Theory HOL-Algebra.Ideal_Product

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

theory Ideal_Product
  imports Ideal
begin

section ‹Product of Ideals›

text ‹In this section, we study the structure of the set of ideals of a given ring.›

inductive_set
  ideal_prod :: "[ ('a, 'b) ring_scheme, 'a set, 'a set ] ⇒ 'a set" (infixl ‹⋅ı› 80)
  for R and I and J (* both I and J are supposed ideals *) where
    prod: "⟦ i ∈ I; j ∈ J ⟧ ⟹ i ⊗⇘R⇙ j ∈ ideal_prod R I J"
  |  sum: "⟦ s1 ∈ ideal_prod R I J; s2 ∈ ideal_prod R I J ⟧ ⟹ s1 ⊕⇘R⇙ s2 ∈ ideal_prod R I J"

definition ideals_set :: "('a, 'b) ring_scheme ⇒ ('a set) ring"
  where "ideals_set R = ⦇ carrier = { I. ideal I R },
                             mult = ideal_prod R,
                              one = carrier R,
                             zero = { 𝟬⇘R⇙ },
                              add = set_add R ⦈"


subsection ‹Basic Properties›

lemma (in ring) ideal_prod_in_carrier:
  assumes "ideal I R" "ideal J R"
  shows "I ⋅ J ⊆ carrier R"
proof
  fix s assume "s ∈ I ⋅ J" thus "s ∈ carrier R"
    by (induct s rule: ideal_prod.induct) (auto, meson assms ideal.I_l_closed ideal.Icarr) 
qed

lemma (in ring) ideal_prod_inter:
  assumes "ideal I R" "ideal J R"
  shows "I ⋅ J ⊆ I ∩ J"
proof
  fix s assume "s ∈ I ⋅ J" thus "s ∈ I ∩ J"
    apply (induct s rule: ideal_prod.induct)
    apply (auto, (meson assms ideal.I_r_closed ideal.I_l_closed ideal.Icarr)+)
    apply (simp_all add: additive_subgroup.a_closed assms ideal.axioms(1))
    done
qed

lemma (in ring) ideal_prod_is_ideal:
  assumes "ideal I R" "ideal J R"
  shows "ideal (I ⋅ J) R"
proof (rule idealI)
  show "ring R" using ring_axioms .
next
  show "subgroup (I ⋅ J) (add_monoid R)"
    unfolding subgroup_def
  proof (auto)
    show "𝟬 ∈ I ⋅ J" using ideal_prod.prod[of 𝟬 I 𝟬 J R]
      by (simp add: additive_subgroup.zero_closed assms ideal.axioms(1))
  next
    fix s1 s2 assume s1: "s1 ∈ I ⋅ J" and s2: "s2 ∈ I ⋅ J"
    have IJcarr: "⋀a. a ∈ I ⋅ J ⟹ a ∈ carrier R"
      by (meson assms subsetD ideal_prod_in_carrier)
    show "s1 ∈ carrier R" using ideal_prod_in_carrier[OF assms] s1 by blast
    show "s1 ⊕ s2 ∈ I ⋅ J" by (simp add: ideal_prod.sum[OF s1 s2])
    show "inv⇘add_monoid R⇙ s1 ∈ I ⋅ J" using s1
    proof (induct s1 rule: ideal_prod.induct)
      case (prod i j)
      hence "inv⇘add_monoid R⇙ (i ⊗ j) = (inv⇘add_monoid R⇙ i) ⊗ j"
        by (metis a_inv_def assms(1) assms(2) ideal.Icarr l_minus)
      thus ?case using ideal_prod.prod[of "inv⇘add_monoid R⇙ i" I j J R] assms
        by (simp add: additive_subgroup.a_subgroup ideal.axioms(1) prod.hyps subgroup.m_inv_closed)
    next
      case (sum s1 s2) thus ?case
        by (metis (no_types) IJcarr a_inv_def add.inv_mult_group ideal_prod.sum sum.hyps)
    qed
  qed
next
  fix s x assume s: "s ∈ I ⋅ J" and x: "x ∈ carrier R"
  show "x ⊗ s ∈ I ⋅ J" using s
  proof (induct s rule: ideal_prod.induct)
    case (prod i j) thus ?case using ideal_prod.prod[of "x ⊗ i" I j J R] assms
      by (simp add: x ideal.I_l_closed ideal.Icarr m_assoc)
  next
    case (sum s1 s2) thus ?case
    proof -
      have IJ: "I ⋅ J ⊆ carrier R"
        by (metis (no_types) assms(1) assms(2) ideal.axioms(2) ring.ideal_prod_in_carrier)
      then have "s2 ∈ carrier R"
        using sum.hyps(3) by blast
      moreover have "s1 ∈ carrier R"
        using IJ sum.hyps(1) by blast
      ultimately show ?thesis
        by (simp add: ideal_prod.sum r_distr sum.hyps x)
    qed
  qed
  show "s ⊗ x ∈ I ⋅ J" using s
  proof (induct s rule: ideal_prod.induct)
    case (prod i j) thus ?case using ideal_prod.prod[of i I "j ⊗ x" J R] assms x
      by (simp add: x ideal.I_r_closed ideal.Icarr m_assoc)
  next
    case (sum s1 s2) thus ?case 
    proof -
      have "s1 ∈ carrier R" "s2 ∈ carrier R"
        by (meson assms subsetD ideal_prod_in_carrier sum.hyps)+
      then show ?thesis
        by (metis ideal_prod.sum l_distr sum.hyps(2) sum.hyps(4) x)
    qed
  qed
qed

