 where
413           N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d" by blast
414         from reals_Archimedean2[of "2/?e"] obtain N2::nat where
415           N2: "2/?e < real N2" by blast
416         have th2: "cmod(poly p (g(f(N1 + N2))) - poly p z) < ?e/2"
417           using N1[rule_format, of "N1 + N2"] th1 by simp
418         {fix a b e2 m :: real
419         have "a < e2 \<Longrightarrow> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a
420           ==> False" by arith}
421       note th0 = this
422       have ath:
423         "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < e" by arith
424       from s1m[OF g(1)[rule_format]]
425       have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
426       from seq_suble[OF fz(1), of "N1+N2"]
427       have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp
428       have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0"
429         using N2 by auto
430       from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp
431       from g(2)[rule_format, of "f (N1 + N2)"]
432       have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
433       from order_less_le_trans[OF th01 th00]
434       have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
435       from N2 have "2/?e < real (Suc (N1 + N2))" by arith
436       with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
437       have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide)
438       with ath[OF th31 th32]
439       have thc1:"\<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar>< ?e/2" by arith
440       have ath2: "\<And>(a::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<bar>a - m\<bar> + c"
441         by arith
442       have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar>
443 \<le> cmod (poly p (g (f (N1 + N2))) - poly p z)"
444         by (simp add: norm_triangle_ineq3)
445       from ath2[OF th22, of ?m]
446       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
447       from th0[OF th2 thc1 thc2] have False .}
448       hence "?e = 0" by auto
449       then have "cmod (poly p z) = ?m" by simp
450       with s1m[OF wr]
451       have "cmod (poly p z) \<le> cmod (poly p w)" by simp }
452     hence ?thesis by blast}
453   ultimately show ?thesis by blast
454 qed
456 lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a"
457   unfolding power2_eq_square
458   apply (simp add: rcis_mult)
459   apply (simp add: power2_eq_square[symmetric])
460   done
462 lemma cispi: "cis pi = -1"
463   unfolding cis_def
464   by simp
466 lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis (- abs r) a"
467   unfolding power2_eq_square
469   apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
470   done
472 text {* Nonzero polynomial in z goes to infinity as z does. *}
474 lemma poly_infinity:
475   assumes ex: "p \<noteq> 0"
476   shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"
477 using ex
478 proof(induct p arbitrary: a d)
479   case (pCons c cs a d)
480   {assume H: "cs \<noteq> 0"
481     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
482     let ?r = "1 + \<bar>r\<bar>"
483     {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"
484       have r0: "r \<le> cmod z" using h by arith
485       from r[rule_format, OF r0]
486       have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith
487       from h have z1: "cmod z \<ge> 1" by arith
488       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
489       have th1: "d \<le> cmod(z * poly (pCons c cs) z) - cmod a"
490         unfolding norm_mult by (simp add: algebra_simps)
491       from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]
492       have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"
493         by (simp add: diff_le_eq algebra_simps)
494       from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}
495     hence ?case by blast}
496   moreover
497   {assume cs0: "\<not> (cs \<noteq> 0)"
498     with pCons.prems have c0: "c \<noteq> 0" by simp
499     from cs0 have cs0': "cs = 0" by simp
500     {fix z
501       assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"
502       from c0 have "cmod c > 0" by simp
503       from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"
504         by (simp add: field_simps norm_mult)
505       have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith
506       from complex_mod_triangle_sub[of "z*c" a ]
507       have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a"
508         by (simp add: algebra_simps)
509       from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"
510         using cs0' by simp}
511     then have ?case  by blast}
512   ultimately show ?case by blast
513 qed simp
515 text {* Hence polynomial's modulus attains its minimum somewhere. *}
516 lemma poly_minimum_modulus:
517   "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
518 proof(induct p)
519   case (pCons c cs)
520   {assume cs0: "cs \<noteq> 0"
521     from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c]
522     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
523     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith
524     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
525     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
526     {fix z assume z: "r \<le> cmod z"
527       from v[of 0] r[OF z]
528       have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
529         by simp }
530     note v0 = this
531     from v0 v ath[of r] have ?case by blast}
532   moreover
533   {assume cs0: "\<not> (cs \<noteq> 0)"
534     hence th:"cs = 0" by simp
535     from th pCons.hyps have ?case by simp}
536   ultimately show ?case by blast
537 qed simp
