# Theory Polynomial_Factorial

theory Polynomial_Factorial
imports Polynomial Normalized_Fraction
```(*  Title:      HOL/Computational_Algebra/Polynomial_Factorial.thy
Author:     Manuel Eberl
*)

section ‹Polynomials, fractions and rings›

theory Polynomial_Factorial
imports
Complex_Main
Polynomial
Normalized_Fraction
begin

subsection ‹Lifting elements into the field of fractions›

definition to_fract :: "'a :: idom ⇒ 'a fract"
where "to_fract x = Fract x 1"
― ‹FIXME: more idiomatic name, abbreviation›

lemma to_fract_0 [simp]: "to_fract 0 = 0"
by (simp add: to_fract_def eq_fract Zero_fract_def)

lemma to_fract_1 [simp]: "to_fract 1 = 1"
by (simp add: to_fract_def eq_fract One_fract_def)

lemma to_fract_add [simp]: "to_fract (x + y) = to_fract x + to_fract y"

lemma to_fract_diff [simp]: "to_fract (x - y) = to_fract x - to_fract y"

lemma to_fract_uminus [simp]: "to_fract (-x) = -to_fract x"

lemma to_fract_mult [simp]: "to_fract (x * y) = to_fract x * to_fract y"

lemma to_fract_eq_iff [simp]: "to_fract x = to_fract y ⟷ x = y"

lemma to_fract_eq_0_iff [simp]: "to_fract x = 0 ⟷ x = 0"
by (simp add: to_fract_def Zero_fract_def eq_fract)

lemma snd_quot_of_fract_nonzero [simp]: "snd (quot_of_fract x) ≠ 0"
by transfer simp

lemma Fract_quot_of_fract [simp]: "Fract (fst (quot_of_fract x)) (snd (quot_of_fract x)) = x"
by transfer (simp del: fractrel_iff, subst fractrel_normalize_quot_left, simp)

lemma to_fract_quot_of_fract:
assumes "snd (quot_of_fract x) = 1"
shows   "to_fract (fst (quot_of_fract x)) = x"
proof -
have "x = Fract (fst (quot_of_fract x)) (snd (quot_of_fract x))" by simp
also note assms
finally show ?thesis by (simp add: to_fract_def)
qed

lemma snd_quot_of_fract_Fract_whole:
assumes "y dvd x"
shows   "snd (quot_of_fract (Fract x y)) = 1"
using assms by transfer (auto simp: normalize_quot_def Let_def gcd_proj2_if_dvd)

lemma Fract_conv_to_fract: "Fract a b = to_fract a / to_fract b"

lemma quot_of_fract_to_fract [simp]: "quot_of_fract (to_fract x) = (x, 1)"
unfolding to_fract_def by transfer (simp add: normalize_quot_def)

lemma fst_quot_of_fract_eq_0_iff [simp]: "fst (quot_of_fract x) = 0 ⟷ x = 0"
by transfer simp

lemma snd_quot_of_fract_to_fract [simp]: "snd (quot_of_fract (to_fract x)) = 1"
unfolding to_fract_def by (rule snd_quot_of_fract_Fract_whole) simp_all

lemma coprime_quot_of_fract:
"coprime (fst (quot_of_fract x)) (snd (quot_of_fract x))"

lemma unit_factor_snd_quot_of_fract: "unit_factor (snd (quot_of_fract x)) = 1"
using quot_of_fract_in_normalized_fracts[of x]

lemma unit_factor_1_imp_normalized: "unit_factor x = 1 ⟹ normalize x = x"
by (subst (2) normalize_mult_unit_factor [symmetric, of x])
(simp del: normalize_mult_unit_factor)

lemma normalize_snd_quot_of_fract: "normalize (snd (quot_of_fract x)) = snd (quot_of_fract x)"
by (intro unit_factor_1_imp_normalized unit_factor_snd_quot_of_fract)

subsection ‹Lifting polynomial coefficients to the field of fractions›

abbreviation (input) fract_poly
where "fract_poly ≡ map_poly to_fract"

abbreviation (input) unfract_poly
where "unfract_poly ≡ map_poly (fst ∘ quot_of_fract)"

lemma fract_poly_smult [simp]: "fract_poly (smult c p) = smult (to_fract c) (fract_poly p)"
by (simp add: smult_conv_map_poly map_poly_map_poly o_def)

lemma fract_poly_0 [simp]: "fract_poly 0 = 0"

lemma fract_poly_1 [simp]: "fract_poly 1 = 1"

"fract_poly (p + q) = fract_poly p + fract_poly q"
by (intro poly_eqI) (simp_all add: coeff_map_poly)

lemma fract_poly_diff [simp]:
"fract_poly (p - q) = fract_poly p - fract_poly q"
by (intro poly_eqI) (simp_all add: coeff_map_poly)

lemma to_fract_sum [simp]: "to_fract (sum f A) = sum (λx. to_fract (f x)) A"
by (cases "finite A", induction A rule: finite_induct) simp_all

lemma fract_poly_mult [simp]:
"fract_poly (p * q) = fract_poly p * fract_poly q"
