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"
by (metis add_diff_cancel norm_triangle_ineq4)

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 "… ≤ ?k"
using mult_mono[OF H th rp norm_ge_zero[of "poly cs z"]]
by (simp add: norm_mult)
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 [simp]: "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)"
by (induct p) (auto simp add: offset_poly_0 offset_poly_pCons algebra_simps)

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 [simp]: "offset_poly p h = 0 ⟷ p = 0"
proof
show "offset_poly p h = 0 ⟹ p = 0"
proof(induction p)
case 0
then show ?case by blast
next
case (pCons a p)
then show ?case
by (metis offset_poly_eq_0_lemma offset_poly_pCons offset_poly_single)
qed
qed (simp add: offset_poly_0)

lemma degree_offset_poly [simp]: "degree (offset_poly p h) = degree p"
proof(induction p)
case 0
then show ?case
by (simp add: offset_poly_0)
next
case (pCons a p)
have "p ≠ 0 ⟹ degree (offset_poly (pCons a p) h) = Suc (degree p)"
by (metis degree_add_eq_right degree_pCons_eq degree_smult_le le_imp_less_Suc offset_poly_eq_0_iff offset_poly_pCons pCons.IH)
then show ?case
by simp
qed

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))"
by (metis degree_offset_poly offset_poly_eq_0_iff poly_offset_poly psize_def)

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: "x⇧2 + y⇧2 = 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 ≤ 1⇧2" "¦2 * y¦⇧2 ≤ 1⇧2"
by (metis abs_square_le_1 one_power2 power2_abs)+
with xy * show ?thesis
by (smt (verit, best) four_x_squared square_le_1)
qed
then show ?thesis
by force
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 obtain m where m: "n = 2 * m" and "m ≠ 0" "m < n"
using n by auto
with IH 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 0: "cmod (complex_of_real (cmod b) / b) = 1"
using b by (simp add: norm_divide)
have "∃v. cmod (complex_of_real (cmod b) / b + v^n) < 1"
proof (cases "cmod (complex_of_real (cmod b) / b + 1) < 1")
case True
then show ?thesis
by (metis power_one)
next
case F1: False
show ?thesis
proof (cases "cmod (complex_of_real (cmod b) / b - 1) < 1")
case True
with ‹odd n› show ?thesis
by (metis add_uminus_conv_diff neg_one_odd_power)
next
case F2: False
show ?thesis
proof (cases "cmod (complex_of_real (cmod b) / b + 𝗂) < 1")
case T1: True
show ?thesis
proof (cases "even m")
case True
with T1 show ?thesis
by (rule_tac x="𝗂" in exI) (simp add: m power_mult)
next
case False
with T1 show ?thesis
by (rule_tac x="- 𝗂" in exI) (simp add: m power_mult)
qed
next
case False
then have lt1: "cmod (of_real (cmod b) / b - 𝗂) < 1"
using "0" F1 F2 unimodular_reduce_norm by blast
show ?thesis
proof (cases "even m")
case True
with m lt1 show ?thesis
by (rule_tac x="- 𝗂" in exI) (simp add: power_mult)
next
case False
with m lt1 show ?thesis
by (rule_tac x="𝗂" in exI) (simp add: power_mult)
qed
qed
qed
qed
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 1: "?w ^ n = v^n / complex_of_real (cmod b)"
by (simp add: power_divide of_real_power[symmetric])
have 2:"cmod (complex_of_real (cmod b) / b) = 1"
using b by (simp add: norm_divide)
then have 3: "cmod (complex_of_real (cmod b) / b) ≥ 0"
by simp
have 4: "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: 2)
done
show ?thesis
by (metis 1 mult_left_less_imp_less[OF 4 3])
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"
have "r ≥ 0"
by (meson norm_ge_zero order_trans r)
have "∀n. r + 1 ≥ ¦Re (s n)¦"
by (smt (verit, ccfv_threshold) abs_Re_le_cmod r)
then have conv1: "convergent (λn. Re (s (f n)))"
by (metis Bseq_monoseq_convergent f(2) BseqI' real_norm_def)
have "∀n. r + 1 ≥ ¦Im (s n)¦"
by (smt (verit) abs_Im_le_cmod r)
then have conv2: "convergent (λn. Im (s (f (g n))))"
by (metis Bseq_monoseq_convergent g(2) BseqI' real_norm_def)

