# Theory Polynomial_Factorization.Square_Free_Factorization

```(*
Author:      René Thiemann
*)
section ‹Square Free Factorization›

text ‹We implemented Yun's algorithm to perform a square-free factorization of a polynomial.
We further show properties of a square-free factorization, namely that the exponents in
the square-free factorization are exactly the orders of the roots. We also show that
factorizing the result of square-free factorization further will again result in a
square-free factorization, and that square-free factorizations can be lifted homomorphically.›

theory Square_Free_Factorization
imports
Matrix.Utility
Polynomial_Irreducibility
Order_Polynomial
Fundamental_Theorem_Algebra_Factorized
Polynomial_Interpolation.Ring_Hom_Poly
begin

definition square_free :: "'a :: comm_semiring_1 poly ⇒ bool" where
"square_free p = (p ≠ 0 ∧ (∀ q. degree q > 0 ⟶ ¬ (q * q dvd p)))"

lemma square_freeI:
assumes "⋀ q. degree q > 0 ⟹ q ≠ 0 ⟹ q * q dvd p ⟹ False"
and p: "p ≠ 0"
shows "square_free p" unfolding square_free_def
proof (intro allI conjI[OF p] impI notI, goal_cases)
case (1 q)
from assms(1)[OF 1(1) _ 1(2)] 1(1) show False by (cases "q = 0", auto)
qed

lemma square_free_multD:
assumes sf: "square_free (f * g)"
shows "h dvd f ⟹ h dvd g ⟹ degree h = 0" "square_free f" "square_free g"
proof -
from sf[unfolded square_free_def] have 0: "f ≠ 0" "g ≠ 0"
and dvd: "⋀ q. q * q dvd f * g ⟹ degree q = 0" by auto
then show "square_free f" "square_free g" by (auto simp: square_free_def)
assume "h dvd f" "h dvd g"
then have "h * h dvd f * g" by (rule mult_dvd_mono)
from dvd[OF this] show "degree h = 0".
qed

lemma irreducible⇩d_square_free:
fixes p :: "'a :: {comm_semiring_1, semiring_no_zero_divisors} poly"
shows "irreducible⇩d p ⟹ square_free p"
by (metis degree_0 degree_mult_eq degree_mult_eq_0 irreducible⇩dD(1) irreducible⇩dD(2) irreducible⇩d_dvd_smult irreducible⇩d_smultI less_add_same_cancel2 not_gr_zero square_free_def)

lemma square_free_factor: assumes dvd: "a dvd p"
and sf: "square_free p"
shows "square_free a"
proof (intro square_freeI)
fix q
assume q: "degree q > 0" and "q * q dvd a"
hence "q * q dvd p" using dvd dvd_trans sf square_free_def by blast
with sf[unfolded square_free_def] q show False by auto
qed (insert dvd sf, auto simp: square_free_def)

lemma square_free_prod_list_distinct:
assumes sf: "square_free (prod_list us :: 'a :: idom poly)"
and us: "⋀ u. u ∈ set us ⟹ degree u > 0"
shows "distinct us"
proof (rule ccontr)
assume "¬ distinct us"
from not_distinct_decomp[OF this] obtain xs ys zs u where
"us = xs @ u # ys @ u # zs" by auto
hence dvd: "u * u dvd prod_list us" and u: "u ∈ set us" by auto
from dvd us[OF u] sf have "prod_list us = 0" unfolding square_free_def by auto
hence "0 ∈ set us" by (simp add: prod_list_zero_iff)
from us[OF this] show False by auto
qed

definition separable where
"separable f = coprime f (pderiv f)"

lemma separable_imp_square_free:
assumes sep: "separable (f :: 'a::{field, factorial_ring_gcd, semiring_gcd_mult_normalize} poly)"
shows "square_free f"
proof (rule ccontr)
note sep = sep[unfolded separable_def]
from sep have f0: "f ≠ 0" by (cases f, auto)
assume "¬ square_free f"
then obtain g where g: "degree g ≠ 0" and "g * g dvd f" using f0 unfolding square_free_def by auto
then obtain h where f: "f = g * (g * h)" unfolding dvd_def by (auto simp: ac_simps)
have "pderiv f = g * ((g * pderiv h + h * pderiv g) + h * pderiv g)"
unfolding f pderiv_mult[of g] by (simp add: field_simps)
hence "g dvd pderiv f" unfolding dvd_def by blast
moreover have "g dvd f" unfolding f dvd_def by blast
ultimately have dvd: "g dvd (gcd f (pderiv f))" by simp
have "gcd f (pderiv f) ≠ 0" using f0 by simp
with g dvd have "degree (gcd f (pderiv f)) ≠ 0"
hence "¬ coprime f (pderiv f)" by auto
with sep show False by simp
qed

lemma square_free_rsquarefree: assumes f: "square_free f"
shows "rsquarefree f"
unfolding rsquarefree_def
proof (intro conjI allI)
fix x
show "order x f = 0 ∨ order x f = 1"
proof (rule ccontr)
assume "¬ ?thesis"
then obtain n where ord: "order x f = Suc (Suc n)"
by (cases "order x f"; cases "order x f - 1"; auto)
define p where "p = [:-x,1:]"
from order_divides[of x "Suc (Suc 0)" f, unfolded ord]
have "p * p dvd f" "degree p ≠ 0" unfolding p_def by auto
hence "¬ square_free f" using f(1) unfolding square_free_def by auto
with assms show False by auto
qed
qed (insert f, auto simp: square_free_def)

lemma square_free_prodD:
fixes fs :: "'a :: {field,euclidean_ring_gcd,semiring_gcd_mult_normalize} poly set"
assumes sf: "square_free (∏ fs)"
and fin: "finite fs"
and f: "f ∈ fs"
and g: "g ∈ fs"
and fg: "f ≠ g"
shows "coprime f g"
proof -
have "(∏ fs) = f * (∏ (fs - {f}))"
by (rule prod.remove[OF fin f])
also have "(∏ (fs - {f})) = g * (∏ (fs - {f} - {g}))"
by (rule prod.remove, insert fin g fg, auto)
finally obtain k where sf: "square_free (f * g * k)" using sf by (simp add: ac_simps)
from sf[unfolded square_free_def] have 0: "f ≠ 0" "g ≠ 0"
and dvd: "⋀ q. q * q dvd f * g * k ⟹ degree q = 0"
by auto
have "gcd f g * gcd f g dvd f * g * k" by (simp add: mult_dvd_mono)
from dvd[OF this] have "degree (gcd f g) = 0" .
