(* Author: Amine Chaieb, TU Muenchen *)

 header{*Fundamental Theorem of Algebra*}

 theory Fundamental_Theorem_Algebra

 imports Polynomial Complex_Main

 begin

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

 text{* The triangle inequality for cmod *}

 lemma complex_mod_triangle_sub: "cmod w \<le> cmod (w + z) + norm z"

   using complex_mod_triangle_ineq2[of "w + z" "-z"] by auto

 subsection {* Basic lemmas about polynomials *}

 lemma poly_bound_exists:

   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"

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

 proof (induct p)

   case 0

   then show ?case by (rule exI[where x=1]) simp

 next

   case (pCons c cs)

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

     by blast

   let ?k = " 1 + norm c + \<bar>r * m\<bar>"

   have kp: "?k > 0"

     using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith

   {

     fix z :: 'a

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

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

       by blast

     from H have rp: "r \<ge> 0"

       using norm_ge_zero[of z] by arith

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

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

     also have "\<dots> \<le> norm c + r * m"

       using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]

       by (simp add: norm_mult)

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

       by simp

     finally have "norm (poly (pCons c cs) z) \<le> ?k" .

   }

   with kp show ?case by blast

 qed

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

 definition offset_poly :: "'a::comm_semiring_0 poly \<Rightarrow> 'a \<Rightarrow> 'a poly"

   where "offset_poly p h = fold_coeffs (\<lambda>a q. smult h q + pCons a q) p 0"

 lemma offset_poly_0: "offset_poly 0 h = 0"

   by (simp add: offset_poly_def)

 lemma offset_poly_pCons:

   "offset_poly (pCons a p) h =

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

   by (cases "p = 0 \<and> a = 0") (auto simp add: offset_poly_def)

 lemma offset_poly_single: "offset_poly [:a:] h = [:a:]"

   by (simp add: offset_poly_pCons offset_poly_0)

 lemma poly_offset_poly: "poly (offset_poly p h) x = poly p (h + x)"

   apply (induct p)

   apply (simp add: offset_poly_0)

   apply (simp add: offset_poly_pCons algebra_simps)

   done

 lemma offset_poly_eq_0_lemma: "smult c p + pCons a p = 0 \<Longrightarrow> p = 0"

   by (induct p arbitrary: a) (simp, force)

 lemma offset_poly_eq_0_iff: "offset_poly p h = 0 \<longleftrightarrow> p = 0"

   apply (safe intro!: offset_poly_0)

   apply (induct p)

   apply simp

   apply (simp add: offset_poly_pCons)

   apply (frule offset_poly_eq_0_lemma, simp)

   done

 lemma degree_offset_poly: "degree (offset_poly p h) = degree p"

   apply (induct p)

   apply (simp add: offset_poly_0)

   apply (case_tac "p = 0")

   apply (simp add: offset_poly_0 offset_poly_pCons)

   apply (simp add: offset_poly_pCons)

   apply (subst degree_add_eq_right)

   apply (rule le_less_trans [OF degree_smult_le])

   apply (simp add: offset_poly_eq_0_iff)

   apply (simp add: offset_poly_eq_0_iff)

   done

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

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

   unfolding psize_def by simp

 lemma poly_offset:

   fixes p :: "'a::comm_ring_1 poly"

   shows "\<exists>q. psize q = psize p \<and> (\<forall>x. poly q x = poly p (a + x))"

 proof (intro exI conjI)

   show "psize (offset_poly p a) = psize p"

     unfolding psize_def

     by (simp add: offset_poly_eq_0_iff degree_offset_poly)

   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"

     by (simp add: poly_offset_poly)

 qed

 text{* An alternative useful formulation of completeness of the reals *}

 lemma real_sup_exists:

   assumes ex: "\<exists>x. P x"

     and bz: "\<exists>z. \<forall>x. P x \<longrightarrow> x < z"

   shows "\<exists>s::real. \<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < s"

 proof

   from bz have "bdd_above (Collect P)"

     by (force intro: less_imp_le)

   then show "\<forall>y. (\<exists>x. P x \<and> y < x) \<longleftrightarrow> y < Sup (Collect P)"

     using ex bz by (subst less_cSup_iff) auto

 qed

 subsection {* Fundamental theorem of algebra *}

 lemma  unimodular_reduce_norm:

   assumes md: "cmod z = 1"

   shows "cmod (z + 1) < 1 \<or> cmod (z - 1) < 1 \<or> cmod (z + ii) < 1 \<or> cmod (z - ii) < 1"

 proof -

   obtain x y where z: "z = Complex x y "

     by (cases z) auto

   from md z have xy: "x\<^sup>2 + y\<^sup>2 = 1"

     by (simp add: cmod_def)

   {

     assume C: "cmod (z + 1) \<ge> 1" "cmod (z - 1) \<ge> 1" "cmod (z + ii) \<ge> 1" "cmod (z - ii) \<ge> 1"

     from C z xy have "2 * x \<le> 1" "2 * x \<ge> -1" "2 * y \<le> 1" "2 * y \<ge> -1"

       by (simp_all add: cmod_def power2_eq_square algebra_simps)

     then have "abs (2 * x) \<le> 1" "abs (2 * y) \<le> 1"

       by simp_all

     then have "(abs (2 * x))\<^sup>2 \<le> 1\<^sup>2" "(abs (2 * y))\<^sup>2 \<le> 1\<^sup>2"

       by - (rule power_mono, simp, simp)+

     then have th0: "4 * x\<^sup>2 \<le> 1" "4 * y\<^sup>2 \<le> 1"

       by (simp_all add: power_mult_distrib)

     from add_mono[OF th0] xy have False by simp

   }

   then show ?thesis

     unfolding linorder_not_le[symmetric] by blast

 qed

 text{* Hence we can always reduce modulus of @{text "1 + b z^n"} if nonzero *}

 lemma reduce_poly_simple:

   assumes b: "b \<noteq> 0"

     and n: "n \<noteq> 0"

   shows "\<exists>z. cmod (1 + b * z^n) < 1"

   using n

 proof (induct n rule: nat_less_induct)

   fix n

   assume IH: "\<forall>m<n. m \<noteq> 0 \<longrightarrow> (\<exists>z. cmod (1 + b * z ^ m) < 1)"

   assume n: "n \<noteq> 0"

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

   {

     assume e: "even n"

     then have "\<exists>m. n = 2 * m"

       by presburger

     then obtain m where m: "n = 2 * m"

       by blast

     from n m have "m \<noteq> 0" "m < n"

       by presburger+

     with IH[rule_format, of m] obtain z where z: "?P z m"

       by blast

     from z have "?P (csqrt z) n"

       by (simp add: m power_mult power2_csqrt)

     then have "\<exists>z. ?P z n" ..

