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(* Author: Amine Chaieb, TU Muenchen *) 
26123  2 

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header{*Fundamental Theorem of Algebra*} 

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theory Fundamental_Theorem_Algebra 

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imports Polynomial Complex 
26123  7 
begin 
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27445  9 
subsection {* Square root of complex numbers *} 
26123  10 
definition csqrt :: "complex \<Rightarrow> complex" where 
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"csqrt z = (if Im z = 0 then 

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if 0 \<le> Re z then Complex (sqrt(Re z)) 0 

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else Complex 0 (sqrt( Re z)) 

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else Complex (sqrt((cmod z + Re z) /2)) 

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((Im z / abs(Im z)) * sqrt((cmod z  Re z) /2)))" 

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27668  17 
lemma csqrt[algebra]: "csqrt z ^ 2 = z" 
26123  18 
proof 
29292  19 
obtain x y where xy: "z = Complex x y" by (cases z) 
26123  20 
{assume y0: "y = 0" 
30488  21 
{assume x0: "x \<ge> 0" 
26123  22 
then have ?thesis using y0 xy real_sqrt_pow2[OF x0] 
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by (simp add: csqrt_def power2_eq_square)} 

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moreover 

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{assume "\<not> x \<ge> 0" hence x0: " x \<ge> 0" by arith 

30488  26 
then have ?thesis using y0 xy real_sqrt_pow2[OF x0] 
26123  27 
by (simp add: csqrt_def power2_eq_square) } 
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ultimately have ?thesis by blast} 

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moreover 

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{assume y0: "y\<noteq>0" 

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{fix x y 

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let ?z = "Complex x y" 

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from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z" by auto 

30488  34 
hence "cmod ?z  x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+ 
26123  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) } 
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note th = this 

30488  37 
have sq4: "\<And>x::real. x^2 / 4 = (x / 2) ^ 2" 
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by (simp add: power2_eq_square) 

26123  39 
from th[of x y] 
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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 

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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" 

30488  42 
unfolding power2_eq_square by simp 
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have "sqrt 4 = sqrt (2^2)" by simp 

26123  44 
hence sqrt4: "sqrt 4 = 2" by (simp only: real_sqrt_abs) 
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have th2: "2 *(y * sqrt ((sqrt (x * x + y * y)  x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y" 

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using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0 

30488  47 
unfolding power2_eq_square 
29667  48 
by (simp add: algebra_simps real_sqrt_divide sqrt4) 
26123  49 
from y0 xy have ?thesis apply (simp add: csqrt_def power2_eq_square) 
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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]) 

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using th1 th2 ..} 

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ultimately show ?thesis by blast 

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qed 

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27445  56 
subsection{* More lemmas about module of complex numbers *} 
26123  57 

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lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)" 

27514  59 
by (rule of_real_power [symmetric]) 
26123  60 

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lemma real_down2: "(0::real) < d1 \<Longrightarrow> 0 < d2 ==> EX e. 0 < e & e < d1 & e < d2" 

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apply (rule exI[where x = "min d1 d2 / 2"]) 
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by (simp add: field_simps min_def) 
26123  64 

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text{* The triangle inequality for cmod *} 

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lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z" 

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using complex_mod_triangle_ineq2[of "w + z" "z"] by auto 

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27445  69 
subsection{* Basic lemmas about complex polynomials *} 
26123  70 

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lemma poly_bound_exists: 

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shows "\<exists>m. m > 0 \<and> (\<forall>z. cmod z <= r \<longrightarrow> cmod (poly p z) \<le> m)" 

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proof(induct p) 

30488  74 
case 0 thus ?case by (rule exI[where x=1], simp) 
26123  75 
next 
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case (pCons c cs) 
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from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m" 
26123  78 
by blast 
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let ?k = " 1 + cmod c + \<bar>r * m\<bar>" 

27514  80 
have kp: "?k > 0" using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith 
26123  81 
{fix z 
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assume H: "cmod z \<le> r" 

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from m H have th: "cmod (poly cs z) \<le> m" by blast 

27514  84 
from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith 
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have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)" 
27514  86 
using norm_triangle_ineq[of c "z* poly cs z"] by simp 
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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) 

26123  88 
also have "\<dots> \<le> ?k" by simp 
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finally have "cmod (poly (pCons c cs) z) \<le> ?k" .} 
26123  90 
with kp show ?case by blast 
91 
qed 

92 

93 

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text{* Offsetting the variable in a polynomial gives another of same degree *} 

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definition 
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"offset_poly p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p" 
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lemma offset_poly_0: "offset_poly 0 h = 0" 
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unfolding offset_poly_def by (simp add: poly_rec_0) 
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lemma offset_poly_pCons: 
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"offset_poly (pCons a p) h = 
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smult h (offset_poly p h) + pCons a (offset_poly p h)" 
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unfolding offset_poly_def by (simp add: poly_rec_pCons) 
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lemma offset_poly_single: "offset_poly [:a:] h = [:a:]" 
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by (simp add: offset_poly_pCons offset_poly_0) 
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lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)" 
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apply (induct p) 
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apply (simp add: offset_poly_0) 
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apply (simp add: offset_poly_pCons algebra_simps) 
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done 
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lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0" 
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by (induct p arbitrary: a, simp, force) 
26123  118 

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lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0" 
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apply (safe intro!: offset_poly_0) 
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apply (induct p, simp) 
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apply (simp add: offset_poly_pCons) 
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apply (frule offset_poly_eq_0_lemma, simp) 
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done 
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lemma degree_offset_poly: "degree (offset_poly p h) = degree p" 
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apply (induct p) 
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apply (simp add: offset_poly_0) 
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apply (case_tac "p = 0") 
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apply (simp add: offset_poly_0 offset_poly_pCons) 
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apply (simp add: offset_poly_pCons) 
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apply (subst degree_add_eq_right) 
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apply (rule le_less_trans [OF degree_smult_le]) 
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apply (simp add: offset_poly_eq_0_iff) 
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apply (simp add: offset_poly_eq_0_iff) 
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done 
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29478  138 
definition 
29538  139 
"psize p = (if p = 0 then 0 else Suc (degree p))" 
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29538  141 
lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0" 
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unfolding psize_def by simp 

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29538  144 
lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))" 
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proof (intro exI conjI) 
29538  146 
show "psize (offset_poly p a) = psize p" 
147 
unfolding psize_def 

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by (simp add: offset_poly_eq_0_iff degree_offset_poly) 
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show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)" 
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by (simp add: poly_offset_poly) 
26123  151 
qed 
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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" 

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shows "\<exists>(s::real). \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s" 

156 
proof 

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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 

30488  159 
from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y" 
26123  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 

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from Y[OF x] have xY: "x < Y" . 