moreover have "gcd f g ≠ 0" using 0 by auto
ultimately show "coprime f g" using is_unit_gcd[of f g] is_unit_iff_degree[of "gcd f g"] by simp
qed

lemma rsquarefree_square_free_complex: assumes "rsquarefree (p :: complex poly)"
shows "square_free p"
proof (rule square_freeI)
fix q
assume d: "degree q > 0" and dvd: "q * q dvd p"
from d have "¬ constant (poly q)" by (simp add: constant_degree)
from fundamental_theorem_of_algebra[OF this] obtain x where "poly q x = 0" by auto
hence "[:-x,1:] dvd q" by (simp add: poly_eq_0_iff_dvd)
then obtain k where q: "q = [:-x,1:] * k" unfolding dvd_def by auto
from dvd obtain l where p: "p = q * q * l" unfolding dvd_def by auto
from p[unfolded q] have "p = [:-x,1:]^2 * (k * k * l)" by algebra
hence "[:-x,1:]^2 dvd p" unfolding dvd_def by blast
from this[unfolded order_divides] have "p = 0 ∨ ¬ order x p ≤ 1" by auto
thus False using assms unfolding rsquarefree_def' by auto
qed (insert assms, auto simp: rsquarefree_def)

lemma square_free_separable_main:
fixes f :: "'a :: {field,factorial_ring_gcd,semiring_gcd_mult_normalize} poly"
assumes "square_free f"
and sep: "¬ separable f"
shows "∃ g k. f = g * k ∧ degree g ≠ 0 ∧ pderiv g = 0"
proof -
note cop = sep[unfolded separable_def]
from assms have f: "f ≠ 0" unfolding square_free_def by auto
let ?g = "gcd f (pderiv f)"
define G where "G = ?g"
from poly_gcd_monic[of f "pderiv f"] f have mon: "monic ?g"
by auto
have deg: "degree G > 0"
proof (cases "degree G")
case 0
from degree0_coeffs[OF this] cop mon show ?thesis
by (auto simp: G_def coprime_iff_gcd_eq_1)
qed auto
have gf: "G dvd f" unfolding G_def by auto
have gf': "G dvd pderiv f" unfolding G_def by auto
from irreducible⇩d_factor[OF deg] obtain g r where g: "irreducible g" and G: "G = g * r" by auto
from gf have gf: "g dvd f" unfolding G by (rule dvd_mult_left)
from gf' have gf': "g dvd pderiv f" unfolding G by (rule dvd_mult_left)
have g0: "degree g ≠ 0" using g unfolding irreducible⇩d_def by auto
from gf obtain k where fgk: "f = g * k" unfolding dvd_def by auto
have id1: "pderiv f = g * pderiv k + k * pderiv g" unfolding fgk pderiv_mult by simp
from gf' obtain h where "pderiv f = g * h" unfolding dvd_def by auto
from id1[unfolded this] have "k * pderiv g = g * (h - pderiv k)" by (simp add: field_simps)
hence dvd: "g dvd k * pderiv g" unfolding dvd_def by auto
{
assume "g dvd k"
then obtain h where k: "k = g * h" unfolding dvd_def by auto
with fgk have "g * g dvd f" by auto
with g0 have "¬ square_free f" unfolding square_free_def using f by auto
with assms have False by simp
}
with  g dvd
have "g dvd pderiv g" by auto
from divides_degree[OF this] degree_pderiv_le[of g] g0
have "pderiv g = 0" by linarith
with fgk g0 show ?thesis by auto
qed

lemma square_free_imp_separable: fixes f :: "'a :: {field_char_0,factorial_ring_gcd,semiring_gcd_mult_normalize} poly"
assumes "square_free f"
shows "separable f"
proof (rule ccontr)
assume "¬ separable f"
from square_free_separable_main[OF assms this]
obtain g k where *: "f = g * k" "degree g ≠ 0" "pderiv g = 0" by auto
hence "g dvd pderiv g" by auto
thus False unfolding dvd_pderiv_iff using * by auto
qed

lemma square_free_iff_separable:
"square_free (f :: 'a :: {field_char_0,factorial_ring_gcd,semiring_gcd_mult_normalize} poly) = separable f"
using separable_imp_square_free[of f] square_free_imp_separable[of f] by auto

context
assumes "SORT_CONSTRAINT('a::{field,factorial_ring_gcd})"
begin
lemma square_free_smult: "c ≠ 0 ⟹ square_free (f :: 'a poly) ⟹ square_free (smult c f)"
by (unfold square_free_def, insert dvd_smult_cancel[of _ c], auto)

lemma square_free_smult_iff[simp]: "c ≠ 0 ⟹ square_free (smult c f) = square_free (f :: 'a poly)"
using square_free_smult[of c f] square_free_smult[of "inverse c" "smult c f"] by auto
end

context
assumes "SORT_CONSTRAINT('a::factorial_ring_gcd)"
begin
definition square_free_factorization :: "'a poly ⇒ 'a × ('a poly × nat) list ⇒ bool" where
"square_free_factorization p cbs ≡ case cbs of (c,bs) ⇒
(p = smult c (∏(a, i)∈ set bs. a ^ i))
∧ (p = 0 ⟶ c = 0 ∧ bs = [])
∧ (∀ a i. (a,i) ∈ set bs ⟶ square_free a ∧ degree a > 0 ∧ i > 0)
∧ (∀ a i b j. (a,i) ∈ set bs ⟶ (b,j) ∈ set bs ⟶ (a,i) ≠ (b,j) ⟶ coprime a b)
∧ distinct bs"

lemma square_free_factorizationD: assumes "square_free_factorization p (c,bs)"
shows "p = smult c (∏(a, i)∈ set bs. a ^ i)"
"(a,i) ∈ set bs ⟹ square_free a ∧ degree a ≠ 0 ∧ i > 0"
"(a,i) ∈ set bs ⟹ (b,j) ∈ set bs ⟹ (a,i) ≠ (b,j) ⟹ coprime a b"
"p = 0 ⟹ c = 0 ∧ bs = []"
"distinct bs"
using assms unfolding square_free_factorization_def split by blast+

lemma square_free_factorization_prod_list: assumes "square_free_factorization p (c,bs)"
shows "p = smult c (prod_list (map (λ (a,i). a ^ i) bs))"
proof -
note sff = square_free_factorizationD[OF assms]
show ?thesis unfolding sff(1)
qed
end

subsection ‹Yun's factorization algorithm›
locale yun_gcd =
fixes Gcd :: "'a :: factorial_ring_gcd poly ⇒ 'a poly ⇒ 'a poly"
begin

partial_function (tailrec) yun_factorization_main ::
"'a poly ⇒ 'a poly ⇒
nat ⇒ ('a poly × nat)list ⇒ ('a poly × nat)list" where
[code]: "yun_factorization_main bn cn i sqr = (
if bn = 1 then sqr
else (
let
dn = cn - pderiv bn;
an = Gcd bn dn
in yun_factorization_main (bn div an) (dn div an) (Suc i) ((an,Suc i) # sqr)))"

definition yun_monic_factorization :: "'a poly ⇒ ('a poly × nat)list" where
"yun_monic_factorization p = (let
pp = pderiv p;
u = Gcd p pp;
b0 = p div u;
c0 = pp div u
in
(filter (λ (a,i). a ≠ 1) (yun_factorization_main b0 c0 0 [])))"

definition square_free_monic_poly :: "'a poly ⇒ 'a poly" where
"square_free_monic_poly p = (p div (Gcd p (pderiv p)))"
end

declare yun_gcd.yun_monic_factorization_def [code]
declare yun_gcd.yun_factorization_main.simps [code]
declare yun_gcd.square_free_monic_poly_def [code]

context
fixes Gcd :: "'a :: {field_char_0,euclidean_ring_gcd} poly ⇒ 'a poly ⇒ 'a poly"
begin
interpretation yun_gcd Gcd .