   }

   moreover

   {

     assume o: "odd n"

     have th0: "cmod (complex_of_real (cmod b) / b) = 1"

       using b by (simp add: norm_divide)

     from o have "\<exists>m. n = Suc (2 * m)"

       by presburger+

     then obtain m where m: "n = Suc (2 * m)"

       by blast

     from unimodular_reduce_norm[OF th0] o

     have "\<exists>v. cmod (complex_of_real (cmod b) / b + v^n) < 1"

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

       apply (rule_tac x="1" in exI)

       apply simp

       apply (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")

       apply (rule_tac x="-1" in exI)

       apply simp

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

       apply (cases "even m")

       apply (rule_tac x="ii" in exI)

       apply (simp add: m power_mult)

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

       apply (simp add: m power_mult)

       apply (cases "even m")

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

       apply (simp add: m power_mult)

       apply (auto simp add: m power_mult)

       apply (rule_tac x="ii" in exI)

       apply (auto simp add: m power_mult)

       done

     then obtain v where v: "cmod (complex_of_real (cmod b) / b + v^n) < 1"

       by blast

     let ?w = "v / complex_of_real (root n (cmod b))"

     from odd_real_root_pow[OF o, of "cmod b"]

     have th1: "?w ^ n = v^n / complex_of_real (cmod b)"

       by (simp add: power_divide of_real_power[symmetric])

     have th2:"cmod (complex_of_real (cmod b) / b) = 1"

       using b by (simp add: norm_divide)

     then have th3: "cmod (complex_of_real (cmod b) / b) \<ge> 0"

       by simp

     have th4: "cmod (complex_of_real (cmod b) / b) *

         cmod (1 + b * (v ^ n / complex_of_real (cmod b))) <

         cmod (complex_of_real (cmod b) / b) * 1"

       apply (simp only: norm_mult[symmetric] distrib_left)

       using b v

       apply (simp add: th2)

       done

     from mult_less_imp_less_left[OF th4 th3]

     have "?P ?w n" unfolding th1 .

     then have "\<exists>z. ?P z n" ..

   }

   ultimately show "\<exists>z. ?P z n" by blast

 qed

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

 lemma metric_bound_lemma: "cmod (x - y) \<le> \<bar>Re x - Re y\<bar> + \<bar>Im x - Im y\<bar>"

   using real_sqrt_sum_squares_triangle_ineq[of "Re x - Re y" 0 0 "Im x - Im y"]

   unfolding cmod_def by simp

 lemma bolzano_weierstrass_complex_disc:

   assumes r: "\<forall>n. cmod (s n) \<le> r"

   shows "\<exists>f z. subseq f \<and> (\<forall>e >0. \<exists>N. \<forall>n \<ge> N. cmod (s (f n) - z) < e)"

 proof-

   from seq_monosub[of "Re \<circ> s"]

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

     unfolding o_def by blast

   from seq_monosub[of "Im \<circ> s \<circ> f"]

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

     unfolding o_def by blast

   let ?h = "f \<circ> g"

   from r[rule_format, of 0] have rp: "r \<ge> 0"

     using norm_ge_zero[of "s 0"] by arith

   have th: "\<forall>n. r + 1 \<ge> \<bar>Re (s n)\<bar>"

   proof

     fix n

     from abs_Re_le_cmod[of "s n"] r[rule_format, of n]

     show "\<bar>Re (s n)\<bar> \<le> r + 1" by arith

   qed

   have conv1: "convergent (\<lambda>n. Re (s (f n)))"

     apply (rule Bseq_monoseq_convergent)

     apply (simp add: Bseq_def)

     apply (metis gt_ex le_less_linear less_trans order.trans th)

     apply (rule f(2))

     done

   have th: "\<forall>n. r + 1 \<ge> \<bar>Im (s n)\<bar>"

   proof

     fix n

     from abs_Im_le_cmod[of "s n"] r[rule_format, of n]

     show "\<bar>Im (s n)\<bar> \<le> r + 1"

       by arith

   qed

   have conv2: "convergent (\<lambda>n. Im (s (f (g n))))"

     apply (rule Bseq_monoseq_convergent)

     apply (simp add: Bseq_def)

     apply (metis gt_ex le_less_linear less_trans order.trans th)

     apply (rule g(2))

     done

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

     by blast

   then have x: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Re (s (f n)) - x\<bar> < r"

     unfolding LIMSEQ_iff real_norm_def .

   from conv2[unfolded convergent_def] obtain y where "LIMSEQ (\<lambda>n. Im (s (f (g n)))) y"

     by blast

   then have y: "\<forall>r>0. \<exists>n0. \<forall>n\<ge>n0. \<bar>Im (s (f (g n))) - y\<bar> < r"

     unfolding LIMSEQ_iff real_norm_def .

