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products/sources/formale Sprachen/Isabelle/HOL/Algebra/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 112 kB

SSL Polynomial_Divisibility.thy Sprache: Isabelle

(*  Title:      HOL/Algebra/Polynomial_Divisibility.thy
    Author:     Paulo Emílio de Vilhena
*)

theory Polynomial_Divisibility
  imports Polynomials Embedded_Algebras "HOL-Library.Multiset"

begin

section \<open>Divisibility of Polynomials\<close>

subsection \<open>Definitions\<close>

abbreviation poly_ring :: "_ \ ('a list) ring"
  where "poly_ring R \ univ_poly R (carrier R)"

abbreviation pirreducible :: "_ \ 'a set \ 'a list \ bool" (\pirreducible\\)
  where "pirreducible\<^bsub>R\<^esub> K p \ ring_irreducible\<^bsub>(univ_poly R K)\<^esub> p"

abbreviation pprime :: "_ \ 'a set \ 'a list \ bool" (\pprime\\)
  where "pprime\<^bsub>R\<^esub> K p \ ring_prime\<^bsub>(univ_poly R K)\<^esub> p"

definition pdivides :: "_ \ 'a list \ 'a list \ bool" (infix \pdivides\\ 65)
  where "p pdivides\<^bsub>R\<^esub> q = p divides\<^bsub>(univ_poly R (carrier R))\<^esub> q"

definition rupture :: "_ \ 'a set \ 'a list \ (('a list) set) ring" (\Rupt\\)
  where "Rupt\<^bsub>R\<^esub> K p = (K[X]\<^bsub>R\<^esub>) Quot (PIdl\<^bsub>K[X]\<^bsub>R\<^esub>\<^esub> p)"

abbreviation (in ring) rupture_surj :: "'a set \ 'a list \ 'a list \ ('a list) set"
  where "rupture_surj K p \ (\q. (PIdl\<^bsub>K[X]\<^esub> p) +>\<^bsub>K[X]\<^esub> q)"

subsection \<open>Basic Properties\<close>

lemma (in ring) carrier_polynomial_shell [intro]:
  assumes "subring K R" and "p \ carrier (K[X])" shows "p \ carrier (poly_ring R)"
  using carrier_polynomial[OF assms(1), of p] assms(2) unfolding sym[OF univ_poly_carrier] by simp

lemma (in domain) pdivides_zero:
  assumes "subring K R" and "p \ carrier (K[X])" shows "p pdivides []"
  using ring.divides_zero[OF univ_poly_is_ring[OF carrier_is_subring]
         carrier_polynomial_shell[OF assms]]
  unfolding univ_poly_zero pdivides_def .

lemma (in domain) zero_pdivides_zero: "[] pdivides []"
  using pdivides_zero[OF carrier_is_subring] univ_poly_carrier by blast

lemma (in domain) zero_pdivides:
  shows "[] pdivides p \ p = []"
  using ring.zero_divides[OF univ_poly_is_ring[OF carrier_is_subring]]
  unfolding univ_poly_zero pdivides_def .

lemma (in domain) pprime_iff_pirreducible:
  assumes "subfield K R" and "p \ carrier (K[X])"
  shows "pprime K p \ pirreducible K p"
  using principal_domain.primeness_condition[OF univ_poly_is_principal] assms by simp

lemma (in domain) pirreducibleE:
  assumes "subring K R" "p \ carrier (K[X])" "pirreducible K p"
  shows "p \ []" "p \ Units (K[X])"
    and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
                 p = q \<otimes>\<^bsub>K[X]\<^esub> r \<Longrightarrow> q \<in> Units (K[X]) \<or> r \<in> Units (K[X])"
  using domain.ring_irreducibleE[OF univ_poly_is_domain[OF assms(1)] _ assms(3)] assms(2)
  by (auto simp add: univ_poly_zero)

lemma (in domain) pirreducibleI:
  assumes "subring K R" "p \ carrier (K[X])" "p \ []" "p \ Units (K[X])"
    and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
                 p = q \<otimes>\<^bsub>K[X]\<^esub> r \<Longrightarrow> q \<in> Units (K[X]) \<or> r \<in> Units (K[X])"
  shows "pirreducible K p"
  using domain.ring_irreducibleI[OF univ_poly_is_domain[OF assms(1)] _ assms(4)] assms(2-3,5)
  by (auto simp add: univ_poly_zero)

lemma (in domain) univ_poly_carrier_units_incl:
  shows "Units ((carrier R) [X]) \ { [ k ] | k. k \ carrier R - { \ } }"
proof
  fix p assume "p \ Units ((carrier R) [X])"
  then obtain q
    where p: "polynomial (carrier R) p" and q: "polynomial (carrier R) q" and pq: "poly_mult p q = [ \ ]"
    unfolding Units_def univ_poly_def by auto
  hence not_nil: "p \ []" and "q \ []"
    using poly_mult_integral[OF carrier_is_subring p q] poly_mult_zero[OF polynomial_incl[OF p]] by auto
  hence "degree p = 0"
    using poly_mult_degree_eq[OF carrier_is_subring p q] unfolding pq by simp
  hence "length p = 1"
    using not_nil by (metis One_nat_def Suc_pred length_greater_0_conv)
  then obtain k where k: "p = [ k ]"
    by (metis One_nat_def length_0_conv length_Suc_conv)
  hence "k \ carrier R - { \ }"
    using p unfolding polynomial_def by auto
  thus "p \ { [ k ] | k. k \ carrier R - { \ } }"
    unfolding k by blast
qed

lemma (in field) univ_poly_carrier_units:
  "Units ((carrier R) [X]) = { [ k ] | k. k \ carrier R - { \ } }"
proof
  show "Units ((carrier R) [X]) \ { [ k ] | k. k \ carrier R - { \ } }"
    using univ_poly_carrier_units_incl by simp
next
  show "{ [ k ] | k. k \ carrier R - { \ } } \ Units ((carrier R) [X])"
  proof (auto)
    fix k assume k: "k \ carrier R" "k \ \"
    hence inv_k: "inv k \ carrier R" "inv k \ \" and "k \ inv k = \" "inv k \ k = \"
      using subfield_m_inv[OF carrier_is_subfield, of k] by auto
    hence "poly_mult [ k ] [ inv k ] = [ \ ]" and "poly_mult [ inv k ] [ k ] = [ \ ]"
      by (auto simp add: k)
    moreover have "polynomial (carrier R) [ k ]" and "polynomial (carrier R) [ inv k ]"
      using const_is_polynomial k inv_k by auto
    ultimately show "[ k ] \ Units ((carrier R) [X])"
      unfolding Units_def univ_poly_def by (auto simp del: poly_mult.simps)
  qed
qed

lemma (in domain) univ_poly_units_incl:
  assumes "subring K R" shows "Units (K[X]) \ { [ k ] | k. k \ K - { \ } }"
  using domain.univ_poly_carrier_units_incl[OF subring_is_domain[OF assms]]
        univ_poly_consistent[OF assms] by auto

lemma (in ring) univ_poly_units:
  assumes "subfield K R" shows "Units (K[X]) = { [ k ] | k. k \ K - { \ } }"
  using field.univ_poly_carrier_units[OF subfield_iff(2)[OF assms]]
        univ_poly_consistent[OF subfieldE(1)[OF assms]] by auto

lemma (in domain) univ_poly_units':
  assumes "subfield K R" shows "p \ Units (K[X]) \ p \ carrier (K[X]) \ p \ [] \ degree p = 0"
  unfolding univ_poly_units[OF assms] sym[OF univ_poly_carrier] polynomial_def
  by (auto, metis hd_in_set le_0_eq le_Suc_eq length_0_conv length_Suc_conv list.sel(1) subsetD)

corollary (in domain) rupture_one_not_zero:
  assumes "subfield K R" and "p \ carrier (K[X])" and "degree p > 0"
  shows "\\<^bsub>Rupt K p\<^esub> \ \\<^bsub>Rupt K p\<^esub>"
proof (rule ccontr)
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assms(1)] .

  assume "\ \\<^bsub>Rupt K p\<^esub> \ \\<^bsub>Rupt K p\<^esub>"
  then have "PIdl\<^bsub>K[X]\<^esub> p +>\<^bsub>K[X]\<^esub> \\<^bsub>K[X]\<^esub> = PIdl\<^bsub>K[X]\<^esub> p"
    unfolding rupture_def FactRing_def by simp
  hence "\\<^bsub>K[X]\<^esub> \ PIdl\<^bsub>K[X]\<^esub> p"
    using ideal.rcos_const_imp_mem[OF UP.cgenideal_ideal[OF assms(2)]] by auto
  then obtain q where "q \ carrier (K[X])" and "\\<^bsub>K[X]\<^esub> = q \\<^bsub>K[X]\<^esub> p"
    using assms(2) unfolding cgenideal_def by auto
  hence "p \ Units (K[X])"
    unfolding Units_def using assms(2) UP.m_comm by auto
  hence "degree p = 0"
    unfolding univ_poly_units[OF assms(1)] by auto
  with \<open>degree p > 0\<close> show False
    by simp
qed