definition square_free_poly :: "'a poly ⇒ 'a poly" where
"square_free_poly p = (if p = 0 then 0 else
square_free_monic_poly (smult (inverse (coeff p (degree p))) p))"

definition yun_factorization :: "'a poly ⇒ 'a × ('a poly × nat)list" where
"yun_factorization p = (if p = 0
then (0,[]) else (let
c = coeff p (degree p);
q = smult (inverse c) p
in (c, yun_monic_factorization q)))"

lemma yun_factorization_0[simp]: "yun_factorization 0 = (0,[])"
unfolding yun_factorization_def by simp
end

locale monic_factorization =
fixes as :: "('a :: {field_char_0,euclidean_ring_gcd,semiring_gcd_mult_normalize} poly × nat) set"
and p :: "'a poly"
assumes p: "p = prod (λ (a,i). a ^ Suc i) as"
and fin: "finite as"
assumes as_distinct: "⋀ a i b j. (a,i) ∈ as ⟹ (b,j) ∈ as ⟹ (a,i) ≠ (b,j) ⟹ a ≠ b"
and as_irred: "⋀ a i. (a,i) ∈ as ⟹ irreducible⇩d a"
and as_monic: "⋀ a i. (a,i) ∈ as ⟹ monic a"
begin

lemma poly_exp_expand:
"p = (prod (λ (a,i). a ^ i) as) * prod (λ (a,i). a) as"
unfolding p prod.distrib[symmetric]
by (rule prod.cong, auto)

lemma pderiv_exp_prod:
"pderiv p = (prod (λ (a,i). a ^ i) as * sum (λ (a,i).
prod (λ (b,j). b) (as - {(a,i)}) * smult (of_nat (Suc i)) (pderiv a)) as)"
unfolding p pderiv_prod sum_distrib_left
proof (rule sum.cong[OF refl])
fix x
assume "x ∈ as"
then obtain a i where x: "x = (a,i)" and mem: "(a,i) ∈ as" by (cases x, auto)
let ?si = "smult (of_nat (Suc i)) :: 'a poly ⇒ 'a poly"
show "(∏(a, i)∈as - {x}. a ^ Suc i) * pderiv (case x of (a, i) ⇒ a ^ Suc i) =
(∏(a, i)∈as. a ^ i) *
(case x of (a, i) ⇒ (∏(a, i)∈as - {(a, i)}. a) * smult (of_nat (Suc i)) (pderiv a))"
unfolding x split pderiv_power_Suc
proof -
let ?prod = "∏(a, i)∈as - {(a, i)}. a ^ Suc i"
let ?l = "?prod * (?si (a ^ i) * pderiv a)"
let ?r = "(∏(a, i)∈as. a ^ i) * ((∏(a, i)∈as - {(a, i)}. a) * ?si (pderiv a))"
have "?r = a ^ i * ((∏(a, i)∈as - {(a, i)}. a ^ i) * (∏(a, i)∈as - {(a, i)}. a) * ?si (pderiv a))"
unfolding prod.remove[OF fin mem] by (simp add: ac_simps)
also have "(∏(a, i)∈as - {(a, i)}. a ^ i) * (∏(a, i)∈as - {(a, i)}. a)
= ?prod" unfolding prod.distrib[symmetric]
by (rule prod.cong[OF refl], auto)
finally show "?l = ?r"
qed
qed

lemma monic_gen: assumes "bs ⊆ as"
shows "monic (∏ (a, i) ∈ bs. a)"
by (rule monic_prod, insert assms as_monic, auto)

lemma nonzero_gen: assumes "bs ⊆ as"
shows "(∏ (a, i) ∈ bs. a) ≠ 0"
using monic_gen[OF assms] by auto

lemma monic_Prod: "monic ((∏(a, i)∈as. a ^ i))"
by (rule monic_prod, insert as_monic, auto intro: monic_power)

lemma coprime_generic:
assumes bs: "bs ⊆ as"
and f: "⋀ a i. (a,i) ∈ bs ⟹ f i > 0"
shows "coprime (∏(a, i) ∈ bs. a)
(∑(a, i)∈ bs. (∏(b, j)∈ bs - {(a, i)} . b) * smult (of_nat (f i)) (pderiv a))"
(is "coprime ?single ?onederiv")
proof -
have single: "?single ≠ 0" by (rule nonzero_gen[OF bs])
show ?thesis
proof (rule gcd_eq_1_imp_coprime, rule gcdI [symmetric])
fix k
assume dvd: "k dvd ?single" "k dvd ?onederiv"
note bs_monic = as_monic[OF subsetD[OF bs]]
from dvd(1) single have k: "k ≠ 0" by auto
show "k dvd 1"
proof (cases "degree k > 0")
case False
with k obtain c where "k = [:c:]"
by (auto dest: degree0_coeffs)
with k have "c ≠ 0"
by auto
with ‹k = [:c:]› show "is_unit k"
using dvdI [of 1 "[:c:]" "[:1 / c:]"] by auto
next
case True
from irreducible⇩d_factor[OF this]
obtain p q where k: "k = p * q" and p: "irreducible p" by auto
from k dvd have dvd: "p dvd ?single" "p dvd ?onederiv" unfolding dvd_def by auto
from irreducible_dvd_prod[OF p dvd(1)] obtain a i where ai: "(a,i) ∈ bs" and pa: "p dvd a"
by force
then obtain q where a: "a = p * q" unfolding dvd_def by auto
from p[unfolded irreducible⇩d_def] have p0: "degree p > 0" by auto
from irreducible⇩d_dvd_smult[OF p0 as_irred pa] ai bs
obtain c where c: "c ≠ 0" and ap: "a = smult c p" by auto
hence ap': "p = smult (1/c) a" by auto
let ?prod = "λ a i. (∏(b, j)∈bs - {(a, i)}. b) * smult (of_nat (f i)) (pderiv a)"
let ?prod' = "λ aa ii a i. (∏(b, j)∈bs - {(a, i),(aa,ii)}. b) * smult (of_nat (f i)) (pderiv a)"
define factor where "factor = sum (λ (b,j). ?prod' a i b j ) (bs - {(a,i)})"
define fac where "fac = q * factor"
from fin finite_subset[OF bs] have fin: "finite bs" by auto
have "?onederiv = ?prod a i + sum (λ (b,j). ?prod b j) (bs - {(a,i)})"
by (subst sum.remove[OF fin ai], auto)
also have "sum (λ (b,j). ?prod b j) (bs - {(a,i)})
= a * factor"
