Theory Fundamental_Theorem_Algebra

(*  Title:      HOL/Computational_Algebra/Fundamental_Theorem_Algebra.thy
    Author:     Amine Chaieb, TU Muenchen
*)

section ‹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  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 "m. m > 0  (z. norm z  r  norm (poly p z)  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: "z. norm z  r  norm (poly cs z)  m"
    by blast
  let ?k = " 1 + norm c + ¦r * m¦"
  have kp: "?k > 0"
    using abs_ge_zero[of "r*m"] norm_ge_zero[of c] by arith
  have "norm (poly (pCons c cs) z)  ?k" if H: "norm z  r" for z
  proof -
    from m H have th: "norm (poly cs z)  m"
      by blast
    from H have rp: "r  0"
      using norm_ge_zero[of z] by arith
    have "norm (poly (pCons c cs) z)  norm c + norm (z * poly cs z)"
      using norm_triangle_ineq[of c "z* poly cs z"] by simp
    also have "  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 "  ?k"
      by simp
    finally show ?thesis .
  qed
  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  'a  'a poly"
  where "offset_poly p h = fold_coeffs (λa 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  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  p = 0"
  by (induct p arbitrary: a) (simp, force)

lemma offset_poly_eq_0_iff: "offset_poly p h = 0  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  p = 0"
  unfolding psize_def by simp

lemma poly_offset:
  fixes p :: "'a::comm_ring_1 poly"
  shows "q. psize q = psize p  (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 "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: "x. P x"
    and bz: "z. x. P x  x < z"
  shows "s::real. y. (x. P x  y < x)  y < s"
proof
  from bz have "bdd_above (Collect P)"
    by (force intro: less_imp_le)
  then show "y. (x. P x  y < x)  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  cmod (z - 1) < 1  cmod (z + 𝗂) < 1  cmod (z - 𝗂) < 1"
proof -
  obtain x y where z: "z = Complex x y "
    by (cases z) auto
  from md z have xy: "x2 + y2 = 1"
    by (simp add: cmod_def)
  have False if "cmod (z + 1)  1" "cmod (z - 1)  1" "cmod (z + 𝗂)  1" "cmod (z - 𝗂)  1"
  proof -
    from that z xy have "2 * x  1" "2 * x  -1" "2 * y  1" "2 * y  -1"
      by (simp_all add: cmod_def power2_eq_square algebra_simps)
    then have "¦2 * x¦  1" "¦2 * y¦  1"
      by simp_all
    then have "¦2 * x¦2  12" "¦2 * y¦2  12"
      by - (rule power_mono, simp, simp)+
    then have th0: "4 * x2  1" "4 * y2  1"
      by (simp_all add: power_mult_distrib)
    from add_mono[OF th0] xy show ?thesis
      by simp
  qed
  then show ?thesis
    unfolding linorder_not_le[symmetric] by blast
qed

text ‹Hence we can always reduce modulus of 1 + b z^n› if nonzero›
lemma reduce_poly_simple:
  assumes b: "b  0"
    and n: "n  0"
  shows "z. cmod (1 + b * z^n) < 1"
  using n
proof (induct n rule: nat_less_induct)
  fix n
  assume IH: "m<n. m  0  (z. cmod (1 + b * z ^ m) < 1)"
  assume n: "n  0"
  let ?P = "λz n. cmod (1 + b * z ^ n) < 1"
  show "z. ?P z n"
  proof cases
    assume "even n"
    then have "m. n = 2 * m"
      by presburger
    then obtain m where m: "n = 2 * m"
      by blast
    from n m have "m  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)
    then show ?thesis ..
  next
    assume "odd n"
    then have "m. n = Suc (2 * m)"
      by presburger+
    then obtain m where m: "n = Suc (2 * m)"
      by blast
    have th0: "cmod (complex_of_real (cmod b) / b) = 1"
      using b by (simp add: norm_divide)
    from unimodular_reduce_norm[OF th0] ‹odd n
    have "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 + 𝗂) < 1")
      apply (cases "even m")
      apply (rule_tac x="𝗂" in exI)
      apply (simp add: m power_mult)
      apply (rule_tac x="- 𝗂" in exI)
      apply (simp add: m power_mult)
      apply (cases "even m")
      apply (rule_tac x="- 𝗂" in exI)
      apply (simp add: m power_mult)
      apply (auto simp add: m power_mult)
      apply (rule_tac x="𝗂" 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 ‹odd n, 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)  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_left_less_imp_less[OF th4 th3]
    have "?P ?w n" unfolding th1 .
    then show ?thesis ..
  qed
qed

