src/HOL/Library/Fundamental_Theorem_Algebra.thy
author wenzelm
Sat Jun 13 13:09:05 2015 +0200 (2015-06-13)
changeset 60449 229bad93377e
parent 60424 c96fff9dcdbc
child 60457 f31f7599ef55
permissions -rw-r--r--
renamed "prems" to "that";
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(* Author: Amine Chaieb, TU Muenchen *)
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section \<open>Fundamental Theorem of Algebra\<close>
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theory Fundamental_Theorem_Algebra
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imports Polynomial Complex_Main
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begin
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subsection \<open>More lemmas about module of complex numbers\<close>
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text \<open>The triangle inequality for cmod\<close>
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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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subsection \<open>Basic lemmas about polynomials\<close>
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lemma poly_bound_exists:
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  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
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  shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
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proof (induct p)
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  case 0
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  then show ?case by (rule exI[where x=1]) simp
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next
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  case (pCons c cs)
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  from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
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    by blast
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  let ?k = " 1 + norm c + \<bar>r * m\<bar>"
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  have kp: "?k > 0"
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    using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
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  have "norm (poly (pCons c cs) z) \<le> ?k" if H: "norm z \<le> r" for z
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  proof -
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    from m H have th: "norm (poly cs z) \<le> m"
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      by blast
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    from H have rp: "r \<ge> 0"
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      using norm_ge_zero[of z] by arith
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    have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)"
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      using norm_triangle_ineq[of c "z* poly cs z"] by simp
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    also have "\<dots> \<le> norm c + r * m"
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      using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
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      by (simp add: norm_mult)
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    also have "\<dots> \<le> ?k"
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      by simp
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    finally show ?thesis .
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  qed
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  with kp show ?case by blast
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qed
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text \<open>Offsetting the variable in a polynomial gives another of same degree\<close>
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definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
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  where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"
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lemma offset_poly_0: "offset_poly 0 h = 0"
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  by (simp add: offset_poly_def)
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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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  by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)
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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)
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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)
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  apply 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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definition "psize p = (if p = 0 then 0 else Suc (degree p))"
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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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lemma poly_offset:
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  fixes p :: "'a::comm_ring_1 poly"
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  shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
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proof (intro exI conjI)
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  show "psize (offset_poly p a) = psize p"
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    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)
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qed
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text \<open>An alternative useful formulation of completeness of the reals\<close>
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lemma real_sup_exists:
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  assumes ex: "\<exists>x. P x"
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    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"
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proof
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  from bz have "bdd_above (Collect P)"
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    by (force intro: less_imp_le)
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  then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
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    using ex bz by (subst less_cSup_iff) auto
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qed
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subsection \<open>Fundamental theorem of algebra\<close>
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lemma unimodular_reduce_norm:
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  assumes md: "cmod z = 1"
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  shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
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proof -
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  obtain x y where z: "z = Complex x y "
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    by (cases z) auto
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  from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
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    by (simp add: cmod_def)
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  {
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    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"
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      by (simp_all add: cmod_def power2_eq_square algebra_simps)
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    then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
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      by simp_all
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    then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
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      by - (rule power_mono, simp, simp)+
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    then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
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      by (simp_all add: power_mult_distrib)
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    from add_mono[OF th0] xy have False by simp
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  }
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  then show ?thesis
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    unfolding linorder_not_le[symmetric] by blast
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qed
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text \<open>Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero\<close>
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lemma reduce_poly_simple:
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  assumes b: "b \<noteq> 0"
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    and n: "n \<noteq> 0"
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  shows "\<exists>z. cmod (1 + b * z^n) < 1"
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  using n
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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)"
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  assume n: "n \<noteq> 0"
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  let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
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  {
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    assume e: "even n"
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    then have "\<exists>m. n = 2 * m"
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      by presburger
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    then obtain m where m: "n = 2 * m"
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      by blast
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    from n m have "m \<noteq> 0" "m < n"
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      by presburger+
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    with IH[rule_format, of m] obtain z where z: "?P z m"
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      by blast
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    from z have "?P (csqrt z) n"
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      by (simp add: m power_mult power2_csqrt)
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    then have "\<exists>z. ?P z n" ..
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  }
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  moreover
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  {
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    assume o: "odd n"
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    have th0: "cmod (complex_of_real (cmod b) / b) = 1"
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      using b by (simp add: norm_divide)
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    from o have "\<exists>m. n = Suc (2 * m)"
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      by presburger+
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    then obtain m where m: "n = Suc (2 * m)"
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      by blast
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    from unimodular_reduce_norm[OF th0] o
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    have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
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      apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
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      apply (rule_tac x="1" in exI)
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      apply simp
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      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")
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      apply (rule_tac x="-1" in exI)
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      apply simp
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      apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
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      apply (cases "even m")
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      apply (rule_tac x="ii" in exI)
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      apply (simp add: m power_mult)
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      apply (rule_tac x="- ii" in exI)
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      apply (simp add: m power_mult)
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      apply (cases "even m")
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      apply (rule_tac x="- ii" in exI)
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      apply (simp add: m power_mult)
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      apply (auto simp add: m power_mult)
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      apply (rule_tac x="ii" in exI)
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      apply (auto simp add: m power_mult)
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      done
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    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
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      by blast
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    let ?w = "v / complex_of_real (root n (cmod b))"
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    from odd_real_root_pow[OF o, of "cmod b"]
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    have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
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      by (simp add: power_divide of_real_power[symmetric])
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    have th2:"cmod (complex_of_real (cmod b) / b) = 1"
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      using b by (simp add: norm_divide)
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    then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
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      by simp
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    have th4: "cmod (complex_of_real (cmod b) / b) *
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        cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
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        cmod (complex_of_real (cmod b) / b) * 1"
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      apply (simp only: norm_mult[symmetric] distrib_left)
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      using b v
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      apply (simp add: th2)
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      done
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    from mult_left_less_imp_less[OF th4 th3]
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    have "?P ?w n" unfolding th1 .
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    then have "\<exists>z. ?P z n" ..
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  }
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  ultimately show "\<exists>z. ?P z n" by blast
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qed
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text \<open>Bolzano-Weierstrass type property for closed disc in complex plane.\<close>
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lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
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  using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]
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  unfolding cmod_def by simp
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lemma bolzano_weierstrass_complex_disc:
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  assumes r: "\<forall>n. cmod (s n) \<le> r"
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  shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
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proof -
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  from seq_monosub[of "Re \<circ> s"]
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  obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
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    unfolding o_def by blast
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  from seq_monosub[of "Im \<circ> s \<circ> f"]
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  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
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    unfolding o_def by blast
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  let ?h = "f \<circ> g"
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  from r[rule_format, of 0] have rp: "r \<ge> 0"
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    using norm_ge_zero[of "s 0"] by arith
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  have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
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  proof
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    fix n
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    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
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    show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
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   254
  qed
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  have conv1: "convergent (\<lambda>n. Re (s (f n)))"
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    apply (rule Bseq_monoseq_convergent)
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    apply (simp add: Bseq_def)
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    apply (metis gt_ex le_less_linear less_trans order.trans th)
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    apply (rule f(2))
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    done
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  have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
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  proof
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    fix n
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    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
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    show "\<bar>Im (s n)\<bar> \<le> r + 1"
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      by arith
chaieb@26123
   267
  qed
chaieb@26123
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chaieb@26123
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  have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
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    apply (rule Bseq_monoseq_convergent)
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    apply (simp add: Bseq_def)
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    apply (metis gt_ex le_less_linear less_trans order.trans th)
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    apply (rule g(2))
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    done
chaieb@26123
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huffman@30488
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  from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
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    by blast
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  then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
huffman@31337
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    unfolding LIMSEQ_iff real_norm_def .
chaieb@26123
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huffman@30488
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  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
huffman@30488
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    by blast
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  then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
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    unfolding LIMSEQ_iff real_norm_def .