lemma (in ring) ideal_prod_eq_genideal:
  assumes "ideal I R" "ideal J R"
  shows "I ⋅ J = Idl (I <#> J)"
proof
  have "I <#> J ⊆ I ⋅ J"
  proof
    fix s assume "s ∈ I <#> J"
    then obtain i j where "i ∈ I" "j ∈ J" "s = i ⊗ j"
      unfolding set_mult_def by blast
    thus "s ∈ I ⋅ J" using ideal_prod.prod by simp
  qed
  thus "Idl (I <#> J) ⊆ I ⋅ J"
    unfolding genideal_def using ideal_prod_is_ideal[OF assms] by blast
next
  show "I ⋅ J ⊆ Idl (I <#> J)"
  proof
    fix s assume "s ∈ I ⋅ J" thus "s ∈ Idl (I <#> J)"
    proof (induct s rule: ideal_prod.induct)
      case (prod i j) hence "i ⊗ j ∈ I <#> J" unfolding set_mult_def by blast
      thus ?case unfolding genideal_def by blast 
    next
      case (sum s1 s2) thus ?case
        by (simp add: additive_subgroup.a_closed additive_subgroup.a_subset
            assms genideal_ideal ideal.axioms(1) set_mult_closed)
    qed
  qed
qed


lemma (in ring) ideal_prod_simp:
  assumes "ideal I R" "ideal J R" (* the second assumption could be suppressed *)
  shows "I = I <+> (I ⋅ J)"
proof
  show "I ⊆ I <+> I ⋅ J"
  proof
    fix i assume "i ∈ I" hence "i ⊕ 𝟬 ∈ I <+> I ⋅ J"
      using set_add_def'[of R I "I ⋅ J"] ideal_prod_is_ideal[OF assms]
            additive_subgroup.zero_closed[OF ideal.axioms(1), of "I ⋅ J" R] by auto
    thus "i ∈ I <+> I ⋅ J"
      using ‹i ∈ I› assms(1) ideal.Icarr by fastforce 
  qed
next
  show "I <+> I ⋅ J ⊆ I"
  proof
    fix s assume "s ∈ I <+> I ⋅ J"
    then obtain i ij where "i ∈ I" "ij ∈ I ⋅ J" "s = i ⊕ ij"
      using set_add_def'[of R I "I ⋅ J"] by auto
    thus "s ∈ I"
      using ideal_prod_inter[OF assms]
      by (meson additive_subgroup.a_closed assms(1) ideal.axioms(1) inf_sup_ord(1) subsetCE) 
  qed
qed

lemma (in ring) ideal_prod_one:
  assumes "ideal I R"
  shows "I ⋅ (carrier R) = I"
proof
  show "I ⋅ (carrier R) ⊆ I"
  proof
    fix s assume "s ∈ I ⋅ (carrier R)" thus "s ∈ I"
      by (induct s rule: ideal_prod.induct)
         (simp_all add: assms ideal.I_r_closed additive_subgroup.a_closed ideal.axioms(1))
  qed
next
  show "I ⊆ I ⋅ (carrier R)"
  proof
    fix i assume "i ∈ I" thus "i ∈  I ⋅ (carrier R)"
      by (metis assms ideal.Icarr ideal_prod.simps one_closed r_one)
  qed
qed