by (intro poly_eqI) (simp_all add: coeff_map_poly coeff_mult)

lemma fract_poly_eq_iff [simp]: "fract_poly p = fract_poly q ⟷ p = q"
by (auto simp: poly_eq_iff coeff_map_poly)

lemma fract_poly_eq_0_iff [simp]: "fract_poly p = 0 ⟷ p = 0"
using fract_poly_eq_iff[of p 0] by (simp del: fract_poly_eq_iff)

lemma fract_poly_dvd: "p dvd q ⟹ fract_poly p dvd fract_poly q"
by (auto elim!: dvdE)

lemma prod_mset_fract_poly:
"(∏x∈#A. map_poly to_fract (f x)) = fract_poly (prod_mset (image_mset f A))"
by (induct A) (simp_all add: ac_simps)

lemma is_unit_fract_poly_iff:
"p dvd 1 ⟷ fract_poly p dvd 1 ∧ content p = 1"
proof safe
assume A: "p dvd 1"
with fract_poly_dvd [of p 1] show "is_unit (fract_poly p)"
by simp
from A show "content p = 1"
by (auto simp: is_unit_poly_iff normalize_1_iff)
next
assume A: "fract_poly p dvd 1" and B: "content p = 1"
from A obtain c where c: "fract_poly p = [:c:]" by (auto simp: is_unit_poly_iff)
{
fix n :: nat assume "n > 0"
have "to_fract (coeff p n) = coeff (fract_poly p) n" by (simp add: coeff_map_poly)
also note c
also from ‹n > 0› have "coeff [:c:] n = 0" by (simp add: coeff_pCons split: nat.splits)
finally have "coeff p n = 0" by simp
}
hence "degree p ≤ 0" by (intro degree_le) simp_all
with B show "p dvd 1" by (auto simp: is_unit_poly_iff normalize_1_iff elim!: degree_eq_zeroE)
qed

lemma fract_poly_is_unit: "p dvd 1 ⟹ fract_poly p dvd 1"
using fract_poly_dvd[of p 1] by simp

lemma fract_poly_smult_eqE:
fixes c :: "'a :: {idom_divide,ring_gcd,semiring_gcd_mult_normalize} fract"
assumes "fract_poly p = smult c (fract_poly q)"
obtains a b
where "c = to_fract b / to_fract a" "smult a p = smult b q" "coprime a b" "normalize a = a"
proof -
define a b where "a = fst (quot_of_fract c)" and "b = snd (quot_of_fract c)"
have "smult (to_fract a) (fract_poly q) = smult (to_fract b) (fract_poly p)"
by (subst smult_eq_iff) (simp_all add: a_def b_def Fract_conv_to_fract [symmetric] assms)
hence "fract_poly (smult a q) = fract_poly (smult b p)" by (simp del: fract_poly_eq_iff)
hence "smult b p = smult a q" by (simp only: fract_poly_eq_iff)
moreover have "c = to_fract a / to_fract b" "coprime b a" "normalize b = b"
by (simp_all add: a_def b_def coprime_quot_of_fract [of c] ac_simps
normalize_snd_quot_of_fract Fract_conv_to_fract [symmetric])
ultimately show ?thesis by (intro that[of a b])
qed

subsection ‹Fractional content›

abbreviation (input) Lcm_coeff_denoms
:: "'a :: {semiring_Gcd,idom_divide,ring_gcd,semiring_gcd_mult_normalize} fract poly ⇒ 'a"
where "Lcm_coeff_denoms p ≡ Lcm (snd ` quot_of_fract ` set (coeffs p))"

definition fract_content ::
"'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} fract poly ⇒ 'a fract" where
"fract_content p =
(let d = Lcm_coeff_denoms p in Fract (content (unfract_poly (smult (to_fract d) p))) d)"

definition primitive_part_fract ::
"'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} fract poly ⇒ 'a poly" where
"primitive_part_fract p =
primitive_part (unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p))"

lemma primitive_part_fract_0 [simp]: "primitive_part_fract 0 = 0"

lemma fract_content_eq_0_iff [simp]:
"fract_content p = 0 ⟷ p = 0"
unfolding fract_content_def Let_def Zero_fract_def
by (subst eq_fract) (auto simp: Lcm_0_iff map_poly_eq_0_iff)

lemma content_primitive_part_fract [simp]:
fixes p :: "'a :: {semiring_gcd_mult_normalize,
factorial_semiring, ring_gcd, semiring_Gcd,idom_divide} fract poly"
shows "p ≠ 0 ⟹ content (primitive_part_fract p) = 1"
unfolding primitive_part_fract_def
by (rule content_primitive_part)
(auto simp: primitive_part_fract_def map_poly_eq_0_iff Lcm_0_iff)

lemma content_times_primitive_part_fract:
"smult (fract_content p) (fract_poly (primitive_part_fract p)) = p"
proof -
define p' where "p' = unfract_poly (smult (to_fract (Lcm_coeff_denoms p)) p)"
have "fract_poly p' =