unfolding factor_def sum_distrib_left
proof (rule sum.cong[OF refl])
fix bj
assume mem: "bj ∈ bs - {(a,i)}"
obtain b j where bj: "bj = (b,j)" by force
from mem bj ai have ai: "(a,i) ∈ bs - {(b,j)}" by auto
have id: "bs - {(b, j)} - {(a, i)} = bs - {(b,j),(a,i)}" by auto
show "(λ (b,j). ?prod b j) bj = a * (λ (b,j). ?prod' a i b j) bj"
unfolding bj split
by (subst prod.remove[OF _ ai], insert fin, auto simp: id ac_simps)
qed
finally have "?onederiv = ?prod a i + p * fac" unfolding fac_def a by simp
from dvd(2)[unfolded this] have "p dvd ?prod a i" by algebra
from this[unfolded field_poly_irreducible_dvd_mult[OF p]]
have False
proof
assume "p dvd (∏(b, j)∈bs - {(a, i)}. b)"
from irreducible_dvd_prod[OF p this] obtain b j where bj': "(b,j) ∈ bs - {(a,i)}"
and pb: "p dvd b" by auto
hence bj: "(b,j) ∈ bs" by auto
from as_irred bj bs have "irreducible⇩d b" by auto
from irreducible⇩d_dvd_smult[OF p0 this pb] obtain d where d: "d ≠ 0"
and b: "b = smult d p" by auto
with ap c have id: "smult (c/d) b = a" and deg: "degree a = degree b" by auto
from coeff_smult[of "c/d" b "degree b", unfolded id] deg bs_monic[OF ai] bs_monic[OF bj]
have "c / d = 1" by simp
from id[unfolded this] have "a = b" by simp
with as_distinct[OF subsetD[OF bs ai] subsetD[OF bs bj]] bj'
show False by auto
next
from f[OF ai] obtain k where fi: "f i = Suc k" by (cases "f i", auto)
assume "p dvd smult (of_nat (f i)) (pderiv a)"
hence "p dvd (pderiv a)" unfolding fi using dvd_smult_cancel of_nat_eq_0_iff by blast
from this[unfolded ap] have "p dvd pderiv p" using c
by (metis ‹p dvd pderiv a› ap' dvd_trans dvd_triv_right mult.left_neutral pderiv_smult smult_dvd_cancel)
with not_dvd_pderiv p0 show False by auto
qed
thus "k dvd 1" by simp
qed
qed (insert ‹?single ≠ 0›, auto)
qed

lemma pderiv_exp_gcd:
"gcd p (pderiv p) = (∏(a, i)∈as. a ^ i)" (is "_ = ?prod")
proof -
let ?sum = "(∑(a, i)∈as. (∏(b, j)∈as - {(a, i)}. b) * smult (of_nat (Suc i)) (pderiv a))"
let ?single = "(∏(a, i)∈as. a)"
let ?prd = "λ a i. (∏(b, j)∈as - {(a, i)}. b) * smult (of_nat (Suc i)) (pderiv a)"
let ?onederiv = "∑(a, i)∈as. ?prd a i"
have pp: "pderiv p = ?prod * ?sum" by (rule pderiv_exp_prod)
have p: "p = ?prod * ?single" by (rule poly_exp_expand)
have monic: "monic ?prod" by (rule monic_Prod)
have gcd: "coprime ?single ?onederiv"
by (rule coprime_generic, auto)
then have gcd: "gcd ?single ?onederiv = 1"
by simp
show ?thesis unfolding pp unfolding p poly_gcd_monic_factor [OF monic] gcd by simp
qed

lemma p_div_gcd_p_pderiv: "p div (gcd p (pderiv p)) = (∏(a, i)∈as. a)"
unfolding pderiv_exp_gcd unfolding poly_exp_expand
by (rule nonzero_mult_div_cancel_left, insert monic_Prod, auto)

fun A B C D :: "nat ⇒ 'a poly" where
"A n = gcd (B n) (D n)"
| "B 0 = p div (gcd p (pderiv p))"
| "B (Suc n) = B n div A n"
| "C 0 = pderiv p div (gcd p (pderiv p))"
| "C (Suc n) = D n div A n"
| "D n = C n - pderiv (B n)"

lemma A_B_C_D: "A n = (∏ (a, i) ∈ as ∩ UNIV × {n}. a)"
"B n = (∏ (a, i) ∈ as - UNIV × {0 ..< n}. a)"
"C n = (∑(a, i)∈as - UNIV × {0 ..< n}.
(∏(b, j)∈as - UNIV × {0 ..< n} - {(a, i)}. b) * smult (of_nat (Suc i - n)) (pderiv a))"
"D n = (∏ (a, i) ∈ as ∩ UNIV × {n}. a) *
(∑ (a,i)∈as - UNIV × {0 ..< Suc n}.
(∏(b, j)∈ as - UNIV × {0 ..< Suc n} - {(a, i)}. b) * (smult (of_nat (i - n)) (pderiv a)))"
proof (induct n and n and n and n rule: A_B_C_D.induct)
case (1 n) (* A *)
note Bn = 1(1)
note Dn = 1(2)
have "(∏(a, i)∈as - UNIV × {0..< n}. a) = (∏(a, i)∈as ∩ UNIV × {n}. a) * (∏(a, i)∈as - UNIV × {0..<Suc n}. a)"
by (subst prod.union_disjoint[symmetric], auto, insert fin, auto intro: prod.cong)
note Bn' = Bn[unfolded this]
let ?an = "(∏ (a, i) ∈ as ∩ UNIV × {n}. a)"
let ?bn = "(∏(a, i)∈as - UNIV × {0..<Suc n}. a)"
show "A n = ?an" unfolding A.simps
proof (rule gcdI[symmetric, OF _ _ _ normalize_monic[OF monic_gen]])
have monB1: "monic (B n)" unfolding Bn by (rule monic_gen, auto)
hence "B n ≠ 0" by auto
let ?dn = "(∑ (a,i)∈as - UNIV × {0 ..< Suc n}.
(∏(b, j)∈ as - UNIV × {0 ..< Suc n} - {(a, i)}. b) * (smult (of_nat (i - n)) (pderiv a)))"
have Dn: "D n = ?an * ?dn" unfolding Dn by auto
show dvd1: "?an dvd B n" unfolding Bn' dvd_def by blast
show dvd2: "?an dvd D n" unfolding Dn dvd_def by blast
{
fix k
assume "k dvd B n" "k dvd D n"
from dvd_gcd_mult[OF this[unfolded Bn' Dn]]
have "k dvd ?an * (gcd ?bn ?dn)" .