   let ?w = "Complex x y"

   from f(1) g(1) have hs: "subseq ?h"

     unfolding subseq_def by auto

   {

     fix e :: real

     assume ep: "e > 0"

     then have e2: "e/2 > 0"

       by simp

     from x[rule_format, OF e2] y[rule_format, OF e2]

     obtain N1 N2 where N1: "\<forall>n\<ge>N1. \<bar>Re (s (f n)) - x\<bar> < e / 2"

       and N2: "\<forall>n\<ge>N2. \<bar>Im (s (f (g n))) - y\<bar> < e / 2"

       by blast

     {

       fix n

       assume nN12: "n \<ge> N1 + N2"

       then have nN1: "g n \<ge> N1" and nN2: "n \<ge> N2"

         using seq_suble[OF g(1), of n] by arith+

       from add_strict_mono[OF N1[rule_format, OF nN1] N2[rule_format, OF nN2]]

       have "cmod (s (?h n) - ?w) < e"

         using metric_bound_lemma[of "s (f (g n))" ?w] by simp

     }

     then have "\<exists>N. \<forall>n\<ge>N. cmod (s (?h n) - ?w) < e"

       by blast

   }

   with hs show ?thesis by blast

 qed

 text{* Polynomial is continuous. *}

 lemma poly_cont:

   fixes p :: "'a::{comm_semiring_0,real_normed_div_algebra} poly"

   assumes ep: "e > 0"

   shows "\<exists>d >0. \<forall>w. 0 < norm (w - z) \<and> norm (w - z) < d \<longrightarrow> norm (poly p w - poly p z) < e"

 proof -

   obtain q where q: "degree q = degree p" "\<And>x. poly q x = poly p (z + x)"

   proof

     show "degree (offset_poly p z) = degree p"

       by (rule degree_offset_poly)

     show "\<And>x. poly (offset_poly p z) x = poly p (z + x)"

       by (rule poly_offset_poly)

   qed

   have th: "\<And>w. poly q (w - z) = poly p w"

     using q(2)[of "w - z" for w] by simp

   show ?thesis unfolding th[symmetric]

   proof (induct q)

     case 0

     then show ?case

       using ep by auto

   next

     case (pCons c cs)

     from poly_bound_exists[of 1 "cs"]

     obtain m where m: "m > 0" "\<And>z. norm z \<le> 1 \<Longrightarrow> norm (poly cs z) \<le> m"

       by blast

     from ep m(1) have em0: "e/m > 0"

       by (simp add: field_simps)

     have one0: "1 > (0::real)"

       by arith

     from real_lbound_gt_zero[OF one0 em0]

     obtain d where d: "d > 0" "d < 1" "d < e / m"

       by blast

     from d(1,3) m(1) have dm: "d * m > 0" "d * m < e"

       by (simp_all add: field_simps)

     show ?case

     proof (rule ex_forward[OF real_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)

       fix d w

       assume H: "d > 0" "d < 1" "d < e/m" "w \<noteq> z" "norm (w - z) < d"

       then have d1: "norm (w-z) \<le> 1" "d \<ge> 0"

         by simp_all

       from H(3) m(1) have dme: "d*m < e"

         by (simp add: field_simps)

       from H have th: "norm (w - z) \<le> d"

         by simp

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

       show "norm (w - z) * norm (poly cs (w - z)) < e"

         by simp

     qed

   qed

 qed

 text{* Hence a polynomial attains minimum on a closed disc

   in the complex plane. *}

 lemma poly_minimum_modulus_disc: "\<exists>z. \<forall>w. cmod w \<le> r \<longrightarrow> cmod (poly p z) \<le> cmod (poly p w)"

 proof -

   {

     assume "\<not> r \<ge> 0"

     then have ?thesis

       by (metis norm_ge_zero order.trans)

   }

   moreover

   {

     assume rp: "r \<ge> 0"

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

       by simp

     then have mth1: "\<exists>x z. cmod z \<le> r \<and> cmod (poly p z) = - x"

       by blast

     {

       fix x z

       assume H: "cmod z \<le> r" "cmod (poly p z) = - x" "\<not> x < 1"

       then have "- x < 0 "

         by arith

       with H(2) norm_ge_zero[of "poly p z"] have False

         by simp

     }

     then have mth2: "\<exists>z. \<forall>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<longrightarrow> x < z"

       by blast

     from real_sup_exists[OF mth1 mth2] obtain s where

       s: "\<forall>y. (\<exists>x. (\<exists>z. cmod z \<le> r \<and> cmod (poly p z) = - x) \<and> y < x) \<longleftrightarrow> y < s" by blast

     let ?m = "- s"

     {

       fix y

       from s[rule_format, of "-y"]

       have "(\<exists>z x. cmod z \<le> r \<and> - (- cmod (poly p z)) < y) \<longleftrightarrow> ?m < y"

         unfolding minus_less_iff[of y ] equation_minus_iff by blast

     }

     note s1 = this[unfolded minus_minus]

     from s1[of ?m] have s1m: "\<And>z x. cmod z \<le> r \<Longrightarrow> cmod (poly p z) \<ge> ?m"

       by auto

     {

       fix n :: nat

       from s1[rule_format, of "?m + 1/real (Suc n)"]

       have "\<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)"

         by simp

     }

     then have th: "\<forall>n. \<exists>z. cmod z \<le> r \<and> cmod (poly p z) < - s + 1 / real (Suc n)" ..

     from choice[OF th] obtain g where

         g: "\<forall>n. cmod (g n) \<le> r" "\<forall>n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"

       by blast

     from bolzano_weierstrass_complex_disc[OF g(1)]

     obtain f z where fz: "subseq f" "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. cmod (g (f n) - z) < e"

       by blast

     {

       fix w

       assume wr: "cmod w \<le> r"

       let ?e = "\<bar>cmod (poly p z) - ?m\<bar>"