map_poly (to_fract ∘ fst ∘ quot_of_fract) (smult (to_fract (Lcm_coeff_denoms p)) p)"
unfolding primitive_part_fract_def p'_def
by (subst map_poly_map_poly) (simp_all add: o_assoc)
also have "… = smult (to_fract (Lcm_coeff_denoms p)) p"
proof (intro map_poly_idI, unfold o_apply)
fix c assume "c ∈ set (coeffs (smult (to_fract (Lcm_coeff_denoms p)) p))"
then obtain c' where c: "c' ∈ set (coeffs p)" "c = to_fract (Lcm_coeff_denoms p) * c'"
by (auto simp add: Lcm_0_iff coeffs_smult split: if_splits)
note c(2)
also have "c' = Fract (fst (quot_of_fract c')) (snd (quot_of_fract c'))"
by simp
also have "to_fract (Lcm_coeff_denoms p) * … =
Fract (Lcm_coeff_denoms p * fst (quot_of_fract c')) (snd (quot_of_fract c'))"
unfolding to_fract_def by (subst mult_fract) simp_all
also have "snd (quot_of_fract …) = 1"
by (intro snd_quot_of_fract_Fract_whole dvd_mult2 dvd_Lcm) (insert c(1), auto)
finally show "to_fract (fst (quot_of_fract c)) = c"
by (rule to_fract_quot_of_fract)
qed
also have "p' = smult (content p') (primitive_part p')"
by (rule content_times_primitive_part [symmetric])
also have "primitive_part p' = primitive_part_fract p"
also have "fract_poly (smult (content p') (primitive_part_fract p)) =
smult (to_fract (content p')) (fract_poly (primitive_part_fract p))" by simp
finally have "smult (to_fract (content p')) (fract_poly (primitive_part_fract p)) =
smult (to_fract (Lcm_coeff_denoms p)) p" .
thus ?thesis
by (subst (asm) smult_eq_iff)
(auto simp add: Let_def p'_def Fract_conv_to_fract field_simps Lcm_0_iff fract_content_def)
qed

lemma fract_content_fract_poly [simp]: "fract_content (fract_poly p) = to_fract (content p)"
proof -
have "Lcm_coeff_denoms (fract_poly p) = 1"
by (auto simp: set_coeffs_map_poly)
hence "fract_content (fract_poly p) =
to_fract (content (map_poly (fst ∘ quot_of_fract ∘ to_fract) p))"
by (simp add: fract_content_def to_fract_def fract_collapse map_poly_map_poly del: Lcm_1_iff)
also have "map_poly (fst ∘ quot_of_fract ∘ to_fract) p = p"
by (intro map_poly_idI) simp_all
finally show ?thesis .
qed

lemma content_decompose_fract:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,
semiring_gcd_mult_normalize} fract poly"
obtains c p' where "p = smult c (map_poly to_fract p')" "content p' = 1"
proof (cases "p = 0")
case True
hence "p = smult 0 (map_poly to_fract 1)" "content 1 = 1" by simp_all
thus ?thesis ..
next
case False
thus ?thesis
by (rule that[OF content_times_primitive_part_fract [symmetric] content_primitive_part_fract])
qed

subsection ‹More properties of content and primitive part›

lemma lift_prime_elem_poly:
assumes "prime_elem (c :: 'a :: semidom)"
shows   "prime_elem [:c:]"
proof (rule prime_elemI)
fix a b assume *: "[:c:] dvd a * b"
from * have dvd: "c dvd coeff (a * b) n" for n
by (subst (asm) const_poly_dvd_iff) blast
{
define m where "m = (GREATEST m. ¬c dvd coeff b m)"
assume "¬[:c:] dvd b"
hence A: "∃i. ¬c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast
have B: "⋀i. ¬c dvd coeff b i ⟹ i ≤ degree b"
by (auto intro: le_degree)
have coeff_m: "¬c dvd coeff b m" unfolding m_def by (rule GreatestI_ex_nat[OF A B])
have "i ≤ m" if "¬c dvd coeff b i" for i
unfolding m_def by (blast intro!: Greatest_le_nat that B)
hence dvd_b: "c dvd coeff b i" if "i > m" for i using that by force

have "c dvd coeff a i" for i
proof (induction i rule: nat_descend_induct[of "degree a"])
case (base i)
thus ?case by (simp add: coeff_eq_0)
next
case (descend i)
let ?A = "{..i+m} - {i}"
have "c dvd coeff (a * b) (i + m)" by (rule dvd)
also have "coeff (a * b) (i + m) = (∑k≤i + m. coeff a k * coeff b (i + m - k))"
also have "{..i+m} = insert i ?A" by auto
also have "(∑k∈…. coeff a k * coeff b (i + m - k)) =
coeff a i * coeff b m + (∑k∈?A. coeff a k * coeff b (i + m - k))"
(is "_ = _ + ?S")
by (subst sum.insert) simp_all
finally have eq: "c dvd coeff a i * coeff b m + ?S" .