text ‹Bolzano-Weierstrass type property for closed disc in complex plane.›

lemma metric_bound_lemma: "cmod (x - y)  ¦Re x - Re y¦ + ¦Im x - Im y¦"
  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: "n. cmod (s n)  r"
  shows "f z. strict_mono (f :: nat  nat)  (e >0. N. n  N. cmod (s (f n) - z) < e)"
proof -
  from seq_monosub[of "Re  s"]
  obtain f where f: "strict_mono f" "monoseq (λn. Re (s (f n)))"
    unfolding o_def by blast
  from seq_monosub[of "Im  s  f"]
  obtain g where g: "strict_mono g" "monoseq (λn. Im (s (f (g n))))"
    unfolding o_def by blast
  let ?h = "f  g"
  from r[rule_format, of 0] have rp: "r  0"
    using norm_ge_zero[of "s 0"] by arith
  have th: "n. r + 1  ¦Re (s n)¦"
  proof
    fix n
    from abs_Re_le_cmod[of "s n"] r[rule_format, of n]
    show "¦Re (s n)¦  r + 1" by arith
  qed
  have conv1: "convergent (λn. 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: "n. r + 1  ¦Im (s n)¦"
  proof
    fix n
    from abs_Im_le_cmod[of "s n"] r[rule_format, of n]
    show "¦Im (s n)¦  r + 1"
      by arith
  qed

  have conv2: "convergent (λn. 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 (λn. Re (s (f n))) x"
    by blast
  then have x: "r>0. n0. nn0. ¦Re (s (f n)) - x¦ < r"
    unfolding LIMSEQ_iff real_norm_def .