obtain x where  x: "∀r>0. ∃n0. ∀n≥n0. ¦Re (s (f n)) - x¦ < r"
using conv1[unfolded convergent_def] LIMSEQ_iff real_norm_def by metis
obtain y where  y: "∀r>0. ∃n0. ∀n≥n0. ¦Im (s (f (g n))) - y¦ < r"
using conv2[unfolded convergent_def] LIMSEQ_iff real_norm_def by metis
let ?w = "Complex x y"
from f(1) g(1) have hs: "strict_mono ?h"
unfolding strict_mono_def by auto
have "∃N. ∀n≥N. cmod (s (?h n) - ?w) < e" if "e > 0" for e
proof -
from that have e2: "e/2 > 0"
by simp
from x y e2
obtain N1 N2 where N1: "∀n≥N1. ¦Re (s (f n)) - x¦ < e / 2"
and N2: "∀n≥N2. ¦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+
show ?thesis
using metric_bound_lemma[of "s (f (g n))" ?w] N1 N2 nN1 nN2 by fastforce
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 "degree q = degree p" and q: "⋀w. poly p w = poly q (w - z)"
by (metis add.commute degree_offset_poly diff_add_cancel poly_offset_poly)
show ?thesis unfolding q
proof (induct q)
case 0
then show ?case
using ep by auto
next
case (pCons c cs)
obtain m where m: "m > 0" "norm z ≤ 1 ⟹ norm (poly cs z) ≤ m" for z
using poly_bound_exists[of 1 "cs"] by blast
with ep have em0: "e/m > 0"
by (simp add: field_simps)
obtain d where d: "d > 0" "d < 1" "d < e / m"
by (meson em0 field_lbound_gt_zero zero_less_one)
then have "⋀w. norm (w - z) < d ⟹ norm (w - z) * norm (poly cs (w - z)) < e"
by (smt (verit, del_insts) m mult_left_mono norm_ge_zero pos_less_divide_eq)
with d show ?case
by (force simp add: norm_mult)
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 mth1: "∃x z. cmod z ≤ r ∧ cmod (poly p z) = - x"
by (metis add.inverse_inverse norm_zero)
obtain s where s: "∀y. (∃x. (∃z. cmod z ≤ r ∧ cmod (poly p z) = - x) ∧ y < x) ⟷ y < s"
by (smt (verit, del_insts) real_sup_exists[OF mth1] norm_zero zero_less_norm_iff)

let ?m = "- s"
have s1: "(∃z. cmod z ≤ r ∧ - (- cmod (poly p z)) < y) ⟷ ?m < y" for y
by (metis add.inverse_inverse minus_less_iff s)
then have s1m: "⋀z. cmod z ≤ r ⟹ cmod (poly p z) ≥ ?m"
by force
have "∃z. cmod z ≤ r ∧ cmod (poly p z) < - s + 1 / real (Suc n)" for n
using s1[of "?m + 1/real (Suc n)"] by simp
then obtain g where g: "∀n. cmod (g n) ≤ r" "∀n. cmod (poly p (g n)) <?m + 1 /real(Suc n)"
by metis
from Bolzano_Weierstrass_complex_disc[OF g(1)]
obtain f::"nat ⇒ nat" and z where fz: "strict_mono f" "∀e>0. ∃N. ∀n≥N. 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
with poly_cont obtain d
where "d > 0" and d: "⋀w. 0<cmod (w - z)∧ cmod(w - z) < d ⟶ cmod(poly p w - poly p z) < ?e/2"
by blast
have 1: "cmod(poly p w - poly p z) < ?e / 2" if w: "cmod (w - z) < d" for w
using d[of w] w e by (cases "w = z") simp_all
from fz(2) ‹d > 0› obtain N1 where N1: "∀n≥N1. cmod (g (f n) - z) < d"
by blast
from reals_Archimedean2 obtain N2 :: nat where N2: "2/?e < real N2"
by blast
have 2: "cmod (poly p (g (f (N1 + N2))) - poly p z) < ?e/2"
using N1 1 by auto
have 0: "a < e2 ⟹ ¦b - m¦ < e2 ⟹ 2 * e2 ≤ ¦b - m¦ + a ⟹ False"
for a b e2 m :: real
by arith
from seq_suble[OF fz(1), of "N1 + N2"]
have 00: "?m + 1 / real (Suc (f (N1 + N2))) ≤ ?m + 1 / real (Suc (N1 + N2))"
by (simp add: frac_le)
from N2 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  order_less_le_trans[OF _ 00]
have 1: "¦cmod (poly p (g (f (N1 + N2)))) - ?m¦ < ?e/2"
using g s1 by (smt (verit))
with 0[OF 2] have False
by (smt (verit) field_sum_of_halves norm_triangle_ineq3)
}
then have "?e = 0"
by auto
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 "d ≤ norm(z * poly (pCons c cs) z) - norm a"
by (smt (verit, best) norm_ge_zero mult_less_cancel_right2 norm_mult r that)
with norm_diff_ineq add.commute
show ?thesis
by (metis order.trans poly_pCons)
qed
then show ?thesis by blast
next
case True
have "d ≤ norm (poly (pCons a (pCons c cs)) z)"
if "(¦d¦ + norm a) / norm c ≤ norm z" for z :: 'a
proof -
have "¦d¦ + norm a ≤ norm (z * c)"
by (metis that True norm_mult pCons.hyps(1) pos_divide_le_eq zero_less_norm_iff)
also have "… ≤ norm (a + z * c) + norm a"
by (simp add: add.commute norm_add_leD)
finally show ?thesis
using True by auto
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
from poly_minimum_modulus_disc[of "¦r¦" "pCons c cs"] show ?thesis
by (smt (verit, del_insts) order.trans linorder_linear r)
qed (use pCons.hyps in auto)
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
qed force
qed