moreover have "c dvd ?S"
proof (rule dvd_sum)
fix k assume k: "k ∈ {..i+m} - {i}"
show "c dvd coeff a k * coeff b (i + m - k)"
proof (cases "k < i")
case False
with k have "c dvd coeff a k" by (intro descend.IH) simp
thus ?thesis by simp
next
case True
hence "c dvd coeff b (i + m - k)" by (intro dvd_b) simp
thus ?thesis by simp
qed
qed
ultimately have "c dvd coeff a i * coeff b m"
with assms coeff_m show "c dvd coeff a i"
qed
hence "[:c:] dvd a" by (subst const_poly_dvd_iff) blast
}
then show "[:c:] dvd a ∨ [:c:] dvd b" by blast
next
from assms show "[:c:] ≠ 0" and "¬ [:c:] dvd 1"
qed

lemma prime_elem_const_poly_iff:
fixes c :: "'a :: semidom"
shows   "prime_elem [:c:] ⟷ prime_elem c"
proof
assume A: "prime_elem [:c:]"
show "prime_elem c"
proof (rule prime_elemI)
fix a b assume "c dvd a * b"
hence "[:c:] dvd [:a:] * [:b:]" by (simp add: mult_ac)
from A and this have "[:c:] dvd [:a:] ∨ [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD)
thus "c dvd a ∨ c dvd b" by simp
qed (insert A, auto simp: prime_elem_def is_unit_poly_iff)
qed (auto intro: lift_prime_elem_poly)

lemma fract_poly_dvdD:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,
semiring_gcd_mult_normalize} poly"
assumes "fract_poly p dvd fract_poly q" "content p = 1"
shows   "p dvd q"
proof -
from assms(1) obtain r where r: "fract_poly q = fract_poly p * r" by (erule dvdE)
from content_decompose_fract[of r] guess c r' . note r' = this
from r r' have eq: "fract_poly q = smult c (fract_poly (p * r'))" by simp
from fract_poly_smult_eqE[OF this] guess a b . note ab = this
have "content (smult a q) = content (smult b (p * r'))" by (simp only: ab(2))
hence eq': "normalize b = a * content q" by (simp add: assms content_mult r' ab(4))
have "1 = gcd a (normalize b)" by (simp add: ab)
also note eq'
also have "gcd a (a * content q) = a" by (simp add: gcd_proj1_if_dvd ab(4))
finally have [simp]: "a = 1" by simp
from eq ab have "q = p * ([:b:] * r')" by simp
thus ?thesis by (rule dvdI)
qed

subsection ‹Polynomials over a field are a Euclidean ring›

context
begin

interpretation field_poly:
normalization_euclidean_semiring_multiplicative where zero = "0 :: 'a :: field poly"
and one = 1 and plus = plus and minus = minus
and times = times
and normalize = "λp. smult (inverse (lead_coeff p)) p"
and unit_factor = "λp. [:lead_coeff p:]"
and euclidean_size = "λp. if p = 0 then 0 else 2 ^ degree p"
and divide = divide and modulo = modulo
rewrites "dvd.dvd (times :: 'a poly ⇒ _) = Rings.dvd"
and "comm_monoid_mult.prod_mset times 1 = prod_mset"
and "comm_semiring_1.irreducible times 1 0 = irreducible"
and "comm_semiring_1.prime_elem times 1 0 = prime_elem"
proof -
show "dvd.dvd (times :: 'a poly ⇒ _) = Rings.dvd"
show "comm_monoid_mult.prod_mset times 1 = prod_mset"
show "comm_semiring_1.irreducible times 1 0 = irreducible"
show "comm_semiring_1.prime_elem times 1 0 = prime_elem"
show "class.normalization_euclidean_semiring_multiplicative divide plus minus (0 :: 'a poly) times 1
modulo (λp. if p = 0 then 0 else 2 ^ degree p)
proof (standard, fold dvd_dict)
fix p :: "'a poly"
by (cases "p = 0") simp_all
next
fix p :: "'a poly" assume "is_unit p"
then show "[:lead_coeff p:] = p"
by (elim is_unit_polyE) (auto simp: monom_0 one_poly_def field_simps)
next
fix p :: "'a poly" assume "p ≠ 0"
next
fix a b :: "'a poly" assume "is_unit a"
by (auto elim!: is_unit_polyE)
qed (auto simp: lead_coeff_mult Rings.div_mult_mod_eq intro!: degree_mod_less' degree_mult_right_le)
qed

lemma field_poly_irreducible_imp_prime:
"prime_elem p" if "irreducible p" for p :: "'a :: field poly"
using that by (fact field_poly.irreducible_imp_prime_elem)
find_theorems name:prod_mset_prime_factorization
lemma field_poly_prod_mset_prime_factorization:
"prod_mset (field_poly.prime_factorization p) = smult (inverse (lead_coeff p)) p"
if "p ≠ 0" for p :: "'a :: field poly"
using that by (fact field_poly.prod_mset_prime_factorization)

lemma field_poly_in_prime_factorization_imp_prime:
"prime_elem p" if "p ∈# field_poly.prime_factorization x"
for p :: "'a :: field poly"
by (rule field_poly.prime_imp_prime_elem, rule field_poly.in_prime_factors_imp_prime)
