(* 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" hence 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

      hence "cmod ?z - x \<ge> 0" "cmod ?z + x \<ge> 0" by arith+

      hence "(sqrt (x * x + y * y) + x) / 2 \<ge> 0" "(sqrt (x * x + y * y) - x) / 2 \<ge> 0" by (simp_all add: power2_eq_square) }

    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

    hence 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

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])

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

  apply (rule exI[where x = "min d1 d2 / 2"])

  by (simp add: field_simps min_def)

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 complex polynomials *}

lemma poly_bound_exists:

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

proof(induct p)

  case 0 thus ?case by (rule exI[where x=1], simp)

next

  case (pCons c cs)

  from pCons.hyps obtain m where m: "\<forall>z. cmod z \<le> r \<longrightarrow> cmod (poly cs z) \<le> m"

    by blast

  let ?k = " 1 + cmod 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

    assume H: "cmod z \<le> r"

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

    from H have rp: "r \<ge> 0" using norm_ge_zero[of z] by arith

    have "cmod (poly (pCons c cs) z) \<le> cmod c + cmod (z* poly cs z)"

      using norm_triangle_ineq[of c "z* poly cs z"] by simp

    also have "\<dots> \<le> cmod c + r*m" using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]] by (simp add: norm_mult)

    also have "\<dots> \<le> ?k" by simp

    finally have "cmod (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, 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: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = 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)

    hence "abs (2*x) \<le> 1" "abs (2*y) \<le> 1" by simp_all

    hence "(abs (2 * x))\<^sup>2 <= 1\<^sup>2" "(abs (2 * y))\<^sup>2 <= 1\<^sup>2"

      by - (rule power_mono, simp, simp)+

    hence 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 }

  thus ?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)" and n: "n \<noteq> 0"

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

  {assume e: "even n"

    hence "\<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)

    hence "\<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", rule_tac x="1" in exI, simp)

      apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1", rule_tac x="-1" in exI, simp del: minus_one add: minus_one [symmetric])

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

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

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

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

      apply (rule_tac x="ii" in exI, 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)

    hence 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 by (simp add: th2)

    from mult_less_imp_less_left[OF th4 th3]

    have "?P ?w n" unfolding th1 .

    hence "\<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) <= \<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 o s"]

  obtain f g where f: "subseq f" "monoseq (\<lambda>n. Re (s (f n)))"

    unfolding o_def by blast

  from seq_monosub[of "Im o s o f"]

  obtain g where g: "subseq g" "monoseq (\<lambda>n. Im (s(f(g n))))" unfolding o_def by blast

  let ?h = "f o 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 (rule exI[where x= "r + 1"])

    using th rp apply simp

    using f(2) .

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

    using th rp apply simp

    using g(2) .

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

    by blast

  hence  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

  hence  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 assume ep: "e > (0::real)"

    hence 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"

      hence 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 }

    hence "\<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:

  assumes ep: "e > 0"

  shows "\<exists>d >0. \<forall>w. 0 < cmod (w - z) \<and> cmod (w - z) < d \<longrightarrow> cmod (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

  {fix w

    note q(2)[of "w - z", simplified]}

  note th = this

  show ?thesis unfolding th[symmetric]

  proof(induct q)

    case 0 thus ?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. cmod z \<le> 1 \<Longrightarrow> cmod (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 mult_pos_pos)

    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" "cmod (w-z) < d"

        hence d1: "cmod (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: "cmod (w-z) \<le> d" by simp

        from mult_mono[OF th m(2)[OF d1(1)] d1(2) norm_ge_zero] dme

        show "cmod (w - z) * cmod (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" hence ?thesis unfolding linorder_not_le

      apply -

      apply (rule exI[where x=0])

      apply auto

      apply (subgoal_tac "cmod w < 0")

      apply simp

      apply arith

      done }

  moreover

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

    from rp have "cmod 0 \<le> r \<and> cmod (poly p 0) = - (- cmod (poly p 0))" by simp

    hence 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"

      hence "- 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}

    hence 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"

        hence 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)[rule_format, OF 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> abs(b - m) < e2 \<Longrightarrow> 2 * e2 <= abs(b - m) + a