       {

         assume e: "?e > 0"

         then have e2: "?e/2 > 0"

           by simp

         from poly_cont[OF e2, of z p] obtain d where

             d: "d > 0" "\<forall>w. 0<cmod (w - z)\<and> cmod(w - z) < d \<longrightarrow> cmod(poly p w - poly p z) < ?e/2"

           by blast

         {

           fix w

           assume w: "cmod (w - z) < d"

           have "cmod(poly p w - poly p z) < ?e / 2"

             using d(2)[rule_format, of w] w e by (cases "w = z") simp_all

         }

         note th1 = this

         from fz(2) d(1) obtain N1 where N1: "\<forall>n\<ge>N1. cmod (g (f n) - z) < d"

           by blast

         from reals_Archimedean2[of "2/?e"] obtain N2 :: nat where N2: "2/?e < real N2"

           by blast

         have th2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"

           using N1[rule_format, of "N1 + N2"] th1 by simp

         {

           fix a b e2 m :: real

           have "a < e2 \<Longrightarrow> \<bar>b - m\<bar> < e2 \<Longrightarrow> 2 * e2 \<le> \<bar>b - m\<bar> + a \<Longrightarrow> False"

             by arith

         }

         note th0 = this

         have ath: "\<And>m x e::real. m \<le> x \<Longrightarrow> x < m + e \<Longrightarrow> \<bar>x - m\<bar> < e"

           by arith

         from s1m[OF g(1)[rule_format]] have th31: "?m \<le> cmod(poly p (g (f (N1 + N2))))" .

         from seq_suble[OF fz(1), of "N1 + N2"]

         have th00: "real (Suc (N1 + N2)) \<le> real (Suc (f (N1 + N2)))"

           by simp

         have th000: "0 \<le> (1::real)" "(1::real) \<le> 1" "real (Suc (N1 + N2)) > 0"

           using N2 by auto

         from frac_le[OF th000 th00]

         have th00: "?m + 1 / real (Suc (f (N1 + N2))) \<le> ?m + 1 / real (Suc (N1 + N2))"

           by simp

         from g(2)[rule_format, of "f (N1 + N2)"]

         have th01:"cmod (poly p (g (f (N1 + N2)))) < - s + 1 / real (Suc (f (N1 + N2)))" .

         from order_less_le_trans[OF th01 th00]

         have th32: "cmod (poly p (g (f (N1 + N2)))) < ?m + (1/ real(Suc (N1 + N2)))" .

         from N2 have "2/?e < real (Suc (N1 + N2))"

           by arith

         with e2 less_imp_inverse_less[of "2/?e" "real (Suc (N1 + N2))"]

         have "?e/2 > 1/ real (Suc (N1 + N2))"

           by (simp add: inverse_eq_divide)

         with ath[OF th31 th32]

         have thc1: "\<bar>cmod (poly p (g (f (N1 + N2)))) - ?m\<bar> < ?e/2"

           by arith

         have ath2: "\<And>a b c m::real. \<bar>a - b\<bar> \<le> c \<Longrightarrow> \<bar>b - m\<bar> \<le> \<bar>a - m\<bar> + c"

           by arith

         have th22: "\<bar>cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)\<bar> \<le>

             cmod (poly p (g (f (N1 + N2))) - poly p z)"

           by (simp add: norm_triangle_ineq3)

         from ath2[OF th22, of ?m]

         have thc2: "2 * (?e/2) \<le>

             \<bar>cmod(poly p (g (f (N1 + N2)))) - ?m\<bar> + cmod (poly p (g (f (N1 + N2))) - poly p z)"

           by simp

         from th0[OF th2 thc1 thc2] have False .

       }

       then have "?e = 0"

         by auto

       then have "cmod (poly p z) = ?m"

         by simp

       with s1m[OF wr] have "cmod (poly p z) \<le> cmod (poly p w)"

         by simp

     }

     then have ?thesis by blast

   }

   ultimately show ?thesis by blast

 qed

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

 lemma poly_infinity:

   fixes p:: "'a::{comm_semiring_0,real_normed_div_algebra} poly"

   assumes ex: "p \<noteq> 0"

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

   using ex

 proof (induct p arbitrary: a d)

   case 0

   then show ?case by simp

 next

   case (pCons c cs a d)

   show ?case

   proof (cases "cs = 0")

     case False

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

       by blast

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

     {

       fix z :: 'a

       assume h: "1 + \<bar>r\<bar> \<le> norm z"

       have r0: "r \<le> norm z"

         using h by arith

       from r[rule_format, OF r0] have th0: "d + norm a \<le> 1 * norm(poly (pCons c cs) z)"

         by arith

       from h have z1: "norm z \<ge> 1"

         by arith

       from order_trans[OF th0 mult_right_mono[OF z1 norm_ge_zero[of "poly (pCons c cs) z"]]]

       have th1: "d \<le> norm(z * poly (pCons c cs) z) - norm a"

         unfolding norm_mult by (simp add: algebra_simps)

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

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

         by (simp add: algebra_simps)

       from th1 th2 have "d \<le> norm (poly (pCons a (pCons c cs)) z)"

         by arith

     }

     then show ?thesis by blast

   next

     case True

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

       by simp

     {

       fix z :: 'a

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

       from c0 have "norm c > 0"

         by simp

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

         by (simp add: field_simps norm_mult)

       have ath: "\<And>mzh mazh ma. mzh \<le> mazh + ma \<Longrightarrow> \<bar>d\<bar> + ma \<le> mzh \<Longrightarrow> d \<le> mazh"

         by arith

       from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c) \<le> norm (a + z * c) + norm a"

         by (simp add: algebra_simps)

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

         using True by simp

     }

     then show ?thesis by blast

   qed

 qed

 text {* Hence polynomial's modulus attains its minimum somewhere. *}

 lemma poly_minimum_modulus: "\<exists>z.\<forall>w. cmod (poly p z) \<le> cmod (poly p w)"

 proof (induct p)

   case 0

   then show ?case by simp

 next

   case (pCons c cs)

   show ?case

   proof (cases "cs = 0")

     case False

     from poly_infinity[OF False, of "cmod (poly (pCons c cs) 0)" c]

     obtain r where r: "\<And>z. r \<le> cmod z \<Longrightarrow> cmod (poly (pCons c cs) 0) \<le> cmod (poly (pCons c cs) z)"

       by blast

     have ath: "\<And>z r. r \<le> cmod z \<or> cmod z \<le> \<bar>r\<bar>"

       by arith

     from poly_minimum_modulus_disc[of "\<bar>r\<bar>" "pCons c cs"]

     obtain v where v: "\<And>w. cmod w \<le> \<bar>r\<bar> \<Longrightarrow> cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) w)"

       by blast

     {

       fix z

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

       from v[of 0] r[OF z] have "cmod (poly (pCons c cs) v) \<le> cmod (poly (pCons c cs) z)"

         by simp

     }

     note v0 = this

     from v0 v ath[of r] show ?thesis

       by blast

   next

     case True

     with pCons.hyps show ?thesis by simp

   qed

 qed

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

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

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

   by (induct p) (auto simp: constant_def psize_def)

