Theory Multirelations

theory Multirelations

imports Power_Allegories_Multirelations

begin

lemma nonempty_set_card:
  assumes "finite S"
  shows "S ≠ {} ⟷ card S ≥ 1"
  using assms card_0_eq by fastforce

no_notation one_class.one (‹1›)
no_notation times_class.times (infixl ‹*› 70)

no_notation rel_fdia (‹(|_⟩_)› [61,81] 82)
no_notation rel_bdia (‹(⟨_|_)› [61,81] 82)
no_notation rel_fbox (‹(|_]_)› [61,81] 82)
no_notation rel_bbox (‹([_|_)› [61,81] 82)

declare s_prod_pa_def [mr_simp]

notation s_prod (infixl ‹*› 70)
notation s_id (‹1›)

lemma sp_oi_subdist:
  "(P ∩ Q) * (R ∩ S) ⊆ P * R"
  unfolding s_prod_def by blast

lemma sp_oi_subdist_2:
  "(P ∩ Q) * (R ∩ S) ⊆ (P * R) ∩ (Q * S)"
  unfolding s_prod_def by blast

section ‹Inner Structure›

subsection ‹Inner union, inner intersection and inner complement›

abbreviation "inner_union" (infixl ‹∪∪› 65)
  where "inner_union ≡ p_prod"

definition inner_intersection :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ ('a,'b) mrel" (infixl ‹∩∩› 65) where
  "R ∩∩ S ≡ { (a,B) . ∃C D . B = C ∩ D ∧ (a,C) ∈ R ∧ (a,D) ∈ S }"

definition inner_complement :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹∼ _› [80] 80) where
  "∼R ≡ { (a,B) . (a,-B) ∈ R }"

abbreviation "iu_unit" (‹1⇩∪⇩∪›)
  where "1⇩∪⇩∪ ≡ p_id"

definition ii_unit :: "('a,'a) mrel" (‹1⇩∩⇩∩›)
  where "1⇩∩⇩∩ ≡ { (a,UNIV) | a . True }"

declare inner_intersection_def [mr_simp] inner_complement_def [mr_simp] ii_unit_def [mr_simp]

lemma iu_assoc:
  "(R ∪∪ S) ∪∪ T = R ∪∪ (S ∪∪ T)"
  by (simp add: p_prod_assoc)

lemma iu_commute:
  "R ∪∪ S = S ∪∪ R"
  by (simp add: p_prod_comm)

lemma iu_unit:
  "R ∪∪ 1⇩∪⇩∪ = R"
  by simp

lemma ii_assoc:
  "(R ∩∩ S) ∩∩ T = R ∩∩ (S ∩∩ T)"
  apply (clarsimp simp: mr_simp)
  by (metis (no_types, opaque_lifting) semilattice_inf_class.inf_assoc)

lemma ii_commute:
  "R ∩∩ S = S ∩∩ R"
  by (auto simp: mr_simp)

lemma ii_unit [simp]:
  "R ∩∩ 1⇩∩⇩∩ = R"
  by (simp add: mr_simp)

lemma pa_ic:
  "∼(R ⊗ ∼S) = R ⊗ S"
  by (clarsimp simp: mr_simp)

lemma ic_involutive [simp]:
  "∼∼R = R"
  by (simp add: mr_simp)

lemma ic_injective:
  "∼R = ∼S ⟹ R = S"
  by (metis ic_involutive)

lemma ic_antidist_iu:
  "∼(R ∪∪ S) = ∼R ∩∩ ∼S"
  apply (clarsimp simp: mr_simp)
  by (metis (no_types, opaque_lifting) compl_sup double_compl)

lemma ic_antidist_ii:
  "∼(R ∩∩ S) = ∼R ∪∪ ∼S"
  by (metis ic_antidist_iu ic_involutive)

lemma ic_iu_unit [simp]:
  "∼1⇩∪⇩∪ = 1⇩∩⇩∩"
  unfolding ii_unit_def p_id_def inner_complement_def by force

lemma ic_ii_unit [simp]:
  "∼1⇩∩⇩∩ = 1⇩∪⇩∪"
  by (metis ic_involutive ic_iu_unit)

lemma ii_unit_split_iu [simp]:
  "1 ∪∪ ∼1 = 1⇩∩⇩∩"
  by (force simp: mr_simp)

lemma aux_1:
  "B = {a} ∩ D ⟹ -D = {a} ⟹ B = {}"
  by auto

lemma iu_unit_split_ii [simp]:
  "1 ∩∩ ∼1 = 1⇩∪⇩∪"
  by (metis ic_antidist_iu ic_ii_unit ic_involutive ii_commute ii_unit_split_iu)

lemma iu_right_dist_ou:
  "(R ∪ S) ∪∪ T = (R ∪∪ T) ∪ (S ∪∪ T)"
  unfolding p_prod_def by auto

lemma ii_right_dist_ou:
  "(R ∪ S) ∩∩ T = (R ∩∩ T) ∪ (S ∩∩ T)"
  by (auto simp: mr_simp)

lemma iu_left_isotone:
  "R ⊆ S ⟹ R ∪∪ T ⊆ S ∪∪ T"
  by (metis iu_right_dist_ou subset_Un_eq)

lemma iu_right_isotone:
  "R ⊆ S ⟹ T ∪∪ R ⊆ T ∪∪ S"
  by (simp add: iu_commute iu_left_isotone)

lemma iu_isotone:
  "R ⊆ S ⟹ P ⊆ Q ⟹ R ∪∪ P ⊆ S ∪∪ Q"
  by (meson dual_order.trans iu_left_isotone iu_right_isotone)

lemma ii_left_isotone:
  "R ⊆ S ⟹ R ∩∩ T ⊆ S ∩∩ T"
  by (metis ii_right_dist_ou subset_Un_eq)

lemma ii_right_isotone:
  "R ⊆ S ⟹ T ∩∩ R ⊆ T ∩∩ S"
  by (simp add: ii_commute ii_left_isotone)