corollary (in ring) pirreducible_degree:
  assumes "subfield K R" "p \ carrier (K[X])" "pirreducible K p"
  shows "degree p \ 1"
proof (rule ccontr)
  assume "\ degree p \ 1" then have "length p \ 1"
    by simp
  moreover have "p \ []" and "p \ Units (K[X])"
    using assms(3) by (auto simp add: ring_irreducible_def irreducible_def univ_poly_zero)
  ultimately obtain k where k: "p = [ k ]"
    by (metis append_butlast_last_id butlast_take diff_is_0_eq le_refl self_append_conv2 take0 take_all)
  hence "k \ K" and "k \ \"
    using assms(2) by (auto simp add: polynomial_def univ_poly_def)
  hence "p \ Units (K[X])"
    using univ_poly_units[OF assms(1)] unfolding k by auto
  from \<open>p \<in> Units (K[X])\<close> and \<open>p \<notin> Units (K[X])\<close> show False by simp
qed

corollary (in domain) univ_poly_not_field:
  assumes "subring K R" shows "\ field (K[X])"
proof -
  have "X \ carrier (K[X]) - { \\<^bsub>(K[X])\<^esub> }" and "X \ { [ k ] | k. k \ K - { \ } }"
    using var_closed(1)[OF assms] unfolding univ_poly_zero var_def by auto
  thus ?thesis
    using field.field_Units[of "K[X]"] univ_poly_units_incl[OF assms] by blast
qed

lemma (in domain) rupture_is_field_iff_pirreducible:
  assumes "subfield K R" and "p \ carrier (K[X])"
  shows "field (Rupt K p) \ pirreducible K p"
proof
  assume "pirreducible K p" thus "field (Rupt K p)"
    using principal_domain.field_iff_prime[OF univ_poly_is_principal[OF assms(1)]] assms(2)
          pprime_iff_pirreducible[OF assms] pirreducibleE(1)[OF subfieldE(1)[OF assms(1)]]
    by (simp add: univ_poly_zero rupture_def)
next
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assms(1)] .

  assume field: "field (Rupt K p)"
  have "p \ []"
  proof (rule ccontr)
    assume "\ p \ []" then have p: "p = []"
      by simp
    hence "Rupt K p \ (K[X])"
      using UP.FactRing_zeroideal(1) UP.genideal_zero
            UP.cgenideal_eq_genideal[OF UP.zero_closed]
      by (simp add: rupture_def univ_poly_zero)
    then obtain h where h: "h \ ring_iso (Rupt K p) (K[X])"
      unfolding is_ring_iso_def by blast
    moreover have "ring (Rupt K p)"
      using field by (simp add: cring_def domain_def field_def)
    ultimately interpret R: ring_hom_ring "Rupt K p" "K[X]" h
      unfolding ring_hom_ring_def ring_hom_ring_axioms_def ring_iso_def
      using UP.ring_axioms by simp
    have "field (K[X])"
      using field.ring_iso_imp_img_field[OF field h] by simp
    thus False
      using univ_poly_not_field[OF subfieldE(1)[OF assms(1)]] by simp
  qed
  thus "pirreducible K p"
    using UP.field_iff_prime pprime_iff_pirreducible[OF assms] assms(2) field
    by (simp add: univ_poly_zero rupture_def)
qed

lemma (in domain) rupture_surj_hom:
  assumes "subring K R" and "p \ carrier (K[X])"
  shows "(rupture_surj K p) \ ring_hom (K[X]) (Rupt K p)"
    and "ring_hom_ring (K[X]) (Rupt K p) (rupture_surj K p)"
proof -
  interpret UP: domain "K[X]"
    using univ_poly_is_domain[OF assms(1)] .
  interpret I: ideal "PIdl\<^bsub>K[X]\<^esub> p" "K[X]"
    using UP.cgenideal_ideal[OF assms(2)] .
  show "(rupture_surj K p) \ ring_hom (K[X]) (Rupt K p)"
   and "ring_hom_ring (K[X]) (Rupt K p) (rupture_surj K p)"
    using ring_hom_ring.intro[OF UP.ring_axioms I.quotient_is_ring] I.rcos_ring_hom
    unfolding symmetric[OF ring_hom_ring_axioms_def] rupture_def by auto
qed

corollary (in domain) rupture_surj_norm_is_hom:
  assumes "subring K R" and "p \ carrier (K[X])"
  shows "((rupture_surj K p) \ poly_of_const) \ ring_hom (R \ carrier := K \) (Rupt K p)"
  using ring_hom_trans[OF canonical_embedding_is_hom[OF assms(1)] rupture_surj_hom(1)[OF assms]] .

lemma (in domain) norm_map_in_poly_ring_carrier:
  assumes "p \ carrier (poly_ring R)" and "\a. a \ carrier R \ f a \ carrier (poly_ring R)"
  shows "ring.normalize (poly_ring R) (map f p) \ carrier (poly_ring (poly_ring R))"
proof -
  have "set p \ carrier R"
    using assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
  hence "set (map f p) \ carrier (poly_ring R)"
    using assms(2) by auto
  thus ?thesis
    using ring.normalize_gives_polynomial[OF univ_poly_is_ring[OF carrier_is_subring]]
    unfolding univ_poly_carrier by simp
qed

lemma (in domain) map_in_poly_ring_carrier:
  assumes "p \ carrier (poly_ring R)" and "\a. a \ carrier R \ f a \ carrier (poly_ring R)"
    and "\a. a \ \ \ f a \ []"
  shows "map f p \ carrier (poly_ring (poly_ring R))"
proof -
  interpret UP: ring "poly_ring R"
    using univ_poly_is_ring[OF carrier_is_subring] .
  have "lead_coeff p \ \" if "p \ []"
    using that assms(1) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
  hence "ring.normalize (poly_ring R) (map f p) = map f p"
    by (cases p) (simp_all add: assms(3) univ_poly_zero)
  thus ?thesis
    using norm_map_in_poly_ring_carrier[of p f] assms(1-2) by simp
qed

lemma (in domain) map_norm_in_poly_ring_carrier:
  assumes "subring K R" and "p \ carrier (K[X])"
  shows "map poly_of_const p \ carrier (poly_ring (K[X]))"
  using domain.map_in_poly_ring_carrier[OF subring_is_domain[OF assms(1)]]
proof -
  have "\a. a \ K \ poly_of_const a \ carrier (K[X])"
   and "\a. a \ \ \ poly_of_const a \ []"
    using ring_hom_memE(1)[OF canonical_embedding_is_hom[OF assms(1)]]
    by (auto simp: poly_of_const_def)
  thus ?thesis
    using domain.map_in_poly_ring_carrier[OF subring_is_domain[OF assms(1)]] assms(2)
    unfolding univ_poly_consistent[OF assms(1)] by simp
qed

lemma (in domain) polynomial_rupture:
  assumes "subring K R" and "p \ carrier (K[X])"
  shows "(ring.eval (Rupt K p)) (map ((rupture_surj K p) \ poly_of_const) p) (rupture_surj K p X) = \\<^bsub>Rupt K p\<^esub>"
proof -
  let ?surj = "rupture_surj K p"

  interpret UP: domain "K[X]"
    using univ_poly_is_domain[OF assms(1)] .
  interpret Hom: ring_hom_ring "K[X]" "Rupt K p" ?surj
    using rupture_surj_hom(2)[OF assms] .

  have "(Hom.S.eval) (map (?surj \ poly_of_const) p) (?surj X) = ?surj ((UP.eval) (map poly_of_const p) X)"
    using Hom.eval_hom[OF UP.carrier_is_subring var_closed(1)[OF assms(1)]
          map_norm_in_poly_ring_carrier[OF assms]] by simp
  also have " ... = ?surj p"
    unfolding sym[OF eval_rewrite[OF assms]] ..
  also have " ... = \\<^bsub>Rupt K p\<^esub>"
    using UP.a_rcos_zero[OF UP.cgenideal_ideal[OF assms(2)] UP.cgenideal_self[OF assms(2)]]
    unfolding rupture_def FactRing_def by simp
  finally show ?thesis .
qed

subsection \<open>Division\<close>

definition (in ring) long_divides :: "'a list \ 'a list \ ('a list \ 'a list) \ bool"
  where "long_divides p q t \
           \<comment> \<open>i\<close>   (t \<in> carrier (poly_ring R) \<times> carrier (poly_ring R)) \<and>
           \<comment> \<open>ii\<close>  (p = (q \<otimes>\<^bsub>poly_ring R\<^esub> (fst t)) \<oplus>\<^bsub>poly_ring R\<^esub> (snd t)) \<and>
           \<comment> \<open>iii\<close> (snd t = [] \<or> degree (snd t) < degree q)"

definition (in ring) long_division :: "'a list \ 'a list \ ('a list \ 'a list)"
  where "long_division p q = (THE t. long_divides p q t)"

definition (in ring) pdiv :: "'a list \ 'a list \ 'a list" (infixl \pdiv\ 65)
  where "p pdiv q = (if q = [] then [] else fst (long_division p q))"

definition (in ring) pmod :: "'a list \ 'a list \ 'a list" (infixl \pmod\ 65)
  where "p pmod q = (if q = [] then p else snd (long_division p q))"

lemma (in ring) long_dividesI:
  assumes "b \ carrier (poly_ring R)" and "r \ carrier (poly_ring R)"
      and "p = (q \\<^bsub>poly_ring R\<^esub> b) \\<^bsub>poly_ring R\<^esub> r" and "r = [] \ degree r < degree q"
    shows "long_divides p q (b, r)"
  using assms unfolding long_divides_def by auto

lemma (in domain) exists_long_division:
  assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
  obtains b r where "b \ carrier (K[X])" and "r \ carrier (K[X])" and "long_divides p q (b, r)"
  using subfield_long_division_theorem_shell[OF assms(1-3)] assms(4)
        carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]]
  unfolding long_divides_def univ_poly_zero univ_poly_add univ_poly_mult by auto

lemma (in domain) exists_unique_long_division:
  assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
  shows "\!t. long_divides p q t"
proof -
  let ?padd   = "\a b. a \\<^bsub>poly_ring R\<^esub> b"
  let ?pmult  = "\a b. a \\<^bsub>poly_ring R\<^esub> b"
  let ?pminus = "\a b. a \\<^bsub>poly_ring R\<^esub> b"

  interpret UP: domain "poly_ring R"
    using univ_poly_is_domain[OF carrier_is_subring] .

