src/HOL/Library/Fundamental_Theorem_Algebra.thy

tuned proofs;

(* Author: Amine Chaieb, TU Muenchen *)

header{*Fundamental Theorem of Algebra*}

theory Fundamental_Theorem_Algebra
imports Polynomial Complex_Main
begin

subsection {* Square root of complex numbers *}

definition csqrt :: "complex \<Rightarrow> complex"
where
  "csqrt z =
    (if Im z = 0 then
       if 0 \<le> Re z then Complex (sqrt(Re z)) 0
       else Complex 0 (sqrt(- Re z))
     else Complex (sqrt((cmod z + Re z) /2)) ((Im z / abs(Im z)) * sqrt((cmod z - Re z) /2)))"

lemma csqrt[algebra]: "(csqrt z)\<^sup>2 = z"
proof -
  obtain x y where xy: "z = Complex x y" by (cases z)
  {
    assume y0: "y = 0"
    {
      assume x0: "x \<ge> 0"
      then have ?thesis
        using y0 xy real_sqrt_pow2[OF x0]
        by (simp add: csqrt_def power2_eq_square)
    }
    moreover
    {
      assume "\<not> x \<ge> 0"
      then have x0: "- x \<ge> 0" by arith
      then have ?thesis
        using y0 xy real_sqrt_pow2[OF x0]
        by (simp add: csqrt_def power2_eq_square)
    }
    ultimately have ?thesis by blast
  }
  moreover
  {
    assume y0: "y \<noteq> 0"
    {
      fix x y
      let ?z = "Complex x y"
      from abs_Re_le_cmod[of ?z] have tha: "abs x \<le> cmod ?z"
        by auto
      then have "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0"
        by arith+
      then have "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0"
        by (simp_all add: power2_eq_square)
    }
    note th = this
    have sq4: "\<And>x::real. x\<^sup>2 / 4 = (x / 2)\<^sup>2"
      by (simp add: power2_eq_square)
    from th[of x y]
    have sq4': "sqrt (((sqrt (x * x + y * y) + x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) + x) / 2"
      "sqrt (((sqrt (x * x + y * y) - x)\<^sup>2 / 4)) = (sqrt (x * x + y * y) - x) / 2"
      unfolding sq4 by simp_all
    then have th1: "sqrt ((sqrt (x * x + y * y) + x) * (sqrt (x * x + y * y) + x) / 4) -
        sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) - x) / 4) = x"
      unfolding power2_eq_square by simp
    have "sqrt 4 = sqrt (2\<^sup>2)"
      by simp
    then have sqrt4: "sqrt 4 = 2"
      by (simp only: real_sqrt_abs)
    have th2: "2 * (y * sqrt ((sqrt (x * x + y * y) - x) * (sqrt (x * x + y * y) + x) / 4)) / \<bar>y\<bar> = y"
      using iffD2[OF real_sqrt_pow2_iff sum_power2_ge_zero[of x y]] y0
      unfolding power2_eq_square
      by (simp add: algebra_simps real_sqrt_divide sqrt4)
    from y0 xy have ?thesis
      apply (simp add: csqrt_def power2_eq_square)
      apply (simp add: real_sqrt_sum_squares_mult_ge_zero[of x y]
        real_sqrt_pow2[OF th(1)[of x y], unfolded power2_eq_square]
        real_sqrt_pow2[OF th(2)[of x y], unfolded power2_eq_square]
        real_sqrt_mult[symmetric])
      using th1 th2  ..
  }
  ultimately show ?thesis by blast
qed

lemma csqrt_Complex: "x \<ge> 0 \<Longrightarrow> csqrt (Complex x 0) = Complex (sqrt x) 0"
  by (simp add: csqrt_def)

lemma csqrt_0 [simp]: "csqrt 0 = 0"
  by (simp add: csqrt_def)

lemma csqrt_1 [simp]: "csqrt 1 = 1"
  by (simp add: csqrt_def)

lemma csqrt_principal: "0 < Re(csqrt(z)) | Re(csqrt(z)) = 0 & 0 \<le> Im(csqrt(z))"
proof (cases z)
  case (Complex x y)
  then show ?thesis
    using real_sqrt_sum_squares_ge1 [of "x" y]
          real_sqrt_sum_squares_ge1 [of "-x" y]
          real_sqrt_sum_squares_eq_cancel [of x y]
    apply (auto simp: csqrt_def intro!: Rings.ordered_ring_class.split_mult_pos_le)
    apply (metis add_commute diff_add_cancel le_add_same_cancel1 real_sqrt_sum_squares_ge1)
    apply (metis add_commute less_eq_real_def power_minus_Bit0
            real_0_less_add_iff real_sqrt_sum_squares_eq_cancel)
    done
qed

lemma Re_csqrt: "0 \<le> Re(csqrt z)"
  by (metis csqrt_principal le_less)

lemma csqrt_square: "0 < Re z \<or> Re z = 0 \<and> 0 \<le> Im z \<Longrightarrow> csqrt (z\<^sup>2) = z"
  using csqrt [of "z\<^sup>2"] csqrt_principal [of "z\<^sup>2"]
  by (cases z) (auto simp: power2_eq_iff)

lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
  by auto (metis csqrt power_eq_0_iff)

lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
  by auto (metis csqrt power2_eq_1_iff)