539 text{* Constant function (non-syntactic characterization). *}
540 definition "constant f = (\<forall>x y. f x = f y)"
542 lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
543   unfolding constant_def psize_def
544   apply (induct p, auto)
545   done
547 lemma poly_replicate_append:
548   "poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x"
549   by (simp add: poly_monom)
551 text {* Decomposition of polynomial, skipping zero coefficients
552   after the first.  *}
554 lemma poly_decompose_lemma:
555  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))"
556   shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and>
557                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
558 unfolding psize_def
559 using nz
560 proof(induct p)
561   case 0 thus ?case by simp
562 next
563   case (pCons c cs)
564   {assume c0: "c = 0"
565     from pCons.hyps pCons.prems c0 have ?case
566       apply (auto)
567       apply (rule_tac x="k+1" in exI)
568       apply (rule_tac x="a" in exI, clarsimp)
569       apply (rule_tac x="q" in exI)
570       by (auto)}
571   moreover
572   {assume c0: "c\<noteq>0"
573     hence ?case apply-
574       apply (rule exI[where x=0])
575       apply (rule exI[where x=c], clarsimp)
576       apply (rule exI[where x=cs])
577       apply auto
578       done}
579   ultimately show ?case by blast
580 qed
582 lemma poly_decompose:
583   assumes nc: "~constant(poly p)"
584   shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and>
585                psize q + k + 1 = psize p \<and>
586               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
587 using nc
588 proof(induct p)
589   case 0 thus ?case by (simp add: constant_def)
590 next
591   case (pCons c cs)
592   {assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
593     {fix x y
594       from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)}
595     with pCons.prems have False by (auto simp add: constant_def)}
596   hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
597   from poly_decompose_lemma[OF th]
598   show ?case
599     apply clarsimp
600     apply (rule_tac x="k+1" in exI)
601     apply (rule_tac x="a" in exI)
602     apply simp
603     apply (rule_tac x="q" in exI)
604     apply (auto simp add: power_Suc)
605     apply (auto simp add: psize_def split: if_splits)
606     done
607 qed
609 text{* Fundamental theorem of algebra *}
611 lemma fundamental_theorem_of_algebra:
612   assumes nc: "~constant(poly p)"
613   shows "\<exists>z::complex. poly p z = 0"
614 using nc
615 proof(induct "psize p" arbitrary: p rule: less_induct)
616   case less
617   let ?p = "poly p"
618   let ?ths = "\<exists>z. ?p z = 0"
620   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
621   from poly_minimum_modulus obtain c where
622     c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
623   {assume pc: "?p c = 0" hence ?ths by blast}
624   moreover
625   {assume pc0: "?p c \<noteq> 0"
626     from poly_offset[of p c] obtain q where
627       q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
628     {assume h: "constant (poly q)"
629       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
630       {fix x y
631         from th have "?p x = poly q (x - c)" by auto
632         also have "\<dots> = poly q (y - c)"
633           using h unfolding constant_def by blast
634         also have "\<dots> = ?p y" using th by auto
635         finally have "?p x = ?p y" .}
636       with less(2) have False unfolding constant_def by blast }
637     hence qnc: "\<not> constant (poly q)" by blast
638     from q(2) have pqc0: "?p c = poly q 0" by simp
639     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
640     let ?a0 = "poly q 0"
641     from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp
642     from a00
643     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
644       by simp
645     let ?r = "smult (inverse ?a0) q"
646     have lgqr: "psize q = psize ?r"
647       using a00 unfolding psize_def degree_def
648       by (simp add: expand_poly_eq)
649     {assume h: "\<And>x y. poly ?r x = poly ?r y"
650       {fix x y
651         from qr[rule_format, of x]
652         have "poly q x = poly ?r x * ?a0" by auto
653         also have "\<dots> = poly ?r y * ?a0" using h by simp
654         also have "\<dots> = poly q y" using qr[rule_format, of y] by simp
655         finally have "poly q x = poly q y" .}
656       with qnc have False unfolding constant_def by blast}
657     hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast
658     from qr[rule_format, of 0] a00  have r01: "poly ?r 0 = 1" by auto
659     {fix w
660       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
661         using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
662       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
663         using a00 unfolding norm_divide by (simp add: field_simps)
664       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
665     note mrmq_eq = this
666     from poly_decompose[OF rnc] obtain k a s where
667       kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
668       "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
669     {assume "psize p = k + 1"
670       with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
671       {fix w
672         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
673           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
674       note hth = this [symmetric]
675         from reduce_poly_simple[OF kas(1,2)]
676       have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
677     moreover
678     {assume kn: "psize p \<noteq> k+1"
679       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
680       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