  from conv2[unfolded convergent_def] obtain y where "LIMSEQ (λn. Im (s (f (g n)))) y"
    by blast
  then have y: "r>0. n0. nn0. ¦Im (s (f (g n))) - y¦ < r"
    unfolding LIMSEQ_iff real_norm_def .
  let ?w = "Complex x y"
  from f(1) g(1) have hs: "strict_mono ?h"
    unfolding strict_mono_def by auto
  have "N. nN. cmod (s (?h n) - ?w) < e" if "e > 0" for e
  proof -
    from that have e2: "e/2 > 0"
      by simp
    from x[rule_format, OF e2] y[rule_format, OF e2]
    obtain N1 N2 where N1: "nN1. ¦Re (s (f n)) - x¦ < e / 2"
      and N2: "nN2. ¦Im (s (f (g n))) - y¦ < e / 2"
      by blast
    have "cmod (s (?h n) - ?w) < e" if "n  N1 + N2" for n
    proof -
      from that have nN1: "g n  N1" and nN2: "n  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]]
      show ?thesis
        using metric_bound_lemma[of "s (f (g n))" ?w] by simp
    qed
    then show ?thesis by blast
  qed
  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 "d >0. w. 0 < norm (w - z)  norm (w - z) < d  norm (poly p w - poly p z) < e"
proof -
  obtain q where q: "degree q = degree p" "poly q x = poly p (z + x)" for x
  proof
    show "degree (offset_poly p z) = degree p"
      by (rule degree_offset_poly)
    show "x. poly (offset_poly p z) x = poly p (z + x)"
      by (rule poly_offset_poly)
  qed
  have th: "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" "norm z  1  norm (poly cs z)  m" for z
      by blast
    from ep m(1) have em0: "e/m > 0"
      by (simp add: field_simps)
    have one0: "1 > (0::real)"
      by arith
    from field_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 field_lbound_gt_zero[OF one0 em0]], clarsimp simp add: norm_mult)
      fix d w
      assume H: "d > 0" "d < 1" "d < e/m" "w  z" "norm (w - z) < d"
      then have d1: "norm (w-z)  1" "d  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)  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: "z. w. cmod w  r  cmod (poly p z)  cmod (poly p w)"
proof -
  show ?thesis
  proof (cases "r  0")
    case False
    then show ?thesis
      by (metis norm_ge_zero order.trans)
  next
    case True
    then have "cmod 0  r  cmod (poly p 0) = - (- cmod (poly p 0))"
      by simp
    then have mth1: "x z. cmod z  r  cmod (poly p z) = - x"
      by blast
    have False if "cmod z  r" "cmod (poly p z) = - x" "¬ x < 1" for x z
    proof -
      from that have "- x < 0 "
        by arith
      with that(2) norm_ge_zero[of "poly p z"] show ?thesis
        by simp
    qed
    then have mth2: "z. x. (z. cmod z  r  cmod (poly p z) = - x)  x < z"
      by blast
    from real_sup_exists[OF mth1 mth2] obtain s where
      s: "y. (x. (z. cmod z  r  cmod (poly p z) = - x)  y < x)  y < s"
      by blast
    let ?m = "- s"
    have s1[unfolded minus_minus]:
      "(z x. cmod z  r  - (- cmod (poly p z)) < y)  ?m < y" for y
      using s[rule_format, of "-y"]
      unfolding minus_less_iff[of y] equation_minus_iff by blast
    from s1[of ?m] have s1m: "z x. cmod z  r  cmod (poly p z)  ?m"
      by auto
    have "z. cmod z  r  cmod (poly p z) < - s + 1 / real (Suc n)" for n
      using s1[rule_format, of "?m + 1/real (Suc n)"] by simp
    then have th: "n. z. cmod z  r  cmod (poly p z) < - s + 1 / real (Suc n)" ..
    from choice[OF th] obtain g where
        g: "n. cmod (g n)  r" "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: "strict_mono (f :: nat  nat)" "e>0. N. nN. cmod (g (f n) - z) < e"
      by blast
    {
      fix w
      assume wr: "cmod w  r"
      let ?e = "¦cmod (poly p z) - ?m¦"
      {
        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" "w. 0<cmod (w - z) cmod(w - z) < d  cmod(poly p w - poly p z) < ?e/2"
          by blast
        have th1: "cmod(poly p w - poly p z) < ?e / 2" if w: "cmod (w - z) < d" for w
          using d(2)[rule_format, of w] w e by (cases "w = z") simp_all
        from fz(2) d(1) obtain N1 where N1: "nN1. 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
        have th0: "a < e2  ¦b - m¦ < e2  2 * e2  ¦b - m¦ + a  False"
          for a b e2 m :: real
          by arith
        have ath: "m  x  x < m + e  ¦x - m¦ < e" for m x e :: real
          by arith
        from s1m[OF g(1)[rule_format]] have th31: "?m  cmod(poly p (g (f (N1 + N2))))" .
        from seq_suble[OF fz(1), of "N1 + N2"]
        have th00: "real (Suc (N1 + N2))  real (Suc (f (N1 + N2)))"
          by simp
        have th000: "0  (1::real)" "(1::real)  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)))  ?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: "¦cmod (poly p (g (f (N1 + N2)))) - ?m¦ < ?e/2"
          by arith
        have ath2: "¦a - b¦  c  ¦b - m¦  ¦a - m¦ + c" for a b c m :: real
          by arith
        have th22: "¦cmod (poly p (g (f (N1 + N2)))) - cmod (poly p z)¦ 
            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) 
            ¦cmod(poly p (g (f (N1 + N2)))) - ?m¦ + 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)  cmod (poly p w)"
        by simp
    }
    then show ?thesis by blast
  qed
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  0"
  shows "r. z. r  norm z  d  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: "z. r  norm z  d + norm a  norm (poly (pCons c cs) z)"
      by blast
    let ?r = "1 + ¦r¦"
    have "d  norm (poly (pCons a (pCons c cs)) z)" if "1 + ¦r¦  norm z" for z
    proof -
      have r0: "r  norm z"
        using that by arith
      from r[rule_format, OF r0] have th0: "d + norm a  1 * norm(poly (pCons c cs) z)"
        by arith
      from that have z1: "norm z  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  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  norm (poly (pCons a (pCons c cs)) z)"
        by (simp add: algebra_simps)
      from th1 th2 show ?thesis
        by arith
    qed
    then show ?thesis by blast
  next
    case True
    with pCons.prems have c0: "c  0"
      by simp
    have "d  norm (poly (pCons a (pCons c cs)) z)"
      if h: "(¦d¦ + norm a) / norm c  norm z" for z :: 'a
    proof -
      from c0 have "norm c > 0"
        by simp
      from h c0 have th0: "¦d¦ + norm a  norm (z * c)"
        by (simp add: field_simps norm_mult)
      have ath: "mzh mazh ma. mzh  mazh + ma  ¦d¦ + ma  mzh  d  mazh"
        by arith
      from norm_diff_ineq[of "z * c" a] have th1: "norm (z * c)  norm (a + z * c) + norm a"
        by (simp add: algebra_simps)
      from ath[OF th1 th0] show ?thesis
        using True by simp
    qed
    then show ?thesis by blast
  qed
qed