lemma ii_isotone:
  "R ⊆ S ⟹ P ⊆ Q ⟹ R ∩∩ P ⊆ S ∩∩ Q"
  by (meson ii_left_isotone ii_right_isotone order_trans)

lemma iu_right_subdist_ii:
  "(R ∩∩ S) ∪∪ T ⊆ (R ∪∪ T) ∩∩ (S ∪∪ T)"
  apply (clarsimp simp: mr_simp)
  by (metis sup_inf_distrib2)

lemma ii_right_subdist_iu:
  "(R ∪∪ S) ∩∩ T ⊆ (R ∩∩ T) ∪∪ (S ∩∩ T)"
  apply (clarsimp simp: mr_simp)
  by (metis inf_sup_distrib2)

lemma ic_isotone:
  "R ⊆ S ⟹ ∼R ⊆ ∼S"
  by (simp add: inner_complement_def subset_eq)

lemma ic_bot [simp]:
  "∼{} = {}"
  by (simp add: mr_simp)

lemma ic_top [simp]:
  "∼U = U"
  by (auto simp: mr_simp)

lemma ic_dist_ou:
  "∼(R ∪ S) = ∼R ∪ ∼S"
  by (auto simp: mr_simp)

lemma ic_dist_oi:
  "∼(R ∩ S) = ∼R ∩ ∼S"
  by (auto simp: mr_simp)

lemma ic_dist_oc:
  "∼-R = -(∼R)"
  by (auto simp: mr_simp)

lemma ii_sub_idempotent:
  "R ⊆ R ∩∩ R"
  unfolding inner_intersection_def by force

definition inner_Union :: "('i ⇒ ('a,'b) mrel) ⇒ 'i set ⇒ ('a,'b) mrel" (‹⋃⋃_|_› [80,80] 80) where
  "⋃⋃X|I ≡ { (a,B) . ∃f . B = (⋃i∈I . f i) ∧ (∀i∈I . (a,f i) ∈ X i) }"

definition inner_Intersection :: "('i ⇒ ('a,'b) mrel) ⇒ 'i set ⇒ ('a,'b) mrel" (‹⋂⋂_|_› [80,80] 80) where
  "⋂⋂X|I ≡ { (a,B) . ∃f . B = (⋂i∈I . f i) ∧ (∀i∈I . (a,f i) ∈ X i) }"

declare inner_Union_def [mr_simp] inner_Intersection_def [mr_simp]

lemma iU_empty:
  "⋃⋃X|{} = 1⇩∪⇩∪"
  by (auto simp: mr_simp)

lemma iI_empty:
  "⋂⋂X|{} = 1⇩∩⇩∩"
  by (auto simp: mr_simp)

lemma ic_antidist_iU:
  "∼⋃⋃X|I = ⋂⋂(inner_complement o X)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis (mono_tags, lifting) Compl_UN double_compl)
  by (clarsimp simp: mr_simp) blast

lemma ic_antidist_iI:
  "∼⋂⋂X|I = ⋃⋃(inner_complement o X)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis Compl_INT double_complement)
  by (clarsimp simp: mr_simp) blast

lemma iu_right_dist_oU:
  "⋃X ∪∪ T = (⋃R∈X . R ∪∪ T)"
  by (clarsimp simp: mr_simp) blast

lemma ii_right_dist_oU:
  "⋃X ∩∩ T = (⋃R∈X . R ∩∩ T)"
  by (clarsimp simp: mr_simp) blast

lemma iu_right_subdist_iI:
  "⋂⋂X|I ∪∪ T ⊆ ⋂⋂(λi . X i ∪∪ T)|I"
  apply (clarsimp simp: mr_simp)
  by (metis INT_simps(6))

lemma ii_right_subdist_iU:
  "⋃⋃X|I ∩∩ T ⊆ ⋃⋃(λi . X i ∩∩ T)|I"
  by (clarsimp simp: mr_simp, metis UN_extend_simps(4))

lemma ic_dist_oU:
  "∼⋃X = ⋃(inner_complement ` X)"
  by (auto simp: mr_simp)

lemma ic_dist_oI:
  "∼⋂X = ⋂(inner_complement ` X)"
  by (auto simp: mr_simp)

lemma sp_left_subdist_iU:
  "R * (⋃⋃X|I) ⊆ ⋃⋃(λi . R * X i)|I"
  apply (clarsimp simp: mr_simp)
  subgoal for a B f proof -
    assume 1: "(a,B) ∈ R"
    assume "∀b∈B . ∃g . f b = ⋃(g ` I) ∧ (∀i∈I . (b,g i) ∈ X i)"
    from this obtain g where 2: "∀b∈B . f b = ⋃(g b ` I) ∧ (∀i∈I . (b,g b i) ∈ X i)"
      by metis
    hence 3: "⋃(f ` B) = (⋃b∈B . ⋃(g b ` I))"
      by (meson SUP_cong)
    let ?h = "λi . ⋃b∈B . g b i"
    have "⋃(f ` B) = ⋃(?h ` I) ∧ (∀i∈I . ∃B . (a,B) ∈ R ∧ (∃f . (∀b∈B . (b,f b) ∈ X i) ∧ ?h i = ⋃(f ` B)))"
      using 1 2 3 by (metis SUP_commute)
    thus ?thesis
      by auto
  qed
  done

lemma sp_right_subdist_iU:
  "(⋃⋃X|I) * R ⊆ ⋃⋃(λi . X i * R)|I"
  by (clarsimp simp: mr_simp, blast)

