src/HOL/Library/Polynomial_Factorial.thy
 author haftmann Wed, 05 Apr 2017 13:47:40 +0200 changeset 65389 6f9c6ae27984 parent 65366 10ca63a18e56 child 65390 83586780598b permissions -rw-r--r--
tuned
```
(*  Title:      HOL/Library/Polynomial_Factorial.thy
Author:     Brian Huffman
Author:     Clemens Ballarin
Author:     Amine Chaieb
Author:     Florian Haftmann
Author:     Manuel Eberl
*)

theory Polynomial_Factorial
imports
Complex_Main
Polynomial
Normalized_Fraction
Field_as_Ring
begin

lemma prod_mset_const_poly: " (\<Prod>x\<in>#A. [:f x:]) = [:prod_mset (image_mset f A):]"
by (induct A) (simp_all add: one_poly_def ac_simps)

lemma irreducible_const_poly_iff:
fixes c :: "'a :: {comm_semiring_1,semiring_no_zero_divisors}"
shows "irreducible [:c:] \<longleftrightarrow> irreducible c"
proof
assume A: "irreducible c"
show "irreducible [:c:]"
proof (rule irreducibleI)
fix a b assume ab: "[:c:] = a * b"
hence "degree [:c:] = degree (a * b)" by (simp only: )
also from A ab have "a \<noteq> 0" "b \<noteq> 0" by auto
hence "degree (a * b) = degree a + degree b" by (simp add: degree_mult_eq)
finally have "degree a = 0" "degree b = 0" by auto
then obtain a' b' where ab': "a = [:a':]" "b = [:b':]" by (auto elim!: degree_eq_zeroE)
from ab have "coeff [:c:] 0 = coeff (a * b) 0" by (simp only: )
hence "c = a' * b'" by (simp add: ab' mult_ac)
from A and this have "a' dvd 1 \<or> b' dvd 1" by (rule irreducibleD)
with ab' show "a dvd 1 \<or> b dvd 1" by (auto simp: one_poly_def)
qed (insert A, auto simp: irreducible_def is_unit_poly_iff)
next
assume A: "irreducible [:c:]"
show "irreducible c"
proof (rule irreducibleI)
fix a b assume ab: "c = a * b"
hence "[:c:] = [:a:] * [:b:]" by (simp add: mult_ac)
from A and this have "[:a:] dvd 1 \<or> [:b:] dvd 1" by (rule irreducibleD)
thus "a dvd 1 \<or> b dvd 1" by (simp add: one_poly_def)
qed (insert A, auto simp: irreducible_def one_poly_def)
qed

subsection \<open>Lifting elements into the field of fractions\<close>

definition to_fract :: "'a :: idom \<Rightarrow> 'a fract" where "to_fract x = Fract x 1"
\<comment> \<open>FIXME: name \<open>of_idom\<close>, abbreviation\<close>

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 \<longleftrightarrow> x = y"

lemma to_fract_eq_0_iff [simp]: "to_fract x = 0 \<longleftrightarrow> 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) \<noteq> 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 \<longleftrightarrow> 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 \<Longrightarrow> 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 \<open>Lifting polynomial coefficients to the field of fractions\<close>

abbreviation (input) fract_poly
where "fract_poly \<equiv> map_poly to_fract"

abbreviation (input) unfract_poly
where "unfract_poly \<equiv> map_poly (fst \<circ> 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 (\<lambda>x. 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 \<longleftrightarrow> p = q"
by (auto simp: poly_eq_iff coeff_map_poly)

lemma fract_poly_eq_0_iff [simp]: "fract_poly p = 0 \<longleftrightarrow> p = 0"
using fract_poly_eq_iff[of p 0] by (simp del: fract_poly_eq_iff)

lemma fract_poly_dvd: "p dvd q \<Longrightarrow> fract_poly p dvd fract_poly q"
by (auto elim!: dvdE)

lemma prod_mset_fract_poly:
"prod_mset (image_mset (\<lambda>x. fract_poly (f x)) A) = fract_poly (prod_mset (image_mset f A))"
by (induct A) (simp_all add: one_poly_def ac_simps)

lemma is_unit_fract_poly_iff:
"p dvd 1 \<longleftrightarrow> fract_poly p dvd 1 \<and> 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 \<open>n > 0\<close> have "coeff [:c:] n = 0" by (simp add: coeff_pCons split: nat.splits)
finally have "coeff p n = 0" by simp
}
hence "degree p \<le> 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 \<Longrightarrow> 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} 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 gcd.commute
normalize_snd_quot_of_fract Fract_conv_to_fract [symmetric])
ultimately show ?thesis by (intro that[of a b])
qed

subsection \<open>Fractional content\<close>

abbreviation (input) Lcm_coeff_denoms
:: "'a :: {semiring_Gcd,idom_divide,ring_gcd} fract poly \<Rightarrow> 'a"
where "Lcm_coeff_denoms p \<equiv> Lcm (snd ` quot_of_fract ` set (coeffs p))"

definition fract_content ::