subsection {* More lemmas about module of complex numbers *}

lemma complex_of_real_power: "complex_of_real x ^ n = complex_of_real (x^n)"
  by (rule of_real_power [symmetric])

text{* The triangle inequality for cmod *}
lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"
  using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto


subsection {* Basic lemmas about polynomials *}

lemma poly_bound_exists:
  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
  shows "\<exists>m. m > 0 \<and> (\<forall>z. norm z \<le> r \<longrightarrow> norm (poly p z) \<le> m)"
proof (induct p)
  case 0
  then show ?case by (rule exI[where x=1]) simp
next
  case (pCons c cs)
  from pCons.hyps obtain m where m: "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
    by blast
  let ?k = " 1 + norm c + \<bar>r * m\<bar>"
  have kp: "?k > 0"
    using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
  {
    fix z :: 'a
    assume H: "norm z \<le> r"
    from m H have th: "norm (poly cs z) \<le> m"
      by blast
    from H have rp: "r \<ge> 0"
      using norm_ge_zero[of z] by arith
    have "norm (poly (pCons c cs) z) \<le> norm c + norm (z * poly cs z)"
      using norm_triangle_ineq[of c "z* poly cs z"] by simp
    also have "\<dots> \<le> norm c + r * m"
      using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
      by (simp add: norm_mult)
    also have "\<dots> \<le> ?k"
      by simp
    finally have "norm (poly (pCons c cs) z) \<le> ?k" .
  }
  with kp show ?case by blast
qed


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

definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"
  where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"

lemma offset_poly_0: "offset_poly 0 h = 0"
  by (simp add: offset_poly_def)

lemma offset_poly_pCons:
  "offset_poly (pCons a p) h =
    smult h (offset_poly p h) + pCons a (offset_poly p h)"
  by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)

lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"
  by (simp add: offset_poly_pCons offset_poly_0)

lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"
  apply (induct p)
  apply (simp add: offset_poly_0)
  apply (simp add: offset_poly_pCons algebra_simps)
  done

lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"
  by (induct p arbitrary: a) (simp, force)

lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"
  apply (safe intro!: offset_poly_0)
  apply (induct p)
  apply simp
  apply (simp add: offset_poly_pCons)
  apply (frule offset_poly_eq_0_lemma, simp)
  done

lemma degree_offset_poly: "degree (offset_poly p h) = degree p"
  apply (induct p)
  apply (simp add: offset_poly_0)
  apply (case_tac "p = 0")
  apply (simp add: offset_poly_0 offset_poly_pCons)
  apply (simp add: offset_poly_pCons)
  apply (subst degree_add_eq_right)
  apply (rule le_less_trans [OF degree_smult_le])
  apply (simp add: offset_poly_eq_0_iff)
  apply (simp add: offset_poly_eq_0_iff)
  done

definition "psize p = (if p = 0 then 0 else Suc (degree p))"

lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
  unfolding psize_def by simp

lemma poly_offset:
  fixes p :: "'a::comm_ring_1 poly"
  shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"
proof (intro exI conjI)
  show "psize (offset_poly p a) = psize p"
    unfolding psize_def
    by (simp add: offset_poly_eq_0_iff degree_offset_poly)
  show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
    by (simp add: poly_offset_poly)
qed

text{* An alternative useful formulation of completeness of the reals *}
lemma real_sup_exists:
  assumes ex: "\<exists>x. P x"
    and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"
  shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"
proof
  from bz have "bdd_above (Collect P)"
    by (force intro: less_imp_le)
  then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"
    using ex bz by (subst less_cSup_iff) auto
qed

subsection {* Fundamental theorem of algebra *}
lemma  unimodular_reduce_norm:
  assumes md: "cmod z = 1"
  shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"
proof -
  obtain x y where z: "z = Complex x y "
    by (cases z) auto
  from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"
    by (simp add: cmod_def)
  {
    assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"
    from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"
      by (simp_all add: cmod_def power2_eq_square algebra_simps)
    then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"
      by simp_all
    then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"
      by - (rule power_mono, simp, simp)+
    then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"
      by (simp_all add: power_mult_distrib)
    from add_mono[OF th0] xy have False by simp
  }
  then show ?thesis
    unfolding linorder_not_le[symmetric] by blast
qed