lemma sp_right_dist_iU:
  assumes "∀J::'a set . J ≠ {} ⟶ (⋃⋃(λj . R)|J) ⊆ R"
  shows "(⋃⋃X|I) * R = ⋃⋃(λi . X i * R)|(I::'a set)"
  apply (rule antisym)
  using sp_right_subdist_iU apply blast
  apply (clarsimp simp: mr_simp)
  subgoal for a f proof -
    assume "∀i∈I . ∃B . (a,B) ∈ X i ∧ (∃g . (∀b∈B . (b,g b) ∈ R) ∧ f i = ⋃(g ` B))"
    from this obtain B g where 1: "∀i∈I . (a,B i) ∈ X i ∧ (∀b∈B i . (b,g i b) ∈ R) ∧ f i = ⋃(g i ` B i)"
      by metis
    let ?B = "⋃(B ` I)"
    let ?g = "λb . ⋃{ g i b | i . i∈I ∧ b ∈ B i }"
    have "∀b∈?B . (b,?g b) ∈ R"
    proof (rule ballI)
      fix b
      let ?I = "{ i | i . i∈I ∧ b ∈ B i }"
      assume 2: "b ∈ ⋃(B ` I)"
      have "(b,?g b) ∈ ⋃⋃(λj . R)|?I"
        apply (clarsimp simp: mr_simp)
        apply (rule exI[of _ "λi . g i b"])
        using 1 by blast
      thus "(b,?g b) ∈ R"
        using 2 by (smt (verit) assms UN_E empty_Collect_eq subset_iff)
    qed
    hence "?B = ⋃(B ` I) ∧ (∀i∈I . (a,B i) ∈ X i) ∧ (∀b∈?B . (b,?g b) ∈ R) ∧ ⋃(f ` I) = ⋃(?g ` ?B)"
      using 1 by auto
    thus "∃B . (∃f . B = ⋃(f ` I) ∧ (∀i∈I . (a,f i) ∈ X i)) ∧ (∃g . (∀b∈B . (b,g b) ∈ R) ∧ ⋃(f ` I) = ⋃(g ` B))"
      by (metis (no_types, lifting))
  qed
  done

subsection ‹Dual›

abbreviation dual :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹_⇧d› [100] 100)
  where "R⇧d ≡ ∼-R"

lemma dual:
  "R⇧d = { (a,B) . (a,-B) ∉ R }"
  by (simp add: inner_complement_def)

declare dual [mr_simp]

lemma dual_antitone:
  "R ⊆ S ⟹ S⇧d ⊆ R⇧d"
  by (simp add: ic_isotone)

lemma ic_oc_dual:
  "∼R = -R⇧d"
  by (simp add: ic_dist_oc)

lemma dual_involutive [simp]:
  "R⇧d⇧d = R"
  by (simp add: ic_dist_oc)

lemma dual_antidist_ou:
  "(R ∪ S)⇧d = R⇧d ∩ S⇧d"
  by (simp add: ic_dist_oi)

lemma dual_antidist_oi:
  "(R ∩ S)⇧d = R⇧d ∪ S⇧d"
  by (simp add: ic_dist_ou)

lemma dual_dist_oc:
  "(-R)⇧d = -R⇧d"
  by (fact ic_dist_oc)

lemma dual_dist_ic:
  "(∼R)⇧d = ∼R⇧d"
  by (simp add: ic_dist_oc)

lemma dual_antidist_oU:
  "(⋃X)⇧d = ⋂(dual ` X)"
  by (simp add: ic_dist_oI uminus_Sup)

lemma dual_antidist_oI:
  "(⋂X)⇧d = ⋃(dual ` X)"
  by (simp add: ic_dist_oU uminus_Inf)

subsection ‹Co-composition›

definition co_prod :: "('a,'b) mrel ⇒ ('b,'c) mrel ⇒ ('a,'c) mrel" (infixl ‹⊙› 70) where
  "R ⊙ S ≡ { (a,C) . ∃B . (a,B) ∈ R ∧ (∃f . (∀b ∈ B . (b,f b) ∈ S) ∧ C = ⋂{ f b | b . b ∈ B }) }"

lemma co_prod_im:
  "R ⊙ S = { (a,C) . ∃B . (a,B) ∈ R ∧ (∃f . (∀b ∈ B . (b,f b) ∈ S) ∧ C = ⋂((λx . f x) ` B)) }"
  by (auto simp: co_prod_def)

lemma co_prod_iff:
  "(a,C) ∈ (R ⊙ S) ⟷ (∃B . (a,B) ∈ R ∧ (∃f . (∀b ∈ B . (b,f b) ∈ S) ∧ C = ⋂{ f b | b . b ∈ B }))"
  by (unfold co_prod_im, auto)

declare co_prod_im [mr_simp]

lemma co_prod:
  "R ⊙ S = ∼(R * ∼S)"
  apply (clarsimp simp: mr_simp)
  by (smt (verit) Collect_cong Compl_INT Compl_UN case_prodI2 double_complement old.prod.case)

lemma cp_left_isotone:
  "R ⊆ S ⟹ R ⊙ T ⊆ S ⊙ T"
  by (simp add: co_prod ic_isotone s_prod_isol)

lemma cp_right_isotone:
  "R ⊆ S ⟹ T ⊙ R ⊆ T ⊙ S"
  by (smt (verit) co_prod_iff in_mono subrelI)

lemma cp_isotone:
  "R ⊆ S ⟹ P ⊆ Q ⟹ R ⊙ P ⊆ S ⊙ Q"
  by (meson cp_left_isotone cp_right_isotone order_trans)

lemma ic_dist_cp:
  "∼(R ⊙ S) = R * ∼S"
  by (simp add: co_prod)

lemma ic_dist_sp:
  "∼(R * S) = R ⊙ ∼S"
  by (simp add: co_prod)

lemma ic_cp_ic_unit:
  "∼R = R ⊙ ∼1"
  by (simp add: co_prod)

lemma cp_left_zero [simp]:
  "{} ⊙ R = {}"
  by (simp add: co_prod_im)

lemma cp_left_unit [simp]:
  "1 ⊙ R = R"
  by (simp add: co_prod)

lemma cp_ic_unit [simp]:
  "∼1 ⊙ ∼1 = 1"
  using ic_cp_ic_unit ic_involutive by blast

lemma cp_right_dist_ou:
  "(R ∪ S) ⊙ T = (R ⊙ T) ∪ (S ⊙ T)"
  by (simp add: co_prod ic_dist_ou s_prod_distr)

lemma cp_left_iu_unit [simp]:
  "1⇩∪⇩∪ ⊙ R = 1⇩∩⇩∩"
  by (simp add: co_prod)