30488  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) 
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from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y" 

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apply (clarsimp, atomize (full)) by auto 

26123  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 

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{fix z assume z: "P z" "y < z" 

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from L' z have "y < L" by auto } 

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moreover 

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{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 

30488  174 
from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def) 
26123  175 
with yL(1) have False by arith} 
176 
ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast} 

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thus ?thesis by blast 

178 
qed 

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27445  180 
subsection {* Fundamental theorem of algebra *} 
26123  181 
lemma unimodular_reduce_norm: 
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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 

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obtain x y where z: "z = Complex x y " by (cases z, auto) 

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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" 

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from C z xy have "2*x \<le> 1" "2*x \<ge> 1" "2*y \<le> 1" "2*y \<ge> 1" 

29667  189 
by (simp_all add: cmod_def power2_eq_square algebra_simps) 
26123  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" 

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by  (rule power_mono, simp, simp)+ 

30488  193 
hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1" 
26123  194 
by (simp_all add: power2_abs power_mult_distrib) 
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from add_mono[OF th0] xy have False by simp } 

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thus ?thesis unfolding linorder_not_le[symmetric] by blast 

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qed 

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26135  199 
text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *} 
26123  200 
lemma reduce_poly_simple: 
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assumes b: "b \<noteq> 0" and n: "n\<noteq>0" 

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shows "\<exists>z. cmod (1 + b * z^n) < 1" 

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using n 

204 
proof(induct n rule: nat_less_induct) 

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fix n 

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assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)" and n: "n \<noteq> 0" 

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let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1" 

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{assume e: "even n" 

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hence "\<exists>m. n = 2*m" by presburger 

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then obtain m where m: "n = 2*m" by blast 

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from n m have "m\<noteq>0" "m < n" by presburger+ 

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with IH[rule_format, of m] obtain z where z: "?P z m" by blast 

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from z have "?P (csqrt z) n" by (simp add: m power_mult csqrt) 

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hence "\<exists>z. ?P z n" ..} 

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moreover 

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{assume o: "odd n" 

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from b have b': "b^2 \<noteq> 0" unfolding power2_eq_square by simp 

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have "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) + 

30488  219 
Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) = 
26123  220 
((Re (inverse b))^2 + (Im (inverse b))^2) * \<bar>Im b * Im b + Re b * Re b\<bar>" by algebra 
30488  221 
also have "\<dots> = cmod (inverse b) ^2 * cmod b ^ 2" 
26123  222 
apply (simp add: cmod_def) using realpow_two_le_add_order[of "Re b" "Im b"] 
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by (simp add: power2_eq_square) 

30488  224 
finally 
26123  225 
have th0: "Im (inverse b) * (Im (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) + 
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Re (inverse b) * (Re (inverse b) * \<bar>Im b * Im b + Re b * Re b\<bar>) = 

30488  227 
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27514  228 
apply (simp add: power2_eq_square norm_mult[symmetric] norm_inverse[symmetric]) 
26123  229 
using right_inverse[OF b'] 
29667  230 
by (simp add: power2_eq_square[symmetric] power_inverse[symmetric] algebra_simps) 
26123  231 
have th0: "cmod (complex_of_real (cmod b) / b) = 1" 
29667  232 
apply (simp add: complex_Re_mult cmod_def power2_eq_square Re_complex_of_real Im_complex_of_real divide_inverse algebra_simps ) 
30488  233 
by (simp add: real_sqrt_mult[symmetric] th0) 
26123  234 
from o have "\<exists>m. n = Suc (2*m)" by presburger+ 
235 
then obtain m where m: "n = Suc (2*m)" by blast 

236 
from unimodular_reduce_norm[OF th0] o 

237 
have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1" 

238 
apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1", rule_tac x="1" in exI, simp) 

239 
apply (cases "cmod (complex_of_real (cmod b) / b  1) < 1", rule_tac x="1" in exI, simp add: diff_def) 

240 
apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1") 

241 
apply (cases "even m", rule_tac x="ii" in exI, simp add: m power_mult) 

242 
apply (rule_tac x=" ii" in exI, simp add: m power_mult) 

243 
apply (cases "even m", rule_tac x=" ii" in exI, simp add: m power_mult diff_def) 

244 
apply (rule_tac x="ii" in exI, simp add: m power_mult diff_def) 

245 
done 

246 
then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1" by blast 

247 
let ?w = "v / complex_of_real (root n (cmod b))" 

248 
from odd_real_root_pow[OF o, of "cmod b"] 

30488  249 
have th1: "?w ^ n = v^n / complex_of_real (cmod b)" 
26123  250 
by (simp add: power_divide complex_of_real_power) 
27514  251 
have th2:"cmod (complex_of_real (cmod b) / b) = 1" using b by (simp add: norm_divide) 
26123  252 
hence th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0" by simp 
253 
have th4: "cmod (complex_of_real (cmod b) / b) * 

254 
cmod (1 + b * (v ^ n / complex_of_real (cmod b))) 

255 
< cmod (complex_of_real (cmod b) / b) * 1" 

27514  256 
apply (simp only: norm_mult[symmetric] right_distrib) 
26123  257 
using b v by (simp add: th2) 
258 

259 
from mult_less_imp_less_left[OF th4 th3] 

30488  260 
have "?P ?w n" unfolding th1 . 
26123  261 
hence "\<exists>z. ?P z n" .. } 
262 
ultimately show "\<exists>z. ?P z n" by blast 

263 
qed 

264 

265 
text{* BolzanoWeierstrass type property for closed disc in complex plane. *} 

266 

267 
lemma metric_bound_lemma: "cmod (x  y) <= \<bar>Re x  Re y\<bar> + \<bar>Im x  Im y\<bar>" 

268 
using real_sqrt_sum_squares_triangle_ineq[of "Re x  Re y" 0 0 "Im x  Im y" ] 

269 
unfolding cmod_def by simp 

270 

271 
lemma bolzano_weierstrass_complex_disc: 

272 
assumes r: "\<forall>n. cmod (s n) \<le> r" 

273 
shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n)  z) < e)" 

274 
proof 

30488  275 
from seq_monosub[of "Re o s"] 
276 
obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))" 

26123  277 
unfolding o_def by blast 
30488  278 
from seq_monosub[of "Im o s o f"] 
279 
obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast 

26123  280 
let ?h = "f o g" 
30488  281 
from r[rule_format, of 0] have rp: "r \<ge> 0" using norm_ge_zero[of "s 0"] by arith 
282 
have th:"\<forall>n. r + 1 \<ge> \<bar> Re (s n)\<bar>" 

26123  283 
proof 
284 
fix n 

285 
from abs_Re_le_cmod[of "s n"] r[rule_format, of n] show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith 

286 
qed 

287 
have conv1: "convergent (\<lambda>n. Re (s ( f n)))" 

288 
apply (rule Bseq_monoseq_convergent) 

289 
apply (simp add: Bseq_def) 

290 
apply (rule exI[where x= "r + 1"]) 

291 
using th rp apply simp 

292 
using f(2) . 

30488  293 
have th:"\<forall>n. r + 1 \<ge> \<bar> Im (s n)\<bar>" 
26123  294 
proof 
295 
fix n 

296 
from abs_Im_le_cmod[of "s n"] r[rule_format, of n] show "\<bar>Im (s n)\<bar> \<le> r + 1" by arith 

297 
qed 

298 

299 
have conv2: "convergent (\<lambda>n. Im (s (f (g n))))" 

300 
apply (rule Bseq_monoseq_convergent) 

301 
apply (simp add: Bseq_def) 

302 
apply (rule exI[where x= "r + 1"]) 

303 
using th rp apply simp 

304 
using g(2) . 