  obtain b r where ldiv: "long_divides p q (b, r)"
    using exists_long_division[OF assms] by metis

  moreover have "(b, r) = (b', r')" if "long_divides p q (b', r')" for b' r'
  proof -
    have q: "q \ carrier (poly_ring R)" "q \ []"
      using assms(3-4) carrier_polynomial[OF subfieldE(1)[OF assms(1)]]
      unfolding univ_poly_carrier by auto
    hence in_carrier: "q \ carrier (poly_ring R)"
      "b \ carrier (poly_ring R)" "r \ carrier (poly_ring R)"
      "b' \ carrier (poly_ring R)" "r' \ carrier (poly_ring R)"
      using assms(3) that ldiv unfolding long_divides_def by auto
    have "?pminus (?padd (?pmult q b) r) r' = ?pminus (?padd (?pmult q b') r') r'"
      using ldiv and that unfolding long_divides_def by auto
    hence eq: "?padd (?pmult q (?pminus b b')) (?pminus r r') = \\<^bsub>poly_ring R\<^esub>"
      using in_carrier by algebra
    have "b = b'"
    proof (rule ccontr)
      assume "b \ b'"
      hence pminus: "?pminus b b' \ \\<^bsub>poly_ring R\<^esub>" "?pminus b b' \ carrier (poly_ring R)"
        using in_carrier(2,4) by (metis UP.add.inv_closed UP.l_neg UP.minus_eq UP.minus_unique, algebra)
      hence degree_ge: "degree (?pmult q (?pminus b b')) \ degree q"
        using poly_mult_degree_eq[OF carrier_is_subring, of q "?pminus b b'"] q
        unfolding univ_poly_zero univ_poly_carrier univ_poly_mult by simp

      have "?pminus b b' = \\<^bsub>poly_ring R\<^esub>" if "?pminus r r' = \\<^bsub>poly_ring R\<^esub>"
        using eq pminus(2) q UP.integral univ_poly_zero unfolding that by auto
      hence "?pminus r r' \ []"
        using pminus(1) unfolding univ_poly_zero by blast
      moreover have "?pminus r r' = []" if "r = []" and "r' = []"
        using univ_poly_a_inv_def'[OF carrier_is_subring UP.zero_closed] that
        unfolding a_minus_def univ_poly_add univ_poly_zero by auto
      ultimately have "r \ [] \ r' \ []"
        by blast
      hence "max (degree r) (degree r') < degree q"
        using ldiv and that unfolding long_divides_def by auto
      moreover have "degree (?pminus r r') \ max (degree r) (degree r')"
        using poly_add_degree[of r "map (a_inv R) r'"]
        unfolding a_minus_def univ_poly_add univ_poly_a_inv_def'[OF carrier_is_subring in_carrier(5)]
        by auto
      ultimately have degree_lt: "degree (?pminus r r') < degree q"
        by linarith
      have is_poly: "polynomial (carrier R) (?pmult q (?pminus b b'))" "polynomial (carrier R) (?pminus r r')"
        using in_carrier pminus(2) unfolding univ_poly_carrier by algebra+

      have "degree (?padd (?pmult q (?pminus b b')) (?pminus r r')) = degree (?pmult q (?pminus b b'))"
        using poly_add_degree_eq[OF carrier_is_subring is_poly] degree_ge degree_lt
        unfolding univ_poly_carrier sym[OF univ_poly_add[of R "carrier R"]] max_def by simp
      hence "degree (?padd (?pmult q (?pminus b b')) (?pminus r r')) > 0"
        using degree_ge degree_lt by simp
      moreover have "degree (?padd (?pmult q (?pminus b b')) (?pminus r r')) = 0"
        using eq unfolding univ_poly_zero by simp
      ultimately show False by simp
    qed
    hence "?pminus r r' = \\<^bsub>poly_ring R\<^esub>"
      using in_carrier eq by algebra
    hence "r = r'"
      using in_carrier by (metis UP.add.inv_closed UP.add.right_cancel UP.minus_eq UP.r_neg)
    with \<open>b = b'\<close> show ?thesis
      by simp
  qed

  ultimately show ?thesis
    by auto
qed

lemma (in domain) long_divisionE:
  assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
  shows "long_divides p q (p pdiv q, p pmod q)"
  using theI'[OF exists_unique_long_division[OF assms]] assms(4)
  unfolding pmod_def pdiv_def long_division_def by auto

lemma (in domain) long_divisionI:
  assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
  shows "long_divides p q (b, r) \ (b, r) = (p pdiv q, p pmod q)"
  using exists_unique_long_division[OF assms] long_divisionE[OF assms] by metis

lemma (in domain) long_division_closed:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
  shows "p pdiv q \ carrier (K[X])" and "p pmod q \ carrier (K[X])"
proof -
  have "p pdiv q \ carrier (K[X]) \ p pmod q \ carrier (K[X])"
    using assms univ_poly_zero_closed[of R] long_divisionI[of K] exists_long_division[OF assms]
    by (cases "q = []") (simp add: pdiv_def pmod_def, metis Pair_inject)+
  thus "p pdiv q \ carrier (K[X])" and "p pmod q \ carrier (K[X])"
    by auto
qed

lemma (in domain) pdiv_pmod:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
  shows "p = (q \\<^bsub>K[X]\<^esub> (p pdiv q)) \\<^bsub>K[X]\<^esub> (p pmod q)"
proof (cases)
  interpret UP: ring "K[X]"
    using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
  assume "q = []" thus ?thesis
    using assms(2) unfolding pdiv_def pmod_def sym[OF univ_poly_zero[of R K]] by simp
next
  assume "q \ []" thus ?thesis
    using long_divisionE[OF assms] unfolding long_divides_def univ_poly_mult univ_poly_add by simp
qed

lemma (in domain) pmod_degree:
  assumes "subfield K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])" "q \ []"
  shows "p pmod q = [] \ degree (p pmod q) < degree q"
  using long_divisionE[OF assms] unfolding long_divides_def by auto

lemma (in domain) pmod_const:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])" and "degree q > degree p"
  shows "p pdiv q = []" and "p pmod q = p"
proof -
  have "p pdiv q = [] \ p pmod q = p"
  proof (cases)
    interpret UP: ring "K[X]"
      using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .

    assume "q \ []"
    have "p = (q \\<^bsub>K[X]\<^esub> []) \\<^bsub>K[X]\<^esub> p"
      using assms(2-3) unfolding sym[OF univ_poly_zero[of R K]] by simp
    moreover have "([], p) \ carrier (poly_ring R) \ carrier (poly_ring R)"
      using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)] assms(2)] by auto
    ultimately have "long_divides p q ([], p)"
      using assms(4) unfolding long_divides_def univ_poly_mult univ_poly_add by auto
    with \<open>q \<noteq> []\<close> show ?thesis
      using long_divisionI[OF assms(1-3)] by auto
  qed (simp add: pmod_def pdiv_def)
  thus "p pdiv q = []" and "p pmod q = p"
    by auto
qed

lemma (in domain) long_division_zero:
  assumes "subfield K R" and "q \ carrier (K[X])" shows "[] pdiv q = []" and "[] pmod q = []"
proof -
  interpret UP: ring "poly_ring R"
    using univ_poly_is_ring[OF carrier_is_subring] .

  have "[] pdiv q = [] \ [] pmod q = []"
  proof (cases)
    assume "q \ []"
    have "q \ carrier (poly_ring R)"
      using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)] assms(2)] .
    hence "long_divides [] q ([], [])"
      unfolding long_divides_def sym[OF univ_poly_zero[of R "carrier R"]] by auto
    with \<open>q \<noteq> []\<close> show ?thesis
      using long_divisionI[OF assms(1) univ_poly_zero_closed assms(2)] by simp
  qed (simp add: pmod_def pdiv_def)
  thus "[] pdiv q = []" and "[] pmod q = []"
    by auto
qed

lemma (in domain) long_division_a_inv:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
  shows "((\\<^bsub>K[X]\<^esub> p) pdiv q) = \\<^bsub>K[X]\<^esub> (p pdiv q)" (is "?pdiv")
    and "((\\<^bsub>K[X]\<^esub> p) pmod q) = \\<^bsub>K[X]\<^esub> (p pmod q)" (is "?pmod")
proof -
  interpret UP: ring "K[X]"
    using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .

