src/HOL/Library/Fundamental_Theorem_Algebra.thy
author wenzelm
Wed Jun 17 11:03:05 2015 +0200 (2015-06-17)
changeset 60500 903bb1495239
parent 60457 f31f7599ef55
child 60557 5854821993d2
permissions -rw-r--r--
isabelle update_cartouches;
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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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  show "\<exists>z. ?P z n"
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  proof cases
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    assume "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)
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    then show ?thesis ..
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  next
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    assume "odd n"
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    then 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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    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 unimodular_reduce_norm[OF th0] \<open>odd n\<close>
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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 \<open>odd n\<close>, 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 show ?thesis ..
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  qed
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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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   252
  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
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  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"
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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
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    from x[rule_format, OF e2] y[rule_format, OF e2]
wenzelm@56778
   292
    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
wenzelm@56795
   293
      and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
wenzelm@56795
   294
      by blast
wenzelm@56778
   295
    {
wenzelm@56778
   296
      fix n
wenzelm@56778
   297
      assume nN12: "n \<ge> N1 + N2"
wenzelm@56778
   298
      then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
wenzelm@56778
   299
        using seq_suble[OF g(1), of n] by arith+
chaieb@26123
   300
      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
huffman@30488
   301
      have "cmod (s (?h n) - ?w) < e"
wenzelm@56778
   302
        using metric_bound_lemma[of "s (f (g n))" ?w] by simp
wenzelm@56778
   303
    }
wenzelm@56795
   304
    then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e"
wenzelm@56795
   305
      by blast
wenzelm@56778
   306
  }
wenzelm@56778
   307
  with hs show ?thesis by blast
chaieb@26123
   308
qed
chaieb@26123
   309
wenzelm@60424
   310
text \<open>Polynomial is continuous.\<close>
chaieb@26123
   311
chaieb@26123
   312
lemma poly_cont:
wenzelm@56778
   313
  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
huffman@30488
   314
  assumes ep: "e > 0"
lp15@55735
   315
  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
   316
proof -
huffman@29464
   317
  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
huffman@29464
   318
  proof
huffman@29464
   319
    show "degree (offset_poly p z) = degree p"
huffman@29464
   320
      by (rule degree_offset_poly)
huffman@29464
   321
    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
huffman@29464
   322
      by (rule poly_offset_poly)
huffman@29464
   323
  qed
wenzelm@56778
   324
  have th: "\<And>w. poly q (w - z) = poly p w"
wenzelm@56778
   325
    using q(2)[of "w - z" for w] by simp
chaieb@26123
   326
  show ?thesis unfolding th[symmetric]
wenzelm@56778
   327
  proof (induct q)
wenzelm@56778
   328
    case 0
wenzelm@56778
   329
    then show ?case
wenzelm@56778
   330
      using ep by auto
chaieb@26123
   331
  next
huffman@29464
   332
    case (pCons c cs)
huffman@30488
   333
    from poly_bound_exists[of 1 "cs"]
wenzelm@56778
   334
    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
wenzelm@56778
   335
      by blast
wenzelm@56778
   336
    from ep m(1) have em0: "e/m > 0"
wenzelm@56778
   337
      by (simp add: field_simps)
wenzelm@56778
   338
    have one0: "1 > (0::real)"
wenzelm@56778
   339
      by arith
huffman@30488
   340
    from real_lbound_gt_zero[OF one0 em0]
wenzelm@56778
   341
    obtain d where d: "d > 0" "d < 1" "d < e / m"
wenzelm@56778
   342
      by blast
wenzelm@56778
   343
    from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
nipkow@56544
   344
      by (simp_all add: field_simps)
huffman@30488
   345
    show ?case
wenzelm@56778
   346
    proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
wenzelm@56778
   347
      fix d w
wenzelm@56778
   348
      assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
wenzelm@56778
   349
      then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
wenzelm@56778
   350
        by simp_all
wenzelm@56778
   351
      from H(3) m(1) have dme: "d*m < e"
wenzelm@56778
   352
        by (simp add: field_simps)
wenzelm@56778
   353
      from H have th: "norm (w - z) \<le> d"
wenzelm@56778
   354
        by simp
wenzelm@56778
   355
      from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
wenzelm@56778
   356
      show "norm (w - z) * norm (poly cs (w - z)) < e"
wenzelm@56778
   357
        by simp
chaieb@26123
   358
    qed
wenzelm@56778
   359
  qed
chaieb@26123
   360
qed
chaieb@26123
   361
wenzelm@60424
   362
text \<open>Hence a polynomial attains minimum on a closed disc
wenzelm@60424
   363
  in the complex plane.\<close>
wenzelm@56778
   364
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
   365
proof -
wenzelm@60424
   366
  show ?thesis
wenzelm@60424
   367
  proof (cases "r \<ge> 0")
wenzelm@60424
   368
    case False
wenzelm@60424
   369
    then show ?thesis
wenzelm@56778
   370
      by (metis norm_ge_zero order.trans)
wenzelm@60424
   371
  next
wenzelm@60424
   372
    case True
wenzelm@60424
   373
    then have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
wenzelm@56778
   374
      by simp
wenzelm@56778
   375
    then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
wenzelm@56778
   376
      by blast
wenzelm@56778
   377
    {
wenzelm@56778
   378
      fix x z
wenzelm@56778
   379
      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
wenzelm@56778
   380
      then have "- x < 0 "
wenzelm@56778
   381
        by arith
wenzelm@56778
   382
      with H(2) norm_ge_zero[of "poly p z"] have False
wenzelm@56778
   383
        by simp
wenzelm@56778
   384
    }
wenzelm@56778
   385
    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
wenzelm@56778
   386
      by blast
huffman@30488
   387
    from real_sup_exists[OF mth1 mth2] obtain s where
wenzelm@56778
   388
      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
   389
    let ?m = "- s"
wenzelm@56778
   390
    {
wenzelm@56778
   391
      fix y
wenzelm@56778
   392
      from s[rule_format, of "-y"]
wenzelm@56778
   393
      have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
wenzelm@56778
   394
        unfolding minus_less_iff[of y ] equation_minus_iff by blast
wenzelm@56778
   395
    }
chaieb@26123
   396
    note s1 = this[unfolded minus_minus]
huffman@30488
   397