(fact that)

subsection ‹Primality and irreducibility in polynomial rings›

lemma nonconst_poly_irreducible_iff:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly"
assumes "degree p ≠ 0"
shows   "irreducible p ⟷ irreducible (fract_poly p) ∧ content p = 1"
proof safe
assume p: "irreducible p"

from content_decompose[of p] guess p' . note p' = this
hence "p = [:content p:] * p'" by simp
from p this have "[:content p:] dvd 1 ∨ p' dvd 1" by (rule irreducibleD)
moreover have "¬p' dvd 1"
proof
assume "p' dvd 1"
hence "degree p = 0" by (subst p') (auto simp: is_unit_poly_iff)
with assms show False by contradiction
qed
ultimately show [simp]: "content p = 1" by (simp add: is_unit_const_poly_iff)

show "irreducible (map_poly to_fract p)"
proof (rule irreducibleI)
have "fract_poly p = 0 ⟷ p = 0" by (intro map_poly_eq_0_iff) auto
with assms show "map_poly to_fract p ≠ 0" by auto
next
show "¬is_unit (fract_poly p)"
proof
assume "is_unit (map_poly to_fract p)"
hence "degree (map_poly to_fract p) = 0"
by (auto simp: is_unit_poly_iff)
hence "degree p = 0" by (simp add: degree_map_poly)
with assms show False by contradiction
qed
next
fix q r assume qr: "fract_poly p = q * r"
from content_decompose_fract[of q] guess cg q' . note q = this
from content_decompose_fract[of r] guess cr r' . note r = this
from qr q r p have nz: "cg ≠ 0" "cr ≠ 0" by auto
from qr have eq: "fract_poly p = smult (cr * cg) (fract_poly (q' * r'))"
from fract_poly_smult_eqE[OF this] guess a b . note ab = this
hence "content (smult a p) = content (smult b (q' * r'))" by (simp only:)
with ab(4) have a: "a = normalize b" by (simp add: content_mult q r)
then have "normalize b = gcd a b"
by simp
with ‹coprime a b› have "normalize b = 1"
by simp
then have "a = 1" "is_unit b"

note eq
also from ab(1) ‹a = 1› have "cr * cg = to_fract b" by simp
also have "smult … (fract_poly (q' * r')) = fract_poly (smult b (q' * r'))" by simp
finally have "p = ([:b:] * q') * r'" by (simp del: fract_poly_smult)
from p and this have "([:b:] * q') dvd 1 ∨ r' dvd 1" by (rule irreducibleD)
hence "q' dvd 1 ∨ r' dvd 1" by (auto dest: dvd_mult_right simp del: mult_pCons_left)
hence "fract_poly q' dvd 1 ∨ fract_poly r' dvd 1" by (auto simp: fract_poly_is_unit)
with q r show "is_unit q ∨ is_unit r"
by (auto simp add: is_unit_smult_iff dvd_field_iff nz)
qed

next

assume irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
show "irreducible p"
proof (rule irreducibleI)
from irred show "p ≠ 0" by auto
next
from irred show "¬p dvd 1"
by (auto simp: irreducible_def dest: fract_poly_is_unit)
next
fix q r assume qr: "p = q * r"
hence "fract_poly p = fract_poly q * fract_poly r" by simp
from irred and this have "fract_poly q dvd 1 ∨ fract_poly r dvd 1"
by (rule irreducibleD)
with primitive qr show "q dvd 1 ∨ r dvd 1"
by (auto simp:  content_prod_eq_1_iff is_unit_fract_poly_iff)
qed
qed

private lemma irreducible_imp_prime_poly:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly"
assumes "irreducible p"
shows   "prime_elem p"
proof (cases "degree p = 0")
case True
with assms show ?thesis
by (auto simp: prime_elem_const_poly_iff irreducible_const_poly_iff
intro!: irreducible_imp_prime_elem elim!: degree_eq_zeroE)
next
case False
from assms False have irred: "irreducible (fract_poly p)" and primitive: "content p = 1"
from irred have prime: "prime_elem (fract_poly p)" by (rule field_poly_irreducible_imp_prime)
show ?thesis
proof (rule prime_elemI)
fix q r assume "p dvd q * r"
hence "fract_poly p dvd fract_poly (q * r)" by (rule fract_poly_dvd)
hence "fract_poly p dvd fract_poly q * fract_poly r" by simp
from prime and this have "fract_poly p dvd fract_poly q ∨ fract_poly p dvd fract_poly r"
by (rule prime_elem_dvd_multD)
with primitive show "p dvd q ∨ p dvd r" by (auto dest: fract_poly_dvdD)
qed (insert assms, auto simp: irreducible_def)
qed

lemma degree_primitive_part_fract [simp]:
"degree (primitive_part_fract p) = degree p"
proof -
have "p = smult (fract_content p) (fract_poly (primitive_part_fract p))"
also have "degree … = degree (primitive_part_fract p)"
by (auto simp: degree_map_poly)
finally show ?thesis ..