text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}
lemma reduce_poly_simple:
  assumes b: "b \<noteq> 0"
    and n: "n \<noteq> 0"
  shows "\<exists>z. cmod (1 + b * z^n) < 1"
  using n
proof (induct n rule: nat_less_induct)
  fix n
  assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"
  assume n: "n \<noteq> 0"
  let ?P = "\<lambda>z n. cmod (1 + b * z ^ n) < 1"
  {
    assume e: "even n"
    then have "\<exists>m. n = 2 * m"
      by presburger
    then obtain m where m: "n = 2 * m"
      by blast
    from n m have "m \<noteq> 0" "m < n"
      by presburger+
    with IH[rule_format, of m] obtain z where z: "?P z m"
      by blast
    from z have "?P (csqrt z) n"
      by (simp add: m power_mult csqrt)
    then have "\<exists>z. ?P z n" ..
  }
  moreover
  {
    assume o: "odd n"
    have th0: "cmod (complex_of_real (cmod b) / b) = 1"
      using b by (simp add: norm_divide)
    from o have "\<exists>m. n = Suc (2 * m)"
      by presburger+
    then obtain m where m: "n = Suc (2 * m)"
      by blast
    from unimodular_reduce_norm[OF th0] o
    have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
      apply (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
      apply (rule_tac x="1" in exI)
      apply simp
      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")
      apply (rule_tac x="-1" in exI)
      apply simp
      apply (cases "cmod (complex_of_real (cmod b) / b + ii) < 1")
      apply (cases "even m")
      apply (rule_tac x="ii" in exI)
      apply (simp add: m power_mult)
      apply (rule_tac x="- ii" in exI)
      apply (simp add: m power_mult)
      apply (cases "even m")
      apply (rule_tac x="- ii" in exI)
      apply (simp add: m power_mult)
      apply (auto simp add: m power_mult)
      apply (rule_tac x="ii" in exI)
      apply (auto simp add: m power_mult)
      done
    then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"
      by blast
    let ?w = "v / complex_of_real (root n (cmod b))"
    from odd_real_root_pow[OF o, of "cmod b"]
    have th1: "?w ^ n = v^n / complex_of_real (cmod b)"
      by (simp add: power_divide complex_of_real_power)
    have th2:"cmod (complex_of_real (cmod b) / b) = 1"
      using b by (simp add: norm_divide)
    then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"
      by simp
    have th4: "cmod (complex_of_real (cmod b) / b) *
        cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <
        cmod (complex_of_real (cmod b) / b) * 1"
      apply (simp only: norm_mult[symmetric] distrib_left)
      using b v
      apply (simp add: th2)
      done
    from mult_less_imp_less_left[OF th4 th3]
    have "?P ?w n" unfolding th1 .
    then have "\<exists>z. ?P z n" ..
  }
  ultimately show "\<exists>z. ?P z n" by blast
qed

text{* Bolzano-Weierstrass type property for closed disc in complex plane. *}

lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"
  using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]
  unfolding cmod_def by simp

lemma bolzano_weierstrass_complex_disc:
  assumes r: "\<forall>n. cmod (s n) \<le> r"
  shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"
proof-
  from seq_monosub[of "Re \<circ> s"]
  obtain f where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"
    unfolding o_def by blast
  from seq_monosub[of "Im \<circ> s \<circ> f"]
  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s (f (g n))))"
    unfolding o_def by blast
  let ?h = "f \<circ> g"
  from r[rule_format, of 0] have rp: "r \<ge> 0"
    using norm_ge_zero[of "s 0"] by arith
  have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"
  proof
    fix n
    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
    show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith
  qed
  have conv1: "convergent (\<lambda>n. Re (s (f n)))"
    apply (rule Bseq_monoseq_convergent)
    apply (simp add: Bseq_def)
    apply (metis gt_ex le_less_linear less_trans order.trans th)
    apply (rule f(2))
    done
  have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"
  proof
    fix n
    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
    show "\<bar>Im (s n)\<bar> \<le> r + 1"
      by arith
  qed

  have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"
    apply (rule Bseq_monoseq_convergent)
    apply (simp add: Bseq_def)
    apply (metis gt_ex le_less_linear less_trans order.trans th)
    apply (rule g(2))
    done

  from conv1[unfolded convergent_def] obtain x where "LIMSEQ (\<lambda>n. Re (s (f n))) x"
    by blast
  then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"
    unfolding LIMSEQ_iff real_norm_def .

  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"
    by blast
  then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"
    unfolding LIMSEQ_iff real_norm_def .
  let ?w = "Complex x y"
  from f(1) g(1) have hs: "subseq ?h"
    unfolding subseq_def by auto
  {
    fix e :: real
    assume ep: "e > 0"
    then have e2: "e/2 > 0"
      by simp
    from x[rule_format, OF e2] y[rule_format, OF e2]
    obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"
      and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"
      by blast
    {
      fix n
      assume nN12: "n \<ge> N1 + N2"
      then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"
        using seq_suble[OF g(1), of n] by arith+
      from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]
      have "cmod (s (?h n) - ?w) < e"
        using metric_bound_lemma[of "s (f (g n))" ?w] by simp
    }
    then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e"
      by blast
  }
  with hs show ?thesis by blast
qed

text{* Polynomial is continuous. *}

lemma poly_cont:
  fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
  assumes ep: "e > 0"
  shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"
proof -
  obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"
  proof
    show "degree (offset_poly p z) = degree p"
      by (rule degree_offset_poly)
    show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"
      by (rule poly_offset_poly)
  qed
  have th: "\<And>w. poly q (w - z) = poly p w"
    using q(2)[of "w - z" for w] by simp
  show ?thesis unfolding th[symmetric]
  proof (induct q)
    case 0
    then show ?case
      using ep by auto
  next
    case (pCons c cs)
    from poly_bound_exists[of 1 "cs"]
    obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"
      by blast
    from ep m(1) have em0: "e/m > 0"
      by (simp add: field_simps)
    have one0: "1 > (0::real)"
      by arith
    from real_lbound_gt_zero[OF one0 em0]
    obtain d where d: "d > 0" "d < 1" "d < e / m"
      by blast
    from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"
      by (simp_all add: field_simps)
    show ?case
    proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
      fix d w
      assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"
      then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"
        by simp_all
      from H(3) m(1) have dme: "d*m < e"
        by (simp add: field_simps)
      from H have th: "norm (w - z) \<le> d"
        by simp
      from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme
      show "norm (w - z) * norm (poly cs (w - z)) < e"
        by simp
    qed
  qed
qed