681         unfolding constant_def poly_pCons poly_monom
682         using kas(1) apply simp
683         by (rule exI[where x=0], rule exI[where x=1], simp)
684       from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
685         by (simp add: psize_def degree_monom_eq)
686       from less(1) [OF k1n [simplified th02] th01]
687       obtain w where w: "1 + w^k * a = 0"
688         unfolding poly_pCons poly_monom
689         using kas(2) by (cases k, auto simp add: algebra_simps)
690       from poly_bound_exists[of "cmod w" s] obtain m where
691         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
692       have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left)
693       from w have "(1 + w ^ k * a) - 1 = 0 - 1" by simp
694       then have wm1: "w^k * a = - 1" by simp
695       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
696         using norm_ge_zero[of w] w0 m(1)
697           by (simp add: inverse_eq_divide zero_less_mult_iff)
698       with real_down2[OF zero_less_one] obtain t where
699         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
700       let ?ct = "complex_of_real t"
701       let ?w = "?ct * w"
702       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)
703       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
704         unfolding wm1 by (simp)
705       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
706         apply -
707         apply (rule cong[OF refl[of cmod]])
708         apply assumption
709         done
710       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
711       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
712       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
713       have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto
714       then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult)
715       from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1"
716         by (simp add: inverse_eq_divide field_simps)
717       with zero_less_power[OF t(1), of k]
718       have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
719         apply - apply (rule mult_strict_left_mono) by simp_all
720       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))"  using w0 t(1)
721         by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult)
722       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
723         using t(1,2) m(2)[rule_format, OF tw] w0
724         apply (simp only: )
725         apply auto
726         done
727       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp
728       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
729         by auto
730       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
731       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
732       from th11 th12
733       have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"  by arith
734       then have "cmod (poly ?r ?w) < 1"
735         unfolding kas(4)[rule_format, of ?w] r01 by simp
736       then have "\<exists>w. cmod (poly ?r w) < 1" by blast}
737     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast
738     from cr0_contr cq0 q(2)
739     have ?ths unfolding mrmq_eq not_less[symmetric] by auto}
740   ultimately show ?ths by blast
741 qed
743 text {* Alternative version with a syntactic notion of constant polynomial. *}
745 lemma fundamental_theorem_of_algebra_alt:
746   assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
747   shows "\<exists>z. poly p z = (0::complex)"
748 using nc
749 proof(induct p)
750   case (pCons c cs)
751   {assume "c=0" hence ?case by auto}
752   moreover
753   {assume c0: "c\<noteq>0"
754     {assume nc: "constant (poly (pCons c cs))"
755       from nc[unfolded constant_def, rule_format, of 0]
756       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
757       hence "cs = 0"
758         proof(induct cs)
759           case (pCons d ds)
760           {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}
761           moreover
762           {assume d0: "d\<noteq>0"
763             from poly_bound_exists[of 1 ds] obtain m where
764               m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
765             have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps)
766             from real_down2[OF dm zero_less_one] obtain x where
767               x: "x > 0" "x < cmod d / m" "x < 1" by blast
768             let ?x = "complex_of_real x"
769             from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1" by simp_all
770             from pCons.prems[rule_format, OF cx(1)]
771             have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric])
772             from m(2)[rule_format, OF cx(2)] x(1)
773             have th0: "cmod (?x*poly ds ?x) \<le> x*m"
774               by (simp add: norm_mult)
775             from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps)
776             with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto
777             with cth  have ?case by blast}
778           ultimately show ?case by blast
779         qed simp}
780       then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0
781         by blast
782       from fundamental_theorem_of_algebra[OF nc] have ?case .}
783   ultimately show ?case by blast
784 qed simp
787 subsection{* Nullstellensatz, degrees and divisibility of polynomials *}
789 lemma nullstellensatz_lemma:
790   fixes p :: "complex poly"
791   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
792   and "degree p = n" and "n \<noteq> 0"
793   shows "p dvd (q ^ n)"
794 using assms
795 proof(induct n arbitrary: p q rule: nat_less_induct)
796   fix n::nat fix p q :: "complex poly"
797   assume IH: "\<forall>m<n. \<forall>p q.