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  let ?w = "Complex x y"
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  from f(1) g(1) have hs: "subseq ?h"
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    unfolding subseq_def by auto
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  {
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    fix e :: real
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    assume ep: "e > 0"
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    then have e2: "e/2 > 0"
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      by simp
chaieb@26123
   293
    from x[rule_format, OF e2] y[rule_format, OF e2]
wenzelm@56778
   294
    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
wenzelm@56795
   295
      and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
wenzelm@56795
   296
      by blast
wenzelm@56778
   297
    {
wenzelm@56778
   298
      fix n
wenzelm@56778
   299
      assume nN12: "n \<ge> N1 + N2"
wenzelm@56778
   300
      then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
wenzelm@56778
   301
        using seq_suble[OF g(1), of n] by arith+
chaieb@26123
   302
      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
huffman@30488
   303
      have "cmod (s (?h n) - ?w) < e"
wenzelm@56778
   304
        using metric_bound_lemma[of "s (f (g n))" ?w] by simp
wenzelm@56778
   305
    }
wenzelm@56795
   306
    then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e"
wenzelm@56795
   307
      by blast
wenzelm@56778
   308
  }
wenzelm@56778
   309
  with hs show ?thesis by blast
chaieb@26123
   310
qed
chaieb@26123
   311
wenzelm@60424
   312
text \<open>Polynomial is continuous.\<close>
chaieb@26123
   313
chaieb@26123
   314
lemma poly_cont:
wenzelm@56778
   315
  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
huffman@30488
   316
  assumes ep: "e > 0"
lp15@55735
   317
  shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
wenzelm@56778
   318
proof -
huffman@29464
   319
  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
huffman@29464
   320
  proof
huffman@29464
   321
    show "degree (offset_poly p z) = degree p"
huffman@29464
   322
      by (rule degree_offset_poly)
huffman@29464
   323
    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
huffman@29464
   324
      by (rule poly_offset_poly)
huffman@29464
   325
  qed
wenzelm@56778
   326
  have th: "\<And>w. poly q (w - z) = poly p w"
wenzelm@56778
   327
    using q(2)[of "w - z" for w] by simp
chaieb@26123
   328
  show ?thesis unfolding th[symmetric]
wenzelm@56778
   329
  proof (induct q)
wenzelm@56778
   330
    case 0
wenzelm@56778
   331
    then show ?case
wenzelm@56778
   332
      using ep by auto
chaieb@26123
   333
  next
huffman@29464
   334
    case (pCons c cs)
huffman@30488
   335
    from poly_bound_exists[of 1 "cs"]
wenzelm@56778
   336
    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
wenzelm@56778
   337
      by blast
wenzelm@56778
   338
    from ep m(1) have em0: "e/m > 0"
wenzelm@56778
   339
      by (simp add: field_simps)
wenzelm@56778
   340
    have one0: "1 > (0::real)"
wenzelm@56778
   341
      by arith
huffman@30488
   342
    from real_lbound_gt_zero[OF one0 em0]
wenzelm@56778
   343
    obtain d where d: "d > 0" "d < 1" "d < e / m"
wenzelm@56778
   344
      by blast
wenzelm@56778
   345
    from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
nipkow@56544
   346
      by (simp_all add: field_simps)
huffman@30488
   347
    show ?case
wenzelm@56778
   348
    proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
wenzelm@56778
   349
      fix d w
wenzelm@56778
   350
      assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
wenzelm@56778
   351
      then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
wenzelm@56778
   352
        by simp_all
wenzelm@56778
   353
      from H(3) m(1) have dme: "d*m < e"
wenzelm@56778
   354
        by (simp add: field_simps)
wenzelm@56778
   355
      from H have th: "norm (w - z) \<le> d"
wenzelm@56778
   356
        by simp
wenzelm@56778
   357
      from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
wenzelm@56778
   358
      show "norm (w - z) * norm (poly cs (w - z)) < e"
wenzelm@56778
   359
        by simp
chaieb@26123
   360
    qed
wenzelm@56778
   361
  qed
chaieb@26123
   362
qed
chaieb@26123
   363
wenzelm@60424
   364
text \<open>Hence a polynomial attains minimum on a closed disc
wenzelm@60424
   365
  in the complex plane.\<close>
wenzelm@56778
   366
lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   367
proof -
wenzelm@60424
   368
  show ?thesis
wenzelm@60424
   369
  proof (cases "r \<ge> 0")
wenzelm@60424
   370
    case False
wenzelm@60424
   371
    then show ?thesis
wenzelm@56778
   372
      by (metis norm_ge_zero order.trans)
wenzelm@60424
   373
  next
wenzelm@60424
   374
    case True
wenzelm@60424
   375
    then have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
wenzelm@56778
   376
      by simp
wenzelm@56778
   377
    then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
wenzelm@56778
   378
      by blast
wenzelm@56778
   379
    {
wenzelm@56778
   380
      fix x z
wenzelm@56778
   381
      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
wenzelm@56778
   382
      then have "- x < 0 "
wenzelm@56778
   383
        by arith
wenzelm@56778
   384
      with H(2) norm_ge_zero[of "poly p z"] have False
wenzelm@56778
   385
        by simp
wenzelm@56778
   386
    }
wenzelm@56778
   387
    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
wenzelm@56778
   388
      by blast
huffman@30488
   389
    from real_sup_exists[OF mth1 mth2] obtain s where
wenzelm@56778
   390
      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
wenzelm@56778
   391
    let ?m = "- s"
wenzelm@56778
   392
    {
wenzelm@56778
   393
      fix y
wenzelm@56778
   394
      from s[rule_format, of "-y"]
wenzelm@56778
   395
      have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
wenzelm@56778
   396
        unfolding minus_less_iff[of y ] equation_minus_iff by blast
wenzelm@56778
   397
    }
chaieb@26123
   398
    note s1 = this[unfolded minus_minus]
huffman@30488
   399
    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
chaieb@26123
   400
      by auto
wenzelm@56778
   401
    {
wenzelm@56778
   402
      fix n :: nat
huffman@30488
   403
      from s1[rule_format, of "?m + 1/real (Suc n)"]
chaieb@26123
   404
      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
wenzelm@56778
   405
        by simp
wenzelm@56778
   406
    }
wenzelm@56778
   407
    then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
huffman@30488
   408
    from choice[OF th] obtain g where
wenzelm@56778
   409
        g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
chaieb@26123
   410
      by blast
huffman@30488
   411
    from bolzano_weierstrass_complex_disc[OF g(1)]
chaieb@26123
   412
    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
huffman@30488
   413
      by blast
wenzelm@56778
   414
    {
wenzelm@56778
   415
      fix w
chaieb@26123
   416
      assume wr: "cmod w \<le> r"
chaieb@26123
   417
      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
wenzelm@56778
   418
      {
wenzelm@56778
   419
        assume e: "?e > 0"
wenzelm@56795
   420
        then have e2: "?e/2 > 0"
wenzelm@56795
   421
          by simp
wenzelm@32960
   422
        from poly_cont[OF e2, of z p] obtain d where
wenzelm@56778
   423
            d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
wenzelm@56778
   424
          by blast
wenzelm@56778
   425