305 

30488  306 
from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x" 
307 
by blast 

308 
hence x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Re (s (f n))  x \<bar> < r" 

31337  309 
unfolding LIMSEQ_iff real_norm_def . 
26123  310 

30488  311 
from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y" 
312 
by blast 

313 
hence y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar> Im (s (f (g n)))  y \<bar> < r" 

31337  314 
unfolding LIMSEQ_iff real_norm_def . 
26123  315 
let ?w = "Complex x y" 
30488  316 
from f(1) g(1) have hs: "subseq ?h" unfolding subseq_def by auto 
26123  317 
{fix e assume ep: "e > (0::real)" 
318 
hence e2: "e/2 > 0" by simp 

319 
from x[rule_format, OF e2] y[rule_format, OF e2] 

320 
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 

321 
{fix n assume nN12: "n \<ge> N1 + N2" 

322 
hence nN1: "g n \<ge> N1" and nN2: "n \<ge> N2" using seq_suble[OF g(1), of n] by arith+ 

323 
from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]] 

30488  324 
have "cmod (s (?h n)  ?w) < e" 
26123  325 
using metric_bound_lemma[of "s (f (g n))" ?w] by simp } 
326 
hence "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n)  ?w) < e" by blast } 

30488  327 
with hs show ?thesis by blast 
26123  328 
qed 
329 

330 
text{* Polynomial is continuous. *} 

331 

332 
lemma poly_cont: 

30488  333 
assumes ep: "e > 0" 
26123  334 
shows "\<exists>d >0. \<forall>w. 0 < cmod (w  z) \<and> cmod (w  z) < d \<longrightarrow> cmod (poly p w  poly p z) < e" 
335 
proof 

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336 
obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)" 
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337 
proof 
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338 
show "degree (offset_poly p z) = degree p" 
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339 
by (rule degree_offset_poly) 
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340 
show "\<And>x. poly (offset_poly p z) x = poly p (z + x)" 
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341 
by (rule poly_offset_poly) 
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342 
qed 
26123  343 
{fix w 
344 
note q(2)[of "w  z", simplified]} 

345 
note th = this 

346 
show ?thesis unfolding th[symmetric] 

347 
proof(induct q) 

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348 
case 0 thus ?case using ep by auto 
26123  349 
next 
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350 
case (pCons c cs) 
30488  351 
from poly_bound_exists[of 1 "cs"] 
26123  352 
obtain m where m: "m > 0" "\<And>z. cmod z \<le> 1 \<Longrightarrow> cmod (poly cs z) \<le> m" by blast 
353 
from ep m(1) have em0: "e/m > 0" by (simp add: field_simps) 

354 
have one0: "1 > (0::real)" by arith 

30488  355 
from real_lbound_gt_zero[OF one0 em0] 
26123  356 
obtain d where d: "d >0" "d < 1" "d < e / m" by blast 
30488  357 
from d(1,3) m(1) have dm: "d*m > 0" "d*m < e" 
26123  358 
by (simp_all add: field_simps real_mult_order) 
30488  359 
show ?case 
27514  360 
proof(rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult) 
26123  361 
fix d w 
362 
assume H: "d > 0" "d < 1" "d < e/m" "w\<noteq>z" "cmod (wz) < d" 

363 
hence d1: "cmod (wz) \<le> 1" "d \<ge> 0" by simp_all 

364 
from H(3) m(1) have dme: "d*m < e" by (simp add: field_simps) 

30488  365 
from H have th: "cmod (wz) \<le> d" by simp 
27514  366 
from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme 
26123  367 
show "cmod (w  z) * cmod (poly cs (w  z)) < e" by simp 
30488  368 
qed 
26123  369 
qed 
370 
qed 

371 

30488  372 
text{* Hence a polynomial attains minimum on a closed disc 
26123  373 
in the complex plane. *} 
374 
lemma poly_minimum_modulus_disc: 

375 
"\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)" 

376 
proof 

377 
{assume "\<not> r \<ge> 0" hence ?thesis unfolding linorder_not_le 

378 
apply  

30488  379 
apply (rule exI[where x=0]) 
26123  380 
apply auto 
381 
apply (subgoal_tac "cmod w < 0") 

382 
apply simp 

383 
apply arith 

384 
done } 

385 
moreover 

386 
{assume rp: "r \<ge> 0" 

30488  387 
from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) =  ( cmod (poly p 0))" by simp 
26123  388 
hence mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) =  x" by blast 
389 
{fix x z 

390 
assume H: "cmod z \<le> r" "cmod (poly p z) =  x" "\<not>x < 1" 

391 
hence " x < 0 " by arith 

27514  392 
with H(2) norm_ge_zero[of "poly p z"] have False by simp } 
26123  393 
then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) =  x) \<longrightarrow> x < z" by blast 
30488  394 
from real_sup_exists[OF mth1 mth2] obtain s where 
26123  395 
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 
396 
let ?m = "s" 

397 
{fix y 

30488  398 
from s[rule_format, of "y"] have 
399 
"(\<exists>z x. cmod z \<le> r \<and> ( cmod (poly p z)) < y) \<longleftrightarrow> ?m < y" 

26123  400 
unfolding minus_less_iff[of y ] equation_minus_iff by blast } 
401 
note s1 = this[unfolded minus_minus] 

30488  402 
from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m" 
26123  403 
by auto 
404 
{fix n::nat 

30488  405 
from s1[rule_format, of "?m + 1/real (Suc n)"] 
26123  406 
have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) <  s + 1 / real (Suc n)" 
407 
by simp} 

408 
hence th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) <  s + 1 / real (Suc n)" .. 

30488  409 
from choice[OF th] obtain g where 
410 
g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m+1 /real(Suc n)" 

26123  411 
by blast 
30488  412 
from bolzano_weierstrass_complex_disc[OF g(1)] 
26123  413 
obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n)  z) < e" 
30488  414 
by blast 
415 
{fix w 

26123  416 
assume wr: "cmod w \<le> r" 
417 
let ?e = "\<bar>cmod (poly p z)  ?m\<bar>" 

418 
{assume e: "?e > 0" 

419 
hence e2: "?e/2 > 0" by simp 

420 
from poly_cont[OF e2, of z p] obtain d where 

421 
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 

422 
{fix w assume w: "cmod (w  z) < d" 

423 
have "cmod(poly p w  poly p z) < ?e / 2" 

424 
using d(2)[rule_format, of w] w e by (cases "w=z", simp_all)} 

425 
note th1 = this 

30488  426 

427 
from fz(2)[rule_format, OF d(1)] obtain N1 where 

26123  428 
N1: "\<forall>n\<ge>N1. cmod (g (f n)  z) < d" by blast 
429 
from reals_Archimedean2[of "2/?e"] obtain N2::nat where 

430 
N2: "2/?e < real N2" by blast 

431 
have th2: "cmod(poly p (g(f(N1 + N2)))  poly p z) < ?e/2" 

432 
using N1[rule_format, of "N1 + N2"] th1 by simp 

433 
{fix a b e2 m :: real 

434 
have "a < e2 \<Longrightarrow> abs(b  m) < e2 \<Longrightarrow> 2 * e2 <= abs(b  m) + a 

435 
==> False" by arith} 

436 
note th0 = this 

30488  437 
have ath: 
26123  438 
"\<And>m x e. m <= x \<Longrightarrow> x < m + e ==> abs(x  m::real) < e" by arith 
439 
from s1m[OF g(1)[rule_format]] 

440 
have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" . 