"'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} fract poly \<Rightarrow> '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} fract poly \<Rightarrow> '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 \<longleftrightarrow> 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]: "p \<noteq> 0 \<Longrightarrow> 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 \<circ> fst \<circ> 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 "\<dots> = smult (to_fract (Lcm_coeff_denoms p)) p"
proof (intro map_poly_idI, unfold o_apply)
fix c assume "c \<in> set (coeffs (smult (to_fract (Lcm_coeff_denoms p)) p))"
then obtain c' where c: "c' \<in> 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) * \<dots> =
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 \<dots>) = 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 \<circ> quot_of_fract \<circ> 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 \<circ> quot_of_fract \<circ> 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} 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 \<open>More properties of content and primitive part\<close>

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. \<not>c dvd coeff b m)"
assume "\<not>[:c:] dvd b"
hence A: "\<exists>i. \<not>c dvd coeff b i" by (subst (asm) const_poly_dvd_iff) blast
have B: "\<forall>i. \<not>c dvd coeff b i \<longrightarrow> i < Suc (degree b)"
by (auto intro: le_degree simp: less_Suc_eq_le)
have coeff_m: "\<not>c dvd coeff b m" unfolding m_def by (rule GreatestI_ex[OF A B])
have "i \<le> m" if "\<not>c dvd coeff b i" for i
unfolding m_def by (rule Greatest_le[OF 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) = (\<Sum>k\<le>i + m. coeff a k * coeff b (i + m - k))"
also have "{..i+m} = insert i ?A" by auto
also have "(\<Sum>k\<in>\<dots>. coeff a k * coeff b (i + m - k)) =
coeff a i * coeff b m + (\<Sum>k\<in>?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 \<in> {..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
}
thus "[:c:] dvd a \<or> [:c:] dvd b" by blast
qed (insert assms, simp_all add: prime_elem_def one_poly_def)

lemma prime_elem_const_poly_iff:
fixes c :: "'a :: semidom"
shows   "prime_elem [:c:] \<longleftrightarrow> 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:] \<or> [:c:] dvd [:b:]" by (rule prime_elem_dvd_multD)
thus "c dvd a \<or> c dvd b" by simp
qed (insert A, auto simp: prime_elem_def is_unit_poly_iff)
qed (auto intro: lift_prime_elem_poly)

context
begin

private lemma content_1_mult:
fixes f g :: "'a :: {semiring_Gcd,factorial_semiring} poly"
assumes "content f = 1" "content g = 1"
shows   "content (f * g) = 1"
proof (cases "f * g = 0")
case False
from assms have "f \<noteq> 0" "g \<noteq> 0" by auto

hence "f * g \<noteq> 0" by auto
{
assume "\<not>is_unit (content (f * g))"
with False have "\<exists>p. p dvd content (f * g) \<and> prime p"
by (intro prime_divisor_exists) simp_all
then obtain p where "p dvd content (f * g)" "prime p" by blast
from \<open>p dvd content (f * g)\<close> have "[:p:] dvd f * g"
moreover from \<open>prime p\<close> have "prime_elem [:p:]" by (simp add: lift_prime_elem_poly)
ultimately have "[:p:] dvd f \<or> [:p:] dvd g"
with assms have "is_unit p" by (simp add: const_poly_dvd_iff_dvd_content)
with \<open>prime p\<close> have False by simp
}
hence "is_unit (content (f * g))" by blast
hence "normalize (content (f * g)) = 1" by (simp add: is_unit_normalize del: normalize_content)
thus ?thesis by simp
qed (insert assms, auto)

lemma content_mult:
fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
shows "content (p * q) = content p * content q"
proof -
from content_decompose[of p] guess p' . note p = this
from content_decompose[of q] guess q' . note q = this
have "content (p * q) = content p * content q * content (p' * q')"
by (subst p, subst q) (simp add: mult_ac normalize_mult)
also from p q have "content (p' * q') = 1" by (intro content_1_mult)
finally show ?thesis by simp
qed

lemma primitive_part_mult:
fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
shows "primitive_part (p * q) = primitive_part p * primitive_part q"
proof -
have "primitive_part (p * q) = p * q div [:content (p * q):]"
also have "\<dots> = (p div [:content p:]) * (q div [:content q:])"
by (subst div_mult_div_if_dvd) (simp_all add: content_mult mult_ac)
also have "\<dots> = primitive_part p * primitive_part q"
finally show ?thesis .