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

   by (simp add: poly_monom)

 text {* Decomposition of polynomial, skipping zero coefficients

   after the first.  *}

 lemma poly_decompose_lemma:

   assumes nz: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly p z = (0::'a::idom))"

   shows "\<exists>k a q. a \<noteq> 0 \<and> Suc (psize q + k) = psize p \<and> (\<forall>z. poly p z = z^k * poly (pCons a q) z)"

   unfolding psize_def

   using nz

 proof (induct p)

   case 0

   then show ?case by simp

 next

   case (pCons c cs)

   show ?case

   proof (cases "c = 0")

     case True

     from pCons.hyps pCons.prems True show ?thesis

       apply (auto)

       apply (rule_tac x="k+1" in exI)

       apply (rule_tac x="a" in exI, clarsimp)

       apply (rule_tac x="q" in exI)

       apply auto

       done

   next

     case False

     show ?thesis

       apply (rule exI[where x=0])

       apply (rule exI[where x=c], auto simp add: False)

       done

   qed

 qed

 lemma poly_decompose:

   assumes nc: "\<not> constant (poly p)"

   shows "\<exists>k a q. a \<noteq> (0::'a::idom) \<and> k \<noteq> 0 \<and>

                psize q + k + 1 = psize p \<and>

               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"

   using nc

 proof (induct p)

   case 0

   then show ?case

     by (simp add: constant_def)

 next

   case (pCons c cs)

   {

     assume C: "\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0"

     {

       fix x y

       from C have "poly (pCons c cs) x = poly (pCons c cs) y"

         by (cases "x = 0") auto

     }

     with pCons.prems have False

       by (auto simp add: constant_def)

   }

   then have th: "\<not> (\<forall>z. z \<noteq> 0 \<longrightarrow> poly cs z = 0)" ..

   from poly_decompose_lemma[OF th]

   show ?case

     apply clarsimp

     apply (rule_tac x="k+1" in exI)

     apply (rule_tac x="a" in exI)

     apply simp

     apply (rule_tac x="q" in exI)

     apply (auto simp add: psize_def split: if_splits)

     done

 qed

 text{* Fundamental theorem of algebra *}

 lemma fundamental_theorem_of_algebra:

   assumes nc: "\<not> constant (poly p)"

   shows "\<exists>z::complex. poly p z = 0"

   using nc

 proof (induct "psize p" arbitrary: p rule: less_induct)

   case less

   let ?p = "poly p"

   let ?ths = "\<exists>z. ?p z = 0"

   from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .

   from poly_minimum_modulus obtain c where c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)"

     by blast

   show ?ths

   proof (cases "?p c = 0")

     case True

     then show ?thesis by blast

   next

     case False

     note pc0 = this

     from poly_offset[of p c] obtain q where q: "psize q = psize p" "\<forall>x. poly q x = ?p (c + x)"

       by blast

     {

       assume h: "constant (poly q)"

       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x"

         by auto

       {

         fix x y

         from th have "?p x = poly q (x - c)"

           by auto

         also have "\<dots> = poly q (y - c)"

           using h unfolding constant_def by blast

         also have "\<dots> = ?p y"

           using th by auto

         finally have "?p x = ?p y" .

       }

       with less(2) have False

         unfolding constant_def by blast

     }

     then have qnc: "\<not> constant (poly q)"

       by blast

     from q(2) have pqc0: "?p c = poly q 0"

       by simp

     from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)"

       by simp

     let ?a0 = "poly q 0"

     from pc0 pqc0 have a00: "?a0 \<noteq> 0"

       by simp

     from a00 have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"

       by simp

     let ?r = "smult (inverse ?a0) q"

     have lgqr: "psize q = psize ?r"

       using a00

       unfolding psize_def degree_def

       by (simp add: poly_eq_iff)

     {

       assume h: "\<And>x y. poly ?r x = poly ?r y"

       {

         fix x y

         from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"

           by auto

         also have "\<dots> = poly ?r y * ?a0"

           using h by simp

         also have "\<dots> = poly q y"

           using qr[rule_format, of y] by simp

         finally have "poly q x = poly q y" .

       }

       with qnc have False

         unfolding constant_def by blast

     }

     then have rnc: "\<not> constant (poly ?r)"

       unfolding constant_def by blast

     from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"

       by auto

     {

       fix w

       have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w / ?a0) < 1"

         using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)

       also have "\<dots> \<longleftrightarrow> cmod (poly q w) < cmod ?a0"

         using a00 unfolding norm_divide by (simp add: field_simps)

       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .

     }

     note mrmq_eq = this

     from poly_decompose[OF rnc] obtain k a s where

       kas: "a \<noteq> 0" "k \<noteq> 0" "psize s + k + 1 = psize ?r"

         "\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast

     {

       assume "psize p = k + 1"

       with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"

         by auto

       {

         fix w

         have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"

           using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)