text{* Hence a polynomial attains minimum on a closed disc
  in the complex plane. *}
lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"
proof -
  {
    assume "\<not> r \<ge> 0"
    then have ?thesis
      by (metis norm_ge_zero order.trans)
  }
  moreover
  {
    assume rp: "r \<ge> 0"
    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))"
      by simp
    then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"
      by blast
    {
      fix x z
      assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"
      then have "- x < 0 "
        by arith
      with H(2) norm_ge_zero[of "poly p z"] have False
        by simp
    }
    then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"
      by blast
    from real_sup_exists[OF mth1 mth2] obtain s where
      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
    let ?m = "- s"
    {
      fix y
      from s[rule_format, of "-y"]
      have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"
        unfolding minus_less_iff[of y ] equation_minus_iff by blast
    }
    note s1 = this[unfolded minus_minus]
    from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"
      by auto
    {
      fix n :: nat
      from s1[rule_format, of "?m + 1/real (Suc n)"]
      have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"
        by simp
    }
    then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..
    from choice[OF th] obtain g where
        g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
      by blast
    from bolzano_weierstrass_complex_disc[OF g(1)]
    obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"
      by blast
    {
      fix w
      assume wr: "cmod w \<le> r"
      let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"
      {
        assume e: "?e > 0"
        then have e2: "?e/2 > 0"
          by simp
        from poly_cont[OF e2, of z p] obtain d where
            d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"
          by blast
        {
          fix w
          assume w: "cmod (w - z) < d"
          have "cmod(poly p w - poly p z) < ?e / 2"
            using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
        }
        note th1 = this

        from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"
          by blast
        from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"
          by blast
        have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
          using N1[rule_format, of "N1 + N2"] th1 by simp
        {
          fix a b e2 m :: real
          have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"
            by arith
        }
        note th0 = this
        have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"
          by arith
        from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .
        from seq_suble[OF fz(1), of "N1 + N2"]
        have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"
          by simp
        have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"
          using N2 by auto
        from frac_le[OF th000 th00]
        have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"
          by simp
        from g(2)[rule_format, of "f (N1 + N2)"]
        have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .
        from order_less_le_trans[OF th01 th00]
        have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .
        from N2 have "2/?e < real (Suc (N1 + N2))"
          by arith
        with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]
        have "?e/2 > 1/ real (Suc (N1 + N2))"
          by (simp add: inverse_eq_divide)
        with ath[OF th31 th32]
        have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"
          by arith
        have ath2: "\<And>a b c m::real. \<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c"
          by arith
        have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>
            cmod (poly p (g (f (N1 + N2))) - poly p z)"
          by (simp add: norm_triangle_ineq3)
        from ath2[OF th22, of ?m]
        have thc2: "2 * (?e/2) \<le>
            \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"
          by simp
        from th0[OF th2 thc1 thc2] have False .
      }
      then have "?e = 0"
        by auto
      then have "cmod (poly p z) = ?m"
        by simp
      with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"
        by simp
    }
    then have ?thesis by blast
  }
  ultimately show ?thesis by blast
qed

lemma "(rcis (sqrt (abs r)) (a/2))\<^sup>2 = rcis (abs r) a"
  unfolding power2_eq_square
  apply (simp add: rcis_mult)
  apply (simp add: power2_eq_square[symmetric])
  done

lemma cispi: "cis pi = -1"
  by (simp add: cis_def)

lemma "(rcis (sqrt (abs r)) ((pi + a) / 2))\<^sup>2 = rcis (- abs r) a"
  unfolding power2_eq_square
  apply (simp add: rcis_mult add_divide_distrib)
  apply (simp add: power2_eq_square[symmetric] rcis_def cispi cis_mult[symmetric])
  done

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

lemma poly_infinity:
  fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"
  assumes ex: "p \<noteq> 0"
  shows "\<exists>r. \<forall>z. r \<le> norm z \<longrightarrow> d \<le> norm (poly (pCons a p) z)"
  using ex
proof (induct p arbitrary: a d)
  case 0
  then show ?case by simp
next
  case (pCons c cs a d)
  show ?case
  proof (cases "cs = 0")
    case False
    with pCons.hyps obtain r where r: "\<forall>z. r \<le> norm z \<longrightarrow> d + norm a \<le> norm (poly (pCons c cs) z)"
      by blast
    let ?r = "1 + \<bar>r\<bar>"
    {
      fix z :: 'a
      assume h: "1 + \<bar>r\<bar> \<le> norm z"
      have r0: "r \<le> norm z"
        using h by arith
      from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"
        by arith
      from h have z1: "norm z \<ge> 1"
        by arith
      from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]
      have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"
        unfolding norm_mult by (simp add: algebra_simps)
      from norm_diff_ineq[of "z * poly (pCons c cs) z" a]
      have th2: "norm (z * poly (pCons c cs) z) - norm a \<le> norm (poly (pCons a (pCons c cs)) z)"
        by (simp add: algebra_simps)
      from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
        by arith
    }
    then show ?thesis by blast
  next
    case True
    with pCons.prems have c0: "c \<noteq> 0"
      by simp
    {
      fix z :: 'a
      assume h: "(\<bar>d\<bar> + norm a) / norm c \<le> norm z"
      from c0 have "norm c > 0"
        by simp
      from h c0 have th0: "\<bar>d\<bar> + norm a \<le> norm (z * c)"
        by (simp add: field_simps norm_mult)
      have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"
        by arith
      from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"
        by (simp add: algebra_simps)
      from ath[OF th1 th0] have "d \<le> norm (poly (pCons a (pCons c cs)) z)"
        using True by simp
    }
    then show ?thesis by blast
  qed
qed