text ‹Hence polynomial's modulus attains its minimum somewhere.›
lemma poly_minimum_modulus: "z.w. cmod (poly p z)  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: "cmod (poly (pCons c cs) 0)  cmod (poly (pCons c cs) z)"
      if "r  cmod z" for z
      by blast
    have ath: "z r. r  cmod z  cmod z  ¦r¦"
      by arith
    from poly_minimum_modulus_disc[of "¦r¦" "pCons c cs"]
    obtain v where v: "cmod (poly (pCons c cs) v)  cmod (poly (pCons c cs) w)"
      if "cmod w  ¦r¦" for w
      by blast
    have "cmod (poly (pCons c cs) v)  cmod (poly (pCons c cs) z)" if z: "r  cmod z" for z
      using v[of 0] r[OF z] by simp
    with 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  (x y. f x = f y)"

lemma nonconstant_length: "¬ constant (poly p)  psize p  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: "¬ (z. z  0  poly p z = (0::'a::idom))"
  shows "k a q. a  0  Suc (psize q + k) = psize p  (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)
      apply 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])
      apply (auto simp: False)
      done
  qed
qed

lemma poly_decompose:
  assumes nc: "¬ constant (poly p)"
  shows "k a q. a  (0::'a::idom)  k  0 
               psize q + k + 1 = psize p 
              (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)
  have "¬ (z. z  0  poly cs z = 0)"
  proof
    assume "z. z  0  poly cs z = 0"
    then have "poly (pCons c cs) x = poly (pCons c cs) y" for x y
      by (cases "x = 0") auto
    with pCons.prems show False
      by (auto simp add: constant_def)
  qed
  from poly_decompose_lemma[OF this]
  show ?case
    apply clarsimp
    apply (rule_tac x="k+1" in exI)
    apply (rule_tac x="a" in exI)
    apply simp
    apply (rule_tac x="q" in exI)
    apply (auto simp add: psize_def split: if_splits)
    done
qed

text ‹Fundamental theorem of algebra›

lemma fundamental_theorem_of_algebra:
  assumes nc: "¬ constant (poly p)"
  shows "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 = "z. ?p z = 0"