  have "?pdiv \ ?pmod"
  proof (cases)
    assume "q = []" thus ?thesis
      unfolding pmod_def pdiv_def sym[OF univ_poly_zero[of R K]] by simp
  next
    assume not_nil: "q \ []"
    have "\\<^bsub>K[X]\<^esub> p = \\<^bsub>K[X]\<^esub> ((q \\<^bsub>K[X]\<^esub> (p pdiv q)) \\<^bsub>K[X]\<^esub> (p pmod q))"
      using pdiv_pmod[OF assms] by simp
    hence "\\<^bsub>K[X]\<^esub> p = (q \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> (p pdiv q))) \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> (p pmod q))"
      using assms(2-3) long_division_closed[OF assms] by algebra
    moreover have "\\<^bsub>K[X]\<^esub> (p pdiv q) \ carrier (K[X])" "\\<^bsub>K[X]\<^esub> (p pmod q) \ carrier (K[X])"
      using long_division_closed[OF assms] by algebra+
    hence "(\\<^bsub>K[X]\<^esub> (p pdiv q), \\<^bsub>K[X]\<^esub> (p pmod q)) \ carrier (poly_ring R) \ carrier (poly_ring R)"
      using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
    moreover have "\\<^bsub>K[X]\<^esub> (p pmod q) = [] \ degree (\\<^bsub>K[X]\<^esub> (p pmod q)) < degree q"
      using univ_poly_a_inv_length[OF subfieldE(1)[OF assms(1)]
            long_division_closed(2)[OF assms]] pmod_degree[OF assms not_nil]
      by auto
    ultimately have "long_divides (\\<^bsub>K[X]\<^esub> p) q (\\<^bsub>K[X]\<^esub> (p pdiv q), \\<^bsub>K[X]\<^esub> (p pmod q))"
      unfolding long_divides_def univ_poly_mult univ_poly_add by simp
    thus ?thesis
      using long_divisionI[OF assms(1) UP.a_inv_closed[OF assms(2)] assms(3) not_nil] by simp
  qed
  thus ?pdiv and ?pmod
    by auto
qed

lemma (in domain) long_division_add:
  assumes "subfield K R" and "a \ carrier (K[X])" "b \ carrier (K[X])" "q \ carrier (K[X])"
  shows "(a \\<^bsub>K[X]\<^esub> b) pdiv q = (a pdiv q) \\<^bsub>K[X]\<^esub> (b pdiv q)" (is "?pdiv")
    and "(a \\<^bsub>K[X]\<^esub> b) pmod q = (a pmod q) \\<^bsub>K[X]\<^esub> (b pmod q)" (is "?pmod")
proof -
  let ?pdiv_add = "(a pdiv q) \\<^bsub>K[X]\<^esub> (b pdiv q)"
  let ?pmod_add = "(a pmod q) \\<^bsub>K[X]\<^esub> (b pmod q)"

  interpret UP: ring "K[X]"
    using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .

  have "?pdiv \ ?pmod"
  proof (cases)
    assume "q = []" thus ?thesis
      using assms(2-3) unfolding pmod_def pdiv_def sym[OF univ_poly_zero[of R K]] by simp
  next
    note in_carrier = long_division_closed[OF assms(1,2,4)]
                      long_division_closed[OF assms(1,3,4)]

    assume "q \ []"
    have "a \\<^bsub>K[X]\<^esub> b = ((q \\<^bsub>K[X]\<^esub> (a pdiv q)) \\<^bsub>K[X]\<^esub> (a pmod q)) \\<^bsub>K[X]\<^esub>
                         ((q \<otimes>\<^bsub>K[X]\<^esub> (b pdiv q)) \<oplus>\<^bsub>K[X]\<^esub> (b pmod q))"
      using assms(2-3)[THEN pdiv_pmod[OF assms(1) _ assms(4)]] by simp
    hence "a \\<^bsub>K[X]\<^esub> b = (q \\<^bsub>K[X]\<^esub> ?pdiv_add) \\<^bsub>K[X]\<^esub> ?pmod_add"
      using assms(4) in_carrier by algebra
    moreover have "(?pdiv_add, ?pmod_add) \ carrier (poly_ring R) \ carrier (poly_ring R)"
      using in_carrier carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
    moreover have "?pmod_add = [] \ degree ?pmod_add < degree q"
    proof (cases)
      assume "?pmod_add \ []"
      hence "a pmod q \ [] \ b pmod q \ []"
        using in_carrier(2,4) unfolding sym[OF univ_poly_zero[of R K]] by auto
      moreover from \<open>q \<noteq> []\<close>
      have "a pmod q = [] \ degree (a pmod q) < degree q" and "b pmod q = [] \ degree (b pmod q) < degree q"
        using assms(2-3)[THEN pmod_degree[OF assms(1) _ assms(4)]] by auto
      ultimately have "max (degree (a pmod q)) (degree (b pmod q)) < degree q"
        by auto
      thus ?thesis
        using poly_add_degree le_less_trans unfolding univ_poly_add by blast
    qed simp
    ultimately have "long_divides (a \\<^bsub>K[X]\<^esub> b) q (?pdiv_add, ?pmod_add)"
      unfolding long_divides_def univ_poly_mult univ_poly_add by simp
    with \<open>q \<noteq> []\<close> show ?thesis
      using long_divisionI[OF assms(1) UP.a_closed[OF assms(2-3)] assms(4)] by simp
  qed
  thus ?pdiv and ?pmod
    by auto
qed

lemma (in domain) long_division_add_iff:
  assumes "subfield K R"
    and "a \ carrier (K[X])" "b \ carrier (K[X])" "c \ carrier (K[X])" "q \ carrier (K[X])"
  shows "a pmod q = b pmod q \ (a \\<^bsub>K[X]\<^esub> c) pmod q = (b \\<^bsub>K[X]\<^esub> c) pmod q"
proof -
  interpret UP: ring "K[X]"
    using univ_poly_is_ring[OF subfieldE(1)[OF assms(1)]] .
  show ?thesis
    using assms(2-4)[THEN long_division_closed(2)[OF assms(1) _ assms(5)]]
    unfolding assms(2-3)[THEN long_division_add(2)[OF assms(1) _ assms(4-5)]] by auto
qed

lemma (in domain) pdivides_iff:
  assumes "subfield K R" and "polynomial K p" "polynomial K q"
  shows "p pdivides q \ p divides\<^bsub>K[X]\<^esub> q"
proof
  show "p divides\<^bsub>K [X]\<^esub> q \ p pdivides q"
    using carrier_polynomial[OF subfieldE(1)[OF assms(1)]]
    unfolding pdivides_def factor_def univ_poly_mult univ_poly_carrier by auto
next
  interpret UP: ring "poly_ring R"
    using univ_poly_is_ring[OF carrier_is_subring] .

  have in_carrier: "p \ carrier (poly_ring R)" "q \ carrier (poly_ring R)"
    using carrier_polynomial[OF subfieldE(1)[OF assms(1)]] assms
    unfolding univ_poly_carrier by auto

  assume "p pdivides q"
  then obtain b where "b \ carrier (poly_ring R)" and "q = p \\<^bsub>poly_ring R\<^esub> b"
      unfolding pdivides_def factor_def by blast
  show "p divides\<^bsub>K[X]\<^esub> q"
  proof (cases)
    assume "p = []"
    with \<open>b \<in> carrier (poly_ring R)\<close> and \<open>q = p \<otimes>\<^bsub>poly_ring R\<^esub> b\<close> have "q = []"
      unfolding univ_poly_mult sym[OF univ_poly_carrier]
      using poly_mult_zero(1)[OF polynomial_incl] by simp
    with \<open>p = []\<close> show ?thesis
      using poly_mult_zero(2)[of "[]"]
      unfolding factor_def univ_poly_mult by auto
  next
    interpret UP: ring "poly_ring R"
      using univ_poly_is_ring[OF carrier_is_subring] .

    assume "p \ []"
    from \<open>p pdivides q\<close> obtain b where "b \<in> carrier (poly_ring R)" and "q = p \<otimes>\<^bsub>poly_ring R\<^esub> b"
      unfolding pdivides_def factor_def by blast
    moreover have "p \ carrier (poly_ring R)" and "q \ carrier (poly_ring R)"
      using assms carrier_polynomial[OF subfieldE(1)[OF assms(1)]] unfolding univ_poly_carrier by auto
    ultimately have "q = (p \\<^bsub>poly_ring R\<^esub> b) \\<^bsub>poly_ring R\<^esub> \\<^bsub>poly_ring R\<^esub>"
      by algebra
    with \<open>b \<in> carrier (poly_ring R)\<close> have "long_divides q p (b, [])"
      unfolding long_divides_def univ_poly_zero by auto
    with \<open>p \<noteq> []\<close> have "b \<in> carrier (K[X])"
      using long_divisionI[of K q p b] long_division_closed[of K q p] assms
      unfolding univ_poly_carrier by auto
    with \<open>q = p \<otimes>\<^bsub>poly_ring R\<^esub> b\<close> show ?thesis
      unfolding factor_def univ_poly_mult by blast
  qed
qed

lemma (in domain) pdivides_iff_shell:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
  shows "p pdivides q \ p divides\<^bsub>K[X]\<^esub> q"
  using pdivides_iff assms by (simp add: univ_poly_carrier)

lemma (in domain) pmod_zero_iff_pdivides:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
  shows "p pmod q = [] \ q pdivides p"
proof -
  interpret UP: domain "K[X]"
    using univ_poly_is_domain[OF subfieldE(1)[OF assms(1)]] .