        {
wenzelm@56778
   426
          fix w
wenzelm@56778
   427
          assume w: "cmod (w - z) < d"
wenzelm@32960
   428
          have "cmod(poly p w - poly p z) < ?e / 2"
wenzelm@56778
   429
            using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
wenzelm@56778
   430
        }
wenzelm@32960
   431
        note th1 = this
huffman@30488
   432
wenzelm@56778
   433
        from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
wenzelm@56778
   434
          by blast
wenzelm@56778
   435
        from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
wenzelm@56778
   436
          by blast
wenzelm@56778
   437
        have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
wenzelm@32960
   438
          using N1[rule_format, of "N1 + N2"] th1 by simp
wenzelm@60424
   439
        have th0: "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
wenzelm@60424
   440
          for a b e2 m :: real
wenzelm@60424
   441
          by arith
wenzelm@60424
   442
        have ath: "m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e" for m x e :: real
wenzelm@56778
   443
          by arith
wenzelm@56778
   444
        from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
wenzelm@56795
   445
        from seq_suble[OF fz(1), of "N1 + N2"]
wenzelm@56778
   446
        have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
wenzelm@56778
   447
          by simp
wenzelm@56778
   448
        have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
wenzelm@56778
   449
          using N2 by auto
wenzelm@56778
   450
        from frac_le[OF th000 th00]
wenzelm@56795
   451
        have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
wenzelm@56778
   452
          by simp
wenzelm@56778
   453
        from g(2)[rule_format, of "f (N1 + N2)"]
wenzelm@56778
   454
        have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
wenzelm@56778
   455
        from order_less_le_trans[OF th01 th00]
wenzelm@56795
   456
        have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
wenzelm@56778
   457
        from N2 have "2/?e < real (Suc (N1 + N2))"
wenzelm@56778
   458
          by arith
wenzelm@56778
   459
        with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
wenzelm@56778
   460
        have "?e/2 > 1/ real (Suc (N1 + N2))"
wenzelm@56778
   461
          by (simp add: inverse_eq_divide)
wenzelm@60424
   462
        with ath[OF th31 th32] have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
wenzelm@56778
   463
          by arith
wenzelm@60424
   464
        have ath2: "\<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c" for a b c m :: real
wenzelm@56778
   465
          by arith
wenzelm@56778
   466
        have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
wenzelm@56778
   467
            cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@56778
   468
          by (simp add: norm_triangle_ineq3)
wenzelm@56778
   469
        from ath2[OF th22, of ?m]
wenzelm@56778
   470
        have thc2: "2 * (?e/2) \<le>
wenzelm@56778
   471
            \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@56778
   472
          by simp
wenzelm@56778
   473
        from th0[OF th2 thc1 thc2] have False .
wenzelm@56778
   474
      }
wenzelm@56778
   475
      then have "?e = 0"
wenzelm@56778
   476
        by auto
wenzelm@56778
   477
      then have "cmod (poly p z) = ?m"
wenzelm@56778
   478
        by simp
wenzelm@56778
   479
      with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   480
        by simp
wenzelm@56778
   481
    }
wenzelm@60424
   482
    then show ?thesis by blast
wenzelm@60424
   483
  qed
chaieb@26123
   484
qed
chaieb@26123
   485
wenzelm@60424
   486
text \<open>Nonzero polynomial in z goes to infinity as z does.\<close>
chaieb@26123
   487
chaieb@26123
   488
lemma poly_infinity:
wenzelm@56778
   489
  fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
huffman@29464
   490
  assumes ex: "p \<noteq> 0"
lp15@55735
   491
  shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
wenzelm@56778
   492
  using ex
wenzelm@56778
   493
proof (induct p arbitrary: a d)
wenzelm@56795
   494
  case 0
wenzelm@56795
   495
  then show ?case by simp
wenzelm@56795
   496
next
huffman@30488
   497
  case (pCons c cs a d)
wenzelm@56795
   498
  show ?case
wenzelm@56795
   499
  proof (cases "cs = 0")
wenzelm@56795
   500
    case False
wenzelm@56778
   501
    with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
wenzelm@56778
   502
      by blast
chaieb@26123
   503
    let ?r = "1 + \<bar>r\<bar>"
wenzelm@56778
   504
    {
wenzelm@56795
   505
      fix z :: 'a
wenzelm@56778
   506
      assume h: "1 + \<bar>r\<bar> \<le> norm z"
wenzelm@56795
   507
      have r0: "r \<le> norm z"
wenzelm@56795
   508
        using h by arith
wenzelm@56778
   509
      from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
wenzelm@56778
   510
        by arith
wenzelm@56778
   511
      from h have z1: "norm z \<ge> 1"
wenzelm@56778
   512
        by arith
huffman@29464
   513
      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
lp15@55735
   514
      have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
wenzelm@32960
   515
        unfolding norm_mult by (simp add: algebra_simps)
lp15@55735
   516
      from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
wenzelm@56795
   517
      have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@51541
   518
        by (simp add: algebra_simps)
wenzelm@56795
   519
      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@56795
   520
        by arith
wenzelm@56778
   521
    }
wenzelm@56795
   522
    then show ?thesis by blast
wenzelm@56795
   523
  next
wenzelm@56795
   524
    case True
wenzelm@56778
   525
    with pCons.prems have c0: "c \<noteq> 0"
wenzelm@56778
   526
      by simp
wenzelm@60424
   527
    have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@60424
   528
      if h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z" for z :: 'a
wenzelm@60424
   529
    proof -
wenzelm@56778
   530
      from c0 have "norm c > 0"
wenzelm@56778
   531
        by simp
blanchet@56403
   532
      from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
wenzelm@32960
   533
        by (simp add: field_simps norm_mult)
wenzelm@56778
   534
      have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
wenzelm@56778
   535
        by arith
wenzelm@56778
   536
      from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
wenzelm@32960
   537
        by (simp add: algebra_simps)
wenzelm@60424
   538
      from ath[OF th1 th0] show ?thesis
wenzelm@56795
   539
        using True by simp
wenzelm@60424
   540
    qed
wenzelm@56795
   541
    then show ?thesis by blast
wenzelm@56795
   542
  qed
wenzelm@56795
   543
qed
chaieb@26123
   544
wenzelm@60424
   545
text \<open>Hence polynomial's modulus attains its minimum somewhere.\<close>
wenzelm@56778
   546
lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   547
proof (induct p)
wenzelm@56778
   548
  case 0
wenzelm@56778
   549
  then show ?case by simp
wenzelm@56778
   550
next
huffman@30488
   551
  case (pCons c cs)
wenzelm@56778
   552
  show ?case
wenzelm@56778
   553
  proof (cases "cs = 0")
wenzelm@56778
   554
    case False
wenzelm@56778
   555
    from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
wenzelm@56778
   556
    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
wenzelm@56778
   557
      by blast
wenzelm@56778
   558
    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
wenzelm@56778
   559
      by arith
huffman@30488
   560