    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
chaieb@26123
   398
      by auto
wenzelm@56778
   399
    {
wenzelm@56778
   400
      fix n :: nat
huffman@30488
   401
      from s1[rule_format, of "?m + 1/real (Suc n)"]
chaieb@26123
   402
      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
wenzelm@56778
   403
        by simp
wenzelm@56778
   404
    }
wenzelm@56778
   405
    then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
huffman@30488
   406
    from choice[OF th] obtain g where
wenzelm@56778
   407
        g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
chaieb@26123
   408
      by blast
huffman@30488
   409
    from bolzano_weierstrass_complex_disc[OF g(1)]
chaieb@26123
   410
    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
huffman@30488
   411
      by blast
wenzelm@56778
   412
    {
wenzelm@56778
   413
      fix w
chaieb@26123
   414
      assume wr: "cmod w \<le> r"
chaieb@26123
   415
      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
wenzelm@56778
   416
      {
wenzelm@56778
   417
        assume e: "?e > 0"
wenzelm@56795
   418
        then have e2: "?e/2 > 0"
wenzelm@56795
   419
          by simp
wenzelm@32960
   420
        from poly_cont[OF e2, of z p] obtain d where
wenzelm@56778
   421
            d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
wenzelm@56778
   422
          by blast
wenzelm@56778
   423
        {
wenzelm@56778
   424
          fix w
wenzelm@56778
   425
          assume w: "cmod (w - z) < d"
wenzelm@32960
   426
          have "cmod(poly p w - poly p z) < ?e / 2"
wenzelm@56778
   427
            using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
wenzelm@56778
   428
        }
wenzelm@32960
   429
        note th1 = this
huffman@30488
   430
wenzelm@56778
   431
        from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
wenzelm@56778
   432
          by blast
wenzelm@56778
   433
        from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
wenzelm@56778
   434
          by blast
wenzelm@56778
   435
        have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
wenzelm@32960
   436
          using N1[rule_format, of "N1 + N2"] th1 by simp
wenzelm@60424
   437
        have th0: "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
wenzelm@60424
   438
          for a b e2 m :: real
wenzelm@60424
   439
          by arith
wenzelm@60424
   440
        have ath: "m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e" for m x e :: real
wenzelm@56778
   441
          by arith
wenzelm@56778
   442
        from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
wenzelm@56795
   443
        from seq_suble[OF fz(1), of "N1 + N2"]
wenzelm@56778
   444
        have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
wenzelm@56778
   445
          by simp
wenzelm@56778
   446
        have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
wenzelm@56778
   447
          using N2 by auto
wenzelm@56778
   448
        from frac_le[OF th000 th00]
wenzelm@56795
   449
        have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
wenzelm@56778
   450
          by simp
wenzelm@56778
   451
        from g(2)[rule_format, of "f (N1 + N2)"]
wenzelm@56778
   452
        have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
wenzelm@56778
   453
        from order_less_le_trans[OF th01 th00]
wenzelm@56795
   454
        have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
wenzelm@56778
   455
        from N2 have "2/?e < real (Suc (N1 + N2))"
wenzelm@56778
   456
          by arith
wenzelm@56778
   457
        with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
wenzelm@56778
   458
        have "?e/2 > 1/ real (Suc (N1 + N2))"
wenzelm@56778
   459
          by (simp add: inverse_eq_divide)
wenzelm@60424
   460
        with ath[OF th31 th32] have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
wenzelm@56778
   461
          by arith
wenzelm@60424
   462
        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
   463
          by arith
wenzelm@56778
   464
        have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
wenzelm@56778
   465
            cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@56778
   466
          by (simp add: norm_triangle_ineq3)
wenzelm@56778
   467
        from ath2[OF th22, of ?m]
wenzelm@56778
   468
        have thc2: "2 * (?e/2) \<le>
wenzelm@56778
   469
            \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
wenzelm@56778
   470
          by simp
wenzelm@56778
   471
        from th0[OF th2 thc1 thc2] have False .
wenzelm@56778
   472
      }
wenzelm@56778
   473
      then have "?e = 0"
wenzelm@56778
   474
        by auto
wenzelm@56778
   475
      then have "cmod (poly p z) = ?m"
wenzelm@56778
   476
        by simp
wenzelm@56778
   477
      with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   478
        by simp
wenzelm@56778
   479
    }
wenzelm@60424
   480
    then show ?thesis by blast
wenzelm@60424
   481
  qed
chaieb@26123
   482
qed
chaieb@26123
   483
wenzelm@60424
   484
text \<open>Nonzero polynomial in z goes to infinity as z does.\<close>
chaieb@26123
   485
chaieb@26123
   486
lemma poly_infinity:
wenzelm@56778
   487
  fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
huffman@29464
   488
  assumes ex: "p \<noteq> 0"
lp15@55735
   489
  shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
wenzelm@56778
   490
  using ex
wenzelm@56778
   491
proof (induct p arbitrary: a d)
wenzelm@56795
   492
  case 0
wenzelm@56795
   493
  then show ?case by simp
wenzelm@56795
   494
next
huffman@30488
   495
  case (pCons c cs a d)
wenzelm@56795
   496
  show ?case
wenzelm@56795
   497
  proof (cases "cs = 0")
wenzelm@56795
   498
    case False
wenzelm@56778
   499
    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
   500
      by blast
chaieb@26123
   501
    let ?r = "1 + \<bar>r\<bar>"
wenzelm@56778
   502
    {
wenzelm@56795
   503
      fix z :: 'a
wenzelm@56778
   504
      assume h: "1 + \<bar>r\<bar> \<le> norm z"
wenzelm@56795
   505
      have r0: "r \<le> norm z"
wenzelm@56795
   506
        using h by arith
wenzelm@56778
   507
      from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
wenzelm@56778
   508
        by arith
wenzelm@56778
   509
      from h have z1: "norm z \<ge> 1"
wenzelm@56778
   510
        by arith
huffman@29464
   511
      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
lp15@55735
   512
      have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
wenzelm@32960
   513
        unfolding norm_mult by (simp add: algebra_simps)
lp15@55735
   514
      from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
wenzelm@56795
   515
      have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@51541
   516
        by (simp add: algebra_simps)
wenzelm@56795
   517
      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@56795
   518
        by arith
wenzelm@56778
   519
    }
wenzelm@56795
   520
    then show ?thesis by blast
wenzelm@56795
   521
  next
wenzelm@56795
   522
    case True
wenzelm@56778
   523
    with pCons.prems have c0: "c \<noteq> 0"
wenzelm@56778
   524
      by simp
wenzelm@60424
   525
    have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
wenzelm@60424
   526
      if h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z" for z :: 'a
wenzelm@60424
   527
    proof -
wenzelm@56778
   528
      from c0 have "norm c > 0"