lemma poly_decompose:
fixes p :: "'a::idom poly"
assumes nc: "¬ constant (poly p)"
shows "∃k a q. a ≠ 0 ∧ 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)"
by (smt (verit) constant_def mult_eq_0_iff pCons.prems poly_pCons)
from poly_decompose_lemma[OF this]
obtain k a q where *: "a ≠ 0 ∧
Suc (psize q + k) = psize cs ∧ (∀z. poly cs z = z ^ k * poly (pCons a q) z)"
by blast
then have "psize q + k + 2 = psize (pCons c cs)"
by (auto simp add: psize_def split: if_splits)
then show ?case
using "*" by force
qed

text ‹Fundamental theorem of algebra›

theorem 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
obtain q where q: "psize q = psize p" "∀x. poly q x = ?p (c + x)"
using poly_offset[of p c] by blast
then have qnc: "¬ constant (poly q)"
by (metis (no_types, opaque_lifting) add.commute constant_def diff_add_cancel less.prems)
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"
by (simp add: a00 psize_def)
have rnc: "¬ constant (poly ?r)"
using constant_def qnc qr by fastforce
have r01: "poly ?r 0 = 1"
by (simp add: a00)
have mrmq_eq: "cmod (poly ?r w) < 1 ⟷ cmod (poly q w) < cmod ?a0" for w
by (smt (verit, del_insts) a00 mult_less_cancel_right2 norm_mult qr zero_less_norm_iff)
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 q have s0: "s = 0"
by (simp add: lgqr)
with reduce_poly_simple kas show ?thesis
by (metis mult.commute mult.right_neutral poly_1 poly_smult r01 smult_one)
next
case False note kn = this
from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p"
by simp
have 01: "¬ constant (poly (pCons 1 (monom a (k - 1))))"
unfolding constant_def poly_pCons poly_monom
by (metis add_cancel_left_right kas(1) mult.commute mult_cancel_right2 power_one)
have 02: "k + 1 = psize (pCons 1 (monom a (k - 1)))"
using kas by (simp add: psize_def degree_monom_eq)
from less(1) [OF _ 01] k1n 02
obtain w where w: "1 + w^k * a = 0"
by (metis kas(2) mult.commute mult.left_commute poly_monom poly_pCons power_eq_if)
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 "w ≠ 0"
using kas(2) w by (auto simp add: power_0_left)
from w have wm1: "w^k * a = - 1"
by (simp add: add_eq_0_iff)
have inv0: "0 < inverse (cmod w ^ (k + 1) * m)"
by (simp add: ‹w ≠ 0› m(1))
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 11: "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 tw: "cmod ?w ≤ cmod w"
by (smt (verit) mult_le_cancel_right2 norm_ge_zero norm_mult norm_of_real t)
have "t * (cmod w ^ (k + 1) * m) < 1"
by (smt (verit, best) inv0 inverse_positive_iff_positive left_inverse mult_strict_right_mono t(3))
with zero_less_power[OF t(1), of k] have 30: "t^k * (t* (cmod w ^ (k + 1) * m)) < t^k"
by simp
have "cmod (?w^k * ?w * poly s ?w) = t^k * (t* (cmod w ^ (k + 1) * cmod (poly s ?w)))"
using ‹w ≠ 0› t(1) by (simp add: algebra_simps norm_power norm_mult)
with 30 have 120: "cmod (?w^k * ?w * poly s ?w) < t^k"
by (smt (verit, ccfv_SIG) m(2) mult_left_mono norm_ge_zero t(1) tw zero_le_power)