qed

lemma primitive_part_smult:
fixes p :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
shows "primitive_part (smult a p) = smult (unit_factor a) (primitive_part p)"
proof -
have "smult a p = [:a:] * p" by simp
also have "primitive_part \<dots> = smult (unit_factor a) (primitive_part p)"
by (subst primitive_part_mult) simp_all
finally show ?thesis .
qed

lemma primitive_part_dvd_primitive_partI [intro]:
fixes p q :: "'a :: {factorial_semiring, semiring_Gcd, ring_gcd, idom_divide} poly"
shows "p dvd q \<Longrightarrow> primitive_part p dvd primitive_part q"
by (auto elim!: dvdE simp: primitive_part_mult)

lemma content_prod_mset:
fixes A :: "'a :: {factorial_semiring, semiring_Gcd} poly multiset"
shows "content (prod_mset A) = prod_mset (image_mset content A)"
by (induction A) (simp_all add: content_mult mult_ac)

lemma fract_poly_dvdD:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} 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

lemma content_prod_eq_1_iff:
fixes p q :: "'a :: {factorial_semiring, semiring_Gcd} poly"
shows "content (p * q) = 1 \<longleftrightarrow> content p = 1 \<and> content q = 1"
proof safe
assume A: "content (p * q) = 1"
{
fix p q :: "'a poly" assume "content p * content q = 1"
hence "1 = content p * content q" by simp
hence "content p dvd 1" by (rule dvdI)
hence "content p = 1" by simp
} note B = this
from A B[of p q] B [of q p] show "content p = 1" "content q = 1"
qed (auto simp: content_mult)

end

subsection \<open>Polynomials over a field are a Euclidean ring\<close>

definition unit_factor_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where

definition normalize_field_poly :: "'a :: field poly \<Rightarrow> 'a poly" where
"normalize_field_poly p = smult (inverse (lead_coeff p)) p"

definition euclidean_size_field_poly :: "'a :: field poly \<Rightarrow> nat" where
"euclidean_size_field_poly p = (if p = 0 then 0 else 2 ^ degree p)"

lemma dvd_field_poly: "dvd.dvd (op * :: 'a :: field poly \<Rightarrow> _) = op dvd"
by (intro ext) (simp_all add: dvd.dvd_def dvd_def)

interpretation field_poly:
unique_euclidean_ring where zero = "0 :: 'a :: field poly"
and one = 1 and plus = plus and uminus = uminus and minus = minus
and times = times
and normalize = normalize_field_poly and unit_factor = unit_factor_field_poly
and euclidean_size = euclidean_size_field_poly
and uniqueness_constraint = top
and divide = divide and modulo = modulo
proof (standard, unfold dvd_field_poly)
fix p :: "'a poly"
show "unit_factor_field_poly p * normalize_field_poly p = p"
by (cases "p = 0")
next
fix p :: "'a poly" assume "is_unit p"
then show "unit_factor_field_poly p = p"
by (elim is_unit_polyE) (auto simp: unit_factor_field_poly_def monom_0 one_poly_def field_simps)
next
fix p :: "'a poly" assume "p \<noteq> 0"
thus "is_unit (unit_factor_field_poly p)"
next
fix p q s :: "'a poly" assume "s \<noteq> 0"
moreover assume "euclidean_size_field_poly p < euclidean_size_field_poly q"
ultimately show "euclidean_size_field_poly (p * s) < euclidean_size_field_poly (q * s)"
by (auto simp add: euclidean_size_field_poly_def degree_mult_eq)
next
fix p q r :: "'a poly" assume "p \<noteq> 0"
moreover assume "euclidean_size_field_poly r < euclidean_size_field_poly p"
ultimately show "(q * p + r) div p = q"
by (cases "r = 0")
(auto simp add: unit_factor_field_poly_def euclidean_size_field_poly_def div_poly_less)
qed (auto simp: unit_factor_field_poly_def normalize_field_poly_def lead_coeff_mult
euclidean_size_field_poly_def Rings.div_mult_mod_eq intro!: degree_mod_less' degree_mult_right_le)

lemma field_poly_irreducible_imp_prime:
assumes "irreducible (p :: 'a :: field poly)"
shows   "prime_elem p"
proof -
have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
from field_poly.irreducible_imp_prime_elem[of p] assms
show ?thesis unfolding irreducible_def prime_elem_def dvd_field_poly
comm_semiring_1.irreducible_def[OF A] comm_semiring_1.prime_elem_def[OF A] by blast
qed

lemma field_poly_prod_mset_prime_factorization:
assumes "(x :: 'a :: field poly) \<noteq> 0"
shows   "prod_mset (field_poly.prime_factorization x) = normalize_field_poly x"
proof -
have A: "class.comm_monoid_mult op * (1 :: 'a poly)" ..