wenzelm@56778
   529
        by simp
blanchet@56403
   530
      from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
wenzelm@32960
   531
        by (simp add: field_simps norm_mult)
wenzelm@56778
   532
      have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
wenzelm@56778
   533
        by arith
wenzelm@56778
   534
      from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
wenzelm@32960
   535
        by (simp add: algebra_simps)
wenzelm@60424
   536
      from ath[OF th1 th0] show ?thesis
wenzelm@56795
   537
        using True by simp
wenzelm@60424
   538
    qed
wenzelm@56795
   539
    then show ?thesis by blast
wenzelm@56795
   540
  qed
wenzelm@56795
   541
qed
chaieb@26123
   542
wenzelm@60424
   543
text \<open>Hence polynomial's modulus attains its minimum somewhere.\<close>
wenzelm@56778
   544
lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
wenzelm@56778
   545
proof (induct p)
wenzelm@56778
   546
  case 0
wenzelm@56778
   547
  then show ?case by simp
wenzelm@56778
   548
next
huffman@30488
   549
  case (pCons c cs)
wenzelm@56778
   550
  show ?case
wenzelm@56778
   551
  proof (cases "cs = 0")
wenzelm@56778
   552
    case False
wenzelm@56778
   553
    from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
wenzelm@56778
   554
    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
   555
      by blast
wenzelm@56778
   556
    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
wenzelm@56778
   557
      by arith
huffman@30488
   558
    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
wenzelm@56778
   559
    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
   560
      by blast
wenzelm@60424
   561
    have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)" if z: "r \<le> cmod z" for z
wenzelm@60424
   562
      using v[of 0] r[OF z] by simp
wenzelm@60424
   563
    with v ath[of r] show ?thesis
wenzelm@56778
   564
      by blast
wenzelm@56778
   565
  next
wenzelm@56778
   566
    case True
wenzelm@60424
   567
    with pCons.hyps show ?thesis
wenzelm@60424
   568
      by simp
wenzelm@56778
   569
  qed
wenzelm@56778
   570
qed
chaieb@26123
   571
wenzelm@60424
   572
text \<open>Constant function (non-syntactic characterization).\<close>
wenzelm@56795
   573
definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"
chaieb@26123
   574
wenzelm@56778
   575
lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
wenzelm@56778
   576
  by (induct p) (auto simp: constant_def psize_def)
huffman@30488
   577
wenzelm@56795
   578
lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
huffman@29464
   579
  by (simp add: poly_monom)
chaieb@26123
   580
wenzelm@60424
   581
text \<open>Decomposition of polynomial, skipping zero coefficients after the first.\<close>
chaieb@26123
   582
chaieb@26123
   583
lemma poly_decompose_lemma:
wenzelm@56778
   584
  assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
wenzelm@56795
   585
  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
   586
  unfolding psize_def
wenzelm@56778
   587
  using nz
wenzelm@56778
   588
proof (induct p)
wenzelm@56778
   589
  case 0
wenzelm@56778
   590
  then show ?case by simp
chaieb@26123
   591
next
huffman@29464
   592
  case (pCons c cs)
wenzelm@56778
   593
  show ?case
wenzelm@56778
   594
  proof (cases "c = 0")
wenzelm@56778
   595
    case True
wenzelm@56778
   596
    from pCons.hyps pCons.prems True show ?thesis
wenzelm@60424
   597
      apply auto
chaieb@26123
   598
      apply (rule_tac x="k+1" in exI)
chaieb@26123
   599
      apply (rule_tac x="a" in exI, clarsimp)
chaieb@26123
   600
      apply (rule_tac x="q" in exI)
wenzelm@56778
   601
      apply auto
wenzelm@56778
   602
      done
wenzelm@56778
   603
  next
wenzelm@56778
   604
    case False
wenzelm@56778
   605
    show ?thesis
chaieb@26123
   606
      apply (rule exI[where x=0])
wenzelm@60424
   607
      apply (rule exI[where x=c])
wenzelm@60424
   608
      apply (auto simp: False)
wenzelm@56778
   609
      done
wenzelm@56778
   610
  qed
chaieb@26123
   611
qed
chaieb@26123
   612
chaieb@26123
   613
lemma poly_decompose:
wenzelm@56776
   614
  assumes nc: "\<not> constant (poly p)"
wenzelm@56778
   615
  shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
huffman@30488
   616
               psize q + k + 1 = psize p \<and>
huffman@29464
   617
              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
wenzelm@56776
   618
  using nc
wenzelm@56776
   619
proof (induct p)
wenzelm@56776
   620
  case 0
wenzelm@56776
   621
  then show ?case
wenzelm@56776
   622
    by (simp add: constant_def)
chaieb@26123
   623
next
huffman@29464
   624
  case (pCons c cs)
wenzelm@56776
   625
  {
wenzelm@60424
   626
    assume "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
wenzelm@60424
   627
    then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y
wenzelm@60424
   628
      by (cases "x = 0") auto
wenzelm@56778
   629
    with pCons.prems have False
wenzelm@56778
   630
      by (auto simp add: constant_def)
wenzelm@56776
   631
  }
wenzelm@56776
   632
  then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
huffman@30488
   633
  from poly_decompose_lemma[OF th]
huffman@30488
   634
  show ?case
huffman@29464
   635
    apply clarsimp
chaieb@26123
   636
    apply (rule_tac x="k+1" in exI)
chaieb@26123
   637
    apply (rule_tac x="a" in exI)
chaieb@26123
   638
    apply simp
chaieb@26123
   639
    apply (rule_tac x="q" in exI)
huffman@29538
   640
    apply (auto simp add: psize_def split: if_splits)
chaieb@26123
   641
    done
chaieb@26123
   642
qed
chaieb@26123
   643
wenzelm@60424
   644
text \<open>Fundamental theorem of algebra\<close>
chaieb@26123
   645
chaieb@26123
   646
lemma fundamental_theorem_of_algebra:
wenzelm@56776
   647
  assumes nc: "\<not> constant (poly p)"
chaieb@26123
   648
  shows "\<exists>z::complex. poly p z = 0"
wenzelm@56776
   649
  using nc
wenzelm@56776
   650
proof (induct "psize p" arbitrary: p rule: less_induct)
berghofe@34915
   651
  case less
chaieb@26123
   652
  let ?p = "poly p"
chaieb@26123
   653
  let ?ths = "\<exists>z. ?p z = 0"
chaieb@26123
   654
berghofe@34915
   655
  from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
wenzelm@56776
   656
  from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
wenzelm@56776
   657
    by blast
wenzelm@56778
   658
wenzelm@56778
   659
  show ?ths
wenzelm@56778
   660
  proof (cases "?p c = 0")
wenzelm@56778
   661
    case True
wenzelm@56778
   662
    then show ?thesis by blast
wenzelm@56778
   663
  next
wenzelm@56778
   664
    case False
wenzelm@56778
   665
    from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
wenzelm@56778
   666
      by blast
wenzelm@60424
   667
    have False if h: "constant (poly q)"
wenzelm@60424
   668
    proof -
wenzelm@56795
   669
      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
wenzelm@56795
   670
        by auto
wenzelm@60424
   671
      have "?p x = ?p y" for x y
wenzelm@60424
   672
      proof -
wenzelm@56795
   673
        from th have "?p x = poly q (x - c)"
wenzelm@56795
   674
          by auto
wenzelm@32960
   675
        also have "\<dots> = poly q (y - c)"
wenzelm@32960
   676
          using h unfolding constant_def by blast
wenzelm@56795
   677
        also have "\<dots> = ?p y"
wenzelm@56795
   678
          using th by auto
wenzelm@60424
   679
        finally show ?thesis .