text {* Hence polynomial's modulus attains its minimum somewhere. *}
lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"
proof (induct p)
  case 0
  then show ?case by simp
next
  case (pCons c cs)
  show ?case
  proof (cases "cs = 0")
    case False
    from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]
    obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"
      by blast
    have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"
      by arith
    from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]
    obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)"
      by blast
    {
      fix z
      assume z: "r \<le> cmod z"
      from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"
        by simp
    }
    note v0 = this
    from v0 v ath[of r] show ?thesis
      by blast
  next
    case True
    with pCons.hyps show ?thesis by simp
  qed
qed

text{* Constant function (non-syntactic characterization). *}
definition "constant f \<longleftrightarrow> (\<forall>x y. f x = f y)"

lemma nonconstant_length: "\<not> constant (poly p) \<Longrightarrow> psize p \<ge> 2"
  by (induct p) (auto simp: constant_def psize_def)

lemma poly_replicate_append: "poly (monom 1 n * p) (x::'a::comm_ring_1) = x^n * poly p x"
  by (simp add: poly_monom)

text {* Decomposition of polynomial, skipping zero coefficients
  after the first.  *}

lemma poly_decompose_lemma:
  assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"
  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)"
  unfolding psize_def
  using nz
proof (induct p)
  case 0
  then show ?case by simp
next
  case (pCons c cs)
  show ?case
  proof (cases "c = 0")
    case True
    from pCons.hyps pCons.prems True show ?thesis
      apply (auto)
      apply (rule_tac x="k+1" in exI)
      apply (rule_tac x="a" in exI, clarsimp)
      apply (rule_tac x="q" in exI)
      apply auto
      done
  next
    case False
    show ?thesis
      apply (rule exI[where x=0])
      apply (rule exI[where x=c], auto simp add: False)
      done
  qed
qed

lemma poly_decompose:
  assumes nc: "\<not> constant (poly p)"
  shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>
               psize q + k + 1 = psize p \<and>
              (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
  using nc
proof (induct p)
  case 0
  then show ?case
    by (simp add: constant_def)
next
  case (pCons c cs)
  {
    assume C: "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"
    {
      fix x y
      from C have "poly (pCons c cs) x = poly (pCons c cs) y"
        by (cases "x = 0") auto
    }
    with pCons.prems have False
      by (auto simp add: constant_def)
  }
  then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..
  from poly_decompose_lemma[OF th]
  show ?case
    apply clarsimp
    apply (rule_tac x="k+1" in exI)
    apply (rule_tac x="a" in exI)
    apply simp
    apply (rule_tac x="q" in exI)
    apply (auto simp add: psize_def split: if_splits)
    done
qed

text{* Fundamental theorem of algebra *}

lemma fundamental_theorem_of_algebra:
  assumes nc: "\<not> constant (poly p)"
  shows "\<exists>z::complex. poly p z = 0"
  using nc
proof (induct "psize p" arbitrary: p rule: less_induct)
  case less
  let ?p = "poly p"
  let ?ths = "\<exists>z. ?p z = 0"

  from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
  from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"
    by blast