    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
wenzelm@56778
   561
    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)"
wenzelm@56778
   562
      by blast
wenzelm@60424
   563
    have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" if z: "r \<le> cmod z" for z
wenzelm@60424
   564
      using v[of 0] r[OF z] by simp
wenzelm@60424
   565
    with v ath[of r] show ?thesis
wenzelm@56778
   566
      by blast
wenzelm@56778
   567
  next
wenzelm@56778
   568
    case True
wenzelm@60424
   569
    with pCons.hyps show ?thesis
wenzelm@60424
   570
      by simp
wenzelm@56778
   571
  qed
wenzelm@56778
   572
qed
chaieb@26123
   573
wenzelm@60424
   574
text \<open>Constant function (non-syntactic characterization).\<close>
wenzelm@56795
   575
definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"
chaieb@26123
   576
wenzelm@56778
   577
lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
wenzelm@56778
   578
  by (induct p) (auto simp: constant_def psize_def)
huffman@30488
   579
wenzelm@56795
   580
lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
huffman@29464
   581
  by (simp add: poly_monom)
chaieb@26123
   582
wenzelm@60424
   583
text \<open>Decomposition of polynomial, skipping zero coefficients after the first.\<close>
chaieb@26123
   584
chaieb@26123
   585
lemma poly_decompose_lemma:
wenzelm@56778
   586
  assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
wenzelm@56795
   587
  shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and> (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
wenzelm@56778
   588
  unfolding psize_def
wenzelm@56778
   589
  using nz
wenzelm@56778
   590
proof (induct p)
wenzelm@56778
   591
  case 0
wenzelm@56778
   592
  then show ?case by simp
chaieb@26123
   593
next
huffman@29464
   594
  case (pCons c cs)
wenzelm@56778
   595
  show ?case
wenzelm@56778
   596
  proof (cases "c = 0")
wenzelm@56778
   597
    case True
wenzelm@56778
   598
    from pCons.hyps pCons.prems True show ?thesis
wenzelm@60424
   599
      apply auto
chaieb@26123
   600
      apply (rule_tac x="k+1" in exI)
chaieb@26123
   601
      apply (rule_tac x="a" in exI, clarsimp)
chaieb@26123
   602
      apply (rule_tac x="q" in exI)
wenzelm@56778
   603
      apply auto
wenzelm@56778
   604
      done
wenzelm@56778
   605
  next
wenzelm@56778
   606
    case False
wenzelm@56778
   607
    show ?thesis
chaieb@26123
   608
      apply (rule exI[where x=0])
wenzelm@60424
   609
      apply (rule exI[where x=c])
wenzelm@60424
   610
      apply (auto simp: False)
wenzelm@56778
   611
      done
wenzelm@56778
   612
  qed
chaieb@26123
   613
qed
chaieb@26123
   614
chaieb@26123
   615
lemma poly_decompose:
wenzelm@56776
   616
  assumes nc: "\<not> constant (poly p)"
wenzelm@56778
   617
  shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
huffman@30488
   618
               psize q + k + 1 = psize p \<and>
huffman@29464
   619
              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
wenzelm@56776
   620
  using nc
wenzelm@56776
   621
proof (induct p)
wenzelm@56776
   622
  case 0
wenzelm@56776
   623
  then show ?case
wenzelm@56776
   624
    by (simp add: constant_def)
chaieb@26123
   625
next
huffman@29464
   626
  case (pCons c cs)
wenzelm@56776
   627
  {
wenzelm@60424
   628
    assume "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
wenzelm@60424
   629
    then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y
wenzelm@60424
   630
      by (cases "x = 0") auto
wenzelm@56778
   631
    with pCons.prems have False
wenzelm@56778
   632
      by (auto simp add: constant_def)
wenzelm@56776
   633
  }
wenzelm@56776
   634
  then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
huffman@30488
   635
  from poly_decompose_lemma[OF th]
huffman@30488
   636
  show ?case
huffman@29464
   637
    apply clarsimp
chaieb@26123
   638
    apply (rule_tac x="k+1" in exI)
chaieb@26123
   639
    apply (rule_tac x="a" in exI)
chaieb@26123
   640
    apply simp
chaieb@26123
   641
    apply (rule_tac x="q" in exI)
huffman@29538
   642
    apply (auto simp add: psize_def split: if_splits)
chaieb@26123
   643
    done
chaieb@26123
   644
qed
chaieb@26123
   645
wenzelm@60424
   646
text \<open>Fundamental theorem of algebra\<close>
chaieb@26123
   647
chaieb@26123
   648
lemma fundamental_theorem_of_algebra:
wenzelm@56776
   649
  assumes nc: "\<not> constant (poly p)"
chaieb@26123
   650
  shows "\<exists>z::complex. poly p z = 0"
wenzelm@56776
   651
  using nc
wenzelm@56776
   652
proof (induct "psize p" arbitrary: p rule: less_induct)
berghofe@34915
   653
  case less
chaieb@26123
   654
  let ?p = "poly p"
chaieb@26123
   655
  let ?ths = "\<exists>z. ?p z = 0"
chaieb@26123
   656
berghofe@34915
   657
  from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
wenzelm@56776
   658
  from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
wenzelm@56776
   659
    by blast
wenzelm@56778
   660
wenzelm@56778
   661
  show ?ths
wenzelm@56778
   662
  proof (cases "?p c = 0")
wenzelm@56778
   663
    case True
wenzelm@56778
   664
    then show ?thesis by blast
wenzelm@56778
   665
  next
wenzelm@56778
   666
    case False
wenzelm@56778
   667
    from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
wenzelm@56778
   668
      by blast
wenzelm@60424
   669
    have False if h: "constant (poly q)"
wenzelm@60424
   670
    proof -
wenzelm@56795
   671
      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
wenzelm@56795
   672
        by auto
wenzelm@60424
   673
      have "?p x = ?p y" for x y
wenzelm@60424
   674
      proof -
wenzelm@56795
   675
        from th have "?p x = poly q (x - c)"
wenzelm@56795
   676
          by auto
wenzelm@32960
   677
        also have "\<dots> = poly q (y - c)"
wenzelm@32960
   678
          using h unfolding constant_def by blast
wenzelm@56795
   679
        also have "\<dots> = ?p y"
wenzelm@56795
   680
          using th by auto
wenzelm@60424
   681
        finally show ?thesis .
wenzelm@60424
   682
      qed
wenzelm@60424
   683
      with less(2) show ?thesis
wenzelm@56778
   684
        unfolding constant_def by blast
wenzelm@60424
   685
    qed
wenzelm@56778
   686
    then have qnc: "\<not> constant (poly q)"
wenzelm@56778
   687
      by blast
wenzelm@56778
   688
    from q(2) have pqc0: "?p c = poly q 0"
wenzelm@56778
   689
      by simp
wenzelm@56778
   690
    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
wenzelm@56778
   691
      by simp
chaieb@26123
   692
    let ?a0 = "poly q 0"
wenzelm@60424
   693
    from False pqc0 have a00: "?a0 \<noteq> 0"
wenzelm@56778
   694
      by simp
wenzelm@56778
   695
    from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
huffman@29464
   696
      by simp
huffman@29464
   697
    let ?r = "smult (inverse ?a0) q"
huffman@29538
   698
    have lgqr: "psize q = psize ?r"
wenzelm@56778
   699
      using a00
wenzelm@56778
   700
      unfolding psize_def degree_def
haftmann@52380
   701
      by (simp add: poly_eq_iff)
wenzelm@60424
   702
    have False if h: "\<And>x y. poly ?r x = poly ?r y"
wenzelm@60424
   703
    proof -
wenzelm@56778
   704
      {
wenzelm@56778
   705
        fix x y
wenzelm@56778
   706
        from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
wenzelm@56778
   707
          by auto
wenzelm@56778
   708
        also have "\<dots> = poly ?r y * ?a0"
wenzelm@56778
   709
          using h by simp
wenzelm@56778
   710
        also have "\<dots> = poly q y"
wenzelm@56778
   711
          using qr[rule_format, of y] by simp
wenzelm@56778
   712
        finally have "poly q x = poly q y" .