wenzelm@60424
   680
      qed
wenzelm@60424
   681
      with less(2) show ?thesis
wenzelm@56778
   682
        unfolding constant_def by blast
wenzelm@60424
   683
    qed
wenzelm@56778
   684
    then have qnc: "\<not> constant (poly q)"
wenzelm@56778
   685
      by blast
wenzelm@56778
   686
    from q(2) have pqc0: "?p c = poly q 0"
wenzelm@56778
   687
      by simp
wenzelm@56778
   688
    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
wenzelm@56778
   689
      by simp
chaieb@26123
   690
    let ?a0 = "poly q 0"
wenzelm@60424
   691
    from False pqc0 have a00: "?a0 \<noteq> 0"
wenzelm@56778
   692
      by simp
wenzelm@56778
   693
    from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
huffman@29464
   694
      by simp
huffman@29464
   695
    let ?r = "smult (inverse ?a0) q"
huffman@29538
   696
    have lgqr: "psize q = psize ?r"
wenzelm@56778
   697
      using a00
wenzelm@56778
   698
      unfolding psize_def degree_def
haftmann@52380
   699
      by (simp add: poly_eq_iff)
wenzelm@60424
   700
    have False if h: "\<And>x y. poly ?r x = poly ?r y"
wenzelm@60424
   701
    proof -
wenzelm@56778
   702
      {
wenzelm@56778
   703
        fix x y
wenzelm@56778
   704
        from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
wenzelm@56778
   705
          by auto
wenzelm@56778
   706
        also have "\<dots> = poly ?r y * ?a0"
wenzelm@56778
   707
          using h by simp
wenzelm@56778
   708
        also have "\<dots> = poly q y"
wenzelm@56778
   709
          using qr[rule_format, of y] by simp
wenzelm@56778
   710
        finally have "poly q x = poly q y" .
wenzelm@56778
   711
      }
wenzelm@60424
   712
      with qnc show ?thesis
wenzelm@56795
   713
        unfolding constant_def by blast
wenzelm@60424
   714
    qed
wenzelm@56778
   715
    then have rnc: "\<not> constant (poly ?r)"
wenzelm@56778
   716
      unfolding constant_def by blast
wenzelm@56778
   717
    from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
wenzelm@56778
   718
      by auto
wenzelm@60424
   719
    have mrmq_eq: "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" for w
wenzelm@60424
   720
    proof -
chaieb@26123
   721
      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
haftmann@57514
   722
        using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)
chaieb@26123
   723
      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
wenzelm@32960
   724
        using a00 unfolding norm_divide by (simp add: field_simps)
wenzelm@60424
   725
      finally show ?thesis .
wenzelm@60424
   726
    qed
huffman@30488
   727
    from poly_decompose[OF rnc] obtain k a s where
wenzelm@56778
   728
      kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
wenzelm@56778
   729
        "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
wenzelm@60424
   730
    have "\<exists>w. cmod (poly ?r w) < 1"
wenzelm@60424
   731
    proof (cases "psize p = k + 1")
wenzelm@60424
   732
      case True
wenzelm@56778
   733
      with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
wenzelm@56778
   734
        by auto
wenzelm@60424
   735
      have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w
wenzelm@60424
   736
        using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
wenzelm@60424
   737
      from reduce_poly_simple[OF kas(1,2)] show ?thesis
wenzelm@56778
   738
        unfolding hth by blast
wenzelm@60424
   739
    next
wenzelm@60424
   740
      case False note kn = this
wenzelm@56778
   741
      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
wenzelm@56778
   742
        by simp
huffman@30488
   743
      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
wenzelm@32960
   744
        unfolding constant_def poly_pCons poly_monom
wenzelm@56795
   745
        using kas(1)
wenzelm@56795
   746
        apply simp
wenzelm@56778
   747
        apply (rule exI[where x=0])
wenzelm@56778
   748
        apply (rule exI[where x=1])
wenzelm@56778
   749
        apply simp
wenzelm@56778
   750
        done
wenzelm@56778
   751
      from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
wenzelm@32960
   752
        by (simp add: psize_def degree_monom_eq)
berghofe@34915
   753
      from less(1) [OF k1n [simplified th02] th01]
chaieb@26123
   754
      obtain w where w: "1 + w^k * a = 0"
wenzelm@32960
   755
        unfolding poly_pCons poly_monom
wenzelm@56778
   756
        using kas(2) by (cases k) (auto simp add: algebra_simps)
huffman@30488
   757
      from poly_bound_exists[of "cmod w" s] obtain m where
wenzelm@32960
   758
        m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
wenzelm@56795
   759
      have w0: "w \<noteq> 0"
wenzelm@56795
   760
        using kas(2) w by (auto simp add: power_0_left)
wenzelm@56778
   761
      from w have "(1 + w ^ k * a) - 1 = 0 - 1"
wenzelm@56778
   762
        by simp
wenzelm@56778
   763
      then have wm1: "w^k * a = - 1"
wenzelm@56778
   764
        by simp
huffman@30488
   765
      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
wenzelm@32960
   766
        using norm_ge_zero[of w] w0 m(1)
wenzelm@56778
   767
        by (simp add: inverse_eq_divide zero_less_mult_iff)
lp15@55358
   768
      with real_lbound_gt_zero[OF zero_less_one] obtain t where
wenzelm@32960
   769
        t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
chaieb@26123
   770
      let ?ct = "complex_of_real t"
chaieb@26123
   771
      let ?w = "?ct * w"
wenzelm@56778
   772
      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
wenzelm@56778
   773
        using kas(1) by (simp add: algebra_simps power_mult_distrib)
chaieb@26123
   774
      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
wenzelm@56778
   775
        unfolding wm1 by simp
wenzelm@56778
   776
      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
