(* Author: Amine Chaieb, TU Muenchen *)

 header{*Fundamental Theorem of Algebra*}

 theory Fundamental_Theorem_Algebra

 imports Polynomial Complex

 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 ^ 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^2 / 4 = (x / 2) ^ 2"

       by (simp add: power2_eq_square)

     from th[of x y]

     have sq4': "sqrt (((sqrt (x * x + y * y) + x)^2 / 4)) = (sqrt (x * x + y * y) + x) / 2" "sqrt (((sqrt (x * x + y * y) - x)^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 (2^2)" 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 p h = poly_rec 0 (\<lambda>a p q. smult h q + pCons a q) p"

 lemma offset_poly_0: "offset_poly 0 h = 0"

   unfolding offset_poly_def by (simp add: poly_rec_0)

 lemma offset_poly_pCons:

   "offset_poly (pCons a p) h =

     smult h (offset_poly p h) + pCons a (offset_poly p h)"

   unfolding offset_poly_def by (simp add: poly_rec_pCons)

 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 ex bz obtain x Y where x: "P x" and Y: "\<And>x. P x \<Longrightarrow> x < Y"  by blast

   from ex have thx:"\<exists>x. x \<in> Collect P" by blast

   from bz have thY: "\<exists>Y. isUb UNIV (Collect P) Y"

     by(auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def order_le_less)

   from reals_complete[OF thx thY] obtain L where L: "isLub UNIV (Collect P) L"

     by blast

   from Y[OF x] have xY: "x < Y" .

   from L have L': "\<forall>x. P x \<longrightarrow> x \<le> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)

   from Y have Y': "\<forall>x. P x \<longrightarrow> x \<le> Y"

     apply (clarsimp, atomize (full)) by auto

   from L Y' have "L \<le> Y" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)

   {fix y

     {fix z assume z: "P z" "y < z"

       from L' z have "y < L" by auto }

     moreover

     {assume yL: "y < L" "\<forall>z. P z \<longrightarrow> \<not> y < z"

       hence nox: "\<forall>z. P z \<longrightarrow> y \<ge> z" by auto

       from nox L have "y \<ge> L" by (auto simp add: isUb_def isLub_def setge_def setle_def leastP_def Ball_def)

       with yL(1) have False  by arith}

     ultimately have "(\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < L" by blast}

   thus ?thesis by blast

 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^2 + y^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))^2 <= 1^2" "(abs (2 * y)) ^2 <= 1^2"

       by - (rule power_mono, simp, simp)+

     hence th0: "4*x^2 \<le> 1" "4*y^2 \<le> 1"

       by (simp_all  add: power2_abs 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 add: diff_minus)

       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 diff_minus)

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

       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] right_distrib)

       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)) ^ 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)) ^ 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: diff_le_eq 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: power_Suc)

     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: expand_poly_eq)

     {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_of_real 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 have "\<exists>k. s = [:k:]"

             by (cases s, simp split: if_splits)

           then obtain k where kpn: "s = [:k:]" by blast

           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 add: smult_smult)

     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"

       apply auto

       apply (rule poly_zero [THEN iffD1])

       by (rule ext, simp)

     {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_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 add: diff_minus)

 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'"

   and qrp': "smult a q - p' \<equiv> r"

   shows "p dvd q \<equiv> p dvd (r::complex poly)" (is "?lhs \<equiv> ?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 mult_smult_right)

      then have ?rhs ..}

   moreover

   {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 mult_smult_right smult_smult)

     hence ?lhs ..}

   ultimately have "?lhs = ?rhs" by blast }

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

 qed

 lemma basic_cqe_conv1:

   "(\<exists>x. poly p x = 0 \<and> poly 0 x \<noteq> 0) \<equiv> False"

   "(\<exists>x. poly 0 x \<noteq> 0) \<equiv> False"

   "(\<exists>x. poly [:c:] x \<noteq> 0) \<equiv> c\<noteq>0"

   "(\<exists>x. poly 0 x = 0) \<equiv> True"

   "(\<exists>x. poly [:c:] x = 0) \<equiv> 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)) \<equiv> True"

 proof-

   {fix h t

     assume h: "h\<noteq>0" "t=0"  "pCons a (pCons b p) = pCons h t"

     with l have False by simp}

   hence 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 "(\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)) \<equiv> True" by auto

 qed

 lemma  basic_cqe_conv_2b: "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> (p \<noteq> 0)"

 proof-

   have "p = 0 \<longleftrightarrow> poly p = poly 0"

     by (simp add: poly_zero)

   also have "\<dots> \<longleftrightarrow> (\<not> (\<exists>x. poly p x \<noteq> 0))" by (auto intro: ext)

   finally show "(\<exists>x. poly p x \<noteq> (0::complex)) \<equiv> p \<noteq> 0"

     by - (atomize (full), blast)

 qed

 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) \<equiv> \<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 "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"

     unfolding dp

     by - (atomize (full), auto)

 qed

 lemma basic_cqe_conv4:

   fixes p q :: "complex poly"

   assumes h: "\<And>x. poly (q ^ n) x \<equiv> poly r x"

   shows "p dvd (q ^ n) \<equiv> p dvd r"

 proof-

   from h have "poly (q ^ n) = poly r" by (auto intro: ext)

   then have "(q ^ n) = r" by (simp add: poly_eq_iff)

   thus "p dvd (q ^ n) \<equiv> p dvd r" by simp

 qed

 lemma pmult_Cons_Cons: "(pCons (a::complex) (pCons b p) * q = (smult a q) + (pCons 0 (pCons b p * q)))"

   by simp

 lemma elim_neg_conv: "- z \<equiv> (-1) * (z::complex)" by simp

 lemma eqT_intr: "PROP P \<Longrightarrow> (True \<Longrightarrow> PROP P )" "PROP P \<Longrightarrow> True" by blast+

 lemma negate_negate_rule: "Trueprop P \<equiv> \<not> P \<equiv> False" by (atomize (full), auto)

 lemma complex_entire: "(z::complex) \<noteq> 0 \<and> w \<noteq> 0 \<equiv> z*w \<noteq> 0" by simp

 lemma resolve_eq_ne: "(P \<equiv> True) \<equiv> (\<not>P \<equiv> False)" "(P \<equiv> False) \<equiv> (\<not>P \<equiv> True)"

   by (atomize (full)) simp_all

 lemma cqe_conv1: "poly 0 x = 0 \<longleftrightarrow> True"  by simp

 lemma cqe_conv2: "(p \<Longrightarrow> (q \<equiv> r)) \<equiv> ((p \<and> q) \<equiv> (p \<and> r))"  (is "?l \<equiv> ?r")

 proof

   assume "p \<Longrightarrow> q \<equiv> r" thus "p \<and> q \<equiv> p \<and> r" apply - apply (atomize (full)) by blast

 next

   assume "p \<and> q \<equiv> p \<and> r" "p"

   thus "q \<equiv> r" apply - apply (atomize (full)) apply blast done

 qed

 lemma poly_const_conv: "poly [:c:] (x::complex) = y \<longleftrightarrow> c = y" by simp

 end