wenzelm@56778
   713
      }
wenzelm@60424
   714
      with qnc show ?thesis
wenzelm@56795
   715
        unfolding constant_def by blast
wenzelm@60424
   716
    qed
wenzelm@56778
   717
    then have rnc: "\<not> constant (poly ?r)"
wenzelm@56778
   718
      unfolding constant_def by blast
wenzelm@56778
   719
    from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
wenzelm@56778
   720
      by auto
wenzelm@60424
   721
    have mrmq_eq: "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" for w
wenzelm@60424
   722
    proof -
chaieb@26123
   723
      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
haftmann@57514
   724
        using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)
chaieb@26123
   725
      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
wenzelm@32960
   726
        using a00 unfolding norm_divide by (simp add: field_simps)
wenzelm@60424
   727
      finally show ?thesis .
wenzelm@60424
   728
    qed
huffman@30488
   729
    from poly_decompose[OF rnc] obtain k a s where
wenzelm@56778
   730
      kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
wenzelm@56778
   731
        "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
wenzelm@60424
   732
    have "\<exists>w. cmod (poly ?r w) < 1"
wenzelm@60424
   733
    proof (cases "psize p = k + 1")
wenzelm@60424
   734
      case True
wenzelm@56778
   735
      with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
wenzelm@56778
   736
        by auto
wenzelm@60424
   737
      have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w
wenzelm@60424
   738
        using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
wenzelm@60424
   739
      from reduce_poly_simple[OF kas(1,2)] show ?thesis
wenzelm@56778
   740
        unfolding hth by blast
wenzelm@60424
   741
    next
wenzelm@60424
   742
      case False note kn = this
wenzelm@56778
   743
      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
wenzelm@56778
   744
        by simp
huffman@30488
   745
      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
wenzelm@32960
   746
        unfolding constant_def poly_pCons poly_monom
wenzelm@56795
   747
        using kas(1)
wenzelm@56795
   748
        apply simp
wenzelm@56778
   749
        apply (rule exI[where x=0])
wenzelm@56778
   750
        apply (rule exI[where x=1])
wenzelm@56778
   751
        apply simp
wenzelm@56778
   752
        done
wenzelm@56778
   753
      from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
wenzelm@32960
   754
        by (simp add: psize_def degree_monom_eq)
berghofe@34915
   755
      from less(1) [OF k1n [simplified th02] th01]
chaieb@26123
   756
      obtain w where w: "1 + w^k * a = 0"
wenzelm@32960
   757
        unfolding poly_pCons poly_monom
wenzelm@56778
   758
        using kas(2) by (cases k) (auto simp add: algebra_simps)
huffman@30488
   759
      from poly_bound_exists[of "cmod w" s] obtain m where
wenzelm@32960
   760
        m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
wenzelm@56795
   761
      have w0: "w \<noteq> 0"
wenzelm@56795
   762
        using kas(2) w by (auto simp add: power_0_left)
wenzelm@56778
   763
      from w have "(1 + w ^ k * a) - 1 = 0 - 1"
wenzelm@56778
   764
        by simp
wenzelm@56778
   765
      then have wm1: "w^k * a = - 1"
wenzelm@56778
   766
        by simp
huffman@30488
   767
      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
wenzelm@32960
   768
        using norm_ge_zero[of w] w0 m(1)
wenzelm@56778
   769
        by (simp add: inverse_eq_divide zero_less_mult_iff)
lp15@55358
   770
      with real_lbound_gt_zero[OF zero_less_one] obtain t where
wenzelm@32960
   771
        t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
chaieb@26123
   772
      let ?ct = "complex_of_real t"
chaieb@26123
   773
      let ?w = "?ct * w"
wenzelm@56778
   774
      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
wenzelm@56778
   775
        using kas(1) by (simp add: algebra_simps power_mult_distrib)
chaieb@26123
   776
      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
wenzelm@56778
   777
        unfolding wm1 by simp
wenzelm@56778
   778
      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
wenzelm@56778
   779
        cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
lp15@55358
   780
        by metis
huffman@30488
   781
      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
wenzelm@56778
   782
      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
wenzelm@56778
   783
        unfolding norm_of_real by simp
wenzelm@56778
   784
      have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
wenzelm@56778
   785
        by arith
wenzelm@56778
   786
      have "t * cmod w \<le> 1 * cmod w"
wenzelm@56778
   787
        apply (rule mult_mono)
wenzelm@56778
   788
        using t(1,2)
wenzelm@56778
   789
        apply auto
wenzelm@56778
   790
        done
wenzelm@56778
   791
      then have tw: "cmod ?w \<le> cmod w"
wenzelm@56778
   792
        using t(1) by (simp add: norm_mult)
wenzelm@56778
   793
      from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
wenzelm@57862
   794
        by (simp add: field_simps)
wenzelm@56778
   795
      with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
haftmann@59557
   796
        by simp
wenzelm@56778
   797
      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
wenzelm@56778
   798
        using w0 t(1)
wenzelm@51541
   799
        by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
chaieb@26123
   800
      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
wenzelm@32960
   801
        using t(1,2) m(2)[rule_format, OF tw] w0
lp15@55358
   802
        by auto
wenzelm@56778
   803
      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
wenzelm@56778
   804
        by simp
huffman@30488
   805
      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
wenzelm@32960
   806
        by auto
huffman@27514
   807
      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
huffman@30488
   808
      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
wenzelm@56778
   809
      from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
wenzelm@56778
   810
        by arith
huffman@30488
   811
      then have "cmod (poly ?r ?w) < 1"
wenzelm@32960
   812
        unfolding kas(4)[rule_format, of ?w] r01 by simp
wenzelm@60424
   813
      then show ?thesis
wenzelm@56778
   814
        by blast
wenzelm@60424
   815
    qed
wenzelm@60424
   816
    with cq0 q(2) show ?thesis
wenzelm@56778
   817
      unfolding mrmq_eq not_less[symmetric] by auto
wenzelm@56778
   818
  qed
chaieb@26123
   819
qed
chaieb@26123
   820
wenzelm@60424
   821
text \<open>Alternative version with a syntactic notion of constant polynomial.\<close>
chaieb@26123
   822
chaieb@26123
   823
lemma fundamental_theorem_of_algebra_alt:
wenzelm@56778
   824
  assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
chaieb@26123
   825
  shows "\<exists>z. poly p z = (0::complex)"
wenzelm@56778
   826
  using nc
wenzelm@56778
   827
proof (induct p)
wenzelm@56778
   828
  case 0
wenzelm@56778
   829
  then show ?case by simp
wenzelm@56778
   830
next
huffman@29464
   831
  case (pCons c cs)
wenzelm@56778
   832
  show ?case
wenzelm@56778
   833
  proof (cases "c = 0")
wenzelm@56778
   834
    case True
wenzelm@56778
   835
    then show ?thesis by auto
wenzelm@56778
   836
  next
wenzelm@56778
   837
    case False
wenzelm@56778
   838
    {
wenzelm@56778
   839
      assume nc: "constant (poly (pCons c cs))"
huffman@30488
   840
      from nc[unfolded constant_def, rule_format, of 0]
huffman@30488
   841
      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
wenzelm@56778
   842