lemma cp_right_ii_unit:
  "R ⊙ 1⇩∩⇩∩ ⊆ R ∪∪ ∼R"
  apply (clarsimp simp: mr_simp)
  by (metis double_compl sup_compl_top)

lemma sp_right_iu_unit:
  "R * 1⇩∪⇩∪ ⊆ R ∩∩ ∼R"
  apply (clarsimp simp: mr_simp)
  by (metis Compl_disjoint double_complement)

lemma cp_left_subdist_ii:
  "R ⊙ (S ∩∩ T) ⊆ (R ⊙ S) ∩∩ (R ⊙ T)"
  by (metis cl3 co_prod ic_antidist_ii ic_antidist_iu ic_isotone)

lemma cp_right_subantidist_iu:
  "(R ∪∪ S) ⊙ T ⊆ (R ⊙ T) ∩∩ (S ⊙ T)"
  by (metis co_prod ic_antidist_iu ic_isotone seq_conc_subdistr)

lemma cp_right_antidist_iu:
  assumes "T ∩∩ T ⊆ T"
  shows "(R ∪∪ S) ⊙ T = (R ⊙ T) ∩∩ (S ⊙ T)"
  by (smt (verit) assms cl4 co_prod cp_right_subantidist_iu ic_antidist_ii ic_involutive ic_isotone subset_antisym)

lemma cp_right_dist_oU:
  "⋃X ⊙ T = (⋃R∈X . R ⊙ T)"
  by (auto simp: mr_simp)

lemma cp_left_subdist_iI:
  "R ⊙ (⋂⋂X|I) ⊆ ⋂⋂(λi . R ⊙ X i)|I"
proof -
  have "R ⊙ (⋂⋂X|I) = ∼(R * (⋃⋃(inner_complement o X)|I))"
    by (simp add: co_prod ic_antidist_iI)
  also have "... ⊆ ∼(⋃⋃(λi . R * ∼(X i))|I)"
    apply (rule ic_isotone)
    using sp_left_subdist_iU by force
  also have "... = ⋂⋂(λi . R ⊙ X i)|I"
    apply (subst ic_antidist_iU)
    by (metis co_prod comp_apply)
  finally show ?thesis
    .
qed

lemma cp_right_subantidist_iU:
  "(⋃⋃X|I) ⊙ R ⊆ ⋂⋂(λi . X i ⊙ R)|I"
proof -
  have "(⋃⋃X|I) ⊙ R = ∼((⋃⋃X|I) * ∼R)"
    by (simp add: co_prod)
  also have "... ⊆ ∼((⋃⋃(λi . X i * ∼R)|I))"
    by (simp add: ic_isotone sp_right_subdist_iU)
  also have "... = ⋂⋂(λi . X i ⊙ R)|I"
    apply (subst ic_antidist_iU)
    by (metis co_prod comp_apply)
  finally show ?thesis
    .
qed

lemma cp_right_antidist_iU:
  assumes "∀J::'a set . J ≠ {} ⟶ (⋂⋂(λj . R)|J) ⊆ R"
  shows "(⋃⋃X|I) ⊙ R = ⋂⋂(λi . X i ⊙ R)|(I::'a set)"
proof -
  have 1: "⋀J . ⋃⋃(λj. ∼ R)|J = ∼⋂⋂(λj . R)|J"
    apply (subst ic_antidist_iI)
    by (metis comp_apply)
  have "(⋃⋃X|I) ⊙ R = ∼((⋃⋃X|I) * ∼R)"
    by (simp add: co_prod)
  also have "... = ∼((⋃⋃(λi . X i * ∼R)|I))"
    by (simp add: 1 assms sp_right_dist_iU ic_isotone)
  also have "... = ⋂⋂(λi . X i ⊙ R)|I"
    apply (subst ic_antidist_iU)
    by (metis co_prod comp_apply)
  finally show ?thesis
    .
qed

subsection ‹Inner order›

definition inner_order_iu :: "'a × 'b set ⇒ 'a × 'b set ⇒ bool" (infix ‹≼⇩∪⇩∪› 50) where
  "x ≼⇩∪⇩∪ y ≡ fst x = fst y ∧ snd x ⊆ snd y"

definition inner_order_ii :: "'a × 'b set ⇒ 'a × 'b set ⇒ bool" (infix ‹≼⇩∩⇩∩› 50) where
  "x ≼⇩∩⇩∩ y ≡ fst x = fst y ∧ snd x ⊇ snd y"

lemma inner_order_dual:
  "x ≼⇩∪⇩∪ y ⟷ y ≼⇩∩⇩∩ x"
  by (metis inner_order_ii_def inner_order_iu_def)

interpretation inner_order_iu: order "(≼⇩∪⇩∪)" "λx y . x ≼⇩∪⇩∪ y ∧ x ≠ y"
  by (unfold_locales, auto simp add: inner_order_iu_def)

subsection ‹Up-closure, down-closure and convex-closure›

abbreviation up :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹_↑› [100] 100)
  where "R↑ ≡ R ∪∪ U"

abbreviation down :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹_↓› [100] 100)
  where "R↓ ≡ R ∩∩ U"

abbreviation convex :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹_↕› [100] 100)
  where "R↕ ≡ R↑ ∩ R↓"

lemma up:
  "R↑ = { (a,B) . ∃C . (a,C) ∈ R ∧ C ⊆ B }"
  by (simp add: p_id_U)

lemma down:
  "R↓ = { (a,B) . ∃C . (a,C) ∈ R ∧ B ⊆ C }"
  by (auto simp: mr_simp)

lemma convex:
  "R↕ = { (a,B) . ∃C D . (a,C) ∈ R ∧ (a,D) ∈ R ∧ C ⊆ B ∧ B ⊆ D }"
  by (auto simp: mr_simp)

declare up [mr_simp] down [mr_simp] convex [mr_simp]

lemma ic_up:
  "∼(R↑) = (∼R)↓"
  by (simp add: ic_antidist_iu)

lemma ic_down:
  "∼(R↓) = (∼R)↑"
  by (simp add: ic_antidist_ii)

lemma ic_convex:
  "∼(R↕) = (∼R)↕"
  by (simp add: ic_dist_oi ic_down ic_up inf_commute)