wenzelm@56778
   777
        cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
lp15@55358
   778
        by metis
huffman@30488
   779
      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
wenzelm@56778
   780
      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
   781
        unfolding norm_of_real by simp
wenzelm@56778
   782
      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
   783
        by arith
wenzelm@56778
   784
      have "t * cmod w \<le> 1 * cmod w"
wenzelm@56778
   785
        apply (rule mult_mono)
wenzelm@56778
   786
        using t(1,2)
wenzelm@56778
   787
        apply auto
wenzelm@56778
   788
        done
wenzelm@56778
   789
      then have tw: "cmod ?w \<le> cmod w"
wenzelm@56778
   790
        using t(1) by (simp add: norm_mult)
wenzelm@56778
   791
      from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
wenzelm@57862
   792
        by (simp add: field_simps)
wenzelm@56778
   793
      with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
haftmann@59557
   794
        by simp
wenzelm@56778
   795
      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
wenzelm@56778
   796
        using w0 t(1)
wenzelm@51541
   797
        by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
chaieb@26123
   798
      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
wenzelm@32960
   799
        using t(1,2) m(2)[rule_format, OF tw] w0
lp15@55358
   800
        by auto
wenzelm@56778
   801
      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
wenzelm@56778
   802
        by simp
huffman@30488
   803
      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
wenzelm@32960
   804
        by auto
huffman@27514
   805
      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
huffman@30488
   806
      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
wenzelm@56778
   807
      from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
wenzelm@56778
   808
        by arith
huffman@30488
   809
      then have "cmod (poly ?r ?w) < 1"
wenzelm@32960
   810
        unfolding kas(4)[rule_format, of ?w] r01 by simp
wenzelm@60424
   811
      then show ?thesis
wenzelm@56778
   812
        by blast
wenzelm@60424
   813
    qed
wenzelm@60424
   814
    with cq0 q(2) show ?thesis
wenzelm@56778
   815
      unfolding mrmq_eq not_less[symmetric] by auto
wenzelm@56778
   816
  qed
chaieb@26123
   817
qed
chaieb@26123
   818
wenzelm@60424
   819
text \<open>Alternative version with a syntactic notion of constant polynomial.\<close>
chaieb@26123
   820
chaieb@26123
   821
lemma fundamental_theorem_of_algebra_alt:
wenzelm@56778
   822
  assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
chaieb@26123
   823
  shows "\<exists>z. poly p z = (0::complex)"
wenzelm@56778
   824
  using nc
wenzelm@56778
   825
proof (induct p)
wenzelm@56778
   826
  case 0
wenzelm@56778
   827
  then show ?case by simp
wenzelm@56778
   828
next
huffman@29464
   829
  case (pCons c cs)
wenzelm@56778
   830
  show ?case
wenzelm@56778
   831
  proof (cases "c = 0")
wenzelm@56778
   832
    case True
wenzelm@56778
   833
    then show ?thesis by auto
wenzelm@56778
   834
  next
wenzelm@56778
   835
    case False
wenzelm@56778
   836
    {
wenzelm@56778
   837
      assume nc: "constant (poly (pCons c cs))"
huffman@30488
   838
      from nc[unfolded constant_def, rule_format, of 0]
huffman@30488
   839
      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
wenzelm@56778
   840
      then have "cs = 0"
wenzelm@56778
   841
      proof (induct cs)
wenzelm@56778
   842
        case 0
wenzelm@56778
   843
        then show ?case by simp
wenzelm@56778
   844
      next
wenzelm@56778
   845
        case (pCons d ds)
wenzelm@56778
   846
        show ?case
wenzelm@56778
   847
        proof (cases "d = 0")
wenzelm@56778
   848
          case True
wenzelm@60424
   849
          then show ?thesis
wenzelm@60424
   850
            using pCons.prems pCons.hyps by simp
wenzelm@56778
   851
        next
wenzelm@56778
   852
          case False
wenzelm@56778
   853
          from poly_bound_exists[of 1 ds] obtain m where
wenzelm@56778
   854
            m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
wenzelm@56795
   855
          have dm: "cmod d / m > 0"
wenzelm@56795
   856
            using False m(1) by (simp add: field_simps)
wenzelm@60424
   857
          from real_lbound_gt_zero[OF dm zero_less_one]
wenzelm@60424
   858
          obtain x where x: "x > 0" "x < cmod d / m" "x < 1"
wenzelm@60424
   859
            by blast
wenzelm@56778
   860
          let ?x = "complex_of_real x"
wenzelm@60424
   861
          from x have cx: "?x \<noteq> 0" "cmod ?x \<le> 1"
wenzelm@56795
   862
            by simp_all
wenzelm@56778
   863
          from pCons.prems[rule_format, OF cx(1)]
wenzelm@56795
   864
          have cth: "cmod (?x*poly ds ?x) = cmod d"
wenzelm@56795
   865
            by (simp add: eq_diff_eq[symmetric])
wenzelm@56778
   866
          from m(2)[rule_format, OF cx(2)] x(1)
wenzelm@56778
   867
          have th0: "cmod (?x*poly ds ?x) \<le> x*m"
wenzelm@56778
   868
            by (simp add: norm_mult)
wenzelm@56795
   869
          from x(2) m(1) have "x * m < cmod d"
wenzelm@56795
   870
            by (simp add: field_simps)
wenzelm@56795
   871
          with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
wenzelm@56795
   872
            by auto
wenzelm@56795
   873
          with cth show ?thesis
wenzelm@56795
   874
            by blast
wenzelm@56778
   875
        qed
wenzelm@56778
   876
      qed
wenzelm@56778
   877
    }
wenzelm@56795
   878
    then have nc: "\<not> constant (poly (pCons c cs))"
wenzelm@56795
   879
      using pCons.prems False by blast
wenzelm@56778
   880
    from fundamental_theorem_of_algebra[OF nc] show ?thesis .