441 
from seq_suble[OF fz(1), of "N1+N2"] 

442 
have th00: "real (Suc (N1+N2)) \<le> real (Suc (f (N1+N2)))" by simp 

30488  443 
have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1+N2)) > 0" 
26123  444 
using N2 by auto 
445 
from frac_le[OF th000 th00] have th00: "?m +1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))" by simp 

446 
from g(2)[rule_format, of "f (N1 + N2)"] 

447 
have th01:"cmod (poly p (g (f (N1 + N2)))) <  s + 1 / real (Suc (f (N1 + N2)))" . 

448 
from order_less_le_trans[OF th01 th00] 

449 
have th32: "cmod(poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" . 

450 
from N2 have "2/?e < real (Suc (N1 + N2))" by arith 

451 
with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"] 

452 
have "?e/2 > 1/ real (Suc (N1 + N2))" by (simp add: inverse_eq_divide) 

453 
with ath[OF th31 th32] 

30488  454 
have thc1:"\<bar>cmod(poly p (g (f (N1 + N2))))  ?m\<bar>< ?e/2" by arith 
455 
have ath2: "\<And>(a::real) b c m. \<bar>a  b\<bar> <= c ==> \<bar>b  m\<bar> <= \<bar>a  m\<bar> + c" 

26123  456 
by arith 
457 
have th22: "\<bar>cmod (poly p (g (f (N1 + N2))))  cmod (poly p z)\<bar> 

30488  458 
\<le> cmod (poly p (g (f (N1 + N2)))  poly p z)" 
27514  459 
by (simp add: norm_triangle_ineq3) 
26123  460 
from ath2[OF th22, of ?m] 
461 
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 

462 
from th0[OF th2 thc1 thc2] have False .} 

463 
hence "?e = 0" by auto 

30488  464 
then have "cmod (poly p z) = ?m" by simp 
26123  465 
with s1m[OF wr] 
466 
have "cmod (poly p z) \<le> cmod (poly p w)" by simp } 

467 
hence ?thesis by blast} 

468 
ultimately show ?thesis by blast 

469 
qed 

470 

471 
lemma "(rcis (sqrt (abs r)) (a/2)) ^ 2 = rcis (abs r) a" 

472 
unfolding power2_eq_square 

473 
apply (simp add: rcis_mult) 

474 
apply (simp add: power2_eq_square[symmetric]) 

475 
done 

476 

30488  477 
lemma cispi: "cis pi = 1" 
26123  478 
unfolding cis_def 
479 
by simp 

480 

481 
lemma "(rcis (sqrt (abs r)) ((pi + a)/2)) ^ 2 = rcis ( abs r) a" 

482 
unfolding power2_eq_square 

483 
apply (simp add: rcis_mult add_divide_distrib) 

484 
apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric]) 

485 
done 

486 

487 
text {* Nonzero polynomial in z goes to infinity as z does. *} 

488 

489 
lemma poly_infinity: 

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490 
assumes ex: "p \<noteq> 0" 
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491 
shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)" 
26123  492 
using ex 
493 
proof(induct p arbitrary: a d) 

30488  494 
case (pCons c cs a d) 
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495 
{assume H: "cs \<noteq> 0" 
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496 
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 
26123  497 
let ?r = "1 + \<bar>r\<bar>" 
498 
{fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z" 

499 
have r0: "r \<le> cmod z" using h by arith 

500 
from r[rule_format, OF r0] 

29464
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501 
have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith 
26123  502 
from h have z1: "cmod z \<ge> 1" by arith 
29464
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503 
from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]] 
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504 
have th1: "d \<le> cmod(z * poly (pCons c cs) z)  cmod a" 
29667  505 
unfolding norm_mult by (simp add: algebra_simps) 
29464
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506 
from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a] 
30488  507 
have th2: "cmod(z * poly (pCons c cs) z)  cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)" 
508 
by (simp add: diff_le_eq algebra_simps) 

29464
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changeset

509 
from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" by arith} 
26123  510 
hence ?case by blast} 
511 
moreover 

29464
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512 
{assume cs0: "\<not> (cs \<noteq> 0)" 
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513 
with pCons.prems have c0: "c \<noteq> 0" by simp 
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514 
from cs0 have cs0': "cs = 0" by simp 
26123  515 
{fix z 
516 
assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z" 

517 
from c0 have "cmod c > 0" by simp 

30488  518 
from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)" 
27514  519 
by (simp add: field_simps norm_mult) 
26123  520 
have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith 
521 
from complex_mod_triangle_sub[of "z*c" a ] 

522 
have th1: "cmod (z * c) \<le> cmod (a + z * c) + cmod a" 

29667  523 
by (simp add: algebra_simps) 
30488  524 
from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)" 
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525 
using cs0' by simp} 
26123  526 
then have ?case by blast} 
527 
ultimately show ?case by blast 

528 
qed simp 

529 

530 
text {* Hence polynomial's modulus attains its minimum somewhere. *} 

531 
lemma poly_minimum_modulus: 

532 
"\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)" 

533 
proof(induct p) 

30488  534 
case (pCons c cs) 
29464
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parents:
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changeset

535 
{assume cs0: "cs \<noteq> 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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parents:
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diff
changeset

536 
from poly_infinity[OF cs0, of "cmod (poly (pCons c cs) 0)" c] 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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parents:
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changeset

537 
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 
26123  538 
have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>" by arith 
30488  539 
from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"] 
29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
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changeset

540 
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 
26123  541 
{fix z assume z: "r \<le> cmod z" 
30488  542 
from v[of 0] r[OF z] 
29464
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diff
changeset

543 
have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" 
26123  544 
by simp } 
545 
note v0 = this 

546 
from v0 v ath[of r] have ?case by blast} 

547 
moreover 

29464
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diff
changeset

548 
{assume cs0: "\<not> (cs \<noteq> 0)" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
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diff
changeset

549 
hence th:"cs = 0" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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parents:
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changeset

550 
from th pCons.hyps have ?case by simp} 
26123  551 
ultimately show ?case by blast 
552 
qed simp 

553 

554 
text{* Constant function (nonsyntactic characterization). *} 

555 
definition "constant f = (\<forall>x y. f x = f y)" 

556 

29538  557 
lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2" 
558 
unfolding constant_def psize_def 

26123  559 
apply (induct p, auto) 
560 
done 

30488  561 

26123  562 
lemma poly_replicate_append: 
31021  563 
"poly (monom 1 n * p) (x::'a::{comm_ring_1}) = x^n * poly p x" 
29464
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changeset

564 
by (simp add: poly_monom) 
26123  565 

30488  566 
text {* Decomposition of polynomial, skipping zero coefficients 
26123  567 
after the first. *} 
568 

569 
lemma poly_decompose_lemma: 

31021  570 
assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{idom}))" 
30488  571 
shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and> 
29464
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572 
(\<forall>z. poly p z = z^k * poly (pCons a q) z)" 
29538  573 
unfolding psize_def 
26123  574 
using nz 
575 
proof(induct p) 