lemma up_isotone:
  "R ⊆ S ⟹ R↑ ⊆ S↑"
  by (fact iu_left_isotone)

lemma up_increasing:
  "R ⊆ R↑"
  by (simp add: upclosed_ext)

lemma up_idempotent [simp]:
  "R↑↑ = R↑"
  by (simp add: iu_assoc)

lemma up_dist_ou:
  "(R ∪ S)↑ = R↑ ∪ S↑"
  by (simp add: iu_right_dist_ou)

lemma up_dist_iu:
  "(R ∪∪ S)↑ = R↑ ∪∪ S↑"
  using cv_hom_par p_prod_assoc by blast

lemma up_dist_ii:
  "(R ∩∩ S)↑ = R↑ ∩∩ S↑"
proof (rule antisym)
  show "(R ∩∩ S)↑ ⊆ R↑ ∩∩ S↑"
    by (simp add: iu_right_subdist_ii)
next
  have "⋀a B C D E . (a,B) ∈ R ⟹ (a,C) ∈ S ⟹ ∃F . (∃G . (B ∪ D) ∩ (C ∪ E) = F ∪ G) ∧ (∃H I . F = H ∩ I ∧ (a,H) ∈ R ∧ (a,I) ∈ S)"
  proof -
    fix a B C D E
    assume 1: "(a,B) ∈ R"
    assume 2: "(a,C) ∈ S"
    let ?F = "B ∩ C"
    let ?G = "(B ∩ E) ∪ (D ∩ C) ∪ (D ∩ E)"
    have "(B ∪ D) ∩ (C ∪ E) = ?F ∪ ?G"
      by auto
    thus "∃F . (∃G . (B ∪ D) ∩ (C ∪ E) = F ∪ G) ∧ (∃H I . F = H ∩ I ∧ (a,H) ∈ R ∧ (a,I) ∈ S)"
      using 1 2 by auto
  qed
  thus "R↑ ∩∩ S↑ ⊆ (R ∩∩ S)↑"
    by (clarsimp simp: mr_simp)
qed

lemma down_isotone:
  "R ⊆ S ⟹ R↓ ⊆ S↓"
  by (fact ii_left_isotone)

lemma down_increasing:
  "R ⊆ R↓"
  by (metis ic_involutive ic_isotone ic_up up_increasing)

lemma down_idempotent [simp]:
  "R↓↓ = R↓"
  by (simp add: ic_down ic_injective)

lemma down_dist_ou:
  "(R ∪ S)↓ = R↓ ∪ S↓"
  by (fact ii_right_dist_ou)

lemma down_dist_iu:
  "(R ∪∪ S)↓ = R↓ ∪∪ S↓"
  by (simp add: ic_antidist_ii ic_antidist_iu ic_injective up_dist_ii)

lemma down_dist_ii:
  "(R ∩∩ S)↓ = R↓ ∩∩ S↓"
  by (metis down_idempotent ii_assoc ii_commute)

lemma convex_isotone:
  "R ⊆ S ⟹ R↕ ⊆ S↕"
  by (meson Int_mono down_isotone up_isotone)

lemma convex_increasing:
  "R ⊆ R↕"
  by (simp add: down_increasing up_increasing)

lemma convex_idempotent [simp]:
  "R↕↕ = R↕"
  by (smt (verit, ccfv_threshold) U_par_idem convex_increasing convex_isotone ic_top ic_up ii_assoc iu_assoc le_inf_iff subsetI subset_antisym)

lemma down_sp:
  "R↓ = R * (1⇩∪⇩∪ ∪ 1)"
proof -
  have "∀a B . (∃C . (∃D . B = C ∩ D) ∧ (a,C) ∈ R) ⟷ (∃C . (a,C) ∈ R ∧ (∃f . (∀c∈C . f c = {} ∨ f c = {c}) ∧ B = (⋃c∈C . f c)))"
  proof (intro allI, rule iffI)
    fix a B
    assume "∃C . (∃D . B = C ∩ D) ∧ (a,C) ∈ R"
    from this obtain C where 1: "∃D . B = C ∩ D" and 2: "(a,C) ∈ R"
      by auto
    let ?f = "λc . if c ∈ B then {c} else {}"
    have "(⋃c∈C . ?f c) = (⋃c∈B . ?f c) ∪ (⋃c∈C∩-B . ?f c)"
      using 1 by blast
    hence 3: "B = (⋃c∈C . ?f c)"
      by auto
    have "∀c∈C . ?f c = {} ∨ ?f c = {c}"
      by auto
    thus "∃C . (a,C) ∈ R ∧ (∃f . (∀c∈C . f c = {} ∨ f c = {c}) ∧ B = (⋃c∈C . f c))"
      using 2 3 by smt
  next
    fix a B
    assume "∃C . (a,C) ∈ R ∧ (∃f . (∀c∈C . f c = {} ∨ f c = {c}) ∧ B = (⋃c∈C . f c))"
    from this obtain C where 4: "(a,C) ∈ R" and "∃f . (∀c∈C . f c = {} ∨ f c = {c}) ∧ B = (⋃c∈C . f c)"
      by auto
    hence "B ⊆ C"
      by auto
    thus "∃C . (∃D . B = C ∩ D) ∧ (a,C) ∈ R"
      using 4 by auto
  qed
  thus ?thesis
    by (clarsimp simp: mr_simp)
qed

lemma up_cp:
  "R↑ = ∼R ⊙ (1⇩∩⇩∩ ∪ ∼1)"
  by (metis co_prod down_sp ic_dist_ou ic_ii_unit ic_involutive ic_up)