have "comm_monoid_mult.prod_mset op * (1 :: 'a poly) = prod_mset"
by (intro ext) (simp add: comm_monoid_mult.prod_mset_def[OF A] prod_mset_def)
with field_poly.prod_mset_prime_factorization[OF assms] show ?thesis by simp
qed

lemma field_poly_in_prime_factorization_imp_prime:
assumes "(p :: 'a :: field poly) \<in># field_poly.prime_factorization x"
shows   "prime_elem p"
proof -
have A: "class.comm_semiring_1 op * 1 op + (0 :: 'a poly)" ..
have B: "class.normalization_semidom op div op + op - (0 :: 'a poly) op * 1
unit_factor_field_poly normalize_field_poly" ..
from field_poly.in_prime_factors_imp_prime [of p x] assms
show ?thesis unfolding prime_elem_def dvd_field_poly
comm_semiring_1.prime_elem_def[OF A] normalization_semidom.prime_def[OF B] by blast
qed

subsection \<open>Primality and irreducibility in polynomial rings\<close>

lemma nonconst_poly_irreducible_iff:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
assumes "degree p \<noteq> 0"
shows   "irreducible p \<longleftrightarrow> irreducible (fract_poly p) \<and> 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 \<or> p' dvd 1" by (rule irreducibleD)
moreover have "\<not>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 \<longleftrightarrow> p = 0" by (intro map_poly_eq_0_iff) auto
with assms show "map_poly to_fract p \<noteq> 0" by auto
next
show "\<not>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 \<noteq> 0" "cr \<noteq> 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)
hence "normalize b = gcd a b" by simp
also from ab(3) have "\<dots> = 1" .
finally have "a = 1" "is_unit b" by (simp_all add: a normalize_1_iff)

note eq
also from ab(1) \<open>a = 1\<close> have "cr * cg = to_fract b" by simp
also have "smult \<dots> (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 \<or> r' dvd 1" by (rule irreducibleD)
hence "q' dvd 1 \<or> r' dvd 1" by (auto dest: dvd_mult_right simp del: mult_pCons_left)
hence "fract_poly q' dvd 1 \<or> fract_poly r' dvd 1" by (auto simp: fract_poly_is_unit)
with q r show "is_unit q \<or> 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 \<noteq> 0" by auto
next
from irred show "\<not>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 \<or> fract_poly r dvd 1"
by (rule irreducibleD)
with primitive qr show "q dvd 1 \<or> r dvd 1"
by (auto simp:  content_prod_eq_1_iff is_unit_fract_poly_iff)
qed
qed

context
begin

private lemma irreducible_imp_prime_poly:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} 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 \<or> fract_poly p dvd fract_poly r"
by (rule prime_elem_dvd_multD)
with primitive show "p dvd q \<or> 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 \<dots> = 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} fract poly"
assumes "irreducible p"
shows   "irreducible (primitive_part_fract p)"
proof -
from assms have deg: "degree (primitive_part_fract p) \<noteq> 0"
by (intro notI)
(auto elim!: degree_eq_zeroE simp: irreducible_def is_unit_poly_iff dvd_field_iff)
hence [simp]: "p \<noteq> 0" by auto

note \<open>irreducible p\<close>
also have "p = [:fract_content p:] * fract_poly (primitive_part_fract p)"
also have "irreducible \<dots> \<longleftrightarrow> 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} fract poly"
shows "irreducible p \<Longrightarrow> 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 \<noteq> 0"
shows "irreducible [:a,b:]"
proof (rule irreducibleI)
fix p q assume pq: "[:a,b:] = p * q"
also from pq assms have "degree \<dots> = degree p + degree q"
by (intro degree_mult_eq) auto
finally have "degree p = 0 \<or> degree q = 0" using assms by auto
with assms pq show "is_unit p \<or> 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) \<noteq> 0 \<Longrightarrow> 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}"
shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> 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}"
shows "b \<noteq> 0 \<Longrightarrow> coprime a b \<Longrightarrow> prime_elem [:a,b:]"
by (rule irreducible_imp_prime_poly, rule irreducible_linear_poly)

end

subsection \<open>Prime factorisation of polynomials\<close>

context
begin

private lemma poly_prime_factorization_exists_content_1:
fixes p :: "'a :: {factorial_semiring,semiring_Gcd,ring_gcd,idom_divide} poly"
assumes "p \<noteq> 0" "content p = 1"
shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> 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 \<circ> primitive_part_fract) ?P"