29464
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changeset

576 
case 0 thus ?case by simp 
26123  577 
next 
29464
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changeset

578 
case (pCons c cs) 
26123  579 
{assume c0: "c = 0" 
32456
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31337
diff
changeset

580 
from pCons.hyps pCons.prems c0 have ?case 
341c83339aeb
tuned the simp rules for Int involving insert and intervals.
nipkow
parents:
31337
diff
changeset

581 
apply (auto) 
26123  582 
apply (rule_tac x="k+1" in exI) 
583 
apply (rule_tac x="a" in exI, clarsimp) 

584 
apply (rule_tac x="q" in exI) 

32456
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parents:
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diff
changeset

585 
by (auto)} 
26123  586 
moreover 
587 
{assume c0: "c\<noteq>0" 

588 
hence ?case apply 

589 
apply (rule exI[where x=0]) 

590 
apply (rule exI[where x=c], clarsimp) 

591 
apply (rule exI[where x=cs]) 

592 
apply auto 

593 
done} 

594 
ultimately show ?case by blast 

595 
qed 

596 

597 
lemma poly_decompose: 

598 
assumes nc: "~constant(poly p)" 

31021  599 
shows "\<exists>k a q. a\<noteq>(0::'a::{idom}) \<and> k\<noteq>0 \<and> 
30488  600 
psize q + k + 1 = psize p \<and> 
29464
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changeset

601 
(\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)" 
30488  602 
using nc 
26123  603 
proof(induct p) 
29464
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diff
changeset

604 
case 0 thus ?case by (simp add: constant_def) 
26123  605 
next 
29464
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parents:
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diff
changeset

606 
case (pCons c cs) 
26123  607 
{assume C:"\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0" 
608 
{fix x y 

29464
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parents:
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diff
changeset

609 
from C have "poly (pCons c cs) x = poly (pCons c cs) y" by (cases "x=0", auto)} 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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parents:
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diff
changeset

610 
with pCons.prems have False by (auto simp add: constant_def)} 
26123  611 
hence th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" .. 
30488  612 
from poly_decompose_lemma[OF th] 
613 
show ?case 

29464
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parents:
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diff
changeset

614 
apply clarsimp 
26123  615 
apply (rule_tac x="k+1" in exI) 
616 
apply (rule_tac x="a" in exI) 

617 
apply simp 

618 
apply (rule_tac x="q" in exI) 

619 
apply (auto simp add: power_Suc) 

29538  620 
apply (auto simp add: psize_def split: if_splits) 
26123  621 
done 
622 
qed 

623 

624 
text{* Fundamental theorem of algebral *} 

625 

626 
lemma fundamental_theorem_of_algebra: 

627 
assumes nc: "~constant(poly p)" 

628 
shows "\<exists>z::complex. poly p z = 0" 

629 
using nc 

29538  630 
proof(induct n\<equiv> "psize p" arbitrary: p rule: nat_less_induct) 
29464
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parents:
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changeset

631 
fix n fix p :: "complex poly" 
26123  632 
let ?p = "poly p" 
29538  633 
assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = psize p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = psize p" 
26123  634 
let ?ths = "\<exists>z. ?p z = 0" 
635 

636 
from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n) 

30488  637 
from poly_minimum_modulus obtain c where 
26123  638 
c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast 
639 
{assume pc: "?p c = 0" hence ?ths by blast} 

640 
moreover 

641 
{assume pc0: "?p c \<noteq> 0" 

642 
from poly_offset[of p c] obtain q where 

29538  643 
q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast 
26123  644 
{assume h: "constant (poly q)" 
645 
from q(2) have th: "\<forall>x. poly q (x  c) = ?p x" by auto 

646 
{fix x y 

30488  647 
from th have "?p x = poly q (x  c)" by auto 
648 
also have "\<dots> = poly q (y  c)" 

26123  649 
using h unfolding constant_def by blast 
650 
also have "\<dots> = ?p y" using th by auto 

651 
finally have "?p x = ?p y" .} 

652 
with nc have False unfolding constant_def by blast } 

653 
hence qnc: "\<not> constant (poly q)" by blast 

654 
from q(2) have pqc0: "?p c = poly q 0" by simp 

30488  655 
from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp 
26123  656 
let ?a0 = "poly q 0" 
30488  657 
from pc0 pqc0 have a00: "?a0 \<noteq> 0" by simp 
658 
from a00 

29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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parents:
29292
diff
changeset

659 
have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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diff
changeset

660 
by simp 
c0d225a7f6ff
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changeset

661 
let ?r = "smult (inverse ?a0) q" 
29538  662 
have lgqr: "psize q = psize ?r" 
663 
using a00 unfolding psize_def degree_def 

29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
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changeset

664 
by (simp add: expand_poly_eq) 
26123  665 
{assume h: "\<And>x y. poly ?r x = poly ?r y" 
666 
{fix x y 

30488  667 
from qr[rule_format, of x] 
26123  668 
have "poly q x = poly ?r x * ?a0" by auto 
669 
also have "\<dots> = poly ?r y * ?a0" using h by simp 

670 
also have "\<dots> = poly q y" using qr[rule_format, of y] by simp 

30488  671 
finally have "poly q x = poly q y" .} 
26123  672 
with qnc have False unfolding constant_def by blast} 
673 
hence rnc: "\<not> constant (poly ?r)" unfolding constant_def by blast 

674 
from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1" by auto 

30488  675 
{fix w 
26123  676 
have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1" 
29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
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diff
changeset

677 
using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac) 
26123  678 
also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0" 
27514  679 
using a00 unfolding norm_divide by (simp add: field_simps) 
26123  680 
finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .} 
681 
note mrmq_eq = this 

30488  682 
from poly_decompose[OF rnc] obtain k a s where 
683 
kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r" 

29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

684 
"\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast 
26123  685 
{assume "k + 1 = n" 
29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

686 
with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto 
26123  687 
{fix w 
30488  688 
have "cmod (poly ?r w) = cmod (1 + a * w ^ k)" 
29667  689 
using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)} 
26123  690 
note hth = this [symmetric] 
30488  691 
from reduce_poly_simple[OF kas(1,2)] 
26123  692 
have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast} 
693 
moreover 

694 
{assume kn: "k+1 \<noteq> n" 

29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

695 
from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp 
30488  696 
have th01: "\<not> constant (poly (pCons 1 (monom a (k  1))))" 
29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
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diff
changeset

697 
unfolding constant_def poly_pCons poly_monom 
30488  698 
using kas(1) apply simp 
26123  699 
by (rule exI[where x=0], rule exI[where x=1], simp) 
29538  700 
from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k  1)))" 
701 
by (simp add: psize_def degree_monom_eq) 

26123  702 
from H[rule_format, OF k1n th01 th02] 
703 
obtain w where w: "1 + w^k * a = 0" 

29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

704 
unfolding poly_pCons poly_monom 
29667  705 
using kas(2) by (cases k, auto simp add: algebra_simps) 
30488  706 
from poly_bound_exists[of "cmod w" s] obtain m where 
26123  707 
m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast 
708 
have w0: "w\<noteq>0" using kas(2) w by (auto simp add: power_0_left) 