  show ?ths
  proof (cases "?p c = 0")
    case True
    then show ?thesis by blast
  next
    case False
    note pc0 = this
    from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"
      by blast
    {
      assume h: "constant (poly q)"
      from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"
        by auto
      {
        fix x y
        from th have "?p x = poly q (x - c)"
          by auto
        also have "\<dots> = poly q (y - c)"
          using h unfolding constant_def by blast
        also have "\<dots> = ?p y"
          using th by auto
        finally have "?p x = ?p y" .
      }
      with less(2) have False
        unfolding constant_def by blast
    }
    then have qnc: "\<not> constant (poly q)"
      by blast
    from q(2) have pqc0: "?p c = poly q 0"
      by simp
    from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"
      by simp
    let ?a0 = "poly q 0"
    from pc0 pqc0 have a00: "?a0 \<noteq> 0"
      by simp
    from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
      by simp
    let ?r = "smult (inverse ?a0) q"
    have lgqr: "psize q = psize ?r"
      using a00
      unfolding psize_def degree_def
      by (simp add: poly_eq_iff)
    {
      assume h: "\<And>x y. poly ?r x = poly ?r y"
      {
        fix x y
        from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
          by auto
        also have "\<dots> = poly ?r y * ?a0"
          using h by simp
        also have "\<dots> = poly q y"
          using qr[rule_format, of y] by simp
        finally have "poly q x = poly q y" .
      }
      with qnc have False
        unfolding constant_def by blast
    }
    then have rnc: "\<not> constant (poly ?r)"
      unfolding constant_def by blast
    from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
      by auto
    {
      fix w
      have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"
        using qr[rule_format, of w] a00 by (simp add: divide_inverse mult_ac)
      also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"
        using a00 unfolding norm_divide by (simp add: field_simps)
      finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .
    }
    note mrmq_eq = this
    from poly_decompose[OF rnc] obtain k a s where
      kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"
        "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
    {
      assume "psize p = k + 1"
      with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
        by auto
      {
        fix w
        have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
          using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
      }
      note hth = this [symmetric]
      from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"
        unfolding hth by blast
    }
    moreover
    {
      assume kn: "psize p \<noteq> k + 1"
      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
        by simp
      have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
        unfolding constant_def poly_pCons poly_monom
        using kas(1)
        apply simp
        apply (rule exI[where x=0])
        apply (rule exI[where x=1])
        apply simp
        done
      from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
        by (simp add: psize_def degree_monom_eq)
      from less(1) [OF k1n [simplified th02] th01]
      obtain w where w: "1 + w^k * a = 0"
        unfolding poly_pCons poly_monom
        using kas(2) by (cases k) (auto simp add: algebra_simps)
      from poly_bound_exists[of "cmod w" s] obtain m where
        m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast
      have w0: "w \<noteq> 0"
        using kas(2) w by (auto simp add: power_0_left)
      from w have "(1 + w ^ k * a) - 1 = 0 - 1"
        by simp
      then have wm1: "w^k * a = - 1"
        by simp
      have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
        using norm_ge_zero[of w] w0 m(1)
        by (simp add: inverse_eq_divide zero_less_mult_iff)
      with real_lbound_gt_zero[OF zero_less_one] obtain t where
        t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast
      let ?ct = "complex_of_real t"
      let ?w = "?ct * w"
      have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"
        using kas(1) by (simp add: algebra_simps power_mult_distrib)
      also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"
        unfolding wm1 by simp
      finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =
        cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"
        by metis
      with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]
      have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"
        unfolding norm_of_real by simp
      have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"
        by arith
      have "t * cmod w \<le> 1 * cmod w"
        apply (rule mult_mono)
        using t(1,2)
        apply auto
        done
      then have tw: "cmod ?w \<le> cmod w"
        using t(1) by (simp add: norm_mult)
      from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"
        by (simp add: inverse_eq_divide field_simps)
      with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"
        by (metis comm_mult_strict_left_mono)
      have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
        using w0 t(1)
        by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)
      then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"
        using t(1,2) m(2)[rule_format, OF tw] w0
        by auto
      with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"
        by simp
      from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"
        by auto
      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
      have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .
      from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"
        by arith
      then have "cmod (poly ?r ?w) < 1"
        unfolding kas(4)[rule_format, of ?w] r01 by simp
      then have "\<exists>w. cmod (poly ?r w) < 1"
        by blast
    }
    ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"
      by blast
    from cr0_contr cq0 q(2) show ?thesis
      unfolding mrmq_eq not_less[symmetric] by auto
  qed
qed

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

lemma fundamental_theorem_of_algebra_alt:
  assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"
  shows "\<exists>z. poly p z = (0::complex)"
  using nc
proof (induct p)
  case 0
  then show ?case by simp
next
  case (pCons c cs)
  show ?case
  proof (cases "c = 0")
    case True
    then show ?thesis by auto
  next
    case False
    {
      assume nc: "constant (poly (pCons c cs))"
      from nc[unfolded constant_def, rule_format, of 0]
      have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto
      then have "cs = 0"
      proof (induct cs)
        case 0
        then show ?case by simp
      next
        case (pCons d ds)
        show ?case
        proof (cases "d = 0")
          case True
          then show ?thesis using pCons.prems pCons.hyps by simp
        next
          case False
          from poly_bound_exists[of 1 ds] obtain m where
            m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast
          have dm: "cmod d / m > 0"
            using False m(1) by (simp add: field_simps)
          from real_lbound_gt_zero[OF dm zero_less_one] obtain x where
            x: "x > 0" "x < cmod d / m" "x < 1" by blast
          let ?x = "complex_of_real x"
          from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1"
            by simp_all
          from pCons.prems[rule_format, OF cx(1)]
          have cth: "cmod (?x*poly ds ?x) = cmod d"
            by (simp add: eq_diff_eq[symmetric])
          from m(2)[rule_format, OF cx(2)] x(1)
          have th0: "cmod (?x*poly ds ?x) \<le> x*m"
            by (simp add: norm_mult)
          from x(2) m(1) have "x * m < cmod d"
            by (simp add: field_simps)
          with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"
            by auto
          with cth show ?thesis
            by blast
        qed
      qed
    }
    then have nc: "\<not> constant (poly (pCons c cs))"
      using pCons.prems False by blast
    from fundamental_theorem_of_algebra[OF nc] show ?thesis .
  qed
qed


