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"
  by (simp add: to_fract_def)

lemma to_fract_diff [simp]: "to_fract (x - y) = to_fract x - to_fract y"
  by (simp add: to_fract_def)
  
lemma to_fract_uminus [simp]: "to_fract (-x) = -to_fract x"
  by (simp add: to_fract_def)
  
lemma to_fract_mult [simp]: "to_fract (x * y) = to_fract x * to_fract y"
  by (simp add: to_fract_def)

lemma to_fract_eq_iff [simp]: "to_fract x = to_fract y ⟷ x = y"
  by (simp add: to_fract_def eq_fract)
  
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"
  by (simp add: to_fract_def)

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))"
  by transfer (simp add: coprime_normalize_quot)

lemma unit_factor_snd_quot_of_fract: "unit_factor (snd (quot_of_fract x)) = 1"
  using quot_of_fract_in_normalized_fracts[of x] 
  by (simp add: normalized_fracts_def case_prod_unfold)  

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"
  by (simp add: poly_eqI coeff_map_poly)

lemma fract_poly_1 [simp]: "fract_poly 1 = 1"
  by (simp add: map_poly_pCons)

lemma fract_poly_add [simp]:
  "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"
  by (simp add: primitive_part_fract_def)

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"
    by (simp add: primitive_part_fract_def p'_def)
  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))"
        by (simp add: coeff_mult)
      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"
        by (simp add: dvd_add_left_iff)
      with assms coeff_m show "c dvd coeff a i"
        by (simp add: prime_elem_dvd_mult_iff)
    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"
    by (simp_all add: prime_elem_def is_unit_const_poly_iff)
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"
    by (simp add: dvd_dict)
  show "comm_monoid_mult.prod_mset times 1 = prod_mset"
    by (simp add: prod_mset_dict)
  show "comm_semiring_1.irreducible times 1 0 = irreducible"
    by (simp add: irreducible_dict)
  show "comm_semiring_1.prime_elem times 1 0 = prime_elem"
    by (simp add: prime_elem_dict)
  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)
    (λp. [:lead_coeff p:]) (λp. smult (inverse (lead_coeff p)) p)"
  proof (standard, fold dvd_dict)
    fix p :: "'a poly"
    show "[:lead_coeff p:] * smult (inverse (lead_coeff p)) p = p"
      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"
    then show "is_unit [:lead_coeff p:]"
      by (simp add: is_unit_pCons_iff)
  next
    fix a b :: "'a poly" assume "is_unit a"
    thus "[:lead_coeff (a * b):] = a * [:lead_coeff b:]"
      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'))"
     by (simp add: 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"
     by (simp_all add: a normalize_1_iff)
   
   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"
    by (simp_all add: nonconst_poly_irreducible_iff)
  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))"
    by (simp add: content_times_primitive_part_fract)
  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)" 
    by (simp add: content_times_primitive_part_fract)
  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
    by (simp add: nonconst_poly_irreducible_iff)
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 
  also have "[:lead_coeff (fract_poly p):] = [:to_fract (lead_coeff p):]" 
    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)"
    by (simp add: c'_def)
  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 ‹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)"
      by (simp add: eucl_rel_poly_iff distrib_right)
    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"
    using gcd_add_mult[of q r s] by (simp add: gcd.commute add_ac mult_ac)
  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