709 
from w have "(1 + w ^ k * a)  1 = 0  1" by simp 

710 
then have wm1: "w^k * a =  1" by simp 

30488  711 
have inv0: "0 < inverse (cmod w ^ (k + 1) * m)" 
27514  712 
using norm_ge_zero[of w] w0 m(1) 
26123  713 
by (simp add: inverse_eq_divide zero_less_mult_iff) 
714 
with real_down2[OF zero_less_one] obtain t where 

715 
t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast 

716 
let ?ct = "complex_of_real t" 

717 
let ?w = "?ct * w" 

29667  718 
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) 
26123  719 
also have "\<dots> = complex_of_real (1  t^k) + ?w^k * ?w * poly s ?w" 
720 
unfolding wm1 by (simp) 

30488  721 
finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) = cmod (complex_of_real (1  t^k) + ?w^k * ?w * poly s ?w)" 
26123  722 
apply  
723 
apply (rule cong[OF refl[of cmod]]) 

724 
apply assumption 

725 
done 

30488  726 
with norm_triangle_ineq[of "complex_of_real (1  t^k)" "?w^k * ?w * poly s ?w"] 
727 
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 

26123  728 
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 
729 
have "t *cmod w \<le> 1 * cmod w" apply (rule mult_mono) using t(1,2) by auto 

30488  730 
then have tw: "cmod ?w \<le> cmod w" using t(1) by (simp add: norm_mult) 
26123  731 
from t inv0 have "t* (cmod w ^ (k + 1) * m) < 1" 
732 
by (simp add: inverse_eq_divide field_simps) 

30488  733 
with zero_less_power[OF t(1), of k] 
734 
have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1" 

26123  735 
apply  apply (rule mult_strict_left_mono) by simp_all 
736 
have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k+1) * cmod (poly s ?w)))" using w0 t(1) 

29667  737 
by (simp add: algebra_simps power_mult_distrib norm_of_real norm_power norm_mult) 
26123  738 
then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))" 
739 
using t(1,2) m(2)[rule_format, OF tw] w0 

740 
apply (simp only: ) 

741 
apply auto 

27514  742 
apply (rule mult_mono, simp_all add: norm_ge_zero)+ 
26123  743 
apply (simp add: zero_le_mult_iff zero_le_power) 
744 
done 

30488  745 
with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k" by simp 
746 
from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1" 

26123  747 
by auto 
27514  748 
from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121] 
30488  749 
have th12: "\<bar>1  t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" . 
26123  750 
from th11 th12 
30488  751 
have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1" by arith 
752 
then have "cmod (poly ?r ?w) < 1" 

753 
unfolding kas(4)[rule_format, of ?w] r01 by simp 

26123  754 
then have "\<exists>w. cmod (poly ?r w) < 1" by blast} 
755 
ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1" by blast 

756 
from cr0_contr cq0 q(2) 

757 
have ?ths unfolding mrmq_eq not_less[symmetric] by auto} 

758 
ultimately show ?ths by blast 

759 
qed 

760 

761 
text {* Alternative version with a syntactic notion of constant polynomial. *} 

762 

763 
lemma fundamental_theorem_of_algebra_alt: 

29464
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huffman
parents:
29292
diff
changeset

764 
assumes nc: "~(\<exists>a l. a\<noteq> 0 \<and> l = 0 \<and> p = pCons a l)" 
26123  765 
shows "\<exists>z. poly p z = (0::complex)" 
766 
using nc 

767 
proof(induct p) 

29464
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convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

768 
case (pCons c cs) 
26123  769 
{assume "c=0" hence ?case by auto} 
770 
moreover 

771 
{assume c0: "c\<noteq>0" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

772 
{assume nc: "constant (poly (pCons c cs))" 
30488  773 
from nc[unfolded constant_def, rule_format, of 0] 
774 
have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

775 
hence "cs = 0" 
26123  776 
proof(induct cs) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

777 
case (pCons d ds) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

778 
{assume "d=0" hence ?case using pCons.prems pCons.hyps by simp} 
26123  779 
moreover 
780 
{assume d0: "d\<noteq>0" 

30488  781 
from poly_bound_exists[of 1 ds] obtain m where 
26123  782 
m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast 
783 
have dm: "cmod d / m > 0" using d0 m(1) by (simp add: field_simps) 

30488  784 
from real_down2[OF dm zero_less_one] obtain x where 
26123  785 
x: "x > 0" "x < cmod d / m" "x < 1" by blast 
786 
let ?x = "complex_of_real x" 

787 
from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1" by simp_all 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

788 
from pCons.prems[rule_format, OF cx(1)] 
26123  789 
have cth: "cmod (?x*poly ds ?x) = cmod d" by (simp add: eq_diff_eq[symmetric]) 
790 
from m(2)[rule_format, OF cx(2)] x(1) 

791 
have th0: "cmod (?x*poly ds ?x) \<le> x*m" 

27514  792 
by (simp add: norm_mult) 
26123  793 
from x(2) m(1) have "x*m < cmod d" by (simp add: field_simps) 
794 
with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d" by auto 

795 
with cth have ?case by blast} 

30488  796 
ultimately show ?case by blast 
26123  797 
qed simp} 
30488  798 
then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0 
26123  799 
by blast 
800 
from fundamental_theorem_of_algebra[OF nc] have ?case .} 

30488  801 
ultimately show ?case by blast 
26123  802 
qed simp 
803 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

804 

27445  805 
subsection{* Nullstellenstatz, degrees and divisibility of polynomials *} 
26123  806 

807 
lemma nullstellensatz_lemma: 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

808 
fixes p :: "complex poly" 
26123  809 
assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" 
810 
and "degree p = n" and "n \<noteq> 0" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

811 
shows "p dvd (q ^ n)" 
26123  812 
using prems 
813 
proof(induct n arbitrary: p q rule: nat_less_induct) 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

814 
fix n::nat fix p q :: "complex poly" 
26123  815 
assume IH: "\<forall>m<n. \<forall>p q. 
816 
(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow> 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

817 
degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)" 
30488  818 
and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0" 
26123  819 
and dpn: "degree p = n" and n0: "n \<noteq> 0" 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

820 
from dpn n0 have pne: "p \<noteq> 0" by auto 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

821 
let ?ths = "p dvd (q ^ n)" 
26123  822 
{fix a assume a: "poly p a = 0" 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

823 
{assume oa: "order a p \<noteq> 0" 
26123  824 
let ?op = "order a p" 
30488  825 
from pne have ap: "([: a, 1:] ^ ?op) dvd p" 
826 
"\<not> [: a, 1:] ^ (Suc ?op) dvd p" using order by blast+ 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

827 
note oop = order_degree[OF pne, unfolded dpn] 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

828 
{assume q0: "q = 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

829 
hence ?ths using n0 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

830 
by (simp add: power_0_left)} 
26123  831 
moreover 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

832 
{assume q0: "q \<noteq> 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

833 
from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd] 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

834 
obtain r where r: "q = [: a, 1:] * r" by (rule dvdE) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

835 
from ap(1) obtain s where 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

836 
s: "p = [: a, 1:] ^ ?op * s" by (rule dvdE) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