lemma (in ring) ideal_prod_zero:
  assumes "ideal I R"
  shows "I ⋅ { 𝟬 } = { 𝟬 }"
proof
  show "I ⋅ { 𝟬 } ⊆ { 𝟬 }"
  proof
    fix s assume "s ∈ I ⋅ {𝟬}" thus "s ∈ { 𝟬 }"
      using assms ideal.Icarr by (induct s rule: ideal_prod.induct) (fastforce, simp)
  qed
next
  show "{ 𝟬 } ⊆ I ⋅ { 𝟬 }"
    by (simp add: additive_subgroup.zero_closed assms
                  ideal.axioms(1) ideal_prod_is_ideal zeroideal)
qed

lemma (in ring) ideal_prod_assoc:
  assumes "ideal I R" "ideal J R" "ideal K R"
  shows "(I ⋅ J) ⋅ K = I ⋅ (J ⋅ K)"
proof
  show "(I ⋅ J) ⋅ K ⊆ I ⋅ (J ⋅ K)"
  proof
    fix s assume "s ∈ (I ⋅ J) ⋅ K" thus "s ∈ I ⋅ (J ⋅ K)"
    proof (induct s rule: ideal_prod.induct)
      case (sum s1 s2) thus ?case
        by (simp add: ideal_prod.sum)
    next
      case (prod i k) thus ?case
      proof (induct i rule: ideal_prod.induct)
        case (prod i j) thus ?case
          using ideal_prod.prod[OF prod(1) ideal_prod.prod[OF prod(2-3),of R], of R]
          by (metis assms ideal.Icarr m_assoc) 
      next
        case (sum s1 s2) thus ?case
        proof -
          have "s1 ∈ carrier R" "s2 ∈ carrier R"
            by (meson assms subsetD ideal.axioms(2) ring.ideal_prod_in_carrier sum.hyps)+
          moreover have "k ∈ carrier R"
            by (meson additive_subgroup.a_Hcarr assms(3) ideal.axioms(1) sum.prems)
          ultimately show ?thesis
            by (metis ideal_prod.sum l_distr sum.hyps(2) sum.hyps(4) sum.prems)
        qed
      qed
    qed
  qed
next
  show "I ⋅ (J ⋅ K) ⊆ (I ⋅ J) ⋅ K"
  proof
    fix s assume "s ∈ I ⋅ (J ⋅ K)" thus "s ∈ (I ⋅ J) ⋅ K"
    proof (induct s rule: ideal_prod.induct)
      case (sum s1 s2) thus ?case by (simp add: ideal_prod.sum)
    next
      case (prod i j) show ?case using prod(2) prod(1)
      proof (induct j rule: ideal_prod.induct)
        case (prod j k) thus ?case
          using ideal_prod.prod[OF ideal_prod.prod[OF prod(3) prod(1), of R] prod (2), of R]
          by (metis assms ideal.Icarr m_assoc)
      next
        case (sum s1 s2) thus ?case
        proof -
          have "⋀a A B. ⟦a ∈ B ⋅ A; ideal A R; ideal B R⟧ ⟹ a ∈ carrier R"
            by (meson subsetD ideal_prod_in_carrier)
          moreover have "i ∈ carrier R"
            by (meson additive_subgroup.a_Hcarr assms(1) ideal.axioms(1) sum.prems)
          ultimately show ?thesis
            by (metis (no_types) assms(2) assms(3) ideal_prod.sum r_distr sum)
        qed
      qed
    qed
  qed
qed