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

  show ?ths
  proof (cases "?p c = 0")
    case True
    then show ?thesis by blast
  next
    case False
    from poly_offset[of p c] obtain q where q: "psize q = psize p" "x. poly q x = ?p (c + x)"
      by blast
    have False if h: "constant (poly q)"
    proof -
      from q(2) have th: "x. poly q (x - c) = ?p x"
        by auto
      have "?p x = ?p y" for x y
      proof -
        from th have "?p x = poly q (x - c)"
          by auto
        also have " = poly q (y - c)"
          using h unfolding constant_def by blast
        also have " = ?p y"
          using th by auto
        finally show ?thesis .
      qed
      with less(2) show ?thesis
        unfolding constant_def by blast
    qed
    then have qnc: "¬ constant (poly q)"
      by blast
    from q(2) have pqc0: "?p c = poly q 0"
      by simp
    from c pqc0 have cq0: "w. cmod (poly q 0)  cmod (?p w)"
      by simp
    let ?a0 = "poly q 0"
    from False pqc0 have a00: "?a0  0"
      by simp
    from a00 have qr: "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)
    have False if h: "x y. poly ?r x = poly ?r y"
    proof -
      have "poly q x = poly q y" for x y
      proof -
        from qr[rule_format, of x] have "poly q x = poly ?r x * ?a0"
          by auto
        also have " = poly ?r y * ?a0"
          using h by simp
        also have " = poly q y"
          using qr[rule_format, of y] by simp
        finally show ?thesis .
      qed
      with qnc show ?thesis
        unfolding constant_def by blast
    qed
    then have rnc: "¬ constant (poly ?r)"
      unfolding constant_def by blast
    from qr[rule_format, of 0] a00 have r01: "poly ?r 0 = 1"
      by auto
    have mrmq_eq: "cmod (poly ?r w) < 1  cmod (poly q w) < cmod ?a0" for w
    proof -
      have "cmod (poly ?r w) < 1  cmod (poly q w / ?a0) < 1"
        using qr[rule_format, of w] a00 by (simp add: divide_inverse ac_simps)
      also have "  cmod (poly q w) < cmod ?a0"
        using a00 unfolding norm_divide by (simp add: field_simps)
      finally show ?thesis .
    qed
    from poly_decompose[OF rnc] obtain k a s where
      kas: "a  0" "k  0" "psize s + k + 1 = psize ?r"
        "z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
    have "w. cmod (poly ?r w) < 1"
    proof (cases "psize p = k + 1")
      case True
      with kas(3) lgqr[symmetric] q(1) have s0: "s = 0"
        by auto
      have hth[symmetric]: "cmod (poly ?r w) = cmod (1 + a * w ^ k)" for w
        using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)
      from reduce_poly_simple[OF kas(1,2)] show ?thesis
        unfolding hth by blast
    next
      case False note kn = this
      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
        by simp
      have th01: "¬ 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" "z. cmod z  cmod w  cmod (poly s z)  m" by blast
      have w0: "w  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 field_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 " = 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))  ¦1 - t^k¦ + cmod (?w^k * ?w * poly s ?w)"
        unfolding norm_of_real by simp
      have ath: "x t::real. 0  x  x < t  t  1  ¦1 - t¦ + x < 1"
        by arith
      have "t * cmod w  1 * cmod w"
        apply (rule mult_mono)
        using t(1,2)
        apply auto
        done
      then have tw: "cmod ?w  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 simp
      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)  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  1"
        by auto
      from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] th120 th121]
      have th12: "¦1 - t^k¦ + 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 show ?thesis
        by blast
    qed
    with 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: "¬ (a l. a  0  l = 0  p = pCons a l)"
  shows "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
    have "¬ constant (poly (pCons c cs))"
    proof
      assume nc: "constant (poly (pCons c cs))"
      from nc[unfolded constant_def, rule_format, of 0]
      have "w. w  0  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" "z. z. cmod z  1  cmod (poly ds z)  m" by blast
          have dm: "cmod d / m > 0"
            using False m(1) by (simp add: field_simps)
          from field_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  0" "cmod ?x  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)  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)  cmod d"
            by auto
          with cth show ?thesis
            by blast
        qed
      qed
      then show False
        using pCons.prems False by blast
    qed
    then show ?thesis
      by (rule fundamental_theorem_of_algebra)
  qed
qed


subsection ‹Nullstellensatz, degrees and divisibility of polynomials›

lemma nullstellensatz_lemma:
  fixes p :: "complex poly"
  assumes "x. poly p x = 0  poly q x = 0"
    and "degree p = n"
    and "n  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: "m<n. p q.
                 (x. poly p x = (0::complex)  poly q x = 0) 
                 degree p = m  m  0  p dvd (q ^ m)"
    and pq0: "x. poly p x = 0  poly q x = 0"
    and dpn: "degree p = n"
    and n0: "n  0"
  from dpn n0 have pne: "p  0" by auto
  show "p dvd (q ^ n)"
  proof (cases "a. poly p a = 0")
    case True
    then obtain a where a: "poly p a = 0" ..
    have ?thesis if oa: "order a p  0"
    proof -
      let ?op = "order a p"
      from pne have ap: "([:- a, 1:] ^ ?op) dvd p" "¬ [:- a, 1:] ^ (Suc ?op) dvd p"
        using order by blast+
      note oop = order_degree[OF pne, unfolded dpn]
      show ?thesis
      proof (cases "q = 0")
        case True
        with n0 show ?thesis by (simp add: power_0_left)
      next
        case False
        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  0"
          using s pne by auto
        show ?thesis
        proof (cases "degree s = 0")
          case True
          then obtain k where kpn: "s = [:k:]"
            by (cases s) (auto split: if_splits)
          from sne kpn have k: "k  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 show ?thesis
            unfolding dvd_def by blast
        next
          case False
          with sne dpn s oa have dsn: "degree s < n"
            apply auto
            apply (erule ssubst)
            apply (simp add: degree_mult_eq degree_linear_power)
            done
          have "poly r x = 0" if h: "poly s x = 0" for x
          proof -
            have xa: "x  a"
            proof
              assume "x = a"
              from h[unfolded this 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] show False
                by blast
            qed
            from h have "poly p x = 0"
              by (subst s) simp
            with pq0 have "poly q x = 0"
              by blast
            with r xa show ?thesis
              by auto
          qed
          with IH[rule_format, OF dsn, of s r] False have "s dvd (r ^ (degree s))"
            by blast
          then obtain u where u: "r ^ (degree s) = s * u" ..
          then have u': "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 show ?thesis
            unfolding dvd_def by blast
        qed
      qed
    qed
    then show ?thesis
      using a order_root pne by blast
  next
    case False
    with fundamental_theorem_of_algebra_alt[of p]
    obtain c where ccs: "c  0" "p = pCons c 0"
      by blast
    then have pp: "poly p x = c" for x
      by simp
    let ?w = "[:1/c:] * (q ^ n)"
    from ccs have "(q ^ n) = (p * ?w)"
      by simp
    then show ?thesis
      unfolding dvd_def by blast
  qed
qed