       }

       note hth = this [symmetric]

       from reduce_poly_simple[OF kas(1,2)] have "\<exists>w. cmod (poly ?r w) < 1"

         unfolding hth by blast

     }

     moreover

     {

       assume kn: "psize p \<noteq> k + 1"

       from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"

         by simp

       have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"

         unfolding constant_def poly_pCons poly_monom

         using kas(1)

         apply simp

         apply (rule exI[where x=0])

         apply (rule exI[where x=1])

         apply simp

         done

       from kas(1) kas(2) have th02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"

         by (simp add: psize_def degree_monom_eq)

       from less(1) [OF k1n [simplified th02] th01]

       obtain w where w: "1 + w^k * a = 0"

         unfolding poly_pCons poly_monom

         using kas(2) by (cases k) (auto simp add: algebra_simps)

       from poly_bound_exists[of "cmod w" s] obtain m where

         m: "m > 0" "\<forall>z. cmod z \<le> cmod w \<longrightarrow> cmod (poly s z) \<le> m" by blast

       have w0: "w \<noteq> 0"

         using kas(2) w by (auto simp add: power_0_left)

       from w have "(1 + w ^ k * a) - 1 = 0 - 1"

         by simp

       then have wm1: "w^k * a = - 1"

         by simp

       have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"

         using norm_ge_zero[of w] w0 m(1)

         by (simp add: inverse_eq_divide zero_less_mult_iff)

       with real_lbound_gt_zero[OF zero_less_one] obtain t where

         t: "t > 0" "t < 1" "t < inverse (cmod w ^ (k + 1) * m)" by blast

       let ?ct = "complex_of_real t"

       let ?w = "?ct * w"

       have "1 + ?w^k * (a + ?w * poly s ?w) = 1 + ?ct^k * (w^k * a) + ?w^k * ?w * poly s ?w"

         using kas(1) by (simp add: algebra_simps power_mult_distrib)

       also have "\<dots> = complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w"

         unfolding wm1 by simp

       finally have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) =

         cmod (complex_of_real (1 - t^k) + ?w^k * ?w * poly s ?w)"

         by metis

       with norm_triangle_ineq[of "complex_of_real (1 - t^k)" "?w^k * ?w * poly s ?w"]

       have th11: "cmod (1 + ?w^k * (a + ?w * poly s ?w)) \<le> \<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w)"

         unfolding norm_of_real by simp

       have ath: "\<And>x t::real. 0 \<le> x \<Longrightarrow> x < t \<Longrightarrow> t \<le> 1 \<Longrightarrow> \<bar>1 - t\<bar> + x < 1"

         by arith

       have "t * cmod w \<le> 1 * cmod w"

         apply (rule mult_mono)

         using t(1,2)

         apply auto

         done

       then have tw: "cmod ?w \<le> cmod w"

         using t(1) by (simp add: norm_mult)

       from t inv0 have "t * (cmod w ^ (k + 1) * m) < 1"

         by (simp add: field_simps)

       with zero_less_power[OF t(1), of k] have th30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k * 1"

         by (metis comm_mult_strict_left_mono)

       have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"

         using w0 t(1)

         by (simp add: algebra_simps power_mult_distrib norm_power norm_mult)

       then have "cmod (?w^k * ?w * poly s ?w) \<le> t^k * (t* (cmod w ^ (k + 1) * m))"

         using t(1,2) m(2)[rule_format, OF tw] w0

         by auto

       with th30 have th120: "cmod (?w^k * ?w * poly s ?w) < t^k"

         by simp

       from power_strict_mono[OF t(2), of k] t(1) kas(2) have th121: "t^k \<le> 1"

         by auto

       from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]

       have th12: "\<bar>1 - t^k\<bar> + cmod (?w^k * ?w * poly s ?w) < 1" .

       from th11 th12 have "cmod (1 + ?w^k * (a + ?w * poly s ?w)) < 1"

         by arith

       then have "cmod (poly ?r ?w) < 1"

         unfolding kas(4)[rule_format, of ?w] r01 by simp

       then have "\<exists>w. cmod (poly ?r w) < 1"

         by blast

     }

     ultimately have cr0_contr: "\<exists>w. cmod (poly ?r w) < 1"

       by blast

     from cr0_contr cq0 q(2) show ?thesis

       unfolding mrmq_eq not_less[symmetric] by auto

   qed

 qed

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

 lemma fundamental_theorem_of_algebra_alt:

   assumes nc: "\<not> (\<exists>a l. a \<noteq> 0 \<and> l = 0 \<and> p = pCons a l)"

   shows "\<exists>z. poly p z = (0::complex)"

   using nc

 proof (induct p)

   case 0

   then show ?case by simp

 next

   case (pCons c cs)

   show ?case

   proof (cases "c = 0")

     case True

     then show ?thesis by auto

   next

     case False

     {

       assume nc: "constant (poly (pCons c cs))"

       from nc[unfolded constant_def, rule_format, of 0]

       have "\<forall>w. w \<noteq> 0 \<longrightarrow> poly cs w = 0" by auto

       then have "cs = 0"

       proof (induct cs)

         case 0

         then show ?case by simp

       next

         case (pCons d ds)

         show ?case

         proof (cases "d = 0")

           case True

           then show ?thesis using pCons.prems pCons.hyps by simp

         next

           case False

           from poly_bound_exists[of 1 ds] obtain m where

             m: "m > 0" "\<forall>z. \<forall>z. cmod z \<le> 1 \<longrightarrow> cmod (poly ds z) \<le> m" by blast

           have dm: "cmod d / m > 0"

             using False m(1) by (simp add: field_simps)

           from real_lbound_gt_zero[OF dm zero_less_one] obtain x where

             x: "x > 0" "x < cmod d / m" "x < 1" by blast

           let ?x = "complex_of_real x"

           from x have cx: "?x \<noteq> 0"  "cmod ?x \<le> 1"

             by simp_all

           from pCons.prems[rule_format, OF cx(1)]

           have cth: "cmod (?x*poly ds ?x) = cmod d"

             by (simp add: eq_diff_eq[symmetric])

           from m(2)[rule_format, OF cx(2)] x(1)

           have th0: "cmod (?x*poly ds ?x) \<le> x*m"

             by (simp add: norm_mult)

           from x(2) m(1) have "x * m < cmod d"

             by (simp add: field_simps)

           with th0 have "cmod (?x*poly ds ?x) \<noteq> cmod d"

             by auto

           with cth show ?thesis

             by blast

         qed

       qed

     }

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

       using pCons.prems False by blast

     from fundamental_theorem_of_algebra[OF nc] show ?thesis .

   qed

 qed

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

 lemma nullstellensatz_lemma:

   fixes p :: "complex poly"

   assumes "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"

     and "degree p = n"

     and "n \<noteq> 0"

   shows "p dvd (q ^ n)"

   using assms

 proof (induct n arbitrary: p q rule: nat_less_induct)

   fix n :: nat

   fix p q :: "complex poly"

   assume IH: "\<forall>m<n. \<forall>p q.