wenzelm@56778
   881
  qed
wenzelm@56778
   882
qed
chaieb@26123
   883
huffman@29464
   884
wenzelm@60424
   885
subsection \<open>Nullstellensatz, degrees and divisibility of polynomials\<close>
chaieb@26123
   886
chaieb@26123
   887
lemma nullstellensatz_lemma:
huffman@29464
   888
  fixes p :: "complex poly"
chaieb@26123
   889
  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56776
   890
    and "degree p = n"
wenzelm@56776
   891
    and "n \<noteq> 0"
huffman@29464
   892
  shows "p dvd (q ^ n)"
wenzelm@56776
   893
  using assms
wenzelm@56776
   894
proof (induct n arbitrary: p q rule: nat_less_induct)
wenzelm@56776
   895
  fix n :: nat
wenzelm@56776
   896
  fix p q :: "complex poly"
chaieb@26123
   897
  assume IH: "\<forall>m<n. \<forall>p q.
chaieb@26123
   898
                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
huffman@29464
   899
                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
huffman@30488
   900
    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
wenzelm@56778
   901
    and dpn: "degree p = n"
wenzelm@56778
   902
    and n0: "n \<noteq> 0"
huffman@29464
   903
  from dpn n0 have pne: "p \<noteq> 0" by auto
huffman@29464
   904
  let ?ths = "p dvd (q ^ n)"
wenzelm@56778
   905
  {
wenzelm@56778
   906
    fix a
wenzelm@56778
   907
    assume a: "poly p a = 0"
wenzelm@60424
   908
    have ?ths if oa: "order a p \<noteq> 0"
wenzelm@60424
   909
    proof -
chaieb@26123
   910
      let ?op = "order a p"
wenzelm@56778
   911
      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
wenzelm@56778
   912
        using order by blast+
huffman@29464
   913
      note oop = order_degree[OF pne, unfolded dpn]
wenzelm@60424
   914
      show ?thesis
wenzelm@60424
   915
      proof (cases "q = 0")
wenzelm@60424
   916
        case True
wenzelm@60424
   917
        with n0 show ?thesis by (simp add: power_0_left)
wenzelm@60424
   918
      next
wenzelm@60424
   919
        case False
wenzelm@32960
   920
        from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
wenzelm@32960
   921
        obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
wenzelm@56778
   922
        from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
wenzelm@56778
   923
          by (rule dvdE)
wenzelm@60424
   924
        have sne: "s \<noteq> 0"
wenzelm@60424
   925
          using s pne by auto
wenzelm@60424
   926
        show ?thesis
wenzelm@60424
   927
        proof (cases "degree s = 0")
wenzelm@60424
   928
          case True
wenzelm@60424
   929
          then obtain k where kpn: "s = [:k:]"
wenzelm@51541
   930
            by (cases s) (auto split: if_splits)
huffman@29464
   931
          from sne kpn have k: "k \<noteq> 0" by simp
wenzelm@32960
   932
          let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
lp15@55358
   933
          have "q ^ n = p * ?w"
wenzelm@56795
   934
            apply (subst r)
wenzelm@56795
   935
            apply (subst s)
wenzelm@56795
   936
            apply (subst kpn)
wenzelm@56778
   937
            using k oop [of a]
wenzelm@56795
   938
            apply (subst power_mult_distrib)
wenzelm@56795
   939
            apply simp
wenzelm@56795
   940
            apply (subst power_add [symmetric])
wenzelm@56795
   941
            apply simp
huffman@29464
   942
            done
wenzelm@60424
   943
          then show ?thesis
wenzelm@56795
   944
            unfolding dvd_def by blast
wenzelm@60424
   945
        next
wenzelm@60424
   946
          case False
wenzelm@60424
   947
          with sne dpn s oa have dsn: "degree s < n"
wenzelm@56778
   948
              apply auto
huffman@29464
   949
              apply (erule ssubst)
huffman@29464
   950
              apply (simp add: degree_mult_eq degree_linear_power)
huffman@29464
   951
              done
wenzelm@56778
   952
            {
wenzelm@56778
   953
              fix x assume h: "poly s x = 0"
wenzelm@56778
   954
              {
wenzelm@56778
   955
                assume xa: "x = a"
wenzelm@56778
   956
                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
wenzelm@56778
   957
                  by (rule dvdE)
wenzelm@32960
   958
                have "p = [:- a, 1:] ^ (Suc ?op) * u"
wenzelm@56795
   959
                  apply (subst s)
wenzelm@56795
   960
                  apply (subst u)
haftmann@57514
   961
                  apply (simp only: power_Suc ac_simps)
wenzelm@56795
   962
                  done
wenzelm@56795
   963
                with ap(2)[unfolded dvd_def] have False
wenzelm@56795
   964
                  by blast
wenzelm@56778
   965
              }
wenzelm@32960
   966
              note xa = this
wenzelm@56795
   967
              from h have "poly p x = 0"
wenzelm@56795
   968
                by (subst s) simp
wenzelm@56795
   969
              with pq0 have "poly q x = 0"
wenzelm@56795
   970
                by blast
wenzelm@32960
   971
              with r xa have "poly r x = 0"
wenzelm@56778
   972
                by auto
wenzelm@56778
   973
            }
wenzelm@32960
   974
            note impth = this
wenzelm@60424
   975
            from IH[rule_format, OF dsn, of s r] impth False
wenzelm@56795
   976
            have "s dvd (r ^ (degree s))"
wenzelm@56795
   977
              by blast
wenzelm@32960
   978
            then obtain u where u: "r ^ (degree s) = s * u" ..