lemma nullstellensatz_univariate:
  "(x. poly p x = (0::complex)  poly q x = 0) 
    p dvd (q ^ (degree p))  (p = 0  q = 0)"
proof -
  consider "p = 0" | "p  0" "degree p = 0" | n where "p  0" "degree p = Suc n"
    by (cases "degree p") auto
  then show ?thesis
  proof cases
    case p: 1
    then have eq: "(x. poly p x = (0::complex)  poly q x = 0)  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 p have False by simp
    }
    with eq p show ?thesis by blast
  next
    case dp: 2
    then obtain k where k: "p = [:k:]" "k  0"
      by (cases p) (simp split: if_splits)
    then have th1: "x. poly p x  0"
      by simp
    from k dp(2) have "q ^ (degree p) = p * [:1/k:]"
      by simp
    then have th2: "p dvd (q ^ (degree p))" ..
    from dp(1) th1 th2 show ?thesis
      by blast
  next
    case dp: 3
    have False if dvd: "p dvd (q ^ (Suc n))" and h: "poly p x = 0" "poly q x  0" for x
    proof -
      from dvd obtain u where u: "q ^ (Suc n) = p * u" ..
      from h have "poly (q ^ (Suc n)) x  0"
        by simp
      with u h(1) show ?thesis
        by (simp only: poly_mult) simp
    qed
    with dp nullstellensatz_lemma[of p q "degree p"] show ?thesis
      by auto
  qed
qed

text ‹Useful lemma›
lemma constant_degree:
  fixes p :: "'a::{idom,ring_char_0} poly"
  shows "constant (poly p)  degree p = 0" (is "?lhs = ?rhs")
proof
  show ?rhs if ?lhs
  proof -
    from that[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 ?thesis
      by simp
  qed
  show ?lhs if ?rhs
  proof -
    from that obtain k where "p = [:k:]"
      by (cases p) (simp split: if_splits)
    then show ?thesis
      unfolding constant_def by auto
  qed
qed

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  poly (pCons 0 p) x = 0"
  by simp

lemma poly_cancel_eq_conv:
  fixes x :: "'a::field"
  shows "x = 0  a  0  y = 0  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  0"
  shows "p dvd q  q = 0" (is "?lhs  ?rhs")
proof
  assume ?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  0"
    from l q0 have "degree p  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  0"
    and pp': "p dvd p'"
    and qrp': "smult a q - p' = r"
  shows "p dvd q  p dvd r" (is "?lhs  ?rhs")
proof
  from pp' obtain t where t: "p' = p * t" ..
  show ?rhs if ?lhs
  proof -
    from that 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 ?thesis ..
  qed
  show ?lhs if ?rhs
  proof -
    from that 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 ?thesis ..
  qed
qed

lemma basic_cqe_conv1:
  "(x. poly p x = 0  poly 0 x  0)  False"
  "(x. poly 0 x  0)  False"
  "(x. poly [:c:] x  0)  c  0"
  "(x. poly 0 x = 0)  True"
  "(x. poly [:c:] x = 0)  c = 0"
  by simp_all

lemma basic_cqe_conv2:
  assumes l: "p  0"
  shows "x. poly (pCons a (pCons b p)) x = (0::complex)"
proof -
  have False if "h  0" "t = 0" and "pCons a (pCons b p) = pCons h t" for h t
    using l that by simp
  then have th: "¬ ( h t. h  0  t = 0  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: "(x. poly p x  (0::complex))  p  0"
  by (metis poly_all_0_iff_0)

lemma basic_cqe_conv3:
  fixes p q :: "complex poly"
  assumes l: "p  0"
  shows "(x. poly (pCons a p) x = 0  poly q x  0)  ¬ (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: "x. poly (q ^ n) x = poly r x"
  shows "p dvd (q ^ n)  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)  p dvd r"
    by simp
qed

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

end