lemma (in ring) ideal_prod_r_distr:
  assumes "ideal I R" "ideal J R" "ideal K R"
  shows "I ⋅ (J <+> K) = (I ⋅ J) <+>  (I ⋅ K)"
proof
  show "I ⋅ (J <+> K) ⊆ I ⋅ J <+> I ⋅ K"
  proof
    fix s assume "s ∈ I ⋅ (J <+> K)" thus "s ∈ I ⋅ J <+> I ⋅ K"
    proof(induct s rule: ideal_prod.induct)
      case (prod i jk)
      then obtain j k where j: "j ∈ J" and k: "k ∈ K" and jk: "jk = j ⊕ k"
        using set_add_def'[of R J K] by auto
      hence "i ⊗ j ⊕ i ⊗ k ∈ I ⋅ J <+> I ⋅ K"
        using ideal_prod.prod[OF prod(1) j,of R]
              ideal_prod.prod[OF prod(1) k,of R]
              set_add_def'[of R "I ⋅ J" "I ⋅ K"] by auto
      thus ?case
        using assms ideal.Icarr r_distr jk j k prod(1) by metis 
    next
      case (sum s1 s2) thus ?case
        by (simp add: add_ideals additive_subgroup.a_closed assms ideal.axioms(1)
                      local.ring_axioms ring.ideal_prod_is_ideal) 
    qed
  qed

  have aux_lemma: "s ∈ I ⋅ (J <+> K) ∧ s ∈ I ⋅ (K <+> J)"
    if A: "ideal J R" "ideal K R" "s ∈ I ⋅ J" for s J K
  proof -
    from ‹s ∈ I ⋅ J› have "s ∈ I ⋅ (J <+> K)"
    proof (induct s rule: ideal_prod.induct)
      case (prod i j)
      hence "(j ⊕ 𝟬) ∈ J <+> K"
        using set_add_def'[of R J K]
          additive_subgroup.zero_closed[OF ideal.axioms(1), of K R] A(2) by auto
      thus ?case
        by (metis A(1) additive_subgroup.a_Hcarr ideal.axioms(1) ideal_prod.prod prod r_zero)  
    next
      case (sum s1 s2) thus ?case
        by (simp add: ideal_prod.sum) 
    qed
    thus ?thesis
      by (metis A(1) A(2) ideal_def ring.union_genideal sup_commute) 
  qed
  show "I ⋅ J <+> I ⋅ K ⊆ I ⋅ (J <+> K)"
  proof
    fix s assume "s ∈ I ⋅ J <+> I ⋅ K"
    then obtain s1 s2 where s1: "s1 ∈ I ⋅ J" and s2: "s2 ∈ I ⋅ K" and  s: "s = s1 ⊕ s2"
      using set_add_def'[of R "I ⋅ J" "I ⋅ K"] by auto
    thus "s ∈ I ⋅ (J <+> K)"
      using aux_lemma[OF assms(2) assms(3) s1]
            aux_lemma[OF assms(3) assms(2) s2] by (simp add: ideal_prod.sum)
  qed
qed

lemma (in cring) ideal_prod_commute:
  assumes "ideal I R" "ideal J R"
  shows "I ⋅ J = J ⋅ I"
proof -
  have "I ⋅ J ⊆ J ⋅ I" if A: "ideal I R" "ideal J R" for I J
  proof
    fix s
    assume "s ∈ I ⋅ J"
    thus "s ∈ J ⋅ I"
    proof (induct s rule: ideal_prod.induct)
      case (prod i j)
      thus ?case
        using m_comm[OF ideal.Icarr[OF A(1) prod(1)] ideal.Icarr[OF A(2) prod(2)]]
        by (simp add: ideal_prod.prod)
    next
      case (sum s1 s2)
      thus ?case by (simp add: ideal_prod.sum) 
    qed
  qed
  with assms show ?thesis by blast 
qed

text ‹The following result would also be true for locale ring›
lemma (in cring) ideal_prod_distr:
  assumes "ideal I R" "ideal J R" "ideal K R"
  shows "I ⋅ (J <+> K) = (I ⋅ J) <+>  (I ⋅ K)"
    and "(J <+> K) ⋅ I = (J ⋅ I) <+>  (K ⋅ I)"
  by (simp_all add: assms ideal_prod_commute local.ring_axioms
                    ring.add_ideals ring.ideal_prod_r_distr)

lemma (in cring) ideal_prod_eq_inter:
  assumes "ideal I R" "ideal J R"
    and "I <+> J = carrier R"
  shows "I ⋅ J = I ∩ J"
proof
  show "I ⋅ J ⊆ I ∩ J"
    using assms ideal_prod_inter by auto
next
  show "I ∩ J ⊆ I ⋅ J"
  proof
    have "𝟭 ∈ I <+> J" using assms(3) one_closed by simp 
    then obtain i j where ij: "i ∈ I" "j ∈ J" "𝟭 = i ⊕ j"
      using set_add_def'[of R I J] by auto