  show ?thesis
  proof
    assume pmod: "p pmod q = []"
    have "p pdiv q \ carrier (K[X])" and "p pmod q \ carrier (K[X])"
      using long_division_closed[OF assms] by auto
    hence "p = q \\<^bsub>K[X]\<^esub> (p pdiv q)"
      using pdiv_pmod[OF assms] assms(3) unfolding pmod sym[OF univ_poly_zero[of R K]] by algebra
    with \<open>p pdiv q \<in> carrier (K[X])\<close> show "q pdivides p"
      unfolding pdivides_iff_shell[OF assms(1,3,2)] factor_def by blast
  next
    assume "q pdivides p" show "p pmod q = []"
    proof (cases)
      assume "q = []" with \<open>q pdivides p\<close> show ?thesis
        using zero_pdivides unfolding pmod_def by simp
    next
      assume "q \ []"
      from \<open>q pdivides p\<close> obtain r where "r \<in> carrier (K[X])" and "p = q \<otimes>\<^bsub>K[X]\<^esub> r"
        unfolding pdivides_iff_shell[OF assms(1,3,2)] factor_def by blast
      hence "p = (q \\<^bsub>K[X]\<^esub> r) \\<^bsub>K[X]\<^esub> []"
        using assms(2) unfolding sym[OF univ_poly_zero[of R K]] by simp
      moreover from \<open>r \<in> carrier (K[X])\<close> have "r \<in> carrier (poly_ring R)"
        using carrier_polynomial_shell[OF subfieldE(1)[OF assms(1)]] by auto
      ultimately have "long_divides p q (r, [])"
        unfolding long_divides_def univ_poly_mult univ_poly_add by auto
      with \<open>q \<noteq> []\<close> show ?thesis
        using long_divisionI[OF assms] by simp
    qed
  qed
qed

lemma (in domain) same_pmod_iff_pdivides:
  assumes "subfield K R" and "a \ carrier (K[X])" "b \ carrier (K[X])" "q \ carrier (K[X])"
  shows "a pmod q = b pmod q \ q pdivides (a \\<^bsub>K[X]\<^esub> b)"
proof -
  interpret UP: domain "K[X]"
    using univ_poly_is_domain[OF subfieldE(1)[OF assms(1)]] .

  have "a pmod q = b pmod q \ (a \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> b)) pmod q = (b \\<^bsub>K[X]\<^esub> (\\<^bsub>K[X]\<^esub> b)) pmod q"
    using long_division_add_iff[OF assms(1-3) UP.a_inv_closed[OF assms(3)] assms(4)] .
  also have " ... \ (a \\<^bsub>K[X]\<^esub> b) pmod q = \\<^bsub>K[X]\<^esub> pmod q"
    using assms(2-3) by algebra
  also have " ... \ q pdivides (a \\<^bsub>K[X]\<^esub> b)"
    using pmod_zero_iff_pdivides[OF assms(1) UP.minus_closed[OF assms(2-3)] assms(4)]
    unfolding univ_poly_zero long_division_zero(2)[OF assms(1,4)] .
  finally show ?thesis .
qed

lemma (in domain) pdivides_imp_degree_le:
  assumes "subring K R" and "p \ carrier (K[X])" "q \ carrier (K[X])" "q \ []"
  shows "p pdivides q \ degree p \ degree q"
proof -
  assume "p pdivides q"
  then obtain r where r: "polynomial (carrier R) r" "q = poly_mult p r"
    unfolding pdivides_def factor_def univ_poly_mult univ_poly_carrier by blast
  moreover have p: "polynomial (carrier R) p"
    using assms(2) carrier_polynomial[OF assms(1)] unfolding univ_poly_carrier by auto
  moreover have "p \ []" and "r \ []"
    using poly_mult_zero(2)[OF polynomial_incl[OF p]] r(2) assms(4) by auto
  ultimately show "degree p \ degree q"
    using poly_mult_degree_eq[OF carrier_is_subring, of p r] by auto
qed

lemma (in domain) pprimeE:
  assumes "subfield K R" "p \ carrier (K[X])" "pprime K p"
  shows "p \ []" "p \ Units (K[X])"
    and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
                 p pdivides (q \<otimes>\<^bsub>K[X]\<^esub> r) \<Longrightarrow> p pdivides q \<or> p pdivides r"
  using assms(2-3) poly_mult_closed[OF subfieldE(1)[OF assms(1)]] pdivides_iff[OF assms(1)]
  unfolding ring_prime_def prime_def
  by (auto simp add: univ_poly_mult univ_poly_carrier univ_poly_zero)

lemma (in domain) pprimeI:
  assumes "subfield K R" "p \ carrier (K[X])" "p \ []" "p \ Units (K[X])"
    and "\q r. \ q \ carrier (K[X]); r \ carrier (K[X])\ \
                 p pdivides (q \<otimes>\<^bsub>K[X]\<^esub> r) \<Longrightarrow> p pdivides q \<or> p pdivides r"
  shows "pprime K p"
  using assms(2-5) poly_mult_closed[OF subfieldE(1)[OF assms(1)]] pdivides_iff[OF assms(1)]
  unfolding ring_prime_def prime_def
  by (auto simp add: univ_poly_mult univ_poly_carrier univ_poly_zero)

lemma (in domain) associated_polynomials_iff:
  assumes "subfield K R" and "p \ carrier (K[X])" "q \ carrier (K[X])"
  shows "p \\<^bsub>K[X]\<^esub> q \ (\k \ K - { \ }. p = [ k ] \\<^bsub>K[X]\<^esub> q)"
  using domain.ring_associated_iff[OF univ_poly_is_domain[OF subfieldE(1)[OF assms(1)]] assms(2-3)]
  unfolding univ_poly_units[OF assms(1)] by auto

corollary (in domain) associated_polynomials_imp_same_length: (* stronger than "imp_same_degree" *)
  assumes "subring K R" and "p \ carrier (K[X])" and "q \ carrier (K[X])"
  shows "p \\<^bsub>K[X]\<^esub> q \ length p = length q"
proof -
  have aux_lemma: "length p \ length q"
    if p: "p \ carrier (K[X])" and q: "q \ carrier (K[X])" and "p \\<^bsub>K[X]\<^esub> q" for p q
  proof (cases "q = []")
    case True with \<open>p \<sim>\<^bsub>K[X]\<^esub> q\<close> have "p = []"
      unfolding associated_def True factor_def univ_poly_def by auto
    thus ?thesis
      using True by simp
  next
    case False
    from \<open>p \<sim>\<^bsub>K[X]\<^esub> q\<close> have "p divides\<^bsub>K [X]\<^esub> q"
      unfolding associated_def by simp
    hence "p divides\<^bsub>poly_ring R\<^esub> q"
      using carrier_polynomial[OF assms(1)]
      unfolding factor_def univ_poly_carrier univ_poly_mult by auto
    with \<open>q \<noteq> []\<close> have "degree p \<le> degree q"
      using pdivides_imp_degree_le[OF assms(1) p q] unfolding pdivides_def by simp
    with \<open>q \<noteq> []\<close> show ?thesis
      by (cases "p = []", auto simp add: Suc_leI le_diff_iff)
  qed

  interpret UP: domain "K[X]"
    using univ_poly_is_domain[OF assms(1)] .

  assume "p \\<^bsub>K[X]\<^esub> q" thus ?thesis
    using aux_lemma[OF assms(2-3)] aux_lemma[OF assms(3,2) UP.associated_sym] by simp
qed

lemma (in ring) divides_pirreducible_condition:
  assumes "pirreducible K q" and "p \ carrier (K[X])"
  shows "p divides\<^bsub>K[X]\<^esub> q \ p \ Units (K[X]) \ p \\<^bsub>K[X]\<^esub> q"
  using divides_irreducible_condition[of "K[X]" q p] assms
  unfolding ring_irreducible_def by auto

subsection \<open>Polynomial Power\<close>

lemma (in domain) polynomial_pow_not_zero:
  assumes "p \ carrier (poly_ring R)" and "p \ []"
  shows "p [^]\<^bsub>poly_ring R\<^esub> (n::nat) \ []"
proof -
  interpret UP: domain "poly_ring R"
    using univ_poly_is_domain[OF carrier_is_subring] .

  from assms UP.integral show ?thesis
    unfolding sym[OF univ_poly_zero[of R "carrier R"]]
    by (induction n, auto)
qed

lemma (in domain) subring_polynomial_pow_not_zero:
  assumes "subring K R" and "p \ carrier (K[X])" and "p \ []"
  shows "p [^]\<^bsub>K[X]\<^esub> (n::nat) \ []"
  using domain.polynomial_pow_not_zero[OF subring_is_domain, of K p n] assms
  unfolding univ_poly_consistent[OF assms(1)] by simp

lemma (in domain) polynomial_pow_degree:
  assumes "p \ carrier (poly_ring R)"
  shows "degree (p [^]\<^bsub>poly_ring R\<^esub> n) = n * degree p"
proof -
  interpret UP: domain "poly_ring R"
    using univ_poly_is_domain[OF carrier_is_subring] .

  show ?thesis
  proof (induction n)
    case 0 thus ?case
      using UP.nat_pow_0 unfolding univ_poly_one by auto
  next
    let ?ppow = "\n. p [^]\<^bsub>poly_ring R\<^esub> n"
    case (Suc n) thus ?case
    proof (cases "p = []")
      case True thus ?thesis
        using univ_poly_zero[of R "carrier R"] UP.r_null assms by auto
    next
      case False
      hence "?ppow n \ carrier (poly_ring R)" and "?ppow n \ []" and "p \ []"
        using polynomial_pow_not_zero[of p n] assms by (auto simp add: univ_poly_one)
      thus ?thesis
        using poly_mult_degree_eq[OF carrier_is_subring, of "?ppow n" p] Suc assms
        unfolding univ_poly_carrier univ_poly_zero
        by (auto simp add: add.commute univ_poly_mult)
    qed
  qed
qed

lemma (in domain) subring_polynomial_pow_degree:
  assumes "subring K R" and "p \ carrier (K[X])"
  shows "degree (p [^]\<^bsub>K[X]\<^esub> n) = n * degree p"
  using domain.polynomial_pow_degree[OF subring_is_domain, of K p n] assms
  unfolding univ_poly_consistent[OF assms(1)] by simp

lemma (in domain) polynomial_pow_division:
  assumes "p \ carrier (poly_ring R)" and "(n::nat) \ m"
  shows "(p [^]\<^bsub>poly_ring R\<^esub> n) pdivides (p [^]\<^bsub>poly_ring R\<^esub> m)"
proof -
  interpret UP: domain "poly_ring R"
    using univ_poly_is_domain[OF carrier_is_subring] .