moreover have "coprime ?bn ?dn"
by (rule coprime_generic, auto)
ultimately show "k dvd ?an" by simp
}
qed auto
next
case 2 (* B 0 *)
have as: "as - UNIV × {0..<0} = as" by auto
show ?case unfolding B.simps as p_div_gcd_p_pderiv by auto
next
case (3 n) (* B n *)
have id: "(∏(a, i)∈as - UNIV × {0..< n}. a) = (∏(a, i)∈as - UNIV × {0..<Suc n}. a) * (∏(a, i)∈as ∩ UNIV × {n}. a)"
by (subst prod.union_disjoint[symmetric], auto, insert fin, auto intro: prod.cong)
show ?case unfolding B.simps 3 id
by (subst nonzero_mult_div_cancel_right[OF nonzero_gen], auto)
next
case 4 (* C 0 *)
have as: "as - UNIV × {0..<0} = as" "⋀ i. Suc i - 0 = Suc i" by auto
show ?case unfolding C.simps pderiv_exp_gcd unfolding pderiv_exp_prod as
by (rule nonzero_mult_div_cancel_left, insert monic_Prod, auto)
next
case (5 n) (* C n *)
show ?case unfolding C.simps 5
by (subst nonzero_mult_div_cancel_left, rule nonzero_gen, auto)
next
case (6 n) (* D n *)
let ?f = "λ (a,i). (∏(b, j)∈as - UNIV × {0 ..< n} - {(a, i)}. b) * (smult (of_nat (i - n)) (pderiv a))"
have "D n = (∑ (a,i)∈as - UNIV × {0 ..< n}. (∏(b, j)∈as - UNIV × {0 ..< n} - {(a, i)}. b) *
(smult (of_nat (Suc i - n)) (pderiv a) - pderiv a))"
unfolding D.simps 6 pderiv_prod sum_subtractf[symmetric] right_diff_distrib
by (rule sum.cong, auto)
also have "… = sum ?f (as - UNIV × {0 ..< n})"
proof (rule sum.cong[OF refl])
fix x
assume "x ∈ as - UNIV × {0 ..< n}"
then obtain a i where x: "x = (a,i)" and i: "Suc i > n" by (cases x, auto)
hence id: "Suc i - n = Suc (i - n)" by arith
have id: "of_nat (Suc i - n) = of_nat (i - n) + (1 :: 'a)" unfolding id by simp
have id: "smult (of_nat (Suc i - n)) (pderiv a) - pderiv a = smult (of_nat (i - n)) (pderiv a)"
have cong: "⋀ x y z :: 'a poly. x = y ⟹ x * z = y * z" by auto
show "(case x of
(a, i) ⇒
(∏(b, j)∈as - UNIV × {0..<n} - {(a, i)}. b) *
(smult (of_nat (Suc i - n)) (pderiv a) - pderiv a)) =
(case x of
(a, i) ⇒ (∏(b, j)∈as - UNIV × {0..<n} - {(a, i)}. b) * smult (of_nat (i - n)) (pderiv a))"
unfolding x split id
by (rule cong, auto)
qed
also have "… = sum ?f (as - UNIV × {0 ..< Suc n}) + sum ?f (as ∩ UNIV × {n})"
by (subst sum.union_disjoint[symmetric], insert fin, auto intro: sum.cong)
also have "sum ?f (as ∩ UNIV × {n}) = 0"
by (rule sum.neutral, auto)
finally have id: "D n = sum ?f (as - UNIV × {0 ..< Suc n})" by simp
show ?case unfolding id sum_distrib_left
proof (rule sum.cong[OF refl])
fix x
assume mem: "x ∈ as - UNIV × {0 ..< Suc n}"
obtain a i where x: "x = (a,i)" by force
with mem have i: "i > n" by auto
have cong: "⋀ x y z v :: 'a poly. x = y * v ⟹ x * z = y * (v * z)" by auto
show "(case x of
(a, i) ⇒ (∏(b, j)∈as - UNIV × {0..<n} - {(a, i)}. b) * smult (of_nat (i - n)) (pderiv a)) =
(∏(a, i)∈as ∩ UNIV × {n}. a) *
(case x of (a, i) ⇒
(∏(b, j)∈as - UNIV × {0..<Suc n} - {(a, i)}. b) * smult (of_nat (i - n)) (pderiv a))"
unfolding x split
by (rule cong, subst prod.union_disjoint[symmetric], insert fin, (auto)[3],
rule prod.cong, insert i, auto)
qed
qed

lemmas A = A_B_C_D(1)
lemmas B = A_B_C_D(2)

lemmas ABCD_simps = A.simps B.simps C.simps D.simps
declare ABCD_simps[simp del]

lemma prod_A:
"(∏i = 0..< n. A i ^ Suc i) = (∏(a, i)∈ as ∩ UNIV × {0 ..< n}. a ^ Suc i)"
proof (induct n)
case (Suc n)
have id: "{0 ..< Suc n} = insert n {0 ..< n}" by auto
have id2: "as ∩ UNIV × {0 ..< Suc n} = as ∩ UNIV × {n} ∪ as ∩ UNIV × {0 ..< n}" by auto
have cong: "⋀ x y z. x = y ⟹ x * z = y * z" by auto
show ?case unfolding id2 unfolding id
proof (subst prod.insert; (subst prod.union_disjoint)?; (unfold Suc)?;
(unfold A, rule cong)?)
show "(∏(a, i)∈as ∩ UNIV × {n}. a) ^ Suc n = (∏(a, i)∈as ∩ UNIV × {n}. a ^ Suc i)"
unfolding prod_power_distrib
by (rule prod.cong, auto)
qed (insert fin, auto)
qed simp

lemma prod_A_is_p_unknown: assumes "⋀ a i. (a,i) ∈ as ⟹ i < n"
shows "p = (∏i = 0..< n. A i ^ Suc i)"
proof -
have "p = (∏(a, i)∈as. a ^ Suc i)" by (rule p)
also have "… = (∏i = 0..< n. A i ^ Suc i)" unfolding prod_A
by (rule prod.cong, insert assms, auto)
finally show ?thesis .
qed

definition bound :: nat where
"bound = Suc (Max (snd ` as))"

lemma bound: assumes m: "m ≥ bound"
shows "B m = 1"
proof -
let ?set = "as - UNIV × {0..<m}"
{
fix a i
assume ai: "(a,i) ∈ ?set"
hence "i ∈ snd ` as" by force
from Max_ge[OF _ this] fin have "i ≤ Max (snd ` as)" by auto
with ai m[unfolded bound_def] have False by auto
}
hence id: "?set = {}" by force
show "B m = 1" unfolding B id by simp
qed

lemma coprime_A_A: assumes "i ≠ j"
shows "coprime (A i) (A j)"
proof (rule coprimeI)
fix k
assume dvd: "k dvd A i" "k dvd A j"
have Ai: "A i ≠ 0" unfolding A
by (rule nonzero_gen, auto)
with dvd have k: "k ≠ 0" by auto
show "is_unit k"
proof (cases "degree k > 0")
case False
then obtain c where kc: "k = [: c :]" by (auto dest: degree0_coeffs)
with k have "1 = k * [:1 / c:]"
by simp
thus ?thesis unfolding dvd_def by blast
next
case True
from irreducible_monic_factor[OF this]
obtain q r where k: "k = q * r" and q: "irreducible q" and mq: "monic q" by auto
with dvd have dvd: "q dvd A i" "q dvd A j" unfolding dvd_def by auto
from q have q0: "degree q > 0" unfolding irreducible⇩d_def by auto
from irreducible_dvd_prod[OF q dvd(1)[unfolded A]]
obtain a where ai: "(a,i) ∈ as" and qa: "q dvd a" by auto
from irreducible_dvd_prod[OF q dvd(2)[unfolded A]]
obtain b where bj: "(b,j) ∈ as" and qb: "q dvd b" by auto
from as_distinct[OF ai bj] assms have neq: "a ≠ b" by auto
from irreducible⇩d_dvd_smult[OF q0 as_irred[OF ai] qa]
irreducible⇩d_dvd_smult[OF q0 as_irred[OF bj] qb]
obtain c d where "c ≠ 0" "d ≠ 0" "a = smult c q" "b = smult d q" by auto
hence ab: "a = smult (c / d) b" and "c / d ≠ 0" by auto
with as_monic[OF bj] as_monic[OF ai] arg_cong[OF ab, of "λ p. coeff p (degree p)"]
have "a = b" unfolding coeff_smult degree_smult_eq by auto
with neq show ?thesis by auto
qed
qed

lemma A_monic: "monic (A i)"
unfolding A by (rule monic_gen, auto)

lemma A_square_free: "square_free (A i)"
proof (rule square_freeI)
fix q k
have mon: "monic (A i)" by (rule A_monic)
hence Ai: "A i ≠ 0" by auto
assume q: "degree q > 0" and dvd: "q * q dvd A i"
from irreducible_monic_factor[OF q] obtain r s where q: "q = r * s" and
irr: "irreducible r" and mr: "monic r" by auto
from dvd[unfolded q] have dvd2: "r * r dvd A i" and dvd1: "r dvd A i" unfolding dvd_def by auto
from irreducible_dvd_prod[OF irr dvd1[unfolded A]]
obtain a where ai: "(a,i) ∈ as" and ra: "r dvd a" by auto
let ?rem = "(∏(a, i)∈as ∩ UNIV × {i} - {(a,i)}. a)"
have a: "irreducible⇩d a" by (rule as_irred[OF ai])
from irreducible⇩d_dvd_smult[OF _ a ra] irr
obtain c where ar: "a = smult c r"  and "c ≠ 0" by force
with mr as_monic[OF ai] arg_cong[OF ar, of "λ p. coeff p (degree p)"]
have "a = r" unfolding coeff_smult degree_smult_eq by auto
with dvd2 have dvd: "a * a dvd A i" by simp
have id: "A i = a * ?rem" unfolding A
by (subst prod.remove[of _ "(a,i)"], insert ai fin, auto)
with dvd have "a dvd ?rem" using a id Ai by auto
from irreducible_dvd_prod[OF _ this] a obtain b where bi: "(b,i) ∈ as"
and neq: "b ≠ a" and ab: "a dvd b" by auto
from as_irred[OF bi] have b: "irreducible⇩d b" .