837 
have sne: "s \<noteq> 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

838 
using s pne by auto 
26123  839 
{assume ds0: "degree s = 0" 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

840 
from ds0 have "\<exists>k. s = [:k:]" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

841 
by (cases s, simp split: if_splits) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

842 
then obtain k where kpn: "s = [:k:]" by blast 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

843 
from sne kpn have k: "k \<noteq> 0" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

844 
let ?w = "([:1/k:] * ([:a,1:] ^ (n  ?op))) * (r ^ n)" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

845 
from k oop [of a] have "q ^ n = p * ?w" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

846 
apply  
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

847 
apply (subst r, subst s, subst kpn) 
29472  848 
apply (subst power_mult_distrib, simp) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

849 
apply (subst power_add [symmetric], simp) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

850 
done 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

851 
hence ?ths unfolding dvd_def by blast} 
26123  852 
moreover 
853 
{assume ds0: "degree s \<noteq> 0" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

854 
from ds0 sne dpn s oa 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

855 
have dsn: "degree s < n" apply auto 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

856 
apply (erule ssubst) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

857 
apply (simp add: degree_mult_eq degree_linear_power) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

858 
done 
26123  859 
{fix x assume h: "poly s x = 0" 
860 
{assume xa: "x = a" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

861 
from h[unfolded xa poly_eq_0_iff_dvd] obtain u where 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

862 
u: "s = [: a, 1:] * u" by (rule dvdE) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

863 
have "p = [: a, 1:] ^ (Suc ?op) * u" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

864 
by (subst s, subst u, simp only: power_Suc mult_ac) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

865 
with ap(2)[unfolded dvd_def] have False by blast} 
26123  866 
note xa = this 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

867 
from h have "poly p x = 0" by (subst s, simp) 
26123  868 
with pq0 have "poly q x = 0" by blast 
869 
with r xa have "poly r x = 0" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

870 
by (auto simp add: uminus_add_conv_diff)} 
26123  871 
note impth = this 
872 
from IH[rule_format, OF dsn, of s r] impth ds0 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

873 
have "s dvd (r ^ (degree s))" by blast 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

874 
then obtain u where u: "r ^ (degree s) = s * u" .. 
26123  875 
hence u': "\<And>x. poly s x * poly u x = poly r x ^ degree s" 
29470
1851088a1f87
convert Deriv.thy to use new Polynomial library (incomplete)
huffman
parents:
29464
diff
changeset

876 
by (simp only: poly_mult[symmetric] poly_power[symmetric]) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

877 
let ?w = "(u * ([:a,1:] ^ (n  ?op))) * (r ^ (n  degree s))" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

878 
from oop[of a] dsn have "q ^ n = p * ?w" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

879 
apply  
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

880 
apply (subst s, subst r) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

881 
apply (simp only: power_mult_distrib) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

882 
apply (subst mult_assoc [where b=s]) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

883 
apply (subst mult_assoc [where a=u]) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

884 
apply (subst mult_assoc [where b=u, symmetric]) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

885 
apply (subst u [symmetric]) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

886 
apply (simp add: mult_ac power_add [symmetric]) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

887 
done 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

888 
hence ?ths unfolding dvd_def by blast} 
26123  889 
ultimately have ?ths by blast } 
890 
ultimately have ?ths by blast} 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

891 
then have ?ths using a order_root pne by blast} 
26123  892 
moreover 
893 
{assume exa: "\<not> (\<exists>a. poly p a = 0)" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

894 
from fundamental_theorem_of_algebra_alt[of p] exa obtain c where 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

895 
ccs: "c\<noteq>0" "p = pCons c 0" by blast 
30488  896 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

897 
then have pp: "\<And>x. poly p x = c" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

898 
let ?w = "[:1/c:] * (q ^ n)" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

899 
from ccs 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

900 
have "(q ^ n) = (p * ?w) " 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

901 
by (simp add: smult_smult) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

902 
hence ?ths unfolding dvd_def by blast} 
26123  903 
ultimately show ?ths by blast 
904 
qed 

905 

906 
lemma nullstellensatz_univariate: 

30488  907 
"(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

908 
p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)" 
26123  909 
proof 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

910 
{assume pe: "p = 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

911 
hence eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0" 
26123  912 
apply auto 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

913 
apply (rule poly_zero [THEN iffD1]) 
26123  914 
by (rule ext, simp) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

915 
{assume "p dvd (q ^ (degree p))" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

916 
then obtain r where r: "q ^ (degree p) = p * r" .. 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

917 
from r pe have False by simp} 
26123  918 
with eq pe have ?thesis by blast} 
919 
moreover 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

920 
{assume pe: "p \<noteq> 0" 
26123  921 
{assume dp: "degree p = 0" 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

922 
then obtain k where k: "p = [:k:]" "k\<noteq>0" using pe 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

923 
by (cases p, simp split: if_splits) 
26123  924 
hence th1: "\<forall>x. poly p x \<noteq> 0" by simp 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

925 
from k dp have "q ^ (degree p) = p * [:1/k:]" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

926 
by (simp add: one_poly_def) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

927 
hence th2: "p dvd (q ^ (degree p))" .. 
26123  928 
from th1 th2 pe have ?thesis by blast} 
929 
moreover 

930 
{assume dp: "degree p \<noteq> 0" 

931 
then obtain n where n: "degree p = Suc n " by (cases "degree p", auto) 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

932 
{assume "p dvd (q ^ (Suc n))" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

933 
then obtain u where u: "q ^ (Suc n) = p * u" .. 
26123  934 
{fix x assume h: "poly p x = 0" "poly q x \<noteq> 0" 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

935 
hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp 
29470
1851088a1f87
convert Deriv.thy to use new Polynomial library (incomplete)
huffman
parents:
29464
diff
changeset

936 
hence False using u h(1) by (simp only: poly_mult) simp}} 
30488  937 
with n nullstellensatz_lemma[of p q "degree p"] dp 
26123  938 
have ?thesis by auto} 
939 
ultimately have ?thesis by blast} 

940 
ultimately show ?thesis by blast 

941 
qed 

942 

943 
text{* Useful lemma *} 

944 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

945 
lemma constant_degree: 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

946 
fixes p :: "'a::{idom,ring_char_0} poly" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

947 
shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs") 
26123  948 
proof 
949 
assume l: ?lhs 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

950 
from l[unfolded constant_def, rule_format, of _ "0"] 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

951 
have th: "poly p = poly [:poly p 0:]" apply  by (rule ext, simp) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

952 
then have "p = [:poly p 0:]" by (simp add: poly_eq_iff) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

953 
then have "degree p = degree [:poly p 0:]" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

954 
then show ?rhs by simp 
26123  955 
next 
956 
assume r: ?rhs 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

957 
then obtain k where "p = [:k:]" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

958 
by (cases p, simp split: if_splits) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

959 
then show ?lhs unfolding constant_def by auto 
26123  960 
qed 
961 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

962 
lemma divides_degree: assumes pq: "p dvd (q:: complex poly)" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

963 
shows "degree p \<le> degree q \<or> q = 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

964 
apply (cases "q = 0", simp_all) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

965 
apply (erule dvd_imp_degree_le [OF pq]) 
26123  966 
done 
967 

968 
(* Arithmetic operations on multivariate polynomials. *) 