  let ?ppow = "\n. p [^]\<^bsub>poly_ring R\<^esub> n"

  have "?ppow n \\<^bsub>poly_ring R\<^esub> ?ppow k = ?ppow (n + k)" for k
    using assms(1) by (simp add: UP.nat_pow_mult)
  thus ?thesis
    using dividesI[of "?ppow (m - n)" "poly_ring R" "?ppow m" "?ppow n"] assms
    unfolding pdivides_def by auto
qed

lemma (in domain) subring_polynomial_pow_division:
  assumes "subring K R" and "p \ carrier (K[X])" and "(n::nat) \ m"
  shows "(p [^]\<^bsub>K[X]\<^esub> n) divides\<^bsub>K[X]\<^esub> (p [^]\<^bsub>K[X]\<^esub> m)"
  using domain.polynomial_pow_division[OF subring_is_domain, of K p n m] assms
  unfolding univ_poly_consistent[OF assms(1)] pdivides_def by simp

lemma (in domain) pirreducible_pow_pdivides_iff:
  assumes "subfield K R" "p \ carrier (K[X])" "q \ carrier (K[X])" "r \ carrier (K[X])"
    and "pirreducible K p" and "\ (p pdivides q)"
  shows "(p [^]\<^bsub>K[X]\<^esub> (n :: nat)) pdivides (q \\<^bsub>K[X]\<^esub> r) \ (p [^]\<^bsub>K[X]\<^esub> n) pdivides r"
proof -
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assms(1)] .
  show ?thesis
  proof (cases "r = []")
    case True with \<open>q \<in> carrier (K[X])\<close> have "q \<otimes>\<^bsub>K[X]\<^esub> r = []" and "r = []"
      unfolding  sym[OF univ_poly_zero[of R K]] by auto
    thus ?thesis
      using pdivides_zero[OF subfieldE(1),of K] assms by auto
  next
    case False then have not_zero: "p \ []" "q \ []" "r \ []" "q \\<^bsub>K[X]\<^esub> r \ []"
      using subfieldE(1) pdivides_zero[OF _ assms(2)] assms(1-2,5-6) pirreducibleE(1)
            UP.integral_iff[OF assms(3-4)] univ_poly_zero[of R K] by auto
    from \<open>p \<noteq> []\<close>
    have ppow: "p [^]\<^bsub>K[X]\<^esub> (n :: nat) \ []" "p [^]\<^bsub>K[X]\<^esub> (n :: nat) \ carrier (K[X])"
      using subring_polynomial_pow_not_zero[OF subfieldE(1)] assms(1-2) by auto
    have not_pdiv: "\ (p divides\<^bsub>mult_of (K[X])\<^esub> q)"
      using assms(6) pdivides_iff_shell[OF assms(1-3)] unfolding pdivides_def by auto
    have prime: "prime (mult_of (K[X])) p"
      using assms(5) pprime_iff_pirreducible[OF assms(1-2)]
      unfolding sym[OF UP.prime_eq_prime_mult[OF assms(2)]] ring_prime_def by simp
    have "a pdivides b \ a divides\<^bsub>mult_of (K[X])\<^esub> b"
      if "a \ carrier (K[X])" "a \ \\<^bsub>K[X]\<^esub>" "b \ carrier (K[X])" "b \ \\<^bsub>K[X]\<^esub>" for a b
      using that UP.divides_imp_divides_mult[of a b] divides_mult_imp_divides[of "K[X]" a b]
      unfolding pdivides_iff_shell[OF assms(1) that(1,3)] by blast
    thus ?thesis
      using UP.mult_of.prime_pow_divides_iff[OF _ _ _ prime not_pdiv, of r] ppow not_zero assms(2-4)
      unfolding nat_pow_mult_of carrier_mult_of mult_mult_of sym[OF univ_poly_zero[of R K]]
      by (metis DiffI UP.m_closed singletonD)
  qed
qed

lemma (in domain) subring_degree_one_imp_pirreducible:
  assumes "subring K R" and "a \ Units (R \ carrier := K \)" and "b \ K"
  shows "pirreducible K [ a, b ]"
proof (rule pirreducibleI[OF assms(1)])
  have "a \ K" and "a \ \"
    using assms(2) subringE(1)[OF assms(1)] unfolding Units_def by auto
  thus "[ a, b ] \ carrier (K[X])" and "[ a, b ] \ []" and "[ a, b ] \ Units (K [X])"
    using univ_poly_units_incl[OF assms(1)] assms(2-3)
    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
next
  interpret UP: domain "K[X]"
    using univ_poly_is_domain[OF assms(1)] .

  have aux_lemma1: "degree q + degree r = 1" "q \ []" "r \ []"
    if q: "q \ carrier (K[X])" and r: "r \ carrier (K[X])" and "[ a, b ] = q \\<^bsub>K[X]\<^esub> r" for q r
  proof -
    from that have not_zero: "q \ []" "r \ []"
      by (metis UP.integral_iff list.distinct(1) univ_poly_zero)+
    have "degree (q \\<^bsub>K[X]\<^esub> r) = degree q + degree r"
      using not_zero poly_mult_degree_eq[OF assms(1)] q r
      by (simp add: univ_poly_carrier univ_poly_mult)
    with sym[OF \<open>[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r\<close>] show "degree q + degree r = 1" and "q \<noteq> []" "r \<noteq> []"
      using not_zero by auto
  qed

  have aux_lemma2: "r \ Units (K[X])"
    if q: "q \ carrier (K[X])" "q \ []" and r: "r \ carrier (K[X])" "r \ []"
    and "[ a, b ] = q \\<^bsub>K[X]\<^esub> r" and "degree q = 1" and "degree r = 0" for q r
  proof -
    from that have "length q = Suc (Suc 0)" and "length r = Suc 0"
      by (linarith, metis add.right_neutral add_eq_if length_0_conv)
    from \<open>length q = Suc (Suc 0)\<close> obtain c d where q_def: "q = [ c, d ]"
      by (metis length_0_conv length_Cons list.exhaust nat.inject)
    from \<open>length r = Suc 0\<close> obtain e where r_def: "r = [ e ]"
      by (metis length_0_conv length_Suc_conv)
    from \<open>r = [ e ]\<close> and \<open>q = [ c, d ]\<close>
    have c: "c \ K" "c \ \" and d: "d \ K" and e: "e \ K" "e \ \"
      using r q subringE(1)[OF assms(1)] unfolding sym[OF univ_poly_carrier] polynomial_def by auto
    with sym[OF \<open>[ a, b ] = q \<otimes>\<^bsub>K[X]\<^esub> r\<close>] have "a = c \<otimes> e"
      using poly_mult_lead_coeff[OF assms(1), of q r]
      unfolding polynomial_def sym[OF univ_poly_mult[of R K]] r_def q_def by auto
    obtain inv_a where a: "a \ K" and inv_a: "inv_a \ K" "a \ inv_a = \" "inv_a \ a = \"
      using assms(2) unfolding Units_def by auto
    hence "a \ \" and "inv_a \ \"
      using subringE(1)[OF assms(1)] integral_iff by auto
    with \<open>c \<in> K\<close> and \<open>c \<noteq> \<zero>\<close> have in_carrier: "[ c \<otimes> inv_a ] \<in> carrier (K[X])"
      using subringE(1,6)[OF assms(1)] inv_a integral
      unfolding sym[OF univ_poly_carrier] polynomial_def
      by (auto, meson subsetD)
    moreover have "[ c \ inv_a ] \\<^bsub>K[X]\<^esub> r = [ \ ]"
      using \<open>a = c \<otimes> e\<close> a inv_a c e subsetD[OF subringE(1)[OF assms(1)]]
      unfolding r_def univ_poly_mult by (auto) (simp add: m_assoc m_lcomm integral_iff)+
    ultimately show ?thesis
      using r(1) UP.m_comm[OF in_carrier r(1)] unfolding sym[OF univ_poly_one[of R K]] Units_def by auto
  qed

  fix q r
  assume q: "q \ carrier (K[X])" and r: "r \ carrier (K[X])" and qr: "[ a, b ] = q \\<^bsub>K[X]\<^esub> r"
  thus "q \ Units (K[X]) \ r \ Units (K[X])"
    using aux_lemma1[OF q r qr] aux_lemma2[of q r] aux_lemma2[of r q] UP.m_comm add_is_1 by auto
qed

lemma (in domain) degree_one_imp_pirreducible:
  assumes "subfield K R" and "p \ carrier (K[X])" and "degree p = 1"
  shows "pirreducible K p"
proof -
  from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
    by simp
  then obtain a b where p: "p = [ a, b ]"
    by (metis length_0_conv length_Cons nat.inject neq_Nil_conv)
  with \<open>p \<in> carrier (K[X])\<close> show ?thesis
    using subring_degree_one_imp_pirreducible[OF subfieldE(1)[OF assms(1)], of a b]
          subfield.subfield_Units[OF assms(1)]
    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
qed