lemma down_dist_sp:
  "(R * S)↓ = R * S↓"
proof (rule antisym)
  show "(R * S)↓ ⊆ R * S↓"
    by (simp add: down_sp s_prod_assoc1)
next
  have "⋀a B f . (a,B) ∈ R ⟹ ∀b∈B . ∃C . (∃D . f b = C ∩ D) ∧ (b,C) ∈ S ⟹ ∃E . (∃F . (⋃b∈B . f b) = E ∩ F) ∧ (∃B . (a,B) ∈ R ∧ (∃g . (∀b∈B . (b,g b) ∈ S) ∧ E = (⋃b∈B . g b)))"
  proof -
    fix a B f
    assume 1: "(a,B) ∈ R"
    assume "∀b∈B . ∃C . (∃D . f b = C ∩ D) ∧ (b,C) ∈ S"
    hence "∃g . ∀b∈B . (∃D . f b = g b ∩ D) ∧ (b,g b) ∈ S"
      by metis
    from this obtain g where 2: "∀b∈B . (∃D . f b = g b ∩ D) ∧ (b,g b) ∈ S"
      by auto
    hence "(⋃b∈B . f b) ⊆ (⋃b∈B . g b)"
      by blast
    thus "∃E . (∃F . (⋃b∈B . f b) = E ∩ F) ∧ (∃B . (a,B) ∈ R ∧ (∃g . (∀b∈B . (b,g b) ∈ S) ∧ E = (⋃b∈B . g b)))"
      using 1 2 by (metis semilattice_inf_class.inf.absorb_iff2)
  qed
  thus "R * S↓ ⊆ (R * S)↓"
    by (clarsimp simp: mr_simp)
qed

lemma up_dist_cp:
  "(R ⊙ S)↑ = R ⊙ S↑"
  by (metis co_prod down_dist_sp ic_down ic_up)

lemma iu_up_oi:
  "R↑ ∪∪ S↑ = R↑ ∩ S↑"
  by (fact up_closed_par_is_meet)

lemma ii_down_oi:
  "R↓ ∩∩ S↓ = R↓ ∩ S↓"
  by (metis ic_antidist_ii ic_dist_oi ic_down ic_involutive up_closed_par_is_meet)

lemma down_dist_ii_oi:
  "R↓ ∩ S↓ = (R ∩∩ S)↓"
  by (simp add: down_dist_ii ii_down_oi)

lemma up_dist_iu_oi:
  "R↑ ∩ S↑ = (R ∪∪ S)↑"
  by (simp add: up_closed_par_is_meet up_dist_iu)

lemma oi_down_sub_up:
  "R↓ ∩ S↑ ⊆ (R↓ ∩ S)↑"
  by (auto simp: mr_simp)

lemma oi_down_up:
  "R↓ ∩ S = {} ⟹ R ∩ S↑ = {}"
  by (metis (no_types, lifting) cp_left_zero down_increasing ic_bot inf.orderE inf_assoc inf_bot_right oi_down_sub_up up_cp)

lemma oi_down_up_iff:
  "R↓ ∩ S = {} ⟷ R ∩ S↑ = {}"
proof (rule iffI)
  show "R↓ ∩ S = {} ⟹ R ∩ S↑ = {}"
    by (simp add: oi_down_up)
next
  assume 1: "R ∩ S↑ = {}"
  have "(∼S)↓ = ∼(S↑)"
    by (metis (no_types) ic_down ic_involutive)
  hence "∼(R↓ ∩ S) = {}"
    using 1 by (metis Int_commute ic_bot ic_dist_oi ic_down oi_down_up)
  thus "R↓ ∩ S = {}"
    by (metis (no_types) ic_bot ic_involutive)
qed

lemma down_double_complement_up:
  "R↓ ⊆ S ⟷ R ⊆ -((-S)↑)"
  by (metis disjoint_eq_subset_Compl double_compl oi_down_up_iff)

lemma up_double_complement_down:
  "R↑ ⊆ S ⟷ R ⊆ -((-S)↓)"
  by (metis Compl_subset_Compl_iff double_compl down_double_complement_up)

lemma below_up_oi_down:
  "R ⊆ R↑ ∩ R↓"
  by (fact convex_increasing)

lemma cp_pa_sim:
  "(R ⊙ S)↓ = R ⊗ S↓"
  by (metis co_prod ic_involutive ic_up pa_ic pe_pa_sim)

lemma domain_up_down_conjugate:
  "(R↑ ∩ S) * 1⇩∪⇩∪ = (R ∩ S↓) * 1⇩∪⇩∪"
  apply (rule set_eqI, clarsimp simp: mr_simp)
  by (smt (verit, del_insts) Int_Un_eq(1) SUP_bot SUP_bot_conv(1) Un_Int_eq(1))

lemma down_below_sp_top:
  "R↓ ⊆ R * U"
  apply (clarsimp simp: mr_simp)
  by (metis Int_Union UN_constant image_empty inf_commute)

lemma down_oi_up_closed:
  assumes "Q↑ = Q"
  shows "R↓ ∩ Q ⊆ (R ∩ Q)↓"
  using assms apply (clarsimp simp: mr_simp)
  by (metis (no_types, lifting) assms inf.cobounded1 ucl_iff)

lemma up_dist_oU:
  "(⋃X)↑ = ⋃(up ` X)"
  by (simp add: iu_right_dist_oU)

lemma up_dist_iU:
  assumes "I ≠ {}"
  shows "(⋃⋃X|I)↑ = ⋃⋃(up o X)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis UN_simps(2) assms)
  apply (clarsimp simp: mr_simp)
  subgoal for a f
  proof -
    fix a f
    assume "∀i∈I . ∃B . (∃C . f i = B ∪ C) ∧ (a,B) ∈ X i"
    from this obtain g where "∀i∈I . (∃C . f i = g i ∪ C) ∧ (a,g i) ∈ X i"
      by metis
    hence "(∃C . ⋃(f ` I) = ⋃(g ` I) ∪ C) ∧ (⋃(g ` I) = ⋃(g ` I) ∧ (∀i∈I . (a,g i) ∈ X i))"
      by auto
    thus "∃B . (∃C . ⋃(f ` I) = B ∪ C) ∧ (∃f . B = ⋃(f ` I) ∧ (∀i∈I . (a,f i) ∈ X i))"
      by auto
  qed
  done