                  (\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longrightarrow>

                  degree p = m \<longrightarrow> m \<noteq> 0 \<longrightarrow> p dvd (q ^ m)"

     and pq0: "\<forall>x. poly p x = 0 \<longrightarrow> poly q x = 0"

     and dpn: "degree p = n"

     and n0: "n \<noteq> 0"

   from dpn n0 have pne: "p \<noteq> 0" by auto

   let ?ths = "p dvd (q ^ n)"

   {

     fix a

     assume a: "poly p a = 0"

     {

       assume oa: "order a p \<noteq> 0"

       let ?op = "order a p"

       from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "\<not> [:- a, 1:] ^ (Suc ?op) dvd p"

         using order by blast+

       note oop = order_degree[OF pne, unfolded dpn]

       {

         assume q0: "q = 0"

         then have ?ths using n0

           by (simp add: power_0_left)

       }

       moreover

       {

         assume q0: "q \<noteq> 0"

         from pq0[rule_format, OF a, unfolded poly_eq_0_iff_dvd]

         obtain r where r: "q = [:- a, 1:] * r" by (rule dvdE)

         from ap(1) obtain s where s: "p = [:- a, 1:] ^ ?op * s"

           by (rule dvdE)

         have sne: "s \<noteq> 0" using s pne by auto

         {

           assume ds0: "degree s = 0"

           from ds0 obtain k where kpn: "s = [:k:]"

             by (cases s) (auto split: if_splits)

           from sne kpn have k: "k \<noteq> 0" by simp

           let ?w = "([:1/k:] * ([:-a,1:] ^ (n - ?op))) * (r ^ n)"

           have "q ^ n = p * ?w"

             apply (subst r)

             apply (subst s)

             apply (subst kpn)

             using k oop [of a]

             apply (subst power_mult_distrib)

             apply simp

             apply (subst power_add [symmetric])

             apply simp

             done

           then have ?ths

             unfolding dvd_def by blast

         }

         moreover

         {

           assume ds0: "degree s \<noteq> 0"

           from ds0 sne dpn s oa

             have dsn: "degree s < n"

               apply auto

               apply (erule ssubst)

               apply (simp add: degree_mult_eq degree_linear_power)

               done

             {

               fix x assume h: "poly s x = 0"

               {

                 assume xa: "x = a"

                 from h[unfolded xa poly_eq_0_iff_dvd] obtain u where u: "s = [:- a, 1:] * u"

                   by (rule dvdE)

                 have "p = [:- a, 1:] ^ (Suc ?op) * u"

                   apply (subst s)

                   apply (subst u)

                   apply (simp only: power_Suc ac_simps)

                   done

                 with ap(2)[unfolded dvd_def] have False

                   by blast

               }

               note xa = this

               from h have "poly p x = 0"

                 by (subst s) simp

               with pq0 have "poly q x = 0"

                 by blast

               with r xa have "poly r x = 0"

                 by auto

             }

             note impth = this

             from IH[rule_format, OF dsn, of s r] impth ds0

             have "s dvd (r ^ (degree s))"

               by blast

             then obtain u where u: "r ^ (degree s) = s * u" ..

             then have u': "\<And>x. poly s x * poly u x = poly r x ^ degree s"

               by (simp only: poly_mult[symmetric] poly_power[symmetric])

             let ?w = "(u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"

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

               apply -

               apply (subst s)

               apply (subst r)

               apply (simp only: power_mult_distrib)

               apply (subst mult.assoc [where b=s])

               apply (subst mult.assoc [where a=u])

               apply (subst mult.assoc [where b=u, symmetric])

               apply (subst u [symmetric])

               apply (simp add: ac_simps power_add [symmetric])

               done

             then have ?ths

               unfolding dvd_def by blast

         }

         ultimately have ?ths by blast

       }

       ultimately have ?ths by blast

     }

     then have ?ths using a order_root pne by blast

   }

   moreover

   {

     assume exa: "\<not> (\<exists>a. poly p a = 0)"

     from fundamental_theorem_of_algebra_alt[of p] exa

     obtain c where ccs: "c \<noteq> 0" "p = pCons c 0"

       by blast

     then have pp: "\<And>x. poly p x = c"

       by simp

     let ?w = "[:1/c:] * (q ^ n)"

     from ccs have "(q ^ n) = (p * ?w)"

       by simp

     then have ?ths

       unfolding dvd_def by blast

   }

   ultimately show ?ths by blast

 qed

 lemma nullstellensatz_univariate:

   "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow>

     p dvd (q ^ (degree p)) \<or> (p = 0 \<and> q = 0)"

 proof -

   {

     assume pe: "p = 0"

     then have eq: "(\<forall>x. poly p x = (0::complex) \<longrightarrow> poly q x = 0) \<longleftrightarrow> q = 0"

       by (auto simp add: poly_all_0_iff_0)

     {

       assume "p dvd (q ^ (degree p))"

       then obtain r where r: "q ^ (degree p) = p * r" ..

       from r pe have False by simp

     }

     with eq pe have ?thesis by blast

   }

   moreover

   {

     assume pe: "p \<noteq> 0"

     {

       assume dp: "degree p = 0"

       then obtain k where k: "p = [:k:]" "k \<noteq> 0" using pe

         by (cases p) (simp split: if_splits)

       then have th1: "\<forall>x. poly p x \<noteq> 0"

         by simp

       from k dp have "q ^ (degree p) = p * [:1/k:]"

         by (simp add: one_poly_def)

       then have th2: "p dvd (q ^ (degree p))" ..

       from th1 th2 pe have ?thesis

         by blast

     }

     moreover

     {

       assume dp: "degree p \<noteq> 0"

       then obtain n where n: "degree p = Suc n "

         by (cases "degree p") auto

       {

         assume "p dvd (q ^ (Suc n))"

         then obtain u where u: "q ^ (Suc n) = p * u" ..