969 

30488  970 
lemma mpoly_base_conv: 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

971 
"(0::complex) \<equiv> poly 0 x" "c \<equiv> poly [:c:] x" "x \<equiv> poly [:0,1:] x" by simp_all 
26123  972 

30488  973 
lemma mpoly_norm_conv: 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

974 
"poly [:0:] (x::complex) \<equiv> poly 0 x" "poly [:poly 0 y:] x \<equiv> poly 0 x" by simp_all 
26123  975 

30488  976 
lemma mpoly_sub_conv: 
26123  977 
"poly p (x::complex)  poly q x \<equiv> poly p x + 1 * poly q x" 
978 
by (simp add: diff_def) 

979 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

980 
lemma poly_pad_rule: "poly p x = 0 ==> poly (pCons 0 p) x = (0::complex)" by simp 
26123  981 

982 
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 

983 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

984 
lemma resolve_eq_raw: "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto 
26123  985 
lemma resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2)) 
30488  986 
\<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast 
26123  987 

30488  988 
lemma poly_divides_pad_rule: 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

989 
fixes p q :: "complex poly" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

990 
assumes pq: "p dvd q" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

991 
shows "p dvd (pCons (0::complex) q)" 
26123  992 
proof 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

993 
have "pCons 0 q = q * [:0,1:]" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

994 
then have "q dvd (pCons 0 q)" .. 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

995 
with pq show ?thesis by (rule dvd_trans) 
26123  996 
qed 
997 

30488  998 
lemma poly_divides_pad_const_rule: 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

999 
fixes p q :: "complex poly" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1000 
assumes pq: "p dvd q" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1001 
shows "p dvd (smult a q)" 
26123  1002 
proof 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1003 
have "smult a q = q * [:a:]" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1004 
then have "q dvd smult a q" .. 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1005 
with pq show ?thesis by (rule dvd_trans) 
26123  1006 
qed 
1007 

1008 

30488  1009 
lemma poly_divides_conv0: 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1010 
fixes p :: "complex poly" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1011 
assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1012 
shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs") 
26123  1013 
proof 
30488  1014 
{assume r: ?rhs 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1015 
hence "q = p * 0" by simp 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1016 
hence ?lhs ..} 
26123  1017 
moreover 
1018 
{assume l: ?lhs 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1019 
{assume q0: "q = 0" 
26123  1020 
hence ?rhs by simp} 
1021 
moreover 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1022 
{assume q0: "q \<noteq> 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1023 
from l q0 have "degree p \<le> degree q" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1024 
by (rule dvd_imp_degree_le) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1025 
with lgpq have ?rhs by simp } 
26123  1026 
ultimately have ?rhs by blast } 
30488  1027 
ultimately show "?lhs \<equiv> ?rhs" by  (atomize (full), blast) 
26123  1028 
qed 
1029 

30488  1030 
lemma poly_divides_conv1: 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1031 
assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1032 
and qrp': "smult a q  p' \<equiv> r" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1033 
shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?rhs") 
26123  1034 
proof 
1035 
{ 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1036 
from pp' obtain t where t: "p' = p * t" .. 
26123  1037 
{assume l: ?lhs 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1038 
then obtain u where u: "q = p * u" .. 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1039 
have "r = p * (smult a u  t)" 
29667  1040 
using u qrp' [symmetric] t by (simp add: algebra_simps mult_smult_right) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1041 
then have ?rhs ..} 
26123  1042 
moreover 
1043 
{assume r: ?rhs 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1044 
then obtain u where u: "r = p * u" .. 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1045 
from u [symmetric] t qrp' [symmetric] a0 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1046 
have "q = p * smult (1/a) (u + t)" 
29667  1047 
by (simp add: algebra_simps mult_smult_right smult_smult) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1048 
hence ?lhs ..} 
26123  1049 
ultimately have "?lhs = ?rhs" by blast } 
30488  1050 
thus "?lhs \<equiv> ?rhs" by  (atomize(full), blast) 
26123  1051 
qed 
1052 

1053 
lemma basic_cqe_conv1: 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1054 
"(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1055 
"(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1056 
"(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1057 
"(\<exists>x. poly 0 x = 0) \<equiv> True" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1058 
"(\<exists>x. poly [:c:] x = 0) \<equiv> c = 0" by simp_all 
26123  1059 

30488  1060 
lemma basic_cqe_conv2: 
1061 
assumes l:"p \<noteq> 0" 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1062 
shows "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" 
26123  1063 
proof 
1064 
{fix h t 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1065 
assume h: "h\<noteq>0" "t=0" "pCons a (pCons b p) = pCons h t" 
26123  1066 
with l have False by simp} 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1067 
hence th: "\<not> (\<exists> h t. h\<noteq>0 \<and> t=0 \<and> pCons a (pCons b p) = pCons h t)" 
26123  1068 
by blast 
30488  1069 
from fundamental_theorem_of_algebra_alt[OF th] 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1070 
show "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto 
26123  1071 
qed 
1072 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1073 
lemma basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)" 
26123  1074 
proof 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1075 
have "p = 0 \<longleftrightarrow> poly p = poly 0" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1076 
by (simp add: poly_zero) 
26123  1077 
also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1078 
finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0" 
26123  1079 
by  (atomize (full), blast) 
1080 
qed 

1081 

1082 
lemma basic_cqe_conv3: 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1083 
fixes p q :: "complex poly" 
30488  1084 
assumes l: "p \<noteq> 0" 
29538  1085 
shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))" 
26123  1086 
proof 
29538  1087 
from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def) 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1088 
from nullstellensatz_univariate[of "pCons a p" q] l 
29538  1089 
show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))" 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1090 
unfolding dp 
26123  1091 
by  (atomize (full), auto) 
1092 
qed 

1093 

1094 
lemma basic_cqe_conv4: 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1095 
fixes p q :: "complex poly" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1096 
assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x" 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1097 
shows "p dvd (q ^ n) \<equiv> p dvd r" 
26123  1098 
proof 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1099 
from h have "poly (q ^ n) = poly r" by (auto intro: ext) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1100 
then have "(q ^ n) = r" by (simp add: poly_eq_iff) 
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1101 
thus "p dvd (q ^ n) \<equiv> p dvd r" by simp 
26123  1102 
qed 
1103 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1104 
lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))" 
26123  1105 
by simp 
1106 

1107 
lemma elim_neg_conv: " z \<equiv> (1) * (z::complex)" by simp 

1108 
lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+ 

1109 
lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto) 

1110 

1111 
lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp 

30488  1112 
lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)" 
26123  1113 
by (atomize (full)) simp_all 
29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1114 
lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True" by simp 
26123  1115 
lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))" (is "?l \<equiv> ?r") 
1116 
proof 

1117 
assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply  apply (atomize (full)) by blast 

1118 
next 

1119 
assume "p \<and> q \<equiv> p \<and> r" "p" 

1120 
thus "q \<equiv> r" apply  apply (atomize (full)) apply blast done 

1121 
qed 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1122 
lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp 
26123  1123 

29464
c0d225a7f6ff
convert Fundamental_Theorem_Algebra.thy to use new Polynomial library
huffman
parents:
29292
diff
changeset

1124 
end 