    fix s assume s: "s ∈ I ∩ J"
    hence "(i ⊗ s ∈ I ⋅ J) ∧ (s ⊗ j ∈ I ⋅ J)"
      using ij(1-2) by (simp add: ideal_prod.prod)
    moreover have "s = (i ⊗ s) ⊕ (s ⊗ j)"
      using ideal.Icarr[OF assms(1) ij(1)]
            ideal.Icarr[OF assms(2) ij(2)]
            ideal.Icarr[OF assms(1), of s]
      by (metis ij(3) s m_comm[of s i] Int_iff r_distr r_one)
    ultimately show "s ∈  I ⋅ J"
      using ideal_prod.sum by fastforce
  qed
qed


subsection ‹Structure of the Set of Ideals›

text ‹We focus on commutative rings for convenience.›

lemma (in cring) ideals_set_is_semiring: "semiring (ideals_set R)"
proof -
  have "abelian_monoid (ideals_set R)"
    apply (rule abelian_monoidI) unfolding ideals_set_def
    apply (simp_all add: add_ideals zeroideal)
    apply (simp add: add.set_mult_assoc additive_subgroup.a_subset ideal.axioms(1) set_add_defs(1))
    apply (metis Un_absorb1 additive_subgroup.a_subset additive_subgroup.zero_closed
        cgenideal_minimal cgenideal_self empty_iff genideal_minimal ideal.axioms(1)
        local.ring_axioms order_refl ring.genideal_self subset_antisym subset_singletonD
        union_genideal zero_closed zeroideal)
    by (metis sup_commute union_genideal)

  moreover have "monoid (ideals_set R)"
    apply (rule monoidI) unfolding ideals_set_def
    apply (simp_all add: ideal_prod_is_ideal oneideal
                         ideal_prod_commute ideal_prod_one)
    by (metis ideal_prod_assoc ideal_prod_commute)

  ultimately show ?thesis
    unfolding semiring_def semiring_axioms_def ideals_set_def
    by (simp_all add: ideal_prod_distr ideal_prod_commute ideal_prod_zero zeroideal) 
qed

lemma (in cring) ideals_set_is_comm_monoid: "comm_monoid (ideals_set R)"
proof -
  have "monoid (ideals_set R)"
    apply (rule monoidI) unfolding ideals_set_def
    apply (simp_all add: ideal_prod_is_ideal oneideal
                         ideal_prod_commute ideal_prod_one)
    by (metis ideal_prod_assoc ideal_prod_commute)
  thus ?thesis
    unfolding comm_monoid_def comm_monoid_axioms_def
    by (simp add: ideal_prod_commute ideals_set_def)
qed

lemma (in cring) ideal_prod_eq_Inter_aux:
  assumes "I: {..(Suc n)} → { J. ideal J R }" 
    and "⋀i j. ⟦ i ≤ Suc n; j ≤ Suc n ⟧ ⟹
                 i ≠ j ⟹ (I i) <+> (I j) = carrier R"
  shows "(⨂⇘(ideals_set R)⇙ k ∈ {..n}. I k) <+> (I (Suc n)) = carrier R" using assms
proof (induct n arbitrary: I)
  case 0
  hence "(⨂⇘(ideals_set R)⇙ k ∈ {..0}. I k) <+> I (Suc 0) = (I 0) <+> (I (Suc 0))"
    using comm_monoid.finprod_0[OF ideals_set_is_comm_monoid, of I]
    by (simp add: atMost_Suc ideals_set_def)
  also have " ... = carrier R"
    using 0(2)[of 0 "Suc 0"] by simp
  finally show ?case .
next
  interpret ISet: comm_monoid "ideals_set R"
    by (simp add: ideals_set_is_comm_monoid)

