src/HOL/Analysis/Poly_Roots.thy
 changeset 63627 6ddb43c6b711 parent 63594 bd218a9320b5 child 63918 6bf55e6e0b75
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/HOL/Analysis/Poly_Roots.thy	Mon Aug 08 14:13:14 2016 +0200
1.3 @@ -0,0 +1,301 @@
1.4 +(*  Author: John Harrison and Valentina Bruno
1.5 +    Ported from "hol_light/Multivariate/complexes.ml" by L C Paulson
1.6 +*)
1.7 +
1.8 +section \<open>polynomial functions: extremal behaviour and root counts\<close>
1.9 +
1.10 +theory Poly_Roots
1.11 +imports Complex_Main
1.12 +begin
1.13 +
1.14 +subsection\<open>Geometric progressions\<close>
1.15 +
1.16 +lemma setsum_gp_basic:
1.17 +  fixes x :: "'a::{comm_ring,monoid_mult}"
1.18 +  shows "(1 - x) * (\<Sum>i\<le>n. x^i) = 1 - x^Suc n"
1.19 +  by (simp only: one_diff_power_eq [of "Suc n" x] lessThan_Suc_atMost)
1.20 +
1.21 +lemma setsum_gp0:
1.22 +  fixes x :: "'a::{comm_ring,division_ring}"
1.23 +  shows   "(\<Sum>i\<le>n. x^i) = (if x = 1 then of_nat(n + 1) else (1 - x^Suc n) / (1 - x))"
1.24 +  using setsum_gp_basic[of x n]
1.25 +  by (simp add: mult.commute divide_simps)
1.26 +
1.28 +  fixes x :: "'a::{comm_ring,monoid_mult}"
1.29 +  shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
1.31 +
1.32 +lemma setsum_power_shift:
1.33 +  fixes x :: "'a::{comm_ring,monoid_mult}"
1.34 +  assumes "m \<le> n"
1.35 +  shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
1.36 +proof -
1.37 +  have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
1.39 +  also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
1.40 +    using \<open>m \<le> n\<close> by (intro setsum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
1.41 +  finally show ?thesis .
1.42 +qed
1.43 +
1.44 +lemma setsum_gp_multiplied:
1.45 +  fixes x :: "'a::{comm_ring,monoid_mult}"
1.46 +  assumes "m \<le> n"
1.47 +  shows "(1 - x) * (\<Sum>i=m..n. x^i) = x^m - x^Suc n"
1.48 +proof -
1.49 +  have  "(1 - x) * (\<Sum>i=m..n. x^i) = x^m * (1 - x) * (\<Sum>i\<le>n-m. x^i)"
1.50 +    by (metis mult.assoc mult.commute assms setsum_power_shift)
1.51 +  also have "... =x^m * (1 - x^Suc(n-m))"
1.52 +    by (metis mult.assoc setsum_gp_basic)
1.53 +  also have "... = x^m - x^Suc n"
1.54 +    using assms
1.56 +  finally show ?thesis .
1.57 +qed
1.58 +
1.59 +lemma setsum_gp:
1.60 +  fixes x :: "'a::{comm_ring,division_ring}"
1.61 +  shows   "(\<Sum>i=m..n. x^i) =
1.62 +               (if n < m then 0
1.63 +                else if x = 1 then of_nat((n + 1) - m)
1.64 +                else (x^m - x^Suc n) / (1 - x))"
1.65 +using setsum_gp_multiplied [of m n x]
1.66 +apply auto
1.67 +by (metis eq_iff_diff_eq_0 mult.commute nonzero_divide_eq_eq)
1.68 +
1.69 +lemma setsum_gp_offset:
1.70 +  fixes x :: "'a::{comm_ring,division_ring}"
1.71 +  shows   "(\<Sum>i=m..m+n. x^i) =
1.72 +       (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
1.73 +  using setsum_gp [of x m "m+n"]
1.74 +  by (auto simp: power_add algebra_simps)
1.75 +
1.76 +lemma setsum_gp_strict:
1.77 +  fixes x :: "'a::{comm_ring,division_ring}"
1.78 +  shows "(\<Sum>i<n. x^i) = (if x = 1 then of_nat n else (1 - x^n) / (1 - x))"
1.79 +  by (induct n) (auto simp: algebra_simps divide_simps)
1.80 +
1.81 +subsection\<open>Basics about polynomial functions: extremal behaviour and root counts.\<close>
1.82 +
1.83 +lemma sub_polyfun:
1.84 +  fixes x :: "'a::{comm_ring,monoid_mult}"
1.85 +  shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
1.86 +           (x - y) * (\<Sum>j<n. \<Sum>k= Suc j..n. a k * y^(k - Suc j) * x^j)"
1.87 +proof -
1.88 +  have "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
1.89 +        (\<Sum>i\<le>n. a i * (x^i - y^i))"
1.90 +    by (simp add: algebra_simps setsum_subtractf [symmetric])
1.91 +  also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
1.92 +    by (simp add: power_diff_sumr2 ac_simps)
1.93 +  also have "... = (x - y) * (\<Sum>i\<le>n. (\<Sum>j<i. a i * y^(i - Suc j) * x^j))"
1.94 +    by (simp add: setsum_right_distrib ac_simps)
1.95 +  also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - Suc j) * x^j))"
1.96 +    by (simp add: nested_setsum_swap')
1.97 +  finally show ?thesis .