subsection{* Nullstellensatz, degrees and divisibility of polynomials *}

lemma nullstellensatz_lemma:
  fixes p :: "complex poly"
  assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
    and "degree p = n"
    and "n \<noteq> 0"
  shows "p dvd (q ^ n)"
  using assms
proof (induct n arbitrary: p q rule: nat_less_induct)
  fix n :: nat
  fix p q :: "complex poly"
  assume IH: "\<forall>m<n. \<forall>p q.
                 (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>
                 degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"
    and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"
    and dpn: "degree p = n"
    and n0: "n \<noteq> 0"
  from dpn n0 have pne: "p \<noteq> 0" by auto
  let ?ths = "p dvd (q ^ n)"
  {
    fix a
    assume a: "poly p a = 0"
    {
      assume oa: "order a p \<noteq> 0"
      let ?op = "order a p"
      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"
        using order by blast+
      note oop = order_degree[OF pne, unfolded dpn]
      {
        assume q0: "q = 0"
        then have ?ths using n0
          by (simp add: power_0_left)
      }
      moreover
      {
        assume q0: "q \<noteq> 0"
        from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]
        obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)
        from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"
          by (rule dvdE)
        have sne: "s \<noteq> 0" using s pne by auto
        {
          assume ds0: "degree s = 0"
          from ds0 obtain k where kpn: "s = [:k:]"
            by (cases s) (auto split: if_splits)
          from sne kpn have k: "k \<noteq> 0" by simp
          let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"
          have "q ^ n = p * ?w"
            apply (subst r)
            apply (subst s)
            apply (subst kpn)
            using k oop [of a]
            apply (subst power_mult_distrib)
            apply simp
            apply (subst power_add [symmetric])
            apply simp
            done
          then have ?ths
            unfolding dvd_def by blast
        }
        moreover
        {
          assume ds0: "degree s \<noteq> 0"
          from ds0 sne dpn s oa
            have dsn: "degree s < n"
              apply auto
              apply (erule ssubst)
              apply (simp add: degree_mult_eq degree_linear_power)
              done
            {
              fix x assume h: "poly s x = 0"
              {
                assume xa: "x = a"
                from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"
                  by (rule dvdE)
                have "p = [:- a, 1:] ^ (Suc ?op) * u"
                  apply (subst s)
                  apply (subst u)
                  apply (simp only: power_Suc mult_ac)
                  done
                with ap(2)[unfolded dvd_def] have False
                  by blast
              }
              note xa = this
              from h have "poly p x = 0"
                by (subst s) simp
              with pq0 have "poly q x = 0"
                by blast
              with r xa have "poly r x = 0"
                by auto
            }
            note impth = this
            from IH[rule_format, OF dsn, of s r] impth ds0
            have "s dvd (r ^ (degree s))"
              by blast
            then obtain u where u: "r ^ (degree s) = s * u" ..
            then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"
              by (simp only: poly_mult[symmetric] poly_power[symmetric])
            let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
            from oop[of a] dsn have "q ^ n = p * ?w"
              apply -
              apply (subst s)
              apply (subst r)
              apply (simp only: power_mult_distrib)
              apply (subst mult_assoc [where b=s])
              apply (subst mult_assoc [where a=u])
              apply (subst mult_assoc [where b=u, symmetric])
              apply (subst u [symmetric])
              apply (simp add: mult_ac power_add [symmetric])
              done
            then have ?ths
              unfolding dvd_def by blast
        }
        ultimately have ?ths by blast
      }
      ultimately have ?ths by blast
    }
    then have ?ths using a order_root pne by blast
  }
  moreover
  {
    assume exa: "\<not> (\<exists>a. poly p a = 0)"
    from fundamental_theorem_of_algebra_alt[of p] exa
    obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"
      by blast
    then have pp: "\<And>x. poly p x = c"
      by simp
    let ?w = "[:1/c:] * (q ^ n)"
    from ccs have "(q ^ n) = (p * ?w)"
      by simp
    then have ?ths
      unfolding dvd_def by blast
  }
  ultimately show ?ths by blast
qed

lemma nullstellensatz_univariate:
  "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>
    p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"
proof -
  {
    assume pe: "p = 0"
    then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"
      by (auto simp add: poly_all_0_iff_0)
    {
      assume "p dvd (q ^ (degree p))"
      then obtain r where r: "q ^ (degree p) = p * r" ..
      from r pe have False by simp
    }
    with eq pe have ?thesis by blast
  }
  moreover
  {
    assume pe: "p \<noteq> 0"
    {
      assume dp: "degree p = 0"
      then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe
        by (cases p) (simp split: if_splits)
      then have th1: "\<forall>x. poly p x \<noteq> 0"
        by simp
      from k dp have "q ^ (degree p) = p * [:1/k:]"
        by (simp add: one_poly_def)
      then have th2: "p dvd (q ^ (degree p))" ..
      from th1 th2 pe have ?thesis
        by blast
    }
    moreover
    {
      assume dp: "degree p \<noteq> 0"
      then obtain n where n: "degree p = Suc n "
        by (cases "degree p") auto
      {
        assume "p dvd (q ^ (Suc n))"
        then obtain u where u: "q ^ (Suc n) = p * u" ..
        {
          fix x
          assume h: "poly p x = 0" "poly q x \<noteq> 0"
          then have "poly (q ^ (Suc n)) x \<noteq> 0"
            by simp
          then have False using u h(1)
            by (simp only: poly_mult) simp
        }
      }
      with n nullstellensatz_lemma[of p q "degree p"] dp
      have ?thesis by auto
    }
    ultimately have ?thesis by blast
  }
  ultimately show ?thesis by blast
qed