  case (Suc n)
  let ?I' = "λi. I (Suc i)"
  have "?I': {..(Suc n)} → { J. ideal J R }"
    using Suc.prems(1) by auto
  moreover have "⋀i j. ⟦ i ≤ Suc n; j ≤ Suc n ⟧ ⟹
                         i ≠ j ⟹ (?I' i) <+> (?I' j) = carrier R"
    by (simp add: Suc.prems(2))
  ultimately have "(⨂⇘(ideals_set R)⇙ k ∈ {..n}. ?I' k) <+> (?I' (Suc n)) = carrier R"
    using Suc.hyps by metis

  moreover have I_carr: "I: {..Suc (Suc n)} → carrier (ideals_set R)"
    unfolding ideals_set_def using Suc by simp
  hence I'_carr: "I ∈ Suc ` {..n} → carrier (ideals_set R)" by auto
  ultimately have "(⨂⇘(ideals_set R)⇙ k ∈ {(Suc 0)..Suc n}. I k) <+> (I (Suc (Suc n))) = carrier R"
    using ISet.finprod_reindex[of I "λi. Suc i" "{..n}"] by (simp add: atMost_atLeast0) 

  hence "(carrier R) ⋅ (I 0) = ((⨂⇘(ideals_set R)⇙ k ∈ {Suc 0..Suc n}. I k) <+> I (Suc (Suc n))) ⋅ (I 0)"
    by auto
  moreover have fprod_cl1: "ideal (⨂⇘(ideals_set R)⇙ k ∈ {Suc 0..Suc n}. I k) R"
    by (metis I'_carr ISet.finprod_closed One_nat_def ideals_set_def image_Suc_atMost
        mem_Collect_eq partial_object.select_convs(1))
  ultimately
  have "I 0 = (⨂⇘(ideals_set R)⇙ k ∈ {Suc 0..Suc n}. I k) ⋅ (I 0) <+> I (Suc (Suc n)) ⋅ (I 0)"
    by (metis PiE Suc.prems(1) atLeast0_atMost_Suc atLeast0_atMost_Suc_eq_insert_0
        atMost_atLeast0 ideal_prod_commute ideal_prod_distr(2) ideal_prod_one insertI1
        mem_Collect_eq oneideal)
  also have " ... = (I 0) ⋅ (⨂⇘(ideals_set R)⇙ k ∈ {Suc 0..Suc n}. I k) <+> I (Suc (Suc n)) ⋅ (I 0)"
    using fprod_cl1 ideal_prod_commute Suc.prems(1)
    by (simp add: atLeast0_atMost_Suc_eq_insert_0 atMost_atLeast0) 
  also have " ... = (I 0) ⊗⇘(ideals_set R)⇙ (⨂⇘(ideals_set R)⇙ k ∈ {Suc 0..Suc n}. I k) <+>
                     I (Suc (Suc n)) ⋅ (I 0)"
    by (simp add: ideals_set_def)
  finally have I0: "I 0 = (⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) <+> I (Suc (Suc n)) ⋅ (I 0)"
    using ISet.finprod_insert[of "{Suc 0..Suc n}" 0 I]
          I_carr I'_carr atMost_atLeast0 ISet.finprod_0' atMost_Suc by auto

  have I_SucSuc_I0: "ideal (I (Suc (Suc n))) R ∧ ideal (I 0) R"
    using Suc.prems(1) by auto
  have fprod_cl2: "ideal (⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) R"
    by (metis (no_types) ISet.finprod_closed I_carr Pi_split_insert_domain atMost_Suc ideals_set_def mem_Collect_eq partial_object.select_convs(1))
  have "carrier R = I (Suc (Suc n)) <+> I 0"
    by (simp add: Suc.prems(2))
  also have " ... = I (Suc (Suc n)) <+>
                    ((⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) <+> I (Suc (Suc n)) ⋅ (I 0))"
    using I0 by auto
  also have " ... = I (Suc (Suc n)) <+>
                    (I (Suc (Suc n)) ⋅ (I 0) <+> (⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k))"
    using fprod_cl2 I_SucSuc_I0 by (metis Un_commute ideal_prod_is_ideal union_genideal)
  also have " ... = (I (Suc (Suc n)) <+> I (Suc (Suc n)) ⋅ (I 0)) <+>
                    (⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k)"
    using fprod_cl2 I_SucSuc_I0 by (metis add.set_mult_assoc ideal_def ideal_prod_in_carrier
                                          oneideal ring.ideal_prod_one set_add_defs(1)) 
  also have " ... = I (Suc (Suc n)) <+> (⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k)"
    using ideal_prod_simp[of "I (Suc (Suc n))" "I 0"] I_SucSuc_I0 by simp 
  also have " ... = (⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) <+> I (Suc (Suc n))"
    using fprod_cl2 I_SucSuc_I0 by (metis Un_commute union_genideal)
  finally show ?case by simp
qed