from irreducible⇩d_dvd_smult[OF _ b ab] a[unfolded irreducible⇩d_def]
obtain c where "c ≠ 0" and ba: "b = smult c a" by auto
with as_monic[OF bi] as_monic[OF ai] arg_cong[OF ba, of "λ p. coeff p (degree p)"]
have "a = b" unfolding coeff_smult degree_smult_eq by auto
with neq show False by auto
qed (insert A_monic[of i], auto)

lemma prod_A_is_p_B_bound: assumes "B n = 1"
shows "p = (∏i = 0..< n. A i ^ Suc i)"
proof (rule prod_A_is_p_unknown)
fix a i
assume ai: "(a,i) ∈ as"
let ?rem = "(∏(a, i)∈as - UNIV × {0..<n} - {(a,i)}. a)"
have rem: "?rem ≠ 0"
by (rule nonzero_gen, auto)
have "irreducible⇩d a" using as_irred[OF ai] .
hence a: "a ≠ 0" "degree a ≠ 0" unfolding irreducible⇩d_def by auto
show "i < n"
proof (rule ccontr)
assume "¬ ?thesis"
hence "i ≥ n" by auto
with ai have mem: "(a,i) ∈ as - UNIV × {0 ..< n}" by auto
have "0 = degree (∏(a, i)∈as - UNIV × {0..<n}. a)" using assms unfolding B by simp
also have "… = degree (a * ?rem)"
by (subst prod.remove[OF _ mem], insert fin, auto)
also have "… = degree a + degree ?rem"
by (rule degree_mult_eq[OF a(1) rem])
finally show False using a(2) by auto
qed
qed

interpretation yun_gcd gcd .

lemma square_free_monic_poly: "(poly (square_free_monic_poly p) x = 0) = (poly p x = 0)"
proof -
show ?thesis unfolding square_free_monic_poly_def unfolding p_div_gcd_p_pderiv
unfolding p poly_prod prod_zero_iff[OF fin] by force
qed

lemma yun_factorization_induct: assumes base: "⋀ bn cn. bn = 1 ⟹ P bn cn"
and step: "⋀ bn cn. bn ≠ 1 ⟹ P (bn div (gcd bn (cn - pderiv bn)))
((cn - pderiv bn) div (gcd bn (cn - pderiv bn))) ⟹ P bn cn"
and id: "bn = p div gcd p (pderiv p)" "cn = pderiv p div gcd p (pderiv p)"
shows "P bn cn"
proof -
define n where "n = (0 :: nat)"
let ?m = "λ n. bound - n"
have "P (B n) (C n)"
proof (induct n rule: wf_induct[OF wf_measure[of ?m]])
case (1 n)
note IH = 1(1)[rule_format]
show ?case
proof (cases "B n = 1")
case True
with base show ?thesis by auto
next
case False note Bn = this
with bound[of n] have "¬ bound ≤ n" by auto
hence "(Suc n, n) ∈ measure ?m" by auto
note IH = IH[OF this]
show ?thesis
by (rule step[OF Bn], insert IH, simp add: D.simps C.simps B.simps A.simps)
qed
qed
thus ?thesis unfolding id n_def B.simps C.simps .
qed

lemma yun_factorization_main: assumes "yun_factorization_main (B n) (C n) n bs = cs"
"set bs = {(A i, Suc i) | i. i < n}" "distinct (map snd bs)"
shows "∃ m. set cs = {(A i, Suc i) | i. i < m} ∧ B m = 1 ∧ distinct (map snd cs)"
using assms
proof -
let ?m = "λ n. bound - n"
show ?thesis using assms
proof (induct n arbitrary: bs rule: wf_induct[OF wf_measure[of ?m]])
case (1 n)
note IH = 1(1)[rule_format]
have res: "yun_factorization_main (B n) (C n) n bs = cs" by fact
note res = res[unfolded yun_factorization_main.simps[of "B n"]]
have bs: "set bs = {(A i, Suc i) |i. i < n}" "distinct (map snd bs)" by fact+
show ?case
proof (cases "B n = 1")
case True
with res have "bs = cs" by auto
with True bs show ?thesis by auto
next
case False note Bn = this
with bound[of n] have "¬ bound ≤ n" by auto
hence "(Suc n, n) ∈ measure ?m" by auto
note IH = IH[OF this]
from Bn res[unfolded Let_def, folded D.simps C.simps B.simps A.simps]
have res: "yun_factorization_main (B (Suc n)) (C (Suc n)) (Suc n) ((A n, Suc n) # bs) = cs"
by simp
note IH = IH[OF this]
{
fix i
assume "i < Suc n" "¬ i < n"
hence "n = i" by arith
} note missing = this
have "set ((A n, Suc n) # bs) = {(A i, Suc i) |i. i < Suc n}"
unfolding list.simps bs by (auto, subst missing, auto)
note IH = IH[OF this]
from bs have "distinct (map snd ((A n, Suc n) # bs))" by auto
note IH = IH[OF this]
show ?thesis by (rule IH)
qed
qed
qed

lemma yun_monic_factorization_res: assumes res: "yun_monic_factorization p = bs"
shows "∃ m. set bs = {(A i, Suc i) | i. i < m ∧ A i ≠ 1} ∧ B m = 1 ∧ distinct (map snd bs)"
proof -
from res[unfolded yun_monic_factorization_def Let_def,
folded B.simps C.simps]
obtain cs where yun: "yun_factorization_main (B 0) (C 0) 0 [] = cs" and bs: "bs = filter (λ (a,i). a ≠ 1) cs"
by auto
from yun_factorization_main[OF yun] obtain m where "set cs = {(A i, Suc i) |i. i < m}"
"B m = 1" "distinct (map snd cs)"
by auto
thus ?thesis unfolding bs by (auto simp: distinct_map_filter)
qed

lemma yun_monic_factorization: assumes yun: "yun_monic_factorization p = bs"
shows "square_free_factorization p (1,bs)" "(b,i) ∈ set bs ⟹ monic b" "distinct (map snd bs)"
proof -
from yun_monic_factorization_res[OF yun]
obtain m where bs: "set bs = {(A i, Suc i) | i. i < m ∧ A i ≠ 1}" and B: "B m = 1"
and dist: "distinct (map snd bs)" by auto
have id: "{0 ..< m} = {i. i < m ∧ A i = 1} ∪ {i. i < m ∧ A i ≠ 1}" (is "_ = ?ignore ∪ _") by auto
have "p = (∏i = 0..<m. A i ^ Suc i)"
by (rule prod_A_is_p_B_bound[OF B])
also have "… = prod (λ i. A i ^ Suc i) {i. i < m ∧ A i ≠ 1}"
unfolding id by (subst prod.union_disjoint, (force+)[3],
subst prod.neutral[of ?ignore], auto)
also have "… = (∏(a, i)∈ set bs. a ^ i)" unfolding bs
by (rule prod.reindex_cong[of "(λ n. n - 1) o snd"], auto simp: inj_on_def, force)
finally have 1: "p = (∏(a, i)∈ set bs. a ^ i)" .