      then have "cs = 0"
wenzelm@56778
   843
      proof (induct cs)
wenzelm@56778
   844
        case 0
wenzelm@56778
   845
        then show ?case by simp
wenzelm@56778
   846
      next
wenzelm@56778
   847
        case (pCons d ds)
wenzelm@56778
   848
        show ?case
wenzelm@56778
   849
        proof (cases "d = 0")
wenzelm@56778
   850
          case True
wenzelm@60424
   851
          then show ?thesis
wenzelm@60424
   852
            using pCons.prems pCons.hyps by simp
wenzelm@56778
   853
        next
wenzelm@56778
   854
          case False
wenzelm@56778
   855
          from poly_bound_exists[of 1 ds] obtain m where
wenzelm@56778
   856
            m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
wenzelm@56795
   857
          have dm: "cmod d / m > 0"
wenzelm@56795
   858
            using False m(1) by (simp add: field_simps)
wenzelm@60424
   859
          from real_lbound_gt_zero[OF dm zero_less_one]
wenzelm@60424
   860
          obtain x where x: "x > 0" "x < cmod d / m" "x < 1"
wenzelm@60424
   861
            by blast
wenzelm@56778
   862
          let ?x = "complex_of_real x"
wenzelm@60424
   863
          from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1"
wenzelm@56795
   864
            by simp_all
wenzelm@56778
   865
          from pCons.prems[rule_format, OF cx(1)]
wenzelm@56795
   866
          have cth: "cmod (?x*poly ds ?x) = cmod d"
wenzelm@56795
   867
            by (simp add: eq_diff_eq[symmetric])
wenzelm@56778
   868
          from m(2)[rule_format, OF cx(2)] x(1)
wenzelm@56778
   869
          have th0: "cmod (?x*poly ds ?x) \<le> x*m"
wenzelm@56778
   870
            by (simp add: norm_mult)
wenzelm@56795
   871
          from x(2) m(1) have "x * m < cmod d"
wenzelm@56795
   872
            by (simp add: field_simps)
wenzelm@56795
   873
          with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
wenzelm@56795
   874
            by auto
wenzelm@56795
   875
          with cth show ?thesis
wenzelm@56795
   876
            by blast
wenzelm@56778
   877
        qed
wenzelm@56778
   878
      qed
wenzelm@56778
   879
    }
wenzelm@56795
   880
    then have nc: "\<not> constant (poly (pCons c cs))"
wenzelm@56795
   881
      using pCons.prems False by blast
wenzelm@56778
   882
    from fundamental_theorem_of_algebra[OF nc] show ?thesis .
wenzelm@56778
   883
  qed
wenzelm@56778
   884
qed
chaieb@26123
   885
huffman@29464
   886
wenzelm@60424
   887
subsection \<open>Nullstellensatz, degrees and divisibility of polynomials\<close>
chaieb@26123
   888
chaieb@26123
   889
lemma nullstellensatz_lemma:
huffman@29464
   890
  fixes p :: "complex poly"
chaieb@26123
   891
  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56776
   892
    and "degree p = n"
wenzelm@56776
   893
    and "n \<noteq> 0"
huffman@29464
   894
  shows "p dvd (q ^ n)"
wenzelm@56776
   895
  using assms
wenzelm@56776
   896
proof (induct n arbitrary: p q rule: nat_less_induct)
wenzelm@56776
   897
  fix n :: nat
wenzelm@56776
   898
  fix p q :: "complex poly"
chaieb@26123
   899
  assume IH: "\<forall>m<n. \<forall>p q.
chaieb@26123
   900
                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
huffman@29464
   901
                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
huffman@30488
   902
    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56778
   903
    and dpn: "degree p = n"
wenzelm@56778
   904
    and n0: "n \<noteq> 0"
huffman@29464
   905
  from dpn n0 have pne: "p \<noteq> 0" by auto
huffman@29464
   906
  let ?ths = "p dvd (q ^ n)"
wenzelm@56778
   907
  {
wenzelm@56778
   908
    fix a
wenzelm@56778
   909
    assume a: "poly p a = 0"
wenzelm@60424
   910
    have ?ths if oa: "order a p \<noteq> 0"
wenzelm@60424
   911
    proof -
chaieb@26123
   912
      let ?op = "order a p"
wenzelm@56778
   913
      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
wenzelm@56778
   914
        using order by blast+
huffman@29464
   915
      note oop = order_degree[OF pne, unfolded dpn]
wenzelm@60424
   916
      show ?thesis
wenzelm@60424
   917
      proof (cases "q = 0")
wenzelm@60424
   918
        case True
wenzelm@60424
   919
        with n0 show ?thesis by (simp add: power_0_left)
wenzelm@60424
   920
      next
wenzelm@60424
   921
        case False
wenzelm@32960
   922
        from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
wenzelm@32960
   923
        obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
wenzelm@56778
   924
        from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
wenzelm@56778
   925
          by (rule dvdE)
wenzelm@60424
   926
        have sne: "s \<noteq> 0"
wenzelm@60424
   927
          using s pne by auto
wenzelm@60424
   928
        show ?thesis
wenzelm@60424
   929
        proof (cases "degree s = 0")
wenzelm@60424
   930
          case True
wenzelm@60424
   931
          then obtain k where kpn: "s = [:k:]"
wenzelm@51541
   932
            by (cases s) (auto split: if_splits)
huffman@29464
   933
          from sne kpn have k: "k \<noteq> 0" by simp
wenzelm@32960
   934
          let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
lp15@55358
   935
          have "q ^ n = p * ?w"
wenzelm@56795
   936
            apply (subst r)
wenzelm@56795
   937
            apply (subst s)
wenzelm@56795
   938
            apply (subst kpn)
wenzelm@56778
   939
            using k oop [of a]
wenzelm@56795
   940
            apply (subst power_mult_distrib)
wenzelm@56795
   941
            apply simp
wenzelm@56795
   942
            apply (subst power_add [symmetric])
wenzelm@56795
   943
            apply simp
huffman@29464
   944
            done
wenzelm@60424
   945
          then show ?thesis
wenzelm@56795
   946
            unfolding dvd_def by blast
wenzelm@60424
   947
        next
wenzelm@60424
   948
          case False
wenzelm@60424
   949
          with sne dpn s oa have dsn: "degree s < n"
wenzelm@56778
   950
              apply auto
huffman@29464
   951
              apply (erule ssubst)
huffman@29464
   952
              apply (simp add: degree_mult_eq degree_linear_power)
huffman@29464
   953
              done
wenzelm@56778
   954
            {
wenzelm@56778
   955
              fix x assume h: "poly s x = 0"
wenzelm@56778
   956
              {
wenzelm@56778
   957
                assume xa: "x = a"
wenzelm@56778
   958
                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
wenzelm@56778
   959
                  by (rule dvdE)
wenzelm@32960
   960
                have "p = [:- a, 1:] ^ (Suc ?op) * u"
wenzelm@56795
   961
                  apply (subst s)
wenzelm@56795
   962
                  apply (subst u)
haftmann@57514
   963
                  apply (simp only: power_Suc ac_simps)
wenzelm@56795
   964
                  done
wenzelm@56795
   965
                with ap(2)[unfolded dvd_def] have False
wenzelm@56795
   966
                  by blast
wenzelm@56778
   967
              }
wenzelm@32960
   968
              note xa = this
wenzelm@56795
   969
              from h have "poly p x = 0"
wenzelm@56795
   970
                by (subst s) simp
wenzelm@56795
   971
              with pq0 have "poly q x = 0"
wenzelm@56795
   972
                by blast
wenzelm@32960
   973
              with r xa have "poly r x = 0"
wenzelm@56778
   974
                by auto
wenzelm@56778
   975
            }
wenzelm@32960
   976
            note impth = this
wenzelm@60424
   977
            from IH[rule_format, OF dsn, of s r] impth False
wenzelm@56795
   978
            have "s dvd (r ^ (degree s))"
wenzelm@56795
   979
              by blast
wenzelm@32960
   980
            then obtain u where u: "r ^ (degree s) = s * u" ..