qed

lemma irreducible_primitive_part_fract:
fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd,semiring_gcd_mult_normalize} fract poly"
assumes "irreducible p"
shows   "irreducible (primitive_part_fract p)"
proof -
from assms have deg: "degree (primitive_part_fract p) ≠ 0"
by (intro notI)
(auto elim!: degree_eq_zeroE simp: irreducible_def is_unit_poly_iff dvd_field_iff)
hence [simp]: "p ≠ 0" by auto

note ‹irreducible p›
also have "p = [:fract_content p:] * fract_poly (primitive_part_fract p)"
also have "irreducible … ⟷ irreducible (fract_poly (primitive_part_fract p))"
by (intro irreducible_mult_unit_left) (simp_all add: is_unit_poly_iff dvd_field_iff)
finally show ?thesis using deg
qed

lemma prime_elem_primitive_part_fract:
fixes p :: "'a :: {idom_divide, ring_gcd, factorial_semiring, semiring_Gcd,semiring_gcd_mult_normalize} fract poly"
shows "irreducible p ⟹ prime_elem (primitive_part_fract p)"
by (intro irreducible_imp_prime_poly irreducible_primitive_part_fract)

lemma irreducible_linear_field_poly:
fixes a b :: "'a::field"
assumes "b ≠ 0"
shows "irreducible [:a,b:]"
proof (rule irreducibleI)
fix p q assume pq: "[:a,b:] = p * q"
also from pq assms have "degree … = degree p + degree q"
by (intro degree_mult_eq) auto
finally have "degree p = 0 ∨ degree q = 0" using assms by auto
with assms pq show "is_unit p ∨ is_unit q"
by (auto simp: is_unit_const_poly_iff dvd_field_iff elim!: degree_eq_zeroE)
qed (insert assms, auto simp: is_unit_poly_iff)

lemma prime_elem_linear_field_poly:
"(b :: 'a :: field) ≠ 0 ⟹ prime_elem [:a,b:]"
by (rule field_poly_irreducible_imp_prime, rule irreducible_linear_field_poly)

lemma irreducible_linear_poly:
fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd,semiring_gcd_mult_normalize}"
shows "b ≠ 0 ⟹ coprime a b ⟹ irreducible [:a,b:]"
by (auto intro!: irreducible_linear_field_poly
simp:   nonconst_poly_irreducible_iff content_def map_poly_pCons)

lemma prime_elem_linear_poly:
fixes a b :: "'a::{idom_divide,ring_gcd,factorial_semiring,semiring_Gcd,semiring_gcd_mult_normalize}"
shows "b ≠ 0 ⟹ coprime a b ⟹ prime_elem [:a,b:]"
by (rule irreducible_imp_prime_poly, rule irreducible_linear_poly)

subsection ‹Prime factorisation of polynomials›

private lemma poly_prime_factorization_exists_content_1:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly"
assumes "p ≠ 0" "content p = 1"
shows   "∃A. (∀p. p ∈# A ⟶ prime_elem p) ∧ prod_mset A = normalize p"
proof -
let ?P = "field_poly.prime_factorization (fract_poly p)"
define c where "c = prod_mset (image_mset fract_content ?P)"
define c' where "c' = c * to_fract (lead_coeff p)"
define e where "e = prod_mset (image_mset primitive_part_fract ?P)"
define A where "A = image_mset (normalize ∘ primitive_part_fract) ?P"
have "content e = (∏x∈#field_poly.prime_factorization (map_poly to_fract p).
content (primitive_part_fract x))"
by (simp add: e_def content_prod_mset multiset.map_comp o_def)
also have "image_mset (λx. content (primitive_part_fract x)) ?P = image_mset (λ_. 1) ?P"
by (intro image_mset_cong content_primitive_part_fract) auto
finally have content_e: "content e = 1"
by simp

from ‹p ≠ 0› have "fract_poly p = [:lead_coeff (fract_poly p):] *
smult (inverse (lead_coeff (fract_poly p))) (fract_poly p)"
by simp
by (simp add: monom_0 degree_map_poly coeff_map_poly)
also from assms have "smult (inverse (lead_coeff (fract_poly p))) (fract_poly p) = prod_mset ?P"
by (subst field_poly_prod_mset_prime_factorization) simp_all
also have "… = prod_mset (image_mset id ?P)" by simp
also have "image_mset id ?P =
image_mset (λx. [:fract_content x:] * fract_poly (primitive_part_fract x)) ?P"
by (intro image_mset_cong) (auto simp: content_times_primitive_part_fract)
also have "prod_mset … = smult c (fract_poly e)"
by (subst prod_mset.distrib) (simp_all add: prod_mset_fract_poly prod_mset_const_poly c_def e_def)
also have "[:to_fract (lead_coeff p):] * … = smult c' (fract_poly e)"
finally have eq: "fract_poly p = smult c' (fract_poly e)" .
also obtain b where b: "c' = to_fract b" "is_unit b"
proof -
from fract_poly_smult_eqE[OF eq] guess a b . note ab = this
from ab(2) have "content (smult a p) = content (smult b e)" by (simp only: )
with assms content_e have "a = normalize b" by (simp add: ab(4))
with ab have ab': "a = 1" "is_unit b"
with ab ab' have "c' = to_fract b" by auto
from this and ‹is_unit b› show ?thesis by (rule that)
qed
hence "smult c' (fract_poly e) = fract_poly (smult b e)" by simp
finally have "p = smult b e" by (simp only: fract_poly_eq_iff)
hence "p = [:b:] * e" by simp
with b have "normalize p = normalize e"
by (simp only: normalize_mult) (simp add: is_unit_normalize is_unit_poly_iff)
also have "normalize e = prod_mset A"
by (simp add: multiset.map_comp e_def A_def normalize_prod_mset)
finally have "prod_mset A = normalize p" ..