1.98 +qed
1.99 +
1.100 +lemma sub_polyfun_alt:
1.101 +  fixes x :: "'a::{comm_ring,monoid_mult}"
1.102 +  shows   "(\<Sum>i\<le>n. a i * x^i) - (\<Sum>i\<le>n. a i * y^i) =
1.103 +           (x - y) * (\<Sum>j<n. \<Sum>k<n-j. a (j+k+1) * y^k * x^j)"
1.104 +proof -
1.105 +  { fix j
1.106 +    have "(\<Sum>k = Suc j..n. a k * y^(k - Suc j) * x^j) =
1.107 +          (\<Sum>k <n - j. a (Suc (j + k)) * y^k * x^j)"
1.108 +      by (rule setsum.reindex_bij_witness[where i="\<lambda>i. i + Suc j" and j="\<lambda>i. i - Suc j"]) auto }
1.109 +  then show ?thesis
1.110 +    by (simp add: sub_polyfun)
1.111 +qed
1.112 +
1.113 +lemma polyfun_linear_factor:
1.114 +  fixes a :: "'a::{comm_ring,monoid_mult}"
1.115 +  shows  "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) =
1.116 +                  (z-a) * (\<Sum>i<n. b i * z^i) + (\<Sum>i\<le>n. c i * a^i)"
1.117 +proof -
1.118 +  { fix z
1.119 +    have "(\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) =
1.120 +          (z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)"
1.121 +      by (simp add: sub_polyfun setsum_left_distrib)
1.122 +    then have "(\<Sum>i\<le>n. c i * z^i) =
1.123 +          (z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)
1.124 +          + (\<Sum>i\<le>n. c i * a^i)"
1.125 +      by (simp add: algebra_simps) }
1.126 +  then show ?thesis
1.127 +    by (intro exI allI)
1.128 +qed
1.129 +
1.130 +lemma polyfun_linear_factor_root:
1.131 +  fixes a :: "'a::{comm_ring,monoid_mult}"
1.132 +  assumes "(\<Sum>i\<le>n. c i * a^i) = 0"
1.133 +  shows  "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z^i) = (z-a) * (\<Sum>i<n. b i * z^i)"
1.134 +  using polyfun_linear_factor [of c n a] assms
1.135 +  by simp
1.136 +
1.137 +lemma adhoc_norm_triangle: "a + norm(y) \<le> b ==> norm(x) \<le> a ==> norm(x + y) \<le> b"
1.138 +  by (metis norm_triangle_mono order.trans order_refl)
1.139 +
1.140 +lemma polyfun_extremal_lemma:
1.141 +  fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
1.142 +  assumes "e > 0"
1.143 +    shows "\<exists>M. \<forall>z. M \<le> norm z \<longrightarrow> norm(\<Sum>i\<le>n. c i * z^i) \<le> e * norm(z) ^ Suc n"
1.144 +proof (induction n)
1.145 +  case 0
1.146 +  show ?case
1.147 +    by (rule exI [where x="norm (c 0) / e"]) (auto simp: mult.commute pos_divide_le_eq assms)
1.148 +next
1.149 +  case (Suc n)
1.150 +  then obtain M where M: "\<forall>z. M \<le> norm z \<longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n" ..
1.151 +  show ?case
1.152 +  proof (rule exI [where x="max 1 (max M ((e + norm(c(Suc n))) / e))"], clarify)
1.153 +    fix z::'a
1.154 +    assume "max 1 (max M ((e + norm (c (Suc n))) / e)) \<le> norm z"
1.155 +    then have norm1: "0 < norm z" "M \<le> norm z" "(e + norm (c (Suc n))) / e \<le> norm z"
1.156 +      by auto
1.157 +    then have norm2: "(e + norm (c (Suc n))) \<le> e * norm z"  "(norm z * norm z ^ n) > 0"
1.158 +      apply (metis assms less_divide_eq mult.commute not_le)
1.159 +      using norm1 apply (metis mult_pos_pos zero_less_power)
1.160 +      done
1.161 +    have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n)) =
1.162 +          (e + norm (c (Suc n))) * (norm z * norm z ^ n)"
1.163 +      by (simp add: norm_mult norm_power algebra_simps)
1.164 +    also have "... \<le> (e * norm z) * (norm z * norm z ^ n)"
1.165 +      using norm2 by (metis real_mult_le_cancel_iff1)
1.166 +    also have "... = e * (norm z * (norm z * norm z ^ n))"
1.167 +      by (simp add: algebra_simps)
1.168 +    finally have "e * (norm z * norm z ^ n) + norm (c (Suc n) * (z * z ^ n))
1.169 +                  \<le> e * (norm z * (norm z * norm z ^ n))" .