wenzelm@56778
   981
            then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
huffman@29470
   982
              by (simp only: poly_mult[symmetric] poly_power[symmetric])
wenzelm@32960
   983
            let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
wenzelm@32960
   984
            from oop[of a] dsn have "q ^ n = p * ?w"
huffman@29464
   985
              apply -
wenzelm@56795
   986
              apply (subst s)
wenzelm@56795
   987
              apply (subst r)
huffman@29464
   988
              apply (simp only: power_mult_distrib)
haftmann@57512
   989
              apply (subst mult.assoc [where b=s])
haftmann@57512
   990
              apply (subst mult.assoc [where a=u])
haftmann@57512
   991
              apply (subst mult.assoc [where b=u, symmetric])
huffman@29464
   992
              apply (subst u [symmetric])
haftmann@57514
   993
              apply (simp add: ac_simps power_add [symmetric])
huffman@29464
   994
              done
wenzelm@60424
   995
            then show ?thesis
wenzelm@56795
   996
              unfolding dvd_def by blast
wenzelm@60424
   997
        qed
wenzelm@60424
   998
      qed
wenzelm@60424
   999
    qed
wenzelm@56778
  1000
    then have ?ths using a order_root pne by blast
wenzelm@56778
  1001
  }
chaieb@26123
  1002
  moreover
wenzelm@56778
  1003
  {
wenzelm@56778
  1004
    assume exa: "\<not> (\<exists>a. poly p a = 0)"
wenzelm@56778
  1005
    from fundamental_theorem_of_algebra_alt[of p] exa
wenzelm@56778
  1006
    obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
wenzelm@56778
  1007
      by blast
wenzelm@56778
  1008
    then have pp: "\<And>x. poly p x = c"
wenzelm@56778
  1009
      by simp
huffman@29464
  1010
    let ?w = "[:1/c:] * (q ^ n)"
wenzelm@56778
  1011
    from ccs have "(q ^ n) = (p * ?w)"
wenzelm@56778
  1012
      by simp
wenzelm@56778
  1013
    then have ?ths
wenzelm@56778
  1014
      unfolding dvd_def by blast
wenzelm@56778
  1015
  }
chaieb@26123
  1016
  ultimately show ?ths by blast
chaieb@26123
  1017
qed
chaieb@26123
  1018
chaieb@26123
  1019
lemma nullstellensatz_univariate:
huffman@30488
  1020
  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
huffman@29464
  1021
    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
wenzelm@56776
  1022
proof -
wenzelm@60424
  1023
  show ?thesis
wenzelm@60424
  1024
  proof (cases "p = 0")
wenzelm@60424
  1025
    case True
wenzelm@56778
  1026
    then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
haftmann@52380
  1027
      by (auto simp add: poly_all_0_iff_0)
wenzelm@56778
  1028
    {
wenzelm@56778
  1029
      assume "p dvd (q ^ (degree p))"
huffman@29464
  1030
      then obtain r where r: "q ^ (degree p) = p * r" ..
wenzelm@60424
  1031
      from r True have False by simp
wenzelm@56778
  1032
    }
wenzelm@60424
  1033
    with eq True show ?thesis by blast
wenzelm@60424
  1034
  next
wenzelm@60424
  1035
    case False
wenzelm@60424
  1036
    show ?thesis
wenzelm@60424
  1037
    proof (cases "degree p = 0")
wenzelm@60424
  1038
      case True
wenzelm@60424
  1039
      with \<open>p \<noteq> 0\<close> obtain k where k: "p = [:k:]" "k \<noteq> 0"
wenzelm@51541
  1040
        by (cases p) (simp split: if_splits)
wenzelm@56778
  1041
      then have th1: "\<forall>x. poly p x \<noteq> 0"
wenzelm@56778
  1042
        by simp
wenzelm@60424
  1043
      from k True have "q ^ (degree p) = p * [:1/k:]"
huffman@29464
  1044
        by (simp add: one_poly_def)
wenzelm@56778
  1045
      then have th2: "p dvd (q ^ (degree p))" ..
wenzelm@60424
  1046
      from False th1 th2 show ?thesis
wenzelm@56795
  1047
        by blast
wenzelm@60424
  1048
    next
wenzelm@60424
  1049
      case False
wenzelm@56778
  1050
      assume dp: "degree p \<noteq> 0"
wenzelm@56778
  1051
      then obtain n where n: "degree p = Suc n "
wenzelm@56778
  1052
        by (cases "degree p") auto
wenzelm@56778
  1053
      {
wenzelm@56778
  1054
        assume "p dvd (q ^ (Suc n))"
wenzelm@32960
  1055
        then obtain u where u: "q ^ (Suc n) = p * u" ..
wenzelm@56778
  1056
        {
wenzelm@56778
  1057
          fix x
wenzelm@56778
  1058
          assume h: "poly p x = 0" "poly q x \<noteq> 0"
wenzelm@56778
  1059
          then have "poly (q ^ (Suc n)) x \<noteq> 0"
wenzelm@56778
  1060
            by simp
wenzelm@56778
  1061
          then have False using u h(1)
wenzelm@56778
  1062
            by (simp only: poly_mult) simp
wenzelm@56778
  1063
        }
wenzelm@56778
  1064
      }
wenzelm@56778
  1065
      with n nullstellensatz_lemma[of p q "degree p"] dp
wenzelm@60424
  1066
      show ?thesis by auto
wenzelm@60424
  1067
    qed
wenzelm@60424
  1068
  qed
chaieb@26123
  1069
qed
chaieb@26123
  1070
wenzelm@60424
  1071
text \<open>Useful lemma\<close>
chaieb@26123
  1072
huffman@29464
  1073
lemma constant_degree:
huffman@29464
  1074
  fixes p :: "'a::{idom,ring_char_0} poly"
huffman@29464
  1075
  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
chaieb@26123
  1076
proof
chaieb@26123
  1077
  assume l: ?lhs
huffman@29464
  1078
  from l[unfolded constant_def, rule_format, of _ "0"]
wenzelm@56776
  1079
  have th: "poly p = poly [:poly p 0:]"
wenzelm@56776
  1080
    by auto
wenzelm@56776
  1081
  then have "p = [:poly p 0:]"
wenzelm@56776
  1082
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1083
  then have "degree p = degree [:poly p 0:]"
wenzelm@56776
  1084
    by simp
wenzelm@56776
  1085
  then show ?rhs
wenzelm@56776
  1086
    by simp
chaieb@26123
  1087
next
chaieb@26123
  1088
  assume r: ?rhs
huffman@29464
  1089
  then obtain k where "p = [:k:]"
wenzelm@51541
  1090
    by (cases p) (simp split: if_splits)
wenzelm@56776
  1091
  then show ?lhs
wenzelm@56776
  1092
    unfolding constant_def by auto
chaieb@26123
  1093
qed
chaieb@26123
  1094
wenzelm@56776
  1095
lemma divides_degree:
wenzelm@56776
  1096
  assumes pq: "p dvd (q:: complex poly)"
huffman@29464
  1097
  shows "degree p \<le> degree q \<or> q = 0"
wenzelm@56776
  1098
  by (metis dvd_imp_degree_le pq)
chaieb@26123
  1099
wenzelm@60424
  1100
text \<open>Arithmetic operations on multivariate polynomials.\<close>
chaieb@26123
  1101
huffman@30488
  1102
lemma mpoly_base_conv:
wenzelm@56778
  1103
  fixes x :: "'a::comm_ring_1"
lp15@55735
  1104
  shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
lp15@55735
  1105
  by simp_all
chaieb@26123
  1106
huffman@30488
  1107
lemma mpoly_norm_conv:
wenzelm@56778
  1108
  fixes x :: "'a::comm_ring_1"
wenzelm@56776
  1109
  shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
wenzelm@56776
  1110
  by simp_all
chaieb@26123
  1111
huffman@30488
  1112
lemma mpoly_sub_conv:
wenzelm@56778
  1113
  fixes x :: "'a::comm_ring_1"
lp15@55735
  1114
  shows "poly p x - poly q x = poly p x + -1 * poly q x"
haftmann@54230
  1115
  by simp
chaieb@26123
  1116
wenzelm@56778
  1117
lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
wenzelm@56778
  1118
  by simp
chaieb@26123
  1119
lp15@55735
  1120
lemma poly_cancel_eq_conv:
wenzelm@56778
  1121
  fixes x :: "'a::field"
wenzelm@56795
  1122
  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
lp15@55735
  1123
  by auto
chaieb@26123
  1124
huffman@30488
  1125
lemma poly_divides_pad_rule:
wenzelm@56778
  1126
  fixes p:: "('a::comm_ring_1) poly"
huffman@29464
  1127
  assumes pq: "p dvd q"
wenzelm@56778
  1128
  shows "p dvd (pCons 0 q)"
wenzelm@56778
  1129
proof -
huffman@29464
  1130
  have "pCons 0 q = q * [:0,1:]" by simp
huffman@29464
  1131
  then have "q dvd (pCons 0 q)" ..