{
fix a i
assume "(a,i) ∈ set bs"
then obtain j where A: "a = A j" "A j ≠ 1" and i: "i ≠ 0" unfolding bs by auto
with A_square_free[of j] A_monic[of j] have "square_free a ∧ degree a ≠ 0" "monic a" "i ≠ 0"
by (auto simp: monic_degree_0)
} note 2 = this
{
fix a i b j
assume ai: "(a,i) ∈ set bs" and bj: "(b,j) ∈ set bs" and neq: "(a,i) ≠ (b,j)"
then obtain i' j' where a: "a = A i'" and b: "b = A j'" and ij': "i = Suc i'" "j = Suc j'"
unfolding bs by auto
from neq dist ai bj have neq: "i' ≠ j'" using a b ij' by blast
from coprime_A_A [OF neq] have "coprime a b" unfolding a b .
} note 3 = this
have "monic p" unfolding p
by (rule monic_prod, insert as_monic, auto intro: monic_power monic_mult)
hence 4: "p ≠ 0" by auto
from dist have 5: "distinct bs" unfolding distinct_map ..
show "square_free_factorization p (1,bs)"
unfolding square_free_factorization_def using 1 2 3 4 5
by auto
show "(b,i) ∈ set bs ⟹ monic b" using 2 by auto
show "distinct (map snd bs)" by fact
qed
end

lemma monic_factorization: assumes "monic p"
shows "∃ as. monic_factorization as p"
proof -
from monic_irreducible_factorization[OF assms]
obtain as f where fin: "finite as" and p: "p = (∏a∈as. a ^ Suc (f a))"
and as: "as ⊆ {q. irreducible⇩d q ∧ monic q}"
by auto
define cs where "cs = {(a, f a) | a. a ∈ as}"
show ?thesis
proof (rule exI, standard)
show "finite cs" unfolding cs_def using fin by auto
{
fix a i
assume "(a,i) ∈ cs"
thus "irreducible⇩d a" "monic a" unfolding cs_def using as by auto
} note irr = this
show "⋀a i b j. (a, i) ∈ cs ⟹ (b, j) ∈ cs ⟹ (a, i) ≠ (b, j) ⟹ a ≠ b" unfolding cs_def by auto
show "p = (∏(a, i)∈cs. a ^ Suc i)" unfolding p cs_def
by (rule prod.reindex_cong, auto, auto simp: inj_on_def)
qed
qed

lemma square_free_monic_poly:
assumes "monic (p :: 'a :: {field_char_0, euclidean_ring_gcd,semiring_gcd_mult_normalize} poly)"
shows "(poly (yun_gcd.square_free_monic_poly gcd p) x = 0) = (poly p x = 0)"
proof -
from monic_factorization[OF assms] obtain as where "monic_factorization as p" ..
from monic_factorization.square_free_monic_poly[OF this] show ?thesis .
qed

lemma yun_factorization_induct:
assumes base: "⋀ bn cn. bn = 1 ⟹ P bn cn"
and step: "⋀ bn cn. bn ≠ 1 ⟹ P (bn div (gcd bn (cn - pderiv bn)))
((cn - pderiv bn) div (gcd bn (cn - pderiv bn))) ⟹ P bn cn"
and id: "bn = p div gcd p (pderiv p)" "cn = pderiv p div gcd p (pderiv p)"
and monic: "monic (p :: 'a :: {field_char_0,euclidean_ring_gcd,semiring_gcd_mult_normalize} poly)"
shows "P bn cn"
proof -
from monic_factorization[OF monic] obtain as where "monic_factorization as p" ..
from monic_factorization.yun_factorization_induct[OF this base step id] show ?thesis .
qed

lemma square_free_poly:
"(poly (square_free_poly gcd p) x = 0) = (poly p x = 0)"
proof (cases "p = 0")
case True
thus ?thesis unfolding square_free_poly_def by auto
next
case False
let ?c = "coeff p (degree p)"
let ?ic = "inverse ?c"
have id: "square_free_poly gcd p = yun_gcd.square_free_monic_poly gcd (smult ?ic p)"
unfolding square_free_poly_def using False by auto
from False have mon: "monic (smult ?ic p)" and ic: "?ic ≠ 0" by auto
show ?thesis unfolding id square_free_monic_poly[OF mon]
using ic by simp
qed

lemma yun_monic_factorization:
fixes p :: "'a :: {field_char_0,euclidean_ring_gcd,semiring_gcd_mult_normalize} poly"
assumes res: "yun_gcd.yun_monic_factorization gcd p = bs"
and monic: "monic p"
shows "square_free_factorization p (1,bs)" "(b,i) ∈ set bs ⟹ monic b" "distinct (map snd bs)"
proof -
from monic_factorization[OF monic] obtain as where "monic_factorization as p" ..
from "monic_factorization.yun_monic_factorization"[OF this res]
show "square_free_factorization p (1,bs)" "(b,i) ∈ set bs ⟹ monic b" "distinct (map snd bs)"
by auto
qed

lemma square_free_factorization_smult: assumes c: "c ≠ 0"
and sf: "square_free_factorization p (d,bs)"
shows "square_free_factorization (smult c p) (c * d, bs)"
proof -
from sf[unfolded square_free_factorization_def split]
have p: "p = smult d (∏(a, i)∈set bs. a ^ i)"
and eq: "p = 0 ⟶ d = 0 ∧ bs = []" by blast+
from eq c have eq: "smult c p = 0 ⟶ c * d = 0 ∧ bs = []" by auto
from p have p: "smult c p = smult (c * d) (∏(a, i)∈set bs. a ^ i)" by auto
from eq p sf show ?thesis unfolding square_free_factorization_def by blast
qed

lemma yun_factorization: assumes res: "yun_factorization gcd p = c_bs"
shows "square_free_factorization p c_bs" "(b,i) ∈ set (snd c_bs) ⟹ monic b"
proof -
interpret yun_gcd gcd .