lemma (in ring) degree_oneE[elim]:
  assumes "p \ carrier (K[X])" and "degree p = 1"
    and "\a b. \ a \ K; a \ \; b \ K; p = [ a, b ] \ \ P"
  shows P
proof -
  from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
    by simp
  then obtain a b where "p = [ a, b ]"
    by (metis length_0_conv length_Cons nat.inject neq_Nil_conv)
  with \<open>p \<in> carrier (K[X])\<close> have "a \<in> K" and "a \<noteq> \<zero>" and "b \<in> K"
    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
  with \<open>p = [ a, b ]\<close> show ?thesis
    using assms(3) by simp
qed

lemma (in domain) subring_degree_one_associatedI:
  assumes "subring K R" and "a \ K" "a' \ K" and "b \ K" and "a \ a' = \"
  shows "[ a , b ] \\<^bsub>K[X]\<^esub> [ \, a' \ b ]"
proof -
  from \<open>a \<otimes> a' = \<one>\<close> have not_zero: "a \<noteq> \<zero>" "a' \<noteq> \<zero>"
    using subringE(1)[OF assms(1)] assms(2-3) by auto
  hence "[ a, b ] = [ a ] \\<^bsub>K[X]\<^esub> [ \, a' \ b ]"
    using assms(2-4)[THEN subsetD[OF subringE(1)[OF assms(1)]]] assms(5) m_assoc
    unfolding univ_poly_mult by fastforce
  moreover have "[ a, b ] \ carrier (K[X])" and "[ \, a' \ b ] \ carrier (K[X])"
    using subringE(1,3,6)[OF assms(1)] not_zero one_not_zero assms
    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
  moreover have "[ a ] \ Units (K[X])"
  proof -
    from \<open>a \<noteq> \<zero>\<close> and \<open>a' \<noteq> \<zero>\<close> have "[ a ] \<in> carrier (K[X])" and "[ a' ] \<in> carrier (K[X])"
      using assms(2-3) unfolding sym[OF univ_poly_carrier] polynomial_def by auto
    moreover have "a' \ a = \"
      using subsetD[OF subringE(1)[OF assms(1)]] assms m_comm by simp
    hence "[ a ] \\<^bsub>K[X]\<^esub> [ a' ] = [ \ ]" and "[ a' ] \\<^bsub>K[X]\<^esub> [ a ] = [ \ ]"
      using assms unfolding univ_poly_mult by auto
    ultimately show ?thesis
      unfolding sym[OF univ_poly_one[of R K]] Units_def by blast
  qed
  ultimately show ?thesis
    using domain.ring_associated_iff[OF univ_poly_is_domain[OF assms(1)]] by blast
qed

lemma (in domain) degree_one_associatedI:
  assumes "subfield K R" and "p \ carrier (K[X])" and "degree p = 1"
  shows "p \\<^bsub>K[X]\<^esub> [ \, inv (lead_coeff p) \ (const_term p) ]"
proof -
  from \<open>p \<in> carrier (K[X])\<close> and \<open>degree p = 1\<close>
  obtain a b where "p = [ a, b ]" and "a \ K" "a \ \" and "b \ K"
    by auto
  thus ?thesis
    using subring_degree_one_associatedI[OF subfieldE(1)[OF assms(1)]]
          subfield_m_inv[OF assms(1)] subsetD[OF subfieldE(3)[OF assms(1)]]
    unfolding const_term_def
    by auto
qed

subsection \<open>Ideals\<close>

lemma (in domain) exists_unique_gen:
  assumes "subfield K R" "ideal I (K[X])" "I \ { [] }"
  shows "\!p \ carrier (K[X]). lead_coeff p = \ \ I = PIdl\<^bsub>K[X]\<^esub> p"
    (is "\!p. ?generator p")
proof -
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assms(1)] .
  obtain q where q: "q \ carrier (K[X])" "I = PIdl\<^bsub>K[X]\<^esub> q"
    using UP.exists_gen[OF assms(2)] by blast
  hence not_nil: "q \ []"
    using UP.genideal_zero UP.cgenideal_eq_genideal[OF UP.zero_closed] assms(3)
    by (auto simp add: univ_poly_zero)
  hence "lead_coeff q \ K - { \ }"
    using q(1) unfolding univ_poly_def polynomial_def by auto
  hence inv_lc_q: "inv (lead_coeff q) \ K - { \ }" "inv (lead_coeff q) \ lead_coeff q = \"
    using subfield_m_inv[OF assms(1)] by auto

  define p where "p = [ inv (lead_coeff q) ] \\<^bsub>K[X]\<^esub> q"
  have is_poly: "polynomial K [ inv (lead_coeff q) ]" "polynomial K q"
    using inv_lc_q(1) q(1) unfolding univ_poly_def polynomial_def by auto
  hence in_carrier: "p \ carrier (K[X])"
    using UP.m_closed unfolding univ_poly_carrier p_def by simp
  have lc_p: "lead_coeff p = \"
    using poly_mult_lead_coeff[OF subfieldE(1)[OF assms(1)] is_poly _ not_nil] inv_lc_q(2)
    unfolding p_def univ_poly_mult[of R K] by simp
  moreover have PIdl_p: "I = PIdl\<^bsub>K[X]\<^esub> p"
    using UP.associated_iff_same_ideal[OF in_carrier q(1)] q(2) inv_lc_q(1) p_def
          associated_polynomials_iff[OF assms(1) in_carrier q(1)]
    by auto
  ultimately have "?generator p"
    using in_carrier by simp

  moreover
  have "\r. \ r \ carrier (K[X]); lead_coeff r = \; I = PIdl\<^bsub>K[X]\<^esub> r \ \ r = p"
  proof -
    fix r assume r: "r \ carrier (K[X])" "lead_coeff r = \" "I = PIdl\<^bsub>K[X]\<^esub> r"
    have "subring K R"
      by (simp add: \<open>subfield K R\<close> subfieldE(1))
    obtain k where k: "k \ K - { \ }" "r = [ k ] \\<^bsub>K[X]\<^esub> p"
      using UP.associated_iff_same_ideal[OF r(1) in_carrier] PIdl_p r(3)
            associated_polynomials_iff[OF assms(1) r(1) in_carrier]
      by auto
    hence "polynomial K [ k ]"
      unfolding polynomial_def by simp
    moreover have "p \ []"
      using not_nil UP.associated_iff_same_ideal[OF in_carrier q(1)] q(2) PIdl_p
            associated_polynomials_imp_same_length[OF \<open>subring K R\<close> in_carrier q(1)] by auto
    ultimately have "lead_coeff r = k \ (lead_coeff p)"
      using poly_mult_lead_coeff[OF subfieldE(1)[OF assms(1)]] in_carrier k(2)
      unfolding univ_poly_def by (auto simp del: poly_mult.simps)
    hence "k = \"
      using lc_p r(2) k(1) subfieldE(3)[OF assms(1)] by auto
    hence "r = map ((\) \) p"
      using poly_mult_const(1)[OF subfieldE(1)[OF assms(1)] _ k(1), of p] in_carrier
      unfolding k(2) univ_poly_carrier[of R K] univ_poly_mult[of R K] by auto
    moreover have "set p \ carrier R"
      using polynomial_in_carrier[OF subfieldE(1)[OF assms(1)]]
            in_carrier univ_poly_carrier[of R K] by auto
    hence "map ((\) \) p = p"
      by (induct p) (auto)
    ultimately show "r = p" by simp
  qed

  ultimately show ?thesis by blast
qed

proposition (in domain) exists_unique_pirreducible_gen:
  assumes "subfield K R" "ring_hom_ring (K[X]) R h"
    and "a_kernel (K[X]) R h \ { [] }" "a_kernel (K[X]) R h \ carrier (K[X])"
  shows "\!p \ carrier (K[X]). pirreducible K p \ lead_coeff p = \ \ a_kernel (K[X]) R h = PIdl\<^bsub>K[X]\<^esub> p"
    (is "\!p. ?generator p")
proof -
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assms(1)] .

  have "ideal (a_kernel (K[X]) R h) (K[X])"
    using ring_hom_ring.kernel_is_ideal[OF assms(2)] .
  then obtain p
    where p: "p \ carrier (K[X])" "lead_coeff p = \" "a_kernel (K[X]) R h = PIdl\<^bsub>K[X]\<^esub> p"
      and unique:
      "\q. \ q \ carrier (K[X]); lead_coeff q = \; a_kernel (K[X]) R h = PIdl\<^bsub>K[X]\<^esub> q \ \ q = p"
    using exists_unique_gen[OF assms(1) _ assms(3)] by metis

  have "p \ carrier (K[X]) - { [] }"
      using UP.genideal_zero UP.cgenideal_eq_genideal[OF UP.zero_closed] assms(3) p(1,3)
      by (auto simp add: univ_poly_zero)
  hence "pprime K p"
    using ring_hom_ring.primeideal_vimage[OF assms(2) UP.is_cring zeroprimeideal]
          UP.primeideal_iff_prime[of p]
    unfolding univ_poly_zero sym[OF p(3)] a_kernel_def' by simp
  hence "pirreducible K p"
    using pprime_iff_pirreducible[OF assms(1) p(1)] by simp
  thus ?thesis
    using p unique by metis
qed

lemma (in domain) cgenideal_pirreducible:
  assumes "subfield K R" and "p \ carrier (K[X])" "pirreducible K p"
  shows "\ pirreducible K q; q \ PIdl\<^bsub>K[X]\<^esub> p \ \ p \\<^bsub>K[X]\<^esub> q"
proof -
  interpret UP: principal_domain "K[X]"
    using univ_poly_is_principal[OF assms(1)] .