798                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
799                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
800     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
801     and dpn: "degree p = n" and n0: "n \<noteq> 0"
802   from dpn n0 have pne: "p \<noteq> 0" by auto
803   let ?ths = "p dvd (q ^ n)"
804   {fix a assume a: "poly p a = 0"
805     {assume oa: "order a p \<noteq> 0"
806       let ?op = "order a p"
807       from pne have ap: "([:- a, 1:] ^ ?op) dvd p"
808         "\<not> [:- a, 1:] ^ (Suc ?op) dvd p" using order by blast+
809       note oop = order_degree[OF pne, unfolded dpn]
810       {assume q0: "q = 0"
811         hence ?ths using n0
812           by (simp add: power_0_left)}
813       moreover
814       {assume q0: "q \<noteq> 0"
815         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
816         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
817         from ap(1) obtain s where
818           s: "p = [:- a, 1:] ^ ?op * s" by (rule dvdE)
819         have sne: "s \<noteq> 0"
820           using s pne by auto
821         {assume ds0: "degree s = 0"
822           from ds0 have "\<exists>k. s = [:k:]"
823             by (cases s, simp split: if_splits)
824           then obtain k where kpn: "s = [:k:]" by blast
825           from sne kpn have k: "k \<noteq> 0" by simp
826           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
827           from k oop [of a] have "q ^ n = p * ?w"
828             apply -
829             apply (subst r, subst s, subst kpn)
830             apply (subst power_mult_distrib, simp)
831             apply (subst power_add [symmetric], simp)
832             done
833           hence ?ths unfolding dvd_def by blast}
834         moreover
835         {assume ds0: "degree s \<noteq> 0"
836           from ds0 sne dpn s oa
837             have dsn: "degree s < n" apply auto
838               apply (erule ssubst)
839               apply (simp add: degree_mult_eq degree_linear_power)
840               done
841             {fix x assume h: "poly s x = 0"
842               {assume xa: "x = a"
843                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where
844                   u: "s = [:- a, 1:] * u" by (rule dvdE)
845                 have "p = [:- a, 1:] ^ (Suc ?op) * u"
846                   by (subst s, subst u, simp only: power_Suc mult_ac)
847                 with ap(2)[unfolded dvd_def] have False by blast}
848               note xa = this
849               from h have "poly p x = 0" by (subst s, simp)
850               with pq0 have "poly q x = 0" by blast
851               with r xa have "poly r x = 0"
853             note impth = this
854             from IH[rule_format, OF dsn, of s r] impth ds0
855             have "s dvd (r ^ (degree s))" by blast
856             then obtain u where u: "r ^ (degree s) = s * u" ..