from power_strict_mono[OF t(2), of k] t(1) kas(2) have 121: "t^k ≤ 1"
by auto
from ath[OF norm_ge_zero[of "?w^k * ?w * poly s ?w"] 120 121]
show ?thesis
by (smt (verit) "11" kas(4) poly_pCons r01)
qed
with cq0 q(2) show ?thesis
by (smt (verit) mrmq_eq)
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)"
proof (rule ccontr)
assume N: "∄z. poly p z = 0"
then have "¬ constant (poly p)"
unfolding constant_def
by (metis (no_types, opaque_lifting) nc poly_pcompose pcompose_0' pcompose_const poly_0_coeff_0
poly_all_0_iff_0 poly_diff right_minus_eq)
then show False
using N fundamental_theorem_of_algebra by blast
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 = [:- a, 1:] ^ n * r ^ n"
using power_mult_distrib r by blast
also have "... = [:- a, 1:] ^ order a p * [:k:] * ([:1 / k:] * [:- a, 1:] ^ (n - order a p) * r ^ n)"
using k oop [of a] by (simp flip: power_add)
also have "... = p * ?w"
by (metis s kpn)
finally show ?thesis
unfolding dvd_def by blast
next
case False
with sne dpn s oa have dsn: "degree s < n"
by (metis add_diff_cancel_right' degree_0 degree_linear_power degree_mult_eq gr0I zero_less_diff)
have "poly r x = 0" if h: "poly s x = 0" for x
proof -
have "x ≠ a"
by (metis ap(2) dvd_refl mult_dvd_mono poly_eq_0_iff_dvd power_Suc power_commutes s that)
moreover have "poly p x = 0"
by (metis (no_types) mult_eq_0_iff poly_mult s that)
ultimately show ?thesis
using pq0 r by auto
qed
with False IH dsn obtain u where u: "r ^ (degree s) = s * u"
by blast
then have u': "⋀x. poly s x * poly u x = poly r x ^ degree s"
by (simp only: poly_mult[symmetric] poly_power[symmetric])
have "q^n = [:- a, 1:] ^ n * r ^ n"
using power_mult_distrib r by blast
also have "... = [:- a, 1:] ^ order a p * (s * u * ([:- a, 1:] ^ (n - order a p) * r ^ (n - degree s)))"
by (smt (verit, del_insts) s u mult_ac power_add add_diff_cancel_right' degree_linear_power degree_mult_eq dpn mult_zero_left)
also have "... = p * (u * ([:-a,1:] ^ (n - ?op))) * (r ^ (n - degree s))"
using s by force
finally show ?thesis
unfolding dvd_def by auto
qed
qed
qed
then show ?thesis
using a order_root pne by blast
next
case False
then show ?thesis
using dpn n0 fundamental_theorem_of_algebra_alt[of p]
by fastforce
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 "(∀x. poly p x = (0::complex) ⟶ poly q x = 0) ⟷ q = 0"
by (auto simp add: poly_all_0_iff_0)
with p show ?thesis
by force
next
case dp: 2
then show ?thesis
by (meson dvd_trans is_unit_iff_degree poly_eq_0_iff_dvd unit_imp_dvd)
next
case dp: 3
have False if "p dvd (q ^ (Suc n))" "poly p x = 0" "poly q x ≠ 0" for x
by (metis dvd_trans poly_eq_0_iff_dvd poly_power power_eq_0_iff that)
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 "poly p = poly [:poly p 0:]"
by auto
then show ?thesis
by (metis degree_pCons_0 poly_eq_poly_eq_iff)
qed
show ?lhs if ?rhs
unfolding constant_def
by (metis degree_eq_zeroE pcompose_const poly_0 poly_pcompose that)
qed