         {

           fix x

           assume h: "poly p x = 0" "poly q x \<noteq> 0"

           then have "poly (q ^ (Suc n)) x \<noteq> 0"

             by simp

           then have False using u h(1)

             by (simp only: poly_mult) simp

         }

       }

       with n nullstellensatz_lemma[of p q "degree p"] dp

       have ?thesis by auto

     }

     ultimately have ?thesis by blast

   }

   ultimately show ?thesis by blast

 qed

 text {* Useful lemma *}

 lemma constant_degree:

   fixes p :: "'a::{idom,ring_char_0} poly"

   shows "constant (poly p) \<longleftrightarrow> degree p = 0" (is "?lhs = ?rhs")

 proof

   assume l: ?lhs

   from l[unfolded constant_def, rule_format, of _ "0"]

   have th: "poly p = poly [:poly p 0:]"

     by auto

   then have "p = [:poly p 0:]"

     by (simp add: poly_eq_poly_eq_iff)

   then have "degree p = degree [:poly p 0:]"

     by simp

   then show ?rhs

     by simp

 next

   assume r: ?rhs

   then obtain k where "p = [:k:]"

     by (cases p) (simp split: if_splits)

   then show ?lhs

     unfolding constant_def by auto

 qed

 lemma divides_degree:

   assumes pq: "p dvd (q:: complex poly)"

   shows "degree p \<le> degree q \<or> q = 0"

   by (metis dvd_imp_degree_le pq)

 text {* Arithmetic operations on multivariate polynomials. *}

 lemma mpoly_base_conv:

   fixes x :: "'a::comm_ring_1"

   shows "0 = poly 0 x" "c = poly [:c:] x" "x = poly [:0,1:] x"

   by simp_all

 lemma mpoly_norm_conv:

   fixes x :: "'a::comm_ring_1"

   shows "poly [:0:] x = poly 0 x" "poly [:poly 0 y:] x = poly 0 x"

   by simp_all

 lemma mpoly_sub_conv:

   fixes x :: "'a::comm_ring_1"

   shows "poly p x - poly q x = poly p x + -1 * poly q x"

   by simp

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

   by simp

 lemma poly_cancel_eq_conv:

   fixes x :: "'a::field"

   shows "x = 0 \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> y = 0 \<longleftrightarrow> a * y - b * x = 0"

   by auto

 lemma poly_divides_pad_rule:

   fixes p:: "('a::comm_ring_1) poly"

   assumes pq: "p dvd q"

   shows "p dvd (pCons 0 q)"

 proof -

   have "pCons 0 q = q * [:0,1:]" by simp

   then have "q dvd (pCons 0 q)" ..

   with pq show ?thesis by (rule dvd_trans)

 qed

 lemma poly_divides_conv0:

   fixes p:: "'a::field poly"

   assumes lgpq: "degree q < degree p"

     and lq: "p \<noteq> 0"

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

 proof

   assume r: ?rhs

   then have "q = p * 0" by simp

   then show ?lhs ..

 next

   assume l: ?lhs

   show ?rhs

   proof (cases "q = 0")

     case True

     then show ?thesis by simp

   next

     assume q0: "q \<noteq> 0"

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

       by (rule dvd_imp_degree_le)

     with lgpq show ?thesis by simp

   qed

 qed

 lemma poly_divides_conv1:

   fixes p :: "'a::field poly"

   assumes a0: "a \<noteq> 0"

     and pp': "p dvd p'"

     and qrp': "smult a q - p' = r"

   shows "p dvd q \<longleftrightarrow> p dvd r" (is "?lhs \<longleftrightarrow> ?rhs")

 proof

   from pp' obtain t where t: "p' = p * t" ..

   {

     assume l: ?lhs

     then obtain u where u: "q = p * u" ..

     have "r = p * (smult a u - t)"

       using u qrp' [symmetric] t by (simp add: algebra_simps)

     then show ?rhs ..

   next

     assume r: ?rhs

     then obtain u where u: "r = p * u" ..

     from u [symmetric] t qrp' [symmetric] a0

     have "q = p * smult (1/a) (u + t)" by (simp add: algebra_simps)

     then show ?lhs ..

   }

 qed

 lemma basic_cqe_conv1:

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

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

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

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

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

   by simp_all

 lemma basic_cqe_conv2:

   assumes l: "p \<noteq> 0"

   shows "\<exists>x. poly (pCons a (pCons b p)) x = (0::complex)"

 proof -

   {

     fix h t

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

     with l have False by simp

   }

   then have th: "\<not> (\<exists> h t. h \<noteq> 0 \<and> t = 0 \<and> pCons a (pCons b p) = pCons h t)"

     by blast

   from fundamental_theorem_of_algebra_alt[OF th] show ?thesis

     by auto

 qed

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

   by (metis poly_all_0_iff_0)

 lemma basic_cqe_conv3:

   fixes p q :: "complex poly"

   assumes l: "p \<noteq> 0"

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

 proof -

   from l have dp: "degree (pCons a p) = psize p"

     by (simp add: psize_def)

   from nullstellensatz_univariate[of "pCons a p" q] l

   show ?thesis

     by (metis dp pCons_eq_0_iff)

 qed

 lemma basic_cqe_conv4:

   fixes p q :: "complex poly"

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

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

 proof -

   from h have "poly (q ^ n) = poly r"

     by auto

   then have "(q ^ n) = r"

     by (simp add: poly_eq_poly_eq_iff)

   then show "p dvd (q ^ n) \<longleftrightarrow> p dvd r"

     by simp

 qed

 lemma poly_const_conv:

   fixes x :: "'a::comm_ring_1"

   shows "poly [:c:] x = y \<longleftrightarrow> c = y"

   by simp

 end