lemma up_dist_iI:
  "(⋂⋂X|I)↑ = ⋂⋂(up o X)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (smt (z3) INT_simps(10) sup_Inf sup_commute)
  apply (clarsimp simp: mr_simp)
  subgoal for a f
  proof -
    assume "∀i∈I . ∃B . (∃C . f i = B ∪ C) ∧ (a,B) ∈ X i"
    from this obtain g where "∀i∈I . (∃C . f i = g i ∪ C) ∧ (a,g i) ∈ X i"
      by metis
    hence "(∃C . ⋂(f ` I) = ⋂(g ` I) ∪ C) ∧ (⋂(g ` I) = ⋂(g ` I) ∧ (∀i∈I . (a,g i) ∈ X i))"
      by auto
    thus "∃B . (∃C . ⋂(f ` I) = B ∪ C) ∧ (∃f . B = ⋂(f ` I) ∧ (∀i∈I . (a,f i) ∈ X i))"
      by auto
  qed
  done

lemma down_dist_oU:
  "(⋃X)↓ = ⋃(down ` X)"
  by (simp add: ii_right_dist_oU)

lemma down_dist_iU:
  "(⋃⋃X|I)↓ = ⋃⋃(down o X)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis UN_extend_simps(4))
  apply (clarsimp simp: mr_simp)
  subgoal for a f
  proof -
    assume "∀i∈I . ∃B . (∃C . f i = B ∩ C) ∧ (a,B) ∈ X i"
    from this obtain g where "∀i∈I . (∃C . f i = g i ∩ C) ∧ (a,g i) ∈ X i"
      by metis
    hence "(∃C . ⋃(f ` I) = ⋃(g ` I) ∩ C) ∧ (⋃(g ` I) = ⋃(g ` I) ∧ (∀i∈I . (a,g i) ∈ X i))"
      by auto
    thus "∃B . (∃C . ⋃(f ` I) = B ∩ C) ∧ (∃f . B = ⋃(f ` I) ∧ (∀i∈I . (a,f i) ∈ X i))"
      by auto
  qed
  done

lemma down_dist_iI:
  assumes "I ≠ {}"
  shows "(⋂⋂X|I)↓ = ⋂⋂(down o X)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (smt (verit, del_insts) INF_const INT_absorb Int_commute assms semilattice_inf_class.inf_left_commute)
  apply (clarsimp simp: mr_simp)
  subgoal for a f
  proof -
    assume "∀i∈I . ∃B . (∃C . f i = B ∩ C) ∧ (a,B) ∈ X i"
    from this obtain g where "∀i∈I . (∃C . f i = g i ∩ C) ∧ (a,g i) ∈ X i"
      by metis
    hence "(∃C . ⋂(f ` I) = ⋂(g ` I) ∩ C) ∧ (⋂(g ` I) = ⋂(g ` I) ∧ (∀i∈I . (a,g i) ∈ X i))"
      by auto
    thus "∃B . (∃C . ⋂(f ` I) = B ∩ C) ∧ (∃f . B = ⋂(f ` I) ∧ (∀i∈I . (a,f i) ∈ X i))"
      by auto
  qed
  done


lemma iU_up_oI:
  assumes "I ≠ {}"
  shows "⋃⋃(up o X)|I = ⋂(up ` X ` I)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis UN_absorb sup_assoc)
  apply (clarsimp simp: mr_simp)
  by (metis UN_constant assms)

lemma iI_down_oI:
  assumes "I ≠ {}"
  shows "⋂⋂(down o X)|I = ⋂(down ` X ` I)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis INF_absorb Int_assoc)
  apply (clarsimp simp: mr_simp)
  using INF_eq_const assms by auto

lemma down_dist_iI_oI:
  "⋂(down ` X ` I) = (⋂⋂X|I)↓"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis INF_const INF_greatest INT_absorb empty_iff semilattice_inf_class.inf.absorb_iff2 semilattice_inf_class.le_inf_iff)
  apply (clarsimp simp: mr_simp)
  by blast

lemma up_dist_iU_oI:
  "⋂(up ` X ` I) = (⋃⋃X|I)↑"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  subgoal for a D proof -
    assume "∀i∈I . ∃B . (∃C . D = B ∪ C) ∧ (a,B) ∈ X i"
    from this obtain f where 1: "∀i∈I . (∃C . D = f i ∪ C) ∧ (a,f i) ∈ X i"
      by metis
    hence "∃C . D = ⋃(f ` I) ∪ C"
      by auto
    thus ?thesis
      using 1 by auto
  qed
  apply (clarsimp simp: mr_simp)
  by blast

lemma iu_up:
  "(R ∪∪ R)↑ = R↑"
  using up_dist_iu_oi by auto

lemma ii_down:
  "(R ∩∩ R)↓ = R↓"
  using down_dist_ii_oi by blast

lemma iU_up:
  assumes "I ≠ {}"
  shows "(⋃⋃(λj . R)|I)↑ = R↑"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  using assms apply blast
  apply (clarsimp simp: mr_simp)
  by (metis UN_constant assms)

lemma iI_down:
  assumes "I ≠ {}"
  shows "(⋂⋂(λj . R)|I)↓ = R↓"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  using assms apply blast
  apply (clarsimp simp: mr_simp)
  by (metis INF_const assms)

lemma iu_unit_up:
  "1⇩∪⇩∪↑ = U"
  by (simp add: iu_commute)

lemma iu_unit_down:
  "1⇩∪⇩∪↓ = 1⇩∪⇩∪"
  by (simp add: down_sp)

lemma iu_unit_convex:
  "1⇩∪⇩∪↕ = 1⇩∪⇩∪"
  by (simp add: iu_unit_down p_id_zero)

lemma ii_unit_up:
  "1⇩∩⇩∩↑ = 1⇩∩⇩∩"
  by (simp add: up_cp)

lemma ii_unit_down:
  "1⇩∩⇩∩↓ = U"
  using ii_commute ii_unit by blast

lemma ii_unit_convex:
  "1⇩∩⇩∩↕ = 1⇩∩⇩∩"
  using down_increasing ii_unit_up by blast

lemma sp_unit_down:
  "1↓ = 1 ∪ 1⇩∪⇩∪"
  by (simp add: down_sp inf_sup_aci(5))