have "content e = (\<Prod>x\<in>#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 (\<lambda>x. content (primitive_part_fract x)) ?P = image_mset (\<lambda>_. 1) ?P"
by (intro image_mset_cong content_primitive_part_fract) auto
finally have content_e: "content e = 1"
by simp

have "fract_poly p = unit_factor_field_poly (fract_poly p) *
normalize_field_poly (fract_poly p)" by simp
also have "unit_factor_field_poly (fract_poly p) = [:to_fract (lead_coeff p):]"
by (simp add: unit_factor_field_poly_def monom_0 degree_map_poly coeff_map_poly)
also from assms have "normalize_field_poly (fract_poly p) = prod_mset ?P"
by (subst field_poly_prod_mset_prime_factorization) simp_all
also have "\<dots> = prod_mset (image_mset id ?P)" by simp
also have "image_mset id ?P =
image_mset (\<lambda>x. [: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 \<dots> = 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):] * \<dots> = 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" by (simp_all add: normalize_1_iff)
with ab ab' have "c' = to_fract b" by auto
from this and \<open>is_unit b\<close> 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 \<in># 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 \<open>prod_mset A = normalize p\<close> 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} poly"
assumes "p \<noteq> 0"
shows   "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> prod_mset A = normalize p"
proof -
define B where "B = image_mset (\<lambda>x. [:x:]) (prime_factorization (content p))"
have "\<exists>A. (\<forall>p. p \<in># A \<longrightarrow> prime_elem p) \<and> 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
moreover from assms have "prod_mset B = [:content p:]"
by (simp add: B_def prod_mset_const_poly prod_mset_prime_factorization)
moreover have "\<forall>p. p \<in># B \<longrightarrow> prime_elem p"
by (auto simp: B_def intro!: lift_prime_elem_poly dest: in_prime_factors_imp_prime)
ultimately show ?thesis by (intro exI[of _ "B + A"]) auto
qed

end

subsection \<open>Typeclass instances\<close>

instance poly :: (factorial_ring_gcd) factorial_semiring
by standard (rule poly_prime_factorization_exists)

instantiation poly :: (factorial_ring_gcd) factorial_ring_gcd
begin

definition gcd_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
[code del]: "gcd_poly = gcd_factorial"

definition lcm_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly" where
[code del]: "lcm_poly = lcm_factorial"

definition Gcd_poly :: "'a poly set \<Rightarrow> 'a poly" where
[code del]: "Gcd_poly = Gcd_factorial"

definition Lcm_poly :: "'a poly set \<Rightarrow> '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

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

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

definition uniqueness_constraint_poly :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
where [simp]: "uniqueness_constraint_poly = top"

instance
by standard
(auto simp: euclidean_size_poly_def Rings.div_mult_mod_eq div_poly_less degree_mult_eq intro!: degree_mod_less' degree_mult_right_le
split: if_splits)

end

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

subsection \<open>Polynomial GCD\<close>

lemma gcd_poly_decompose:
fixes p q :: "'a :: factorial_ring_gcd 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 poly"
assumes nz: "q \<noteq> 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 \<noteq> 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 \<noteq> 0" "pseudo_mod p q \<noteq> 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 \<Rightarrow> 'a poly \<Rightarrow> '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 ((\<lambda>p. if p = 0 then 0 else Suc (degree p)) \<circ> 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 \<or> 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 \<or>
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 "\<dots> = 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 \<Rightarrow> 'a poly \<Rightarrow> '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 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 poly"]
lemmas Lcm_poly_set_eq_fold [code] = Lcm_set_eq_fold [where ?'a = "'a :: factorial_ring_gcd poly"]

text \<open>Example:
@{lemma "Lcm {[:1, 2, 3:], [:2, 3, 4:]} = [:[:2:], [:7:], [:16:], [:17:], [:12 :: int:]:]" by eval}
\<close>

end
```