1.170 +    then show "norm (\<Sum>i\<le>Suc n. c i * z^i) \<le> e * norm z ^ Suc (Suc n)" using M norm1
1.171 +      by (drule_tac x=z in spec) (auto simp: intro!: adhoc_norm_triangle)
1.172 +    qed
1.173 +qed
1.174 +
1.175 +lemma norm_lemma_xy: assumes "\<bar>b\<bar> + 1 \<le> norm(y) - a" "norm(x) \<le> a" shows "b \<le> norm(x + y)"
1.176 +proof -
1.177 +  have "b \<le> norm y - norm x"
1.178 +    using assms by linarith
1.179 +  then show ?thesis
1.180 +    by (metis (no_types) add.commute norm_diff_ineq order_trans)
1.181 +qed
1.182 +
1.183 +lemma polyfun_extremal:
1.184 +  fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
1.185 +  assumes "\<exists>k. k \<noteq> 0 \<and> k \<le> n \<and> c k \<noteq> 0"
1.186 +    shows "eventually (\<lambda>z. norm(\<Sum>i\<le>n. c i * z^i) \<ge> B) at_infinity"
1.187 +using assms
1.188 +proof (induction n)
1.189 +  case 0 then show ?case
1.190 +    by simp
1.191 +next
1.192 +  case (Suc n)
1.193 +  show ?case
1.194 +  proof (cases "c (Suc n) = 0")
1.195 +    case True
1.196 +    with Suc show ?thesis
1.197 +      by auto (metis diff_is_0_eq diffs0_imp_equal less_Suc_eq_le not_less_eq)
1.198 +  next
1.199 +    case False
1.200 +    with polyfun_extremal_lemma [of "norm(c (Suc n)) / 2" c n]
1.201 +    obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow>
1.202 +               norm (\<Sum>i\<le>n. c i * z^i) \<le> norm (c (Suc n)) / 2 * norm z ^ Suc n"
1.203 +      by auto
1.204 +    show ?thesis
1.205 +    unfolding eventually_at_infinity
1.206 +    proof (rule exI [where x="max M (max 1 ((\<bar>B\<bar> + 1) / (norm (c (Suc n)) / 2)))"], clarsimp)
1.207 +      fix z::'a
1.208 +      assume les: "M \<le> norm z"  "1 \<le> norm z"  "(\<bar>B\<bar> * 2 + 2) / norm (c (Suc n)) \<le> norm z"
1.209 +      then have "\<bar>B\<bar> * 2 + 2 \<le> norm z * norm (c (Suc n))"
1.210 +        by (metis False pos_divide_le_eq zero_less_norm_iff)
1.211 +      then have "\<bar>B\<bar> * 2 + 2 \<le> norm z ^ (Suc n) * norm (c (Suc n))"
1.212 +        by (metis \<open>1 \<le> norm z\<close> order.trans mult_right_mono norm_ge_zero self_le_power zero_less_Suc)
1.213 +      then show "B \<le> norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * (z * z ^ n))" using M les
1.214 +        apply auto
1.215 +        apply (rule norm_lemma_xy [where a = "norm (c (Suc n)) * norm z ^ (Suc n) / 2"])
1.216 +        apply (simp_all add: norm_mult norm_power)
1.217 +        done
1.218 +    qed
1.219 +  qed
1.220 +qed
1.221 +
1.222 +lemma polyfun_rootbound:
1.223 + fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
1.224 + assumes "\<exists>k. k \<le> n \<and> c k \<noteq> 0"
1.225 +   shows "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<and> card {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<le> n"
1.226 +using assms
1.227 +proof (induction n arbitrary: c)
1.228 + case (Suc n) show ?case
1.229 + proof (cases "{z. (\<Sum>i\<le>Suc n. c i * z^i) = 0} = {}")
1.230 +   case False
1.231 +   then obtain a where a: "(\<Sum>i\<le>Suc n. c i * a^i) = 0"
1.232 +     by auto
1.233 +   from polyfun_linear_factor_root [OF this]
1.234 +   obtain b where "\<And>z. (\<Sum>i\<le>Suc n. c i * z^i) = (z - a) * (\<Sum>i< Suc n. b i * z^i)"
1.235 +     by auto
1.236 +   then have b: "\<And>z. (\<Sum>i\<le>Suc n. c i * z^i) = (z - a) * (\<Sum>i\<le>n. b i * z^i)"