theorem (in cring) ideal_prod_eq_Inter:
  assumes "I: {..n :: nat} → { J. ideal J R }" 
    and "⋀i j. ⟦ i ∈ {..n}; j ∈ {..n} ⟧ ⟹ i ≠ j ⟹ (I i) <+> (I j) = carrier R"
  shows "(⨂⇘(ideals_set R)⇙ k ∈ {..n}. I k) = (⋂ k ∈ {..n}. I k)" using assms
proof (induct n)
  case 0 thus ?case
    using comm_monoid.finprod_0[OF ideals_set_is_comm_monoid] by (simp add: ideals_set_def) 
next
  interpret ISet: comm_monoid "ideals_set R"
    by (simp add: ideals_set_is_comm_monoid)

  case (Suc n)
  hence IH: "(⨂⇘(ideals_set R)⇙ k ∈ {..n}. I k) = (⋂ k ∈ {..n}. I k)"
    by (simp add: atMost_Suc)
  hence "(⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) = I (Suc n) ⊗⇘(ideals_set R)⇙ (⋂ k ∈ {..n}. I k)"
    using ISet.finprod_insert[of "{Suc 0..Suc n}" 0 I] atMost_Suc_eq_insert_0[of n]
    by (metis ISet.finprod_Suc Suc.prems(1) ideals_set_def partial_object.select_convs(1))
  hence "(⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) = I (Suc n) ⋅ (⋂ k ∈ {..n}. I k)"
    by (simp add: ideals_set_def)
  moreover have "(⋂ k ∈ {..n}. I k) <+> I (Suc n) = carrier R"
    using ideal_prod_eq_Inter_aux[of I n] by (simp add: Suc.prems IH)
  moreover have "ideal (⋂ k ∈ {..n}. I k) R"
    using ring.i_Intersect[of R "I ` {..n}"]
    by (metis IH ISet.finprod_closed Pi_split_insert_domain Suc.prems(1) atMost_Suc
              ideals_set_def mem_Collect_eq partial_object.select_convs(1))
  ultimately
  have "(⨂⇘(ideals_set R)⇙ k ∈ {..Suc n}. I k) = (⋂ k ∈ {..n}. I k) ∩ I (Suc n)"
    using ideal_prod_eq_inter[of "⋂ k ∈ {..n}. I k" "I (Suc n)"]
          ideal_prod_commute[of "⋂ k ∈ {..n}. I k" "I (Suc n)"]
    by (metis PiE Suc.prems(1) atMost_iff mem_Collect_eq order_refl)
  thus ?case by (simp add: Int_commute atMost_Suc) 
qed

corollary (in cring) inter_plus_ideal_eq_carrier:
  assumes "⋀i. i ≤ Suc n ⟹ ideal (I i) R" 
      and "⋀i j. ⟦ i ≤ Suc n; j ≤ Suc n; i ≠ j ⟧ ⟹ I i <+> I j = carrier R"
  shows "(⋂ i ≤ n. I i) <+> (I (Suc n)) = carrier R"
  using ideal_prod_eq_Inter[of I n] ideal_prod_eq_Inter_aux[of I n] by (auto simp add: assms)

corollary (in cring) inter_plus_ideal_eq_carrier_arbitrary:
  assumes "⋀i. i ≤ Suc n ⟹ ideal (I i) R" 
      and "⋀i j. ⟦ i ≤ Suc n; j ≤ Suc n; i ≠ j ⟧ ⟹ I i <+> I j = carrier R"
      and "j ≤ Suc n"
  shows "(⋂ i ∈ ({..(Suc n)} - { j }). I i) <+> (I j) = carrier R"
proof -
  define I' where "I' = (λi. if i = Suc n then (I j) else
                             if i = j     then (I (Suc n))
                                          else (I i))"
  have "⋀i. i ≤ Suc n ⟹ ideal (I' i) R"
    using I'_def assms(1) assms(3) by auto
  moreover have "⋀i j. ⟦ i ≤ Suc n; j ≤ Suc n; i ≠ j ⟧ ⟹ I' i <+> I' j = carrier R"
    using I'_def assms(2-3) by force
  ultimately have "(⋂ i ≤ n. I' i) <+> (I' (Suc n)) = carrier R"
    using inter_plus_ideal_eq_carrier by simp