note res = res[unfolded yun_factorization_def Let_def]
have "square_free_factorization p c_bs ∧ ((b,i) ∈ set (snd c_bs) ⟶ monic b)"
proof (cases "p = 0")
case True
with res have "c_bs = (0, [])" by auto
thus ?thesis unfolding True by (auto simp: square_free_factorization_def)
next
case False
let ?c = "coeff p (degree p)"
let ?ic = "inverse ?c"
obtain c bs where cbs: "c_bs = (c,bs)" by force
with False res
have c: "c = ?c" "?c ≠ 0" and fact: "yun_monic_factorization (smult ?ic p) = bs" by auto
from False have mon: "monic (smult ?ic p)" by auto
from yun_monic_factorization[OF fact mon]
have sff: "square_free_factorization (smult ?ic p) (1, bs)" "(b, i) ∈ set bs ⟹ monic b" by auto
have id: "smult ?c (smult ?ic p) = p" using False by auto
from square_free_factorization_smult[OF c(2) sff(1), unfolded id] sff
show ?thesis unfolding cbs c by simp
qed
thus "square_free_factorization p c_bs" "(b,i) ∈ set (snd c_bs) ⟹ monic b" by blast+
qed

lemma prod_list_pow: "(∏x←bs. (x :: 'a :: comm_monoid_mult) ^ i)
= prod_list bs ^ i"
by (induct bs, auto simp: field_simps)

declare irreducible_linear_field_poly[intro!]

context
assumes "SORT_CONSTRAINT('a :: {field, factorial_ring_gcd,semiring_gcd_mult_normalize})"
begin
lemma square_free_factorization_order_root_mem:
assumes sff: "square_free_factorization p (c,bs)"
and p: "p ≠ (0 :: 'a poly)"
and ai: "(a,i) ∈ set bs" and rt: "poly a x = 0"
shows "order x p = i"
proof -
note sff = square_free_factorizationD[OF sff]
let ?prod = "(∏(a, i)∈set bs. a ^ i)"
from sff have pf: "p = smult c ?prod" by blast
with p have c: "c ≠ 0" by auto
have ord: "order x p = order x ?prod" unfolding pf
using order_smult[OF c] by auto
define q where "q = [: -x, 1 :]"
have q0: "q ≠ 0" unfolding q_def by auto
have iq: "irreducible q" by (auto simp: q_def)
from rt have qa: "q dvd a" unfolding q_def poly_eq_0_iff_dvd .
then obtain b where aqb: "a = q * b" unfolding dvd_def by auto
from sff(2)[OF ai] have sq: "square_free a" and mon: "degree a ≠ 0" by auto
let ?rem = "(∏(a, i)∈set bs - {(a,i)}. a ^ i)"
have p0: "?prod ≠ 0" using p pf by auto
have "?prod = a ^ i * ?rem"
by (subst prod.remove[OF _ ai], auto)
also have "a ^ i = q ^ i * b ^ i" unfolding aqb by (simp add: field_simps)
finally have id: "?prod = q ^ i * (b ^ i * ?rem)" by simp
hence dvd: "q ^ i dvd ?prod" by auto
{
assume "q ^ Suc i dvd ?prod"
hence "q dvd ?prod div q ^ i"
by (metis dvd dvd_0_left_iff dvd_div_iff_mult p0 power_Suc)
also have "?prod div q ^ i = b ^ i * ?rem"
unfolding id by (rule nonzero_mult_div_cancel_left, insert q0, auto)
finally have "q dvd b ∨ q dvd ?rem"
using iq irreducible_dvd_pow[OF iq] by auto
hence False
proof
assume "q dvd b"
with aqb have "q * q dvd a" by auto
with sq[unfolded square_free_def] mon iq show False
unfolding irreducible⇩d_def by auto
next
assume "q dvd ?rem"
from irreducible_dvd_prod[OF iq this]
obtain b j where bj: "(b,j) ∈ set bs" and neq: "(a,i) ≠ (b,j)" and dvd: "q dvd b ^ j" by auto
from irreducible_dvd_pow[OF iq dvd] have qb: "q dvd b" .
from sff(3)[OF ai bj neq] have gcd: "coprime a b" .
from qb qa have "q dvd gcd a b" by simp
from dvd_imp_degree_le[OF this[unfolded gcd]] iq q0 show False
using gcd by auto
qed
}
hence ndvd: "¬ q ^ Suc i dvd ?prod" by blast
with dvd have "order x ?prod = i" unfolding q_def
by (metis order_unique_lemma)
thus ?thesis unfolding ord .
qed

lemma square_free_factorization_order_root_no_mem:
assumes sff: "square_free_factorization p (c,bs)"
and p: "p ≠ (0 :: 'a poly)"
and no_root: "⋀ a i. (a,i) ∈ set bs ⟹ poly a x ≠ 0"
shows "order x p = 0"
proof (rule ccontr)
assume o0: "order x p ≠ 0"
with order_root[of p x] p have 0: "poly p x = 0" by auto
note sff = square_free_factorizationD[OF sff]
let ?prod = "(∏(a, i)∈set bs. a ^ i)"
from sff have pf: "p = smult c ?prod" by blast
with p have c: "c ≠ 0" by auto
with 0 have 0: "poly ?prod x = 0" unfolding pf by auto
define q where "q = [: -x, 1 :]"
from 0 have dvd: "q dvd ?prod" unfolding poly_eq_0_iff_dvd by (simp add: q_def)
have q0: "q ≠ 0" unfolding q_def by auto
have iq: "irreducible q" by (unfold q_def, auto intro:)
from irreducible_dvd_prod[OF iq dvd]
obtain a i where ai: "(a,i) ∈ set bs" and dvd: "q dvd a ^ Suc i" by auto
from irreducible_dvd_pow[OF iq dvd] have dvd: "q dvd a" .
hence "poly a x = 0" unfolding q_def by (simp add: poly_eq_0_iff_dvd q_def)
with no_root[OF ai] show False by simp
qed

lemma square_free_factorization_order_root:
assumes sff: "square_free_factorization p (c,bs)"
and p: "p ≠ (0 :: 'a poly)"
shows "order x p = i ⟷ (i = 0 ∧ (∀ a j. (a,j) ∈ set bs ⟶ poly a x ≠ 0)
∨ (∃ a j. (a,j) ∈ set bs ∧ poly a x = 0 ∧ i = j))" (is "?l = (?r1 ∨ ?r2)")
proof -
note mem = square_free_factorization_order_root_mem[OF sff p]
note no_mem = square_free_factorization_order_root_no_mem[OF sff p]
show ?thesis
proof
assume "?r1 ∨ ?r2"
thus ?l
proof
assume ?r2
then obtain a j where aj: "(a,j) ∈ set bs" "poly a x = 0" and i: "i = j" by auto
from mem[OF aj] i show ?l by simp
next
assume ?r1
with no_mem[of x] show ?l by auto
qed
next
assume ?l
show "?r1 ∨ ?r2"
proof (cases "∃a j. (a, j) ∈ set bs ∧ poly a x = 0")
case True
then obtain a j where "(a, j) ∈ set bs" "poly a x = 0" by auto
with mem[OF this] ‹?l›
have ?r2 by auto
thus ?thesis ..
next
case False
with no_mem[of x] ‹?l› have ?r1 by auto
thus ?thesis ..
qed
qed
qed

lemma square_free_factorization_root:
assumes sff: "square_free_factorization p (c,bs)"
and p: "p ≠ (0 :: 'a poly)"
shows "{x. poly p x ```