1.237 +     by (metis lessThan_Suc_atMost)
1.238 +   then have ins_ab: "{z. (\<Sum>i\<le>Suc n. c i * z^i) = 0} = insert a {z. (\<Sum>i\<le>n. b i * z^i) = 0}"
1.239 +     by auto
1.240 +   have c0: "c 0 = - (a * b 0)" using  b [of 0]
1.241 +     by simp
1.242 +   then have extr_prem: "~ (\<exists>k\<le>n. b k \<noteq> 0) \<Longrightarrow> \<exists>k. k \<noteq> 0 \<and> k \<le> Suc n \<and> c k \<noteq> 0"
1.243 +     by (metis Suc.prems le0 minus_zero mult_zero_right)
1.244 +   have "\<exists>k\<le>n. b k \<noteq> 0"
1.245 +     apply (rule ccontr)
1.246 +     using polyfun_extremal [OF extr_prem, of 1]
1.247 +     apply (auto simp: eventually_at_infinity b simp del: setsum_atMost_Suc)
1.248 +     apply (drule_tac x="of_real ba" in spec, simp)
1.249 +     done
1.250 +   then show ?thesis using Suc.IH [of b] ins_ab
1.251 +     by (auto simp: card_insert_if)
1.252 +   qed simp
1.253 +qed simp
1.254 +
1.255 +corollary
1.256 +  fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
1.257 +  assumes "\<exists>k. k \<le> n \<and> c k \<noteq> 0"
1.258 +    shows polyfun_rootbound_finite: "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0}"
1.259 +      and polyfun_rootbound_card:   "card {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<le> n"
1.260 +using polyfun_rootbound [OF assms] by auto
1.261 +
1.262 +lemma polyfun_finite_roots:
1.263 +  fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
1.264 +    shows  "finite {z. (\<Sum>i\<le>n. c i * z^i) = 0} \<longleftrightarrow> (\<exists>k. k \<le> n \<and> c k \<noteq> 0)"
1.265 +proof (cases " \<exists>k\<le>n. c k \<noteq> 0")
1.266 +  case True then show ?thesis
1.267 +    by (blast intro: polyfun_rootbound_finite)
1.268 +next
1.269 +  case False then show ?thesis
1.270 +    by (auto simp: infinite_UNIV_char_0)
1.271 +qed
1.272 +
1.273 +lemma polyfun_eq_0:
1.274 +  fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
1.275 +    shows  "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = 0) \<longleftrightarrow> (\<forall>k. k \<le> n \<longrightarrow> c k = 0)"
1.276 +proof (cases "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = 0)")
1.277 +  case True
1.278 +  then have "~ finite {z. (\<Sum>i\<le>n. c i * z^i) = 0}"
1.279 +    by (simp add: infinite_UNIV_char_0)
1.280 +  with True show ?thesis
1.281 +    by (metis (poly_guards_query) polyfun_rootbound_finite)
1.282 +next
1.283 +  case False
1.284 +  then show ?thesis
1.285 +    by auto
1.286 +qed
1.287 +
1.288 +lemma polyfun_eq_const:
1.289 +  fixes c :: "nat \<Rightarrow> 'a::{comm_ring,real_normed_div_algebra}"
1.290 +    shows  "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = k) \<longleftrightarrow> c 0 = k \<and> (\<forall>k. k \<noteq> 0 \<and> k \<le> n \<longrightarrow> c k = 0)"
1.291 +proof -
1.292 +  {fix z
1.293 +    have "(\<Sum>i\<le>n. c i * z^i) = (\<Sum>i\<le>n. (if i = 0 then c 0 - k else c i) * z^i) + k"
1.294 +      by (induct n) auto
1.295 +  } then
1.296 +  have "(\<forall>z. (\<Sum>i\<le>n. c i * z^i) = k) \<longleftrightarrow> (\<forall>z. (\<Sum>i\<le>n. (if i = 0 then c 0 - k else c i) * z^i) = 0)"
1.297 +    by auto
1.298 +  also have "... \<longleftrightarrow>  c 0 = k \<and> (\<forall>k. k \<noteq> 0 \<and> k \<le> n \<longrightarrow> c k = 0)"
1.299 +    by (auto simp: polyfun_eq_0)
1.300 +  finally show ?thesis .
1.301 +qed
1.302 +
1.303 +end
1.304 +
```