          ==> False" by arith}

      note th0 = this

      have ath:

        "\<And>m x e. m <= x \<Longrightarrow>  x < m + e ==> abs(x - m::real) < 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::real) b c m. \<bar>a - b\<bar> <= c ==> \<bar>b - m\<bar> <= \<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 .}

      hence "?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 }

    hence ?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"

  unfolding cis_def

  by simp

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:

  assumes ex: "p \<noteq> 0"

  shows "\<exists>r. \<forall>z. r \<le> cmod z \<longrightarrow> d \<le> cmod (poly (pCons a p) z)"

using ex

proof(induct p arbitrary: a d)

  case (pCons c cs a d)

  {assume H: "cs \<noteq> 0"

    with pCons.hyps obtain r where r: "\<forall>z. r \<le> cmod z \<longrightarrow> d + cmod a \<le> cmod (poly (pCons c cs) z)" by blast

    let ?r = "1 + \<bar>r\<bar>"

    {fix z assume h: "1 + \<bar>r\<bar> \<le> cmod z"

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

      from r[rule_format, OF r0]

      have th0: "d + cmod a \<le> 1 * cmod(poly (pCons c cs) z)" by arith

      from h have z1: "cmod 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> cmod(z * poly (pCons c cs) z) - cmod a"

        unfolding norm_mult by (simp add: algebra_simps)

      from complex_mod_triangle_sub[of "z * poly (pCons c cs) z" a]

      have th2: "cmod(z * poly (pCons c cs) z) - cmod a \<le> cmod (poly (pCons a (pCons c cs)) z)"

        by (simp add: algebra_simps)

      from th1 th2 have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"  by arith}

    hence ?case by blast}

  moreover

  {assume cs0: "\<not> (cs \<noteq> 0)"

    with pCons.prems have c0: "c \<noteq> 0" by simp

    from cs0 have cs0': "cs = 0" by simp

    {fix z

      assume h: "(\<bar>d\<bar> + cmod a) / cmod c \<le> cmod z"

      from c0 have "cmod c > 0" by simp

      from h c0 have th0: "\<bar>d\<bar> + cmod a \<le> cmod (z*c)"

        by (simp add: field_simps norm_mult)

      have ath: "\<And>mzh mazh ma. mzh <= mazh + ma ==> abs(d) + ma <= mzh ==> d <= mazh" by arith

      from complex_mod_triangle_sub[of "z*c" a ]

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

        by (simp add: algebra_simps)

      from ath[OF th1 th0] have "d \<le> cmod (poly (pCons a (pCons c cs)) z)"

        using cs0' by simp}

    then have ?case  by blast}

  ultimately show ?case by blast

qed simp

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 (pCons c cs)

  {assume cs0: "cs \<noteq> 0"

    from poly_infinity[OF cs0, 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] have ?case by blast}

  moreover

  {assume cs0: "\<not> (cs \<noteq> 0)"

    hence th:"cs = 0" by simp

    from th pCons.hyps have ?case by simp}

  ultimately show ?case by blast

qed simp

text{* Constant function (non-syntactic characterization). *}

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

lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"

  unfolding constant_def psize_def

  apply (induct p, auto)

  done

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 thus ?case by simp

next

  case (pCons c cs)

  {assume c0: "c = 0"

    from pCons.hyps pCons.prems c0 have ?case

      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)

      by (auto)}

  moreover

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

    hence ?case apply-

      apply (rule exI[where x=0])

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

      apply (rule exI[where x=cs])

      apply auto

      done}

  ultimately show ?case by blast

qed

lemma poly_decompose:

  assumes nc: "~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 thus ?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)}

  hence 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: "~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

  {assume pc: "?p c = 0" hence ?ths by blast}

  moreover

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

    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 }

    hence 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}

    hence 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

        by (rule exI[where x=0], rule exI[where x=1], simp)

      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_down2[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)"

        apply -

        apply (rule cong[OF refl[of cmod]])

        apply assumption

        done

      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) by auto

      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"

        apply - apply (rule mult_strict_left_mono) by simp_all

      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

        apply (simp only: )

        apply auto

        done

      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)