wenzelm@56778
   979
            then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
huffman@29470
   980
              by (simp only: poly_mult[symmetric] poly_power[symmetric])
wenzelm@32960
   981
            let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
wenzelm@32960
   982
            from oop[of a] dsn have "q ^ n = p * ?w"
huffman@29464
   983
              apply -
wenzelm@56795
   984
              apply (subst s)
wenzelm@56795
   985
              apply (subst r)
huffman@29464
   986
              apply (simp only: power_mult_distrib)
haftmann@57512
   987
              apply (subst mult.assoc [where b=s])
haftmann@57512
   988
              apply (subst mult.assoc [where a=u])
haftmann@57512
   989
              apply (subst mult.assoc [where b=u, symmetric])
huffman@29464
   990
              apply (subst u [symmetric])
haftmann@57514
   991
              apply (simp add: ac_simps power_add [symmetric])
huffman@29464
   992
              done
wenzelm@60424
   993
            then show ?thesis
wenzelm@56795
   994
              unfolding dvd_def by blast
wenzelm@60424
   995
        qed
wenzelm@60424
   996
      qed
wenzelm@60424
   997
    qed
wenzelm@56778
   998
    then have ?ths using a order_root pne by blast
wenzelm@56778
   999
  }
chaieb@26123
  1000
  moreover
wenzelm@56778
  1001
  {
wenzelm@56778
  1002
    assume exa: "\<not> (\<exists>a. poly p a = 0)"
wenzelm@56778
  1003
    from fundamental_theorem_of_algebra_alt[of p] exa
wenzelm@56778
  1004
    obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
wenzelm@56778
  1005
      by blast
wenzelm@56778
  1006
    then have pp: "\<And>x. poly p x = c"
wenzelm@56778
  1007
      by simp
huffman@29464
  1008
    let ?w = "[:1/c:] * (q ^ n)"
wenzelm@56778
  1009
    from ccs have "(q ^ n) = (p * ?w)"
wenzelm@56778
  1010
      by simp
wenzelm@56778
  1011
    then have ?ths
wenzelm@56778
  1012
      unfolding dvd_def by blast
wenzelm@56778
  1013
  }
chaieb@26123
  1014
  ultimately show ?ths by blast
chaieb@26123
  1015
qed
chaieb@26123
  1016
chaieb@26123
  1017
lemma nullstellensatz_univariate:
huffman@30488
  1018
  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
huffman@29464
  1019
    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
wenzelm@56776
  1020
proof -
wenzelm@60457
  1021
  consider "p = 0" | "p \<noteq> 0" "degree p = 0" | n where "p \<noteq> 0" "degree p = Suc n"
wenzelm@60457
  1022
    by (cases "degree p") auto
wenzelm@60457
  1023
  then show ?thesis
wenzelm@60457
  1024
  proof cases
wenzelm@60457
  1025
    case 1
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@60457
  1031
      from r 1 have False by simp
wenzelm@56778
  1032
    }
wenzelm@60457
  1033
    with eq 1 show ?thesis by blast
wenzelm@60424
  1034
  next
wenzelm@60457
  1035
    case 2
wenzelm@60457
  1036
    then obtain k where k: "p = [:k:]" "k \<noteq> 0"
wenzelm@60457
  1037
      by (cases p) (simp split: if_splits)
wenzelm@60457
  1038
    then have th1: "\<forall>x. poly p x \<noteq> 0"
wenzelm@60457
  1039
      by simp
wenzelm@60457
  1040
    from k 2(2) have "q ^ (degree p) = p * [:1/k:]"
wenzelm@60457
  1041
      by (simp add: one_poly_def)
wenzelm@60457
  1042
    then have th2: "p dvd (q ^ (degree p))" ..
wenzelm@60457
  1043
    from 2(1) th1 th2 show ?thesis
wenzelm@60457
  1044
      by blast
wenzelm@60457
  1045
  next
wenzelm@60457
  1046
    case 3
wenzelm@60457
  1047
    {
wenzelm@60457
  1048
      assume "p dvd (q ^ (Suc n))"
wenzelm@60457
  1049
      then obtain u where u: "q ^ (Suc n) = p * u" ..
wenzelm@60457
  1050
      fix x
wenzelm@60457
  1051
      assume h: "poly p x = 0" "poly q x \<noteq> 0"
wenzelm@60457
  1052
      then have "poly (q ^ (Suc n)) x \<noteq> 0"
wenzelm@56778
  1053
        by simp
wenzelm@60457
  1054
      then have False using u h(1)
wenzelm@60457
  1055
        by (simp only: poly_mult) simp
wenzelm@60457
  1056
    }
wenzelm@60457
  1057
    with 3 nullstellensatz_lemma[of p q "degree p"]
wenzelm@60457
  1058
    show ?thesis by auto
wenzelm@60424
  1059
  qed
chaieb@26123
  1060
qed
chaieb@26123
  1061
wenzelm@60424
  1062
text \<open>Useful lemma\<close>
chaieb@26123
  1063
huffman@29464
  1064
lemma constant_degree:
huffman@29464
  1065
  fixes p :: "'a::{idom,ring_char_0} poly"
huffman@29464
  1066
  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
chaieb@26123
  1067
proof
chaieb@26123
  1068
  assume l: ?lhs
huffman@29464
  1069
  from l[unfolded constant_def, rule_format, of _ "0"]
wenzelm@56776
  1070
  have th: "poly p = poly [:poly p 0:]"
wenzelm@56776
  1071
    by auto
wenzelm@56776
  1072
  then have "p = [:poly p 0:]"
wenzelm@56776
  1073
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1074
  then have "degree p = degree [:poly p 0:]"
wenzelm@56776
  1075
    by simp
wenzelm@56776
  1076
  then show ?rhs
wenzelm@56776
  1077
    by simp
chaieb@26123
  1078
next
chaieb@26123
  1079
  assume r: ?rhs
huffman@29464
  1080
  then obtain k where "p = [:k:]"
wenzelm@51541
  1081
    by (cases p) (simp split: if_splits)
wenzelm@56776
  1082
  then show ?lhs
wenzelm@56776
  1083
    unfolding constant_def by auto
chaieb@26123
  1084
qed
chaieb@26123
  1085
wenzelm@56776
  1086
lemma divides_degree:
wenzelm@56776
  1087
  assumes pq: "p dvd (q:: complex poly)"
huffman@29464
  1088
  shows "degree p \<le> degree q \<or> q = 0"
wenzelm@56776
  1089
  by (metis dvd_imp_degree_le pq)
chaieb@26123
  1090
wenzelm@60424
  1091
text \<open>Arithmetic operations on multivariate polynomials.\<close>
chaieb@26123
  1092
huffman@30488
  1093
lemma mpoly_base_conv:
wenzelm@56778
  1094
  fixes x :: "'a::comm_ring_1"
lp15@55735
  1095
  shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
lp15@55735
  1096
  by simp_all
chaieb@26123
  1097
huffman@30488
  1098
lemma mpoly_norm_conv:
wenzelm@56778
  1099
  fixes x :: "'a::comm_ring_1"
wenzelm@56776
  1100
  shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
wenzelm@56776
  1101
  by simp_all
chaieb@26123
  1102
huffman@30488
  1103
lemma mpoly_sub_conv:
wenzelm@56778
  1104
  fixes x :: "'a::comm_ring_1"
lp15@55735
  1105
  shows "poly p x - poly q x = poly p x + -1 * poly q x"
haftmann@54230
  1106
  by simp
chaieb@26123
  1107
wenzelm@56778
  1108
lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
wenzelm@56778
  1109
  by simp
chaieb@26123
  1110
lp15@55735
  1111
lemma poly_cancel_eq_conv:
wenzelm@56778
  1112
  fixes x :: "'a::field"
wenzelm@56795
  1113
  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
lp15@55735
  1114
  by auto
chaieb@26123
  1115
huffman@30488
  1116
lemma poly_divides_pad_rule:
wenzelm@56778
  1117
  fixes p:: "('a::comm_ring_1) poly"
huffman@29464
  1118
  assumes pq: "p dvd q"
wenzelm@56778
  1119
  shows "p dvd (pCons 0 q)"
wenzelm@56778
  1120
proof -
huffman@29464
  1121
  have "pCons 0 q = q * [:0,1:]" by simp
huffman@29464
  1122
  then have "q dvd (pCons 0 q)" ..