857             hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
858               by (simp only: poly_mult[symmetric] poly_power[symmetric])
859             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
860             from oop[of a] dsn have "q ^ n = p * ?w"
861               apply -
862               apply (subst s, subst r)
863               apply (simp only: power_mult_distrib)
864               apply (subst mult_assoc [where b=s])
865               apply (subst mult_assoc [where a=u])
866               apply (subst mult_assoc [where b=u, symmetric])
867               apply (subst u [symmetric])
868               apply (simp add: mult_ac power_add [symmetric])
869               done
870             hence ?ths unfolding dvd_def by blast}
871       ultimately have ?ths by blast }
872       ultimately have ?ths by blast}
873     then have ?ths using a order_root pne by blast}
874   moreover
875   {assume exa: "\<not> (\<exists>a. poly p a = 0)"
876     from fundamental_theorem_of_algebra_alt[of p] exa obtain c where
877       ccs: "c\<noteq>0" "p = pCons c 0" by blast
879     then have pp: "\<And>x. poly p x =  c" by simp
880     let ?w = "[:1/c:] * (q ^ n)"
881     from ccs
882     have "(q ^ n) = (p * ?w) "
883       by (simp add: smult_smult)
884     hence ?ths unfolding dvd_def by blast}
885   ultimately show ?ths by blast
886 qed
888 lemma nullstellensatz_univariate:
889   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
890     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
891 proof-
892   {assume pe: "p = 0"
893     hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
894       apply auto
895       apply (rule poly_zero [THEN iffD1])
896       by (rule ext, simp)
897     {assume "p dvd (q ^ (degree p))"
898       then obtain r where r: "q ^ (degree p) = p * r" ..
899       from r pe have False by simp}
900     with eq pe have ?thesis by blast}
901   moreover
902   {assume pe: "p \<noteq> 0"
903     {assume dp: "degree p = 0"
904       then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe
905         by (cases p, simp split: if_splits)
906       hence th1: "\<forall>x. poly p x \<noteq> 0" by simp
907       from k dp have "q ^ (degree p) = p * [:1/k:]"
908         by (simp add: one_poly_def)
909       hence th2: "p dvd (q ^ (degree p))" ..
910       from th1 th2 pe have ?thesis by blast}
911     moreover
912     {assume dp: "degree p \<noteq> 0"
913       then obtain n where n: "degree p = Suc n " by (cases "degree p", auto)
914       {assume "p dvd (q ^ (Suc n))"
915         then obtain u where u: "q ^ (Suc n) = p * u" ..
916         {fix x assume h: "poly p x = 0" "poly q x \<noteq> 0"
917           hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp
918           hence False using u h(1) by (simp only: poly_mult) simp}}
919         with n nullstellensatz_lemma[of p q "degree p"] dp
920         have ?thesis by auto}
921     ultimately have ?thesis by blast}
922   ultimately show ?thesis by blast
923 qed
925 text{* Useful lemma *}
927 lemma constant_degree:
928   fixes p :: "'a::{idom,ring_char_0} poly"
929   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
930 proof
931   assume l: ?lhs
932   from l[unfolded constant_def, rule_format, of _ "0"]
933   have th: "poly p = poly [:poly p 0:]" apply - by (rule ext, simp)
934   then have "p = [:poly p 0:]" by (simp add: poly_eq_iff)
935   then have "degree p = degree [:poly p 0:]" by simp
936   then show ?rhs by simp
937 next
938   assume r: ?rhs
939   then obtain k where "p = [:k:]"
940     by (cases p, simp split: if_splits)
941   then show ?lhs unfolding constant_def by auto
942 qed
944 lemma divides_degree: assumes pq: "p dvd (q:: complex poly)"
945   shows "degree p \<le> degree q \<or> q = 0"
946 apply (cases "q = 0", simp_all)
947 apply (erule dvd_imp_degree_le [OF pq])
948 done
950 (* Arithmetic operations on multivariate polynomials.                        *)
952 lemma mpoly_base_conv:
953   "(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all
955 lemma mpoly_norm_conv:
956   "poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all
958 lemma mpoly_sub_conv:
959   "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"
960   by (simp add: diff_minus)
962 lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp
964 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
966 lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto
967 lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))
968   \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast
971   fixes p q :: "complex poly"
972   assumes pq: "p dvd q"
973   shows "p dvd (pCons (0::complex) q)"
974 proof-
975   have "pCons 0 q = q * [:0,1:]" by simp
976   then have "q dvd (pCons 0 q)" ..