have "prime_elem p" if "p ∈# A" for p
using that by (auto simp: A_def prime_elem_primitive_part_fract prime_elem_imp_irreducible
dest!: field_poly_in_prime_factorization_imp_prime )
from this and ‹prod_mset A = normalize p› show ?thesis
by (intro exI[of _ A]) blast
qed

lemma poly_prime_factorization_exists:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide,semiring_gcd_mult_normalize} poly"
assumes "p ≠ 0"
shows   "∃A. (∀p. p ∈# A ⟶ prime_elem p) ∧ normalize (prod_mset A) = normalize p"
proof -
define B where "B = image_mset (λx. [:x:]) (prime_factorization (content p))"
have "∃A. (∀p. p ∈# A ⟶ prime_elem p) ∧ prod_mset A = normalize (primitive_part p)"
by (rule poly_prime_factorization_exists_content_1) (insert assms, simp_all)
then guess A by (elim exE conjE) note A = this
have "normalize (prod_mset (A + B)) = normalize (prod_mset A * normalize (prod_mset B))"
by simp
also from assms have "normalize (prod_mset B) = normalize [:content p:]"
by (simp add: prod_mset_const_poly normalize_const_poly prod_mset_prime_factorization_weak B_def)
also have "prod_mset A = normalize (primitive_part p)"
using A by simp
finally have "normalize (prod_mset (A + B)) = normalize (primitive_part p * [:content p:])"
by simp
moreover have "∀p. p ∈# B ⟶ prime_elem p"
by (auto simp: B_def intro!: lift_prime_elem_poly dest: in_prime_factors_imp_prime)
ultimately show ?thesis using A by (intro exI[of _ "A + B"]) (auto)
qed

end

subsection ‹Typeclass instances›

instance poly :: ("{factorial_ring_gcd,semiring_gcd_mult_normalize}") factorial_semiring
by standard (rule poly_prime_factorization_exists)

instantiation poly :: ("{factorial_ring_gcd, semiring_gcd_mult_normalize}") factorial_ring_gcd
begin

definition gcd_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" where
[code del]: "gcd_poly = gcd_factorial"

definition lcm_poly :: "'a poly ⇒ 'a poly ⇒ 'a poly" where
[code del]: "lcm_poly = lcm_factorial"

definition Gcd_poly :: "'a poly set ⇒ 'a poly" where
[code del]: "Gcd_poly = Gcd_factorial"

definition Lcm_poly :: "'a poly set ⇒ 'a poly" where
[code del]: "Lcm_poly = Lcm_factorial"

instance by standard (simp_all add: gcd_poly_def lcm_poly_def Gcd_poly_def Lcm_poly_def)

end

instance poly :: ("{factorial_ring_gcd, semiring_gcd_mult_normalize}") semiring_gcd_mult_normalize ..

instantiation poly :: ("{field,factorial_ring_gcd,semiring_gcd_mult_normalize}")
"{unique_euclidean_ring, normalization_euclidean_semiring}"
begin

definition euclidean_size_poly :: "'a poly ⇒ nat"
where "euclidean_size_poly p = (if p = 0 then 0 else 2 ^ degree p)"

definition division_segment_poly :: "'a poly ⇒ 'a poly"
where [simp]: "division_segment_poly p = 1"

instance proof
show "(q * p + r) div p = q" if "p ≠ 0"
and "euclidean_size r < euclidean_size p" for q p r :: "'a poly"
proof -
from ‹p ≠ 0› eucl_rel_poly [of r p] have "eucl_rel_poly (r + q * p) p (q + r div p, r mod p)"
then have "(r + q * p) div p = q + r div p"
by (rule div_poly_eq)
with that show ?thesis
by (cases "r = 0") (simp_all add: euclidean_size_poly_def div_poly_less ac_simps)
qed
qed (auto simp: euclidean_size_poly_def Rings.div_mult_mod_eq div_poly_less degree_mult_eq power_add
intro!: degree_mod_less' split: if_splits)

end

instance poly :: ("{field, normalization_euclidean_semiring, factorial_ring_gcd,
semiring_gcd_mult_normalize}") euclidean_ring_gcd
by (rule euclidean_ring_gcd_class.intro, rule factorial_euclidean_semiring_gcdI) standard

subsection ‹Polynomial GCD›

lemma gcd_poly_decompose:
fixes p q :: "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize} poly"
shows "gcd p q =
smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
proof (rule sym, rule gcdI)
have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
[:content p:] * primitive_part p" by (intro mult_dvd_mono) simp_all
thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd p"
by simp
next
have "[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q) dvd