huffman@29464
  1123
  with pq show ?thesis by (rule dvd_trans)
chaieb@26123
  1124
qed
chaieb@26123
  1125
huffman@30488
  1126
lemma poly_divides_conv0:
wenzelm@56778
  1127
  fixes p:: "'a::field poly"
wenzelm@56776
  1128
  assumes lgpq: "degree q < degree p"
wenzelm@56776
  1129
    and lq: "p \<noteq> 0"
wenzelm@56776
  1130
  shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1131
proof
wenzelm@56776
  1132
  assume r: ?rhs
wenzelm@56776
  1133
  then have "q = p * 0" by simp
wenzelm@56776
  1134
  then show ?lhs ..
wenzelm@56776
  1135
next
wenzelm@56776
  1136
  assume l: ?lhs
wenzelm@56778
  1137
  show ?rhs
wenzelm@56778
  1138
  proof (cases "q = 0")
wenzelm@56778
  1139
    case True
wenzelm@56778
  1140
    then show ?thesis by simp
wenzelm@56778
  1141
  next
wenzelm@56776
  1142
    assume q0: "q \<noteq> 0"
wenzelm@56776
  1143
    from l q0 have "degree p \<le> degree q"
wenzelm@56776
  1144
      by (rule dvd_imp_degree_le)
wenzelm@56778
  1145
    with lgpq show ?thesis by simp
wenzelm@56778
  1146
  qed
chaieb@26123
  1147
qed
chaieb@26123
  1148
huffman@30488
  1149
lemma poly_divides_conv1:
wenzelm@56778
  1150
  fixes p :: "'a::field poly"
wenzelm@56776
  1151
  assumes a0: "a \<noteq> 0"
wenzelm@56776
  1152
    and pp': "p dvd p'"
wenzelm@56776
  1153
    and qrp': "smult a q - p' = r"
wenzelm@56776
  1154
  shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@56776
  1155
proof
huffman@29464
  1156
  from pp' obtain t where t: "p' = p * t" ..
wenzelm@56776
  1157
  {
wenzelm@56776
  1158
    assume l: ?lhs
huffman@29464
  1159
    then obtain u where u: "q = p * u" ..
wenzelm@56776
  1160
    have "r = p * (smult a u - t)"
wenzelm@56776
  1161
      using u qrp' [symmetric] t by (simp add: algebra_simps)
wenzelm@56776
  1162
    then show ?rhs ..
wenzelm@56776
  1163
  next
wenzelm@56776
  1164
    assume r: ?rhs
huffman@29464
  1165
    then obtain u where u: "r = p * u" ..
huffman@29464
  1166
    from u [symmetric] t qrp' [symmetric] a0
wenzelm@51541
  1167
    have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
wenzelm@56776
  1168
    then show ?lhs ..
wenzelm@56776
  1169
  }
chaieb@26123
  1170
qed
chaieb@26123
  1171
chaieb@26123
  1172
lemma basic_cqe_conv1:
lp15@55358
  1173
  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
lp15@55358
  1174
  "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
wenzelm@56776
  1175
  "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
lp15@55358
  1176
  "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
wenzelm@56776
  1177
  "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
wenzelm@56776
  1178
  by simp_all
chaieb@26123
  1179
huffman@30488
  1180
lemma basic_cqe_conv2:
wenzelm@56795
  1181
  assumes l: "p \<noteq> 0"
wenzelm@56795
  1182
  shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
wenzelm@56776
  1183
proof -
wenzelm@60424
  1184
  have False if "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t
wenzelm@60449
  1185
    using l that by simp
wenzelm@56776
  1186
  then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
chaieb@26123
  1187
    by blast
wenzelm@56776
  1188
  from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
wenzelm@56776
  1189
    by auto
chaieb@26123
  1190
qed
chaieb@26123
  1191
wenzelm@56776
  1192
lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
wenzelm@56776
  1193
  by (metis poly_all_0_iff_0)
chaieb@26123
  1194
chaieb@26123
  1195
lemma basic_cqe_conv3:
huffman@29464
  1196
  fixes p q :: "complex poly"
huffman@30488
  1197
  assumes l: "p \<noteq> 0"
wenzelm@56795
  1198
  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
  1199
proof -
wenzelm@56776
  1200
  from l have dp: "degree (pCons a p) = psize p"
wenzelm@56776
  1201
    by (simp add: psize_def)
huffman@29464
  1202
  from nullstellensatz_univariate[of "pCons a p" q] l
lp15@55358
  1203
  show ?thesis
lp15@55358
  1204
    by (metis dp pCons_eq_0_iff)
chaieb@26123
  1205
qed
chaieb@26123
  1206
chaieb@26123
  1207
lemma basic_cqe_conv4:
huffman@29464
  1208
  fixes p q :: "complex poly"
lp15@55358
  1209
  assumes h: "\<And>x. poly (q ^ n) x = poly r x"
lp15@55358
  1210
  shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1211
proof -
wenzelm@56776
  1212
  from h have "poly (q ^ n) = poly r"
wenzelm@56776
  1213
    by auto
wenzelm@56776
  1214
  then have "(q ^ n) = r"
wenzelm@56776
  1215
    by (simp add: poly_eq_poly_eq_iff)
wenzelm@56776
  1216
  then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
wenzelm@56776
  1217
    by simp
chaieb@26123
  1218
qed
chaieb@26123
  1219
lp15@55735
  1220
lemma poly_const_conv:
wenzelm@56778
  1221
  fixes x :: "'a::comm_ring_1"
wenzelm@56776
  1222
  shows "poly [:c:] x = y \<longleftrightarrow> c = y"
wenzelm@56776
  1223
  by simp
chaieb@26123
  1224
huffman@29464
  1225
end