977   with pq show ?thesis by (rule dvd_trans)
978 qed
981   fixes p q :: "complex poly"
982   assumes pq: "p dvd q"
983   shows "p dvd (smult a q)"
984 proof-
985   have "smult a q = q * [:a:]" by simp
986   then have "q dvd smult a q" ..
987   with pq show ?thesis by (rule dvd_trans)
988 qed
991 lemma poly_divides_conv0:
992   fixes p :: "complex poly"
993   assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"
994   shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")
995 proof-
996   {assume r: ?rhs
997     hence "q = p * 0" by simp
998     hence ?lhs ..}
999   moreover
1000   {assume l: ?lhs
1001     {assume q0: "q = 0"
1002       hence ?rhs by simp}
1003     moreover
1004     {assume q0: "q \<noteq> 0"
1005       from l q0 have "degree p \<le> degree q"
1006         by (rule dvd_imp_degree_le)
1007       with lgpq have ?rhs by simp }
1008     ultimately have ?rhs by blast }
1009   ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)
1010 qed
1012 lemma poly_divides_conv1:
1013   assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"
1014   and qrp': "smult a q - p' \<equiv> r"
1015   shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs")
1016 proof-
1017   {
1018   from pp' obtain t where t: "p' = p * t" ..
1019   {assume l: ?lhs
1020     then obtain u where u: "q = p * u" ..
1021      have "r = p * (smult a u - t)"
1022        using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right)
1023      then have ?rhs ..}
1024   moreover
1025   {assume r: ?rhs
1026     then obtain u where u: "r = p * u" ..
1027     from u [symmetric] t qrp' [symmetric] a0
1028     have "q = p * smult (1/a) (u + t)"
1029       by (simp add: algebra_simps mult_smult_right smult_smult)
1030     hence ?lhs ..}
1031   ultimately have "?lhs = ?rhs" by blast }
1032 thus "?lhs \<equiv> ?rhs"  by - (atomize(full), blast)
1033 qed
1035 lemma basic_cqe_conv1:
1036   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"
1037   "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"
1038   "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"
1039   "(\<exists>x. poly 0 x = 0) \<equiv> True"
1040   "(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all
1042 lemma basic_cqe_conv2:
1043   assumes l:"p \<noteq> 0"
1044   shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True"
1045 proof-
1046   {fix h t
1047     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"
1048     with l have False by simp}
1049   hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)"
1050     by blast
1051   from fundamental_theorem_of_algebra_alt[OF th]
1052   show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto
1053 qed
1055 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"
1056 proof-
1057   have "p = 0 \<longleftrightarrow> poly p = poly 0"
1058     by (simp add: poly_zero)
1059   also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)
1060   finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"
1061     by - (atomize (full), blast)
1062 qed
1064 lemma basic_cqe_conv3:
1065   fixes p q :: "complex poly"
1066   assumes l: "p \<noteq> 0"
1067   shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
1068 proof-
1069   from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
1070   from nullstellensatz_univariate[of "pCons a p" q] l
1071   show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
1072     unfolding dp
1073     by - (atomize (full), auto)
1074 qed
1076 lemma basic_cqe_conv4:
1077   fixes p q :: "complex poly"
1078   assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"
1079   shows "p dvd (q ^ n) \<equiv> p dvd r"
1080 proof-
1081   from h have "poly (q ^ n) = poly r" by (auto intro: ext)
1082   then have "(q ^ n) = r" by (simp add: poly_eq_iff)
1083   thus "p dvd (q ^ n) \<equiv> p dvd r" by simp
1084 qed
1086 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"
1087   by simp
1089 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp
1090 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+
1091 lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)
1093 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp
1094 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"
1095   by (atomize (full)) simp_all
1096 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp
1097 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")
1098 proof
1099   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast
1100 next
1101   assume "p \<and> q \<equiv> p \<and> r" "p"
1102   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done
1103 qed
1104 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp
1106 end