  assume q: "pirreducible K q" "q \ PIdl\<^bsub>K[X]\<^esub> p"
  hence in_carrier: "q \ carrier (K[X])"
    using additive_subgroup.a_subset[OF ideal.axioms(1)[OF UP.cgenideal_ideal[OF assms(2)]]] by auto
  hence "p divides\<^bsub>K[X]\<^esub> q"
    by (meson q assms(2) UP.cgenideal_ideal UP.cgenideal_minimal UP.to_contain_is_to_divide)
  then obtain r where r: "r \ carrier (K[X])" "q = p \\<^bsub>K[X]\<^esub> r"
    by auto
  hence "r \ Units (K[X])"
    using pirreducibleE(3)[OF _ in_carrier q(1) assms(2) r(1)] subfieldE(1)[OF assms(1)]
          pirreducibleE(2)[OF _ assms(2-3)] by auto
  thus "p \\<^bsub>K[X]\<^esub> q"
    using UP.ring_associated_iff[OF in_carrier assms(2)] r(2) UP.associated_sym
    unfolding UP.m_comm[OF assms(2) r(1)] by auto
qed

subsection \<open>Roots and Multiplicity\<close>

definition (in ring) is_root :: "'a list \ 'a \ bool"
  where "is_root p x \ (x \ carrier R \ eval p x = \ \ p \ [])"

definition (in ring) alg_mult :: "'a list \ 'a \ nat"
  where "alg_mult p x =
           (if p = [] then 0 else
             (if x \<in> carrier R then Greatest (\<lambda> n. ([ \<one>, \<ominus> x ] [^]\<^bsub>poly_ring R\<^esub> n) pdivides p) else 0))"

definition (in ring) roots :: "'a list \ 'a multiset"
  where "roots p = Abs_multiset (alg_mult p)"

definition (in ring) roots_on :: "'a set \ 'a list \ 'a multiset"
  where "roots_on K p = roots p \# mset_set K"

definition (in ring) splitted :: "'a list \ bool"
  where "splitted p \ size (roots p) = degree p"

definition (in ring) splitted_on :: "'a set \ 'a list \ bool"
  where "splitted_on K p \ size (roots_on K p) = degree p"

lemma (in domain) pdivides_imp_root_sharing:
  assumes "p \ carrier (poly_ring R)" "p pdivides q" and "a \ carrier R"
  shows "eval p a = \ \ eval q a = \"
proof -
  from \<open>p pdivides q\<close> obtain r where r: "q = p \<otimes>\<^bsub>poly_ring R\<^esub> r" "r \<in> carrier (poly_ring R)"
    unfolding pdivides_def factor_def by auto
  hence "eval q a = (eval p a) \ (eval r a)"
    using ring_hom_memE(2)[OF eval_is_hom[OF carrier_is_subring assms(3)] assms(1) r(2)] by simp
  thus "eval p a = \ \ eval q a = \"
    using ring_hom_memE(1)[OF eval_is_hom[OF carrier_is_subring assms(3)] r(2)] by auto
qed

lemma (in domain) degree_one_root:
  assumes "subfield K R" and "p \ carrier (K[X])" and "degree p = 1"
  shows "eval p (\ (inv (lead_coeff p) \ (const_term p))) = \"
    and "inv (lead_coeff p) \ (const_term p) \ K"
proof -
  from \<open>degree p = 1\<close> have "length p = Suc (Suc 0)"
    by simp
  then obtain a b where p: "p = [ a, b ]"
    by (metis (no_types, opaque_lifting) Suc_length_conv length_0_conv)
  hence "a \ K - { \ }" "b \ K" and in_carrier: "a \ carrier R" "b \ carrier R"
    using assms(2) subfieldE(3)[OF assms(1)] unfolding sym[OF univ_poly_carrier] polynomial_def by auto
  hence inv_a: "inv a \ carrier R" "a \ inv a = \" and "inv a \ K"
    using subfield_m_inv(1-2)[OF assms(1), of a] subfieldE(3)[OF assms(1)] by auto
  hence "eval p (\ (inv a \ b)) = a \ (\ (inv a \ b)) \ b"
    using in_carrier unfolding p by simp
  also have " ... = \ (a \ (inv a \ b)) \ b"
    using inv_a in_carrier by (simp add: r_minus)
  also have " ... = \"
    using in_carrier(2) unfolding sym[OF m_assoc[OF in_carrier(1) inv_a(1) in_carrier(2)]] inv_a(2) by algebra
  finally have "eval p (\ (inv a \ b)) = \" .
  moreover have ct: "const_term p = b"
    using in_carrier unfolding p const_term_def by auto
  ultimately show "eval p (\ (inv (lead_coeff p) \ (const_term p))) = \"
    unfolding p by simp
  from \<open>inv a \<in> K\<close> and \<open>b \<in> K\<close>
  show "inv (lead_coeff p) \ (const_term p) \ K"
    using p subringE(6)[OF subfieldE(1)[OF assms(1)]] unfolding ct by auto
qed
lemma (in domain) is_root_imp_pdivides:
  assumes "p \ carrier (poly_ring R)"
  shows "is_root p x \ [ \, \ x ] pdivides p"
proof -
  let ?b = "[ \ , \ x ]"

  interpret UP: domain "poly_ring R"
    using univ_poly_is_domain[OF carrier_is_subring] .

  assume "is_root p x" hence x: "x \ carrier R" and is_root: "eval p x = \"
    unfolding is_root_def by auto
  hence b: "?b \ carrier (poly_ring R)"
    unfolding sym[OF univ_poly_carrier] polynomial_def by auto
  then obtain q r where q: "q \ carrier (poly_ring R)" and r: "r \ carrier (poly_ring R)"
    and long_divides: "p = (?b \\<^bsub>poly_ring R\<^esub> q) \\<^bsub>poly_ring R\<^esub> r" "r = [] \ degree r < degree ?b"
    using long_division_theorem[OF carrier_is_subring, of p ?b] assms by (auto simp add: univ_poly_carrier)

  show ?thesis
  proof (cases "r = []")
    case True then have "r = \\<^bsub>poly_ring R\<^esub>"
      unfolding univ_poly_zero[of R "carrier R"] .
    thus ?thesis
      using long_divides(1) q r b dividesI[OF q, of p ?b] by (simp add: pdivides_def)
  next
    case False then have "length r = Suc 0"
      using long_divides(2) le_SucE by fastforce
    then obtain a where "r = [ a ]" and a: "a \ carrier R" and "a \ \"
      using r unfolding sym[OF univ_poly_carrier] polynomial_def
      by (metis False length_0_conv length_Suc_conv list.sel(1) list.set_sel(1) subset_code(1))

    have "eval p x = ((eval ?b x) \ (eval q x)) \ (eval r x)"
      using long_divides(1) ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] by (simp add: b q r)
    also have " ... = eval r x"
      using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring x]] x b q r by (auto, algebra)
    finally have "a = \"
      using a unfolding \<open>r = [ a ]\<close> is_root by simp
    with \<open>a \<noteq> \<zero>\<close> have False .. thus ?thesis ..
  qed
qed

lemma (in domain) pdivides_imp_is_root:
  assumes "p \ []" and "x \ carrier R"
  shows "[ \, \ x ] pdivides p \ is_root p x"
proof -
  assume "[ \, \ x ] pdivides p"
  then obtain q where q: "q \ carrier (poly_ring R)" and pdiv: "p = [ \, \ x ] \\<^bsub>poly_ring R\<^esub> q"
    unfolding pdivides_def by auto
  moreover have "[ \, \ x ] \ carrier (poly_ring R)"
    using assms(2) unfolding sym[OF univ_poly_carrier] polynomial_def by simp
  ultimately have "eval p x = \"
    using ring_hom_memE[OF eval_is_hom[OF carrier_is_subring, of x]] assms(2) by (auto, algebra)
  with \<open>p \<noteq> []\<close> and \<open>x \<in> carrier R\<close> show "is_root p x"
    unfolding is_root_def by simp
qed

lemma (in domain) associated_polynomials_imp_same_is_root:
  assumes "p \ carrier (poly_ring R)" and "q \ carrier (poly_ring R)" and "p \\<^bsub>poly_ring R\<^esub> q"
  shows "is_root p x \ is_root q x"
proof (cases "p = []")
  case True with \<open>p \<sim>\<^bsub>poly_ring R\<^esub> q\<close> have "q = []"
    unfolding associated_def True factor_def univ_poly_def by auto
  thus ?thesis
    using True unfolding is_root_def by simp
next
  case False
  interpret UP: domain "poly_ring R"
    using univ_poly_is_domain[OF carrier_is_subring] .

  have "is_root q x"
    if p: "p \ carrier (poly_ring R)"
    and q: "q \ carrier (poly_ring R)"
    and pq: "p \\<^bsub>poly_ring R\<^esub> q"
    and is_root: "is_root p x"
    for p q
  proof -
    from is_root have "[ \, \ x ] pdivides p" and "p \ []" and "x \ carrier R"
      using is_root_imp_pdivides[OF p] unfolding is_root_def by auto
    moreover have "[ \, \ x ] \ carrier (poly_ring R)"
      using is_root unfolding is_root_def sym[OF univ_poly_carrier] polynomial_def by simp
    ultimately have "[ \, \ x ] pdivides q"
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