huffman@29464
  1132
  with pq show ?thesis by (rule dvd_trans)
chaieb@26123
  1133
qed
chaieb@26123
  1134
huffman@30488
  1135
lemma poly_divides_conv0:
wenzelm@56778
  1136
  fixes p:: "'a::field poly"
wenzelm@56776
  1137
  assumes lgpq: "degree q < degree p"
wenzelm@56776
  1138
    and lq: "p \<noteq> 0"
wenzelm@56776
  1139
  shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1140
proof
wenzelm@56776
  1141
  assume r: ?rhs
wenzelm@56776
  1142
  then have "q = p * 0" by simp
wenzelm@56776
  1143
  then show ?lhs ..
wenzelm@56776
  1144
next
wenzelm@56776
  1145
  assume l: ?lhs
wenzelm@56778
  1146
  show ?rhs
wenzelm@56778
  1147
  proof (cases "q = 0")
wenzelm@56778
  1148
    case True
wenzelm@56778
  1149
    then show ?thesis by simp
wenzelm@56778
  1150
  next
wenzelm@56776
  1151
    assume q0: "q \<noteq> 0"
wenzelm@56776
  1152
    from l q0 have "degree p \<le> degree q"
wenzelm@56776
  1153
      by (rule dvd_imp_degree_le)
wenzelm@56778
  1154
    with lgpq show ?thesis by simp
wenzelm@56778
  1155
  qed
chaieb@26123
  1156
qed
chaieb@26123
  1157
huffman@30488
  1158
lemma poly_divides_conv1:
wenzelm@56778
  1159
  fixes p :: "'a::field poly"
wenzelm@56776
  1160
  assumes a0: "a \<noteq> 0"
wenzelm@56776
  1161
    and pp': "p dvd p'"
wenzelm@56776
  1162
    and qrp': "smult a q - p' = r"
wenzelm@56776
  1163
  shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1164
proof
huffman@29464
  1165
  from pp' obtain t where t: "p' = p * t" ..
wenzelm@56776
  1166
  {
wenzelm@56776
  1167
    assume l: ?lhs
huffman@29464
  1168
    then obtain u where u: "q = p * u" ..
wenzelm@56776
  1169
    have "r = p * (smult a u - t)"
wenzelm@56776
  1170
      using u qrp' [symmetric] t by (simp add: algebra_simps)
wenzelm@56776
  1171
    then show ?rhs ..
wenzelm@56776
  1172
  next
wenzelm@56776
  1173
    assume r: ?rhs
huffman@29464
  1174
    then obtain u where u: "r = p * u" ..
huffman@29464
  1175
    from u [symmetric] t qrp' [symmetric] a0
wenzelm@51541
  1176
    have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
wenzelm@56776
  1177
    then show ?lhs ..
wenzelm@56776
  1178
  }
chaieb@26123
  1179
qed
chaieb@26123
  1180
chaieb@26123
  1181
lemma basic_cqe_conv1:
lp15@55358
  1182
  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
lp15@55358
  1183
  "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
wenzelm@56776
  1184
  "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
lp15@55358
  1185
  "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
wenzelm@56776
  1186
  "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
wenzelm@56776
  1187
  by simp_all
chaieb@26123
  1188
huffman@30488
  1189
lemma basic_cqe_conv2:
wenzelm@56795
  1190
  assumes l: "p \<noteq> 0"
wenzelm@56795
  1191
  shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
wenzelm@56776
  1192
proof -
wenzelm@60424
  1193
  have False if "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t
wenzelm@60449
  1194
    using l that by simp
wenzelm@56776
  1195
  then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
chaieb@26123
  1196
    by blast
wenzelm@56776
  1197
  from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
wenzelm@56776
  1198
    by auto
chaieb@26123
  1199
qed
chaieb@26123
  1200
wenzelm@56776
  1201
lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
wenzelm@56776
  1202
  by (metis poly_all_0_iff_0)
chaieb@26123
  1203
chaieb@26123
  1204
lemma basic_cqe_conv3:
huffman@29464
  1205
  fixes p q :: "complex poly"
huffman@30488
  1206
  assumes l: "p \<noteq> 0"
wenzelm@56795
  1207
  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)"
lp15@55358
  1208
proof -
wenzelm@56776
  1209
  from l have dp: "degree (pCons a p) = psize p"
wenzelm@56776
  1210
    by (simp add: psize_def)
huffman@29464
  1211
  from nullstellensatz_univariate[of "pCons a p" q] l
lp15@55358
  1212
  show ?thesis
lp15@55358
  1213
    by (metis dp pCons_eq_0_iff)
chaieb@26123
  1214
qed
chaieb@26123
  1215
chaieb@26123
  1216
lemma basic_cqe_conv4:
huffman@29464
  1217
  fixes p q :: "complex poly"
lp15@55358
  1218
  assumes h: "\<And>x. poly (q ^ n) x = poly r x"
lp15@55358
  1219
  shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1220
proof -
wenzelm@56776
  1221
  from h have "poly (q ^ n) = poly r"
wenzelm@56776
  1222
    by auto
wenzelm@56776
  1223
  then have "(q ^ n) = r"
wenzelm@56776
  1224
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1225
  then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1226
    by simp
chaieb@26123
  1227
qed
chaieb@26123
  1228
lp15@55735
  1229
lemma poly_const_conv:
wenzelm@56778
  1230
  fixes x :: "'a::comm_ring_1"
wenzelm@56776
  1231
  shows "poly [:c:] x = y \<longleftrightarrow> c = y"
wenzelm@56776
  1232
  by simp
chaieb@26123
  1233
huffman@29464
  1234
end