text {* Useful lemma *}

lemma constant_degree:
  fixes p :: "'a::{idom,ring_char_0} poly"
  shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")
proof
  assume l: ?lhs
  from l[unfolded constant_def, rule_format, of _ "0"]
  have th: "poly p = poly [:poly p 0:]"
    by auto
  then have "p = [:poly p 0:]"
    by (simp add: poly_eq_poly_eq_iff)
  then have "degree p = degree [:poly p 0:]"
    by simp
  then show ?rhs
    by simp
next
  assume r: ?rhs
  then obtain k where "p = [:k:]"
    by (cases p) (simp split: if_splits)
  then show ?lhs
    unfolding constant_def by auto
qed

lemma divides_degree:
  assumes pq: "p dvd (q:: complex poly)"
  shows "degree p \<le> degree q \<or> q = 0"
  by (metis dvd_imp_degree_le pq)

text {* Arithmetic operations on multivariate polynomials. *}

lemma mpoly_base_conv:
  fixes x :: "'a::comm_ring_1"
  shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"
  by simp_all

lemma mpoly_norm_conv:
  fixes x :: "'a::comm_ring_1"
  shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"
  by simp_all

lemma mpoly_sub_conv:
  fixes x :: "'a::comm_ring_1"
  shows "poly p x - poly q x = poly p x + -1 * poly q x"
  by simp

lemma poly_pad_rule: "poly p x = 0 \<Longrightarrow> poly (pCons 0 p) x = 0"
  by simp

lemma poly_cancel_eq_conv:
  fixes x :: "'a::field"
  shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"
  by auto

lemma poly_divides_pad_rule:
  fixes p:: "('a::comm_ring_1) poly"
  assumes pq: "p dvd q"
  shows "p dvd (pCons 0 q)"
proof -
  have "pCons 0 q = q * [:0,1:]" by simp
  then have "q dvd (pCons 0 q)" ..
  with pq show ?thesis by (rule dvd_trans)
qed

lemma poly_divides_conv0:
  fixes p:: "'a::field poly"
  assumes lgpq: "degree q < degree p"
    and lq: "p \<noteq> 0"
  shows "p dvd q \<longleftrightarrow> q = 0" (is "?lhs \<longleftrightarrow> ?rhs")
proof
  assume r: ?rhs
  then have "q = p * 0" by simp
  then show ?lhs ..
next
  assume l: ?lhs
  show ?rhs
  proof (cases "q = 0")
    case True
    then show ?thesis by simp
  next
    assume q0: "q \<noteq> 0"
    from l q0 have "degree p \<le> degree q"
      by (rule dvd_imp_degree_le)
    with lgpq show ?thesis by simp
  qed
qed

lemma poly_divides_conv1:
  fixes p :: "'a::field poly"
  assumes a0: "a \<noteq> 0"
    and pp': "p dvd p'"
    and qrp': "smult a q - p' = r"
  shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")
proof
  from pp' obtain t where t: "p' = p * t" ..
  {
    assume l: ?lhs
    then obtain u where u: "q = p * u" ..
    have "r = p * (smult a u - t)"
      using u qrp' [symmetric] t by (simp add: algebra_simps)
    then show ?rhs ..
  next
    assume r: ?rhs
    then obtain u where u: "r = p * u" ..
    from u [symmetric] t qrp' [symmetric] a0
    have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)
    then show ?lhs ..
  }
qed

lemma basic_cqe_conv1:
  "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<longleftrightarrow> False"
  "(\<exists>x. poly 0 x \<noteq> 0) \<longleftrightarrow> False"
  "(\<exists>x. poly [:c:] x \<noteq> 0) \<longleftrightarrow> c \<noteq> 0"
  "(\<exists>x. poly 0 x = 0) \<longleftrightarrow> True"
  "(\<exists>x. poly [:c:] x = 0) \<longleftrightarrow> c = 0"
  by simp_all

lemma basic_cqe_conv2:
  assumes l: "p \<noteq> 0"
  shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"
proof -
  {
    fix h t
    assume h: "h \<noteq> 0" "t = 0" and "pCons a (pCons b p) = pCons h t"
    with l have False by simp
  }
  then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"
    by blast
  from fundamental_theorem_of_algebra_alt[OF th] show ?thesis
    by auto
qed

lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<longleftrightarrow> p \<noteq> 0"
  by (metis poly_all_0_iff_0)

lemma basic_cqe_conv3:
  fixes p q :: "complex poly"
  assumes l: "p \<noteq> 0"
  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<longleftrightarrow> \<not> (pCons a p) dvd (q ^ psize p)"
proof -
  from l have dp: "degree (pCons a p) = psize p"
    by (simp add: psize_def)
  from nullstellensatz_univariate[of "pCons a p" q] l
  show ?thesis
    by (metis dp pCons_eq_0_iff)
qed

lemma basic_cqe_conv4:
  fixes p q :: "complex poly"
  assumes h: "\<And>x. poly (q ^ n) x = poly r x"
  shows "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
proof -
  from h have "poly (q ^ n) = poly r"
    by auto
  then have "(q ^ n) = r"
    by (simp add: poly_eq_poly_eq_iff)
  then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"
    by simp
qed

lemma poly_const_conv:
  fixes x :: "'a::comm_ring_1"
  shows "poly [:c:] x = y \<longleftrightarrow> c = y"
  by simp

end