  moreover have "I' ` {..n} = I ` ({..(Suc n)} - { j })"
  proof
    show "I' ` {..n} ⊆ I ` ({..Suc n} - {j})"
    proof
      fix x assume "x ∈ I' ` {..n}"
      then obtain i where i: "i ∈ {..n}" "I' i = x" by blast
      thus "x ∈ I ` ({..Suc n} - {j})"
      proof (cases)
        assume "i = j" thus ?thesis using i I'_def by auto
      next
        assume "i ≠ j" thus ?thesis using I'_def i insert_iff by auto
      qed
    qed
  next
    show "I ` ({..Suc n} - {j}) ⊆ I' ` {..n}"
    proof
      fix x assume "x ∈ I ` ({..Suc n} - {j})"
      then obtain i where i: "i ∈ {..Suc n}" "i ≠ j" "I i = x" by blast
      thus "x ∈ I' ` {..n}"
      proof (cases)
        assume "i = Suc n" thus ?thesis using I'_def assms(3) i(2-3) by auto
      next
        assume "i ≠ Suc n" thus ?thesis using I'_def i by auto
      qed
    qed
  qed
  ultimately show ?thesis using I'_def by metis 
qed


subsection ‹Another Characterization of Prime Ideals›

text ‹With product of ideals being defined, we can give another definition of a prime ideal›

lemma (in ring) primeideal_divides_ideal_prod:
  assumes "primeideal P R" "ideal I R" "ideal J R"
      and "I ⋅ J ⊆ P"
    shows "I ⊆ P ∨ J ⊆ P"
proof (cases)
  assume "∃ i ∈ I. i ∉ P"
  then obtain i where i: "i ∈ I" "i ∉ P" by blast
  have "J ⊆ P"
  proof
    fix j assume j: "j ∈ J"
    hence "i ⊗ j ∈ P"
      using ideal_prod.prod[OF i(1) j, of R] assms(4) by auto
    thus "j ∈ P"
      using primeideal.I_prime[OF assms(1), of i j] i j
      by (meson assms(2-3) ideal.Icarr) 
  qed
  thus ?thesis by blast
next
  assume "¬ (∃ i ∈ I. i ∉ P)" thus ?thesis by blast
qed

lemma (in cring) divides_ideal_prod_imp_primeideal:
  assumes "ideal P R"
    and "P ≠ carrier R"
    and "⋀I J. ⟦ ideal I R; ideal J R; I ⋅ J ⊆ P ⟧ ⟹ I ⊆ P ∨ J ⊆ P"
  shows "primeideal P R"
proof -
  have "⋀a b. ⟦ a ∈ carrier R; b ∈ carrier R; a ⊗ b ∈ P ⟧ ⟹ a ∈ P ∨ b ∈ P"
  proof -
    fix a b assume A: "a ∈ carrier R" "b ∈ carrier R" "a ⊗ b ∈ P"
    have "(PIdl a) ⋅ (PIdl b) = Idl (PIdl (a ⊗ b))"
      using ideal_prod_eq_genideal[of "Idl { a }" "Idl { b }"]
            A(1-2) cgenideal_eq_genideal cgenideal_ideal cgenideal_prod by auto
    hence "(PIdl a) ⋅ (PIdl b) = PIdl (a ⊗ b)"
      by (simp add: A Idl_subset_ideal cgenideal_ideal cgenideal_minimal
                    genideal_self oneideal subset_antisym)
    hence "(PIdl a) ⋅ (PIdl b) ⊆ P"
      by (simp add: A(3) assms(1) cgenideal_minimal)
    hence "(PIdl a) ⊆ P ∨ (PIdl b) ⊆ P"
      by (simp add: A assms(3) cgenideal_ideal)
    thus "a ∈ P ∨ b ∈ P"
      using A cgenideal_self by blast
  qed
  thus ?thesis
    using assms is_cring by (simp add: primeidealI)
qed

end