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

  ultimately show ?ths by blast

qed

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

lemma fundamental_theorem_of_algebra_alt:

  assumes nc: "~(\<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 (pCons c cs)

  {assume "c=0" hence ?case by auto}

  moreover

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

    {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

      hence "cs = 0"

        proof(induct cs)

          case (pCons d ds)

          {assume "d=0" hence ?case using pCons.prems pCons.hyps by simp}

          moreover

          {assume d0: "d\<noteq>0"

            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 d0 m(1) by (simp add: field_simps)

            from real_down2[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  have ?case by blast}

          ultimately show ?case by blast

        qed simp}

      then have nc: "\<not> constant (poly (pCons c cs))" using pCons.prems c0

        by blast

      from fundamental_theorem_of_algebra[OF nc] have ?case .}

  ultimately show ?case by blast

qed simp

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"

        hence ?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)"

          from k oop [of a] have "q ^ n = p * ?w"

            apply -

            apply (subst r, subst s, subst kpn)

            apply (subst power_mult_distrib, simp)

            apply (subst power_add [symmetric], simp)

            done

          hence ?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"

                  by (subst s, subst u, simp only: power_Suc mult_ac)

                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 simp add: uminus_add_conv_diff)}

            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" ..

            hence 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, 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

            hence ?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

    hence ?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"

    hence 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)

      hence 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)

      hence 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"

          hence "poly (q ^ (Suc n)) x \<noteq> 0" by simp

          hence 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:]" apply - by (rule ext, simp)

  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"

apply (cases "q = 0", simp_all)

apply (erule dvd_imp_degree_le [OF pq])

done

(* Arithmetic operations on multivariate polynomials.                        *)

lemma mpoly_base_conv:

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

lemma mpoly_norm_conv:

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

lemma mpoly_sub_conv:

  "poly p (x::complex) - poly q x \<equiv> poly p x + -1 * poly q x"

  by simp

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

lemma poly_cancel_eq_conv: "p = (0::complex) \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> (q = 0) \<equiv> (a * q - b * p = 0)" apply (atomize (full)) by auto

lemma resolve_eq_raw:  "poly 0 x \<equiv> 0" "poly [:c:] x \<equiv> (c::complex)" by auto

lemma  resolve_eq_then: "(P \<Longrightarrow> (Q \<equiv> Q1)) \<Longrightarrow> (\<not>P \<Longrightarrow> (Q \<equiv> Q2))

  \<Longrightarrow> Q \<equiv> P \<and> Q1 \<or> \<not>P\<and> Q2" apply (atomize (full)) by blast

lemma poly_divides_pad_rule:

  fixes p q :: "complex poly"

  assumes pq: "p dvd q"

  shows "p dvd (pCons (0::complex) 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_pad_const_rule:

  fixes p q :: "complex poly"

  assumes pq: "p dvd q"

  shows "p dvd (smult a q)"

proof-

  have "smult a q = q * [:a:]" by simp

  then have "q dvd smult a q" ..

  with pq show ?thesis by (rule dvd_trans)

qed

lemma poly_divides_conv0:

  fixes p :: "complex poly"

  assumes lgpq: "degree q < degree p" and lq:"p \<noteq> 0"

  shows "p dvd q \<equiv> q = 0" (is "?lhs \<equiv> ?rhs")

proof-

  {assume r: ?rhs

    hence "q = p * 0" by simp

    hence ?lhs ..}

  moreover

  {assume l: ?lhs

    {assume q0: "q = 0"

      hence ?rhs by simp}

    moreover

    {assume q0: "q \<noteq> 0"

      from l q0 have "degree p \<le> degree q"

        by (rule dvd_imp_degree_le)

      with lgpq have ?rhs by simp }

    ultimately have ?rhs by blast }

  ultimately show "?lhs \<equiv> ?rhs" by - (atomize (full), blast)

qed

lemma poly_divides_conv1:

  assumes a0: "a\<noteq> (0::complex)" and pp': "(p::complex poly) dvd p'"