lemma complex_poly_decompose:
"smult (lead_coeff p) (∏z|poly p z = 0. [:-z, 1:] ^ order z p) = (p :: complex poly)"
proof (induction p rule: poly_root_order_induct)
case (no_roots p)
show ?case
proof (cases "degree p = 0")
case False
hence "¬constant (poly p)" by (subst constant_degree)
with fundamental_theorem_of_algebra and no_roots show ?thesis by blast
qed (auto elim!: degree_eq_zeroE)
next
case (root p x n)
from root have *: "{z. poly ([:- x, 1:] ^ n * p) z = 0} = insert x {z. poly p z = 0}"
by auto
have "smult (lead_coeff ([:-x, 1:] ^ n * p))
(∏z|poly ([:-x,1:] ^ n * p) z = 0. [:-z, 1:] ^ order z ([:- x, 1:] ^ n * p)) =
[:- x, 1:] ^ order x ([:- x, 1:] ^ n * p) *
smult (lead_coeff p) (∏z∈{z. poly p z = 0}. [:- z, 1:] ^ order z ([:- x, 1:] ^ n * p))"
by (subst *, subst prod.insert)
(insert root, auto intro: poly_roots_finite simp: mult_ac lead_coeff_mult lead_coeff_power)
also have "order x ([:- x, 1:] ^ n * p) = n"
using root by (subst order_mult) (auto simp: order_power_n_n order_0I)
also have "(∏z∈{z. poly p z = 0}. [:- z, 1:] ^ order z ([:- x, 1:] ^ n * p)) =
(∏z∈{z. poly p z = 0}. [:- z, 1:] ^ order z p)"
proof (intro prod.cong refl, goal_cases)
case (1 y)
with root have "order y ([:-x,1:] ^ n) = 0" by (intro order_0I) auto
thus ?case using root by (subst order_mult) auto
qed
also note root.IH
finally show ?case .
qed simp_all

instance complex :: alg_closed_field
by standard (use fundamental_theorem_of_algebra constant_degree neq0_conv in blast)

lemma size_proots_complex: "size (proots (p :: complex poly)) = degree p"
proof (cases "p = 0")
case [simp]: False
show "size (proots p) = degree p"
by (subst (1 2) complex_poly_decompose [symmetric])
(simp add: proots_prod proots_power degree_prod_sum_eq degree_power_eq)
qed auto

lemma complex_poly_decompose_multiset:
"smult (lead_coeff p) (∏x∈#proots p. [:-x, 1:]) = (p :: complex poly)"
proof (cases "p = 0")
case False
hence "(∏x∈#proots p. [:-x, 1:]) = (∏x | poly p x = 0. [:-x, 1:] ^ order x p)"
by (subst image_prod_mset_multiplicity) simp_all
also have "smult (lead_coeff p) … = p"
by (rule complex_poly_decompose)
finally show ?thesis .
qed auto

lemma complex_poly_decompose':
obtains root where "smult (lead_coeff p) (∏i<degree p. [:-root i, 1:]) = (p :: complex poly)"
proof -
obtain roots where roots: "mset roots = proots p"
using ex_mset by blast

have "p = smult (lead_coeff p) (∏x∈#proots p. [:-x, 1:])"
by (rule complex_poly_decompose_multiset [symmetric])
also have "(∏x∈#proots p. [:-x, 1:]) = (∏x←roots. [:-x, 1:])"
by (subst prod_mset_prod_list [symmetric]) (simp add: roots)
also have "… = (∏i<length roots. [:-roots ! i, 1:])"
by (subst prod.list_conv_set_nth) (auto simp: atLeast0LessThan)
finally have eq: "p = smult (lead_coeff p) (∏i<length roots. [:-roots ! i, 1:])" .
also have [simp]: "degree p = length roots"
using roots by (subst eq) (auto simp: degree_prod_sum_eq)
finally show ?thesis by (intro that[of "λi. roots ! i"]) auto
qed

lemma complex_poly_decompose_rsquarefree:
assumes "rsquarefree p"
shows   "smult (lead_coeff p) (∏z|poly p z = 0. [:-z, 1:]) = (p :: complex poly)"
proof (cases "p = 0")
case False
have "(∏z|poly p z = 0. [:-z, 1:]) = (∏z|poly p z = 0. [:-z, 1:] ^ order z p)"
using assms False by (intro prod.cong) (auto simp: rsquarefree_root_order)
also have "smult (lead_coeff p) … = p"
by (rule complex_poly_decompose)
finally show ?thesis .
qed auto

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)"
by (metis add_0 dvd_def mult_pCons_right pq smult_0_left)

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"
using lgpq mod_poly_less by fastforce

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"
by (metis a0 diff_add_cancel dvd_add_left_iff dvd_smult_iff pp' qrp')

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)"
by (meson fundamental_theorem_of_algebra_alt l pCons_eq_0_iff pCons_eq_iff)

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)"
by (metis degree_pCons_eq_if l nullstellensatz_univariate pCons_eq_0_iff psize_def)

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"
by (metis (no_types) basic_cqe_conv_2b h poly_diff right_minus_eq)

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

end
```