[:content q:] * primitive_part q" by (intro mult_dvd_mono) simp_all
thus "smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q)) dvd q"
by simp
next
fix d assume "d dvd p" "d dvd q"
hence "[:content d:] * primitive_part d dvd
[:gcd (content p) (content q):] * gcd (primitive_part p) (primitive_part q)"
by (intro mult_dvd_mono) auto
thus "d dvd smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
by simp
qed (auto simp: normalize_smult)

lemma gcd_poly_pseudo_mod:
fixes p q :: "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize} poly"
assumes nz: "q ≠ 0" and prim: "content p = 1" "content q = 1"
shows   "gcd p q = gcd q (primitive_part (pseudo_mod p q))"
proof -
define r s where "r = fst (pseudo_divmod p q)" and "s = snd (pseudo_divmod p q)"
define a where "a = [:coeff q (degree q) ^ (Suc (degree p) - degree q):]"
have [simp]: "primitive_part a = unit_factor a"
by (simp add: a_def unit_factor_poly_def unit_factor_power monom_0)
from nz have [simp]: "a ≠ 0" by (auto simp: a_def)

have rs: "pseudo_divmod p q = (r, s)" by (simp add: r_def s_def)
have "gcd (q * r + s) q = gcd q s"
with pseudo_divmod(1)[OF nz rs]
have "gcd (p * a) q = gcd q s" by (simp add: a_def)
also from prim have "gcd (p * a) q = gcd p q"
by (subst gcd_poly_decompose)
(auto simp: primitive_part_mult gcd_mult_unit1 primitive_part_prim
simp del: mult_pCons_right )
also from prim have "gcd q s = gcd q (primitive_part s)"
by (subst gcd_poly_decompose) (simp_all add: primitive_part_prim)
also have "s = pseudo_mod p q" by (simp add: s_def pseudo_mod_def)
finally show ?thesis .
qed

lemma degree_pseudo_mod_less:
assumes "q ≠ 0" "pseudo_mod p q ≠ 0"
shows   "degree (pseudo_mod p q) < degree q"
using pseudo_mod(2)[of q p] assms by auto

function gcd_poly_code_aux :: "'a :: factorial_ring_gcd poly ⇒ 'a poly ⇒ 'a poly" where
"gcd_poly_code_aux p q =
(if q = 0 then normalize p else gcd_poly_code_aux q (primitive_part (pseudo_mod p q)))"
by auto
termination
by (relation "measure ((λp. if p = 0 then 0 else Suc (degree p)) ∘ snd)")
(auto simp: degree_pseudo_mod_less)

declare gcd_poly_code_aux.simps [simp del]

lemma gcd_poly_code_aux_correct:
assumes "content p = 1" "q = 0 ∨ content q = 1"
shows   "gcd_poly_code_aux p q = gcd p q"
using assms
proof (induction p q rule: gcd_poly_code_aux.induct)
case (1 p q)
show ?case
proof (cases "q = 0")
case True
thus ?thesis by (subst gcd_poly_code_aux.simps) auto
next
case False
hence "gcd_poly_code_aux p q = gcd_poly_code_aux q (primitive_part (pseudo_mod p q))"
by (subst gcd_poly_code_aux.simps) simp_all
also from "1.prems" False
have "primitive_part (pseudo_mod p q) = 0 ∨
content (primitive_part (pseudo_mod p q)) = 1"
by (cases "pseudo_mod p q = 0") auto
with "1.prems" False
have "gcd_poly_code_aux q (primitive_part (pseudo_mod p q)) =
gcd q (primitive_part (pseudo_mod p q))"
by (intro 1) simp_all
also from "1.prems" False
have "… = gcd p q" by (intro gcd_poly_pseudo_mod [symmetric]) auto
finally show ?thesis .
qed
qed

definition gcd_poly_code
:: "'a :: factorial_ring_gcd poly ⇒ 'a poly ⇒ 'a poly"
where "gcd_poly_code p q =
(if p = 0 then normalize q else if q = 0 then normalize p else
smult (gcd (content p) (content q))
(gcd_poly_code_aux (primitive_part p) (primitive_part q)))"

lemma gcd_poly_code [code]: "gcd p q = gcd_poly_code p q"
by (simp add: gcd_poly_code_def gcd_poly_code_aux_correct gcd_poly_decompose [symmetric])

lemma lcm_poly_code [code]:
fixes p q :: "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize} poly"
shows "lcm p q = normalize (p * q div gcd p q)"
by (fact lcm_gcd)

lemmas Gcd_poly_set_eq_fold [code] =
Gcd_set_eq_fold [where ?'a = "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize} poly"]
lemmas Lcm_poly_set_eq_fold [code] =
Lcm_set_eq_fold [where ?'a = "'a :: {factorial_ring_gcd,semiring_gcd_mult_normalize} poly"]

text ‹Example:
@{lemma "Lcm {[:1, 2, 3:], [:2, 3, 4:]} = [:[:2:], [:7:], [:16:], [:17:], [:12 :: int:]:]" by eval}
›

end
```