lemma sp_unit_convex:
  "1↕ = 1"
  unfolding convex s_id_def by force

lemma top_up:
  "U↑ = U"
  by simp

lemma top_down:
  "U↓ = U"
  by (metis U_par_idem ic_top ic_up)

lemma top_convex:
  "U↕ = U"
  by (simp add: top_down)

lemma bot_up:
  "{}↑ = {}"
  by (simp add: p_prod_comm)

lemma bot_down:
  "{}↓ = {}"
  using oi_down_up_iff by fastforce

lemma bot_convex:
  "{}↕ = {}"
  by (simp add: bot_down)

lemma down_oi_up_convex:
  "(R↓ ∩ S↑)↕ = R↓ ∩ S↑"
  unfolding up down convex by blast

lemma convex_iff_down_oi_up:
  "Q = Q↕ ⟷ (∃R S . Q = R↓ ∩ S↑)"
  using down_oi_up_convex by blast

lemma convex_closed_oI:
  "(⋂R∈X . R↕)↕ = (⋂R∈X . R↕)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (smt (verit, best) semilattice_inf_class.inf_commute semilattice_inf_class.inf_left_commute sup_commute sup_left_commute)
  by (meson convex_increasing)

lemma convex_closed_oi:
  "(R↕ ∩ S↕)↕ = R↕ ∩ S↕"
  using convex_closed_oI[of "{R,S}"] by simp

lemma
  "(R↕ ∪∪ S↕)↕ = R↕ ∪∪ S↕"
  nitpick[expect=genuine,card=1,3]
  oops

section ‹Powerdomain Preorders›

abbreviation lower_less_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹⊑↓› 50) where
  "R ⊑↓ S ≡ R ⊆ S↓"

abbreviation upper_less_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹⊑↑› 50) where
  "R ⊑↑ S ≡ S ⊆ R↑"

abbreviation convex_less_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹⊑↕› 50) where
  "R ⊑↕ S ≡ R ⊑↓ S ∧ R ⊑↑ S"

abbreviation Convex_less_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹⊑⇕› 50) where
  "R ⊑⇕ S ≡ R ⊆ S↕"

lemma lower_less_eq:
  "R ⊑↓ S ⟷ (∀a B . (a,B) ∈ R ⟶ (∃C . (a,C) ∈ S ∧ B ⊆ C))"
  apply (clarsimp simp: mr_simp)
  apply safe
   apply blast
  by (metis inf.absorb_iff2)

lemma upper_less_eq:
  "R ⊑↑ S ⟷ (∀a C . (a,C) ∈ S ⟶ (∃B . (a,B) ∈ R ∧ B ⊆ C))"
  by (meson U_par_st subrelI subsetD)

lemma Convex_less_eq:
  "R ⊑⇕ S ⟷ (∀a C . (a,C) ∈ R ⟶ (∃B D . (a,B) ∈ S ∧ (a,D) ∈ S ∧ B ⊆ C ∧ C ⊆ D))"
  by (meson lower_less_eq semilattice_inf_class.le_inf_iff upper_less_eq)

lemma Convex_lower_upper:
  "R ⊑⇕ S ⟷ R ⊑↓ S ∧ S ⊑↑ R"
  by auto

lemma lower_reflexive:
  "R ⊑↓ R"
  by (fact down_increasing)

lemma upper_reflexive:
  "R ⊑↑ R"
  by (fact up_increasing)

lemma convex_reflexive:
  "R ⊑↕ R"
  by (simp add: lower_reflexive upper_reflexive)

lemma Convex_reflexive:
  "R ⊑⇕ R"
  by (fact convex_increasing)

lemma lower_transitive:
  "R ⊑↓ S ⟹ S ⊑↓ T ⟹ R ⊑↓ T"
  using down_idempotent down_isotone by blast

lemma upper_transitive:
  "R ⊑↑ S ⟹ S ⊑↑ T ⟹ R ⊑↑ T"
  using up_idempotent up_isotone by blast

lemma convex_transitive:
  "R ⊑↕ S ⟹ S ⊑↕ T ⟹ R ⊑↕ T"
  by (meson lower_transitive upper_transitive)

lemma Convex_transitive:
  "R ⊑⇕ S ⟹ S ⊑⇕ T ⟹ R ⊑⇕ T"
  by (metis le_inf_iff lower_transitive upper_transitive)

lemma bot_lower_least:
  "{} ⊑↓ R"
  by simp

lemma top_upper_least:
  "U ⊑↑ R"
  by (metis U_par_idem iu_assoc le_inf_iff up_dist_iu_oi upper_reflexive)

lemma bot_Convex_least:
  "{} ⊑⇕ R"
  by simp

lemma top_lower_greatest:
  "R ⊑↓ U"
  using U_par_idem top_down top_upper_least by blast

lemma bot_upper_greatest:
  "R ⊑↑ {}"
  by simp

lemma top_Convex_greatest:
  "R ⊑⇕ U"
  using U_par_idem top_down top_upper_least by auto

lemma lower_iu_increasing:
  "R ⊑↓ R ∪∪ R"
  by (meson dual_order.trans lower_reflexive subidem_par)

lemma upper_iu_increasing:
  "R ⊑↑ R ∪∪ S"
  using p_prod_isor top_upper_least by auto

lemma convex_iu_increasing:
  "R ⊑↕ R ∪∪ R"
  by (simp add: lower_iu_increasing upper_iu_increasing)

lemma Convex_iu_increasing:
  "R ⊑⇕ R ∪∪ R"
  by (simp add: iu_up lower_iu_increasing upper_reflexive)

lemma lower_ii_decreasing:
  "R ∩∩ S ⊑↓ R"
  by (metis ii_right_isotone top_down top_lower_greatest)

lemma upper_ii_decreasing:
  "R ∩∩ R ⊑↑ R"
  using convex_reflexive ii_sub_idempotent by fastforce

lemma convex_ii_decreasing:
  "R ∩∩ R ⊑↕ R"
  by (simp add: lower_ii_decreasing upper_ii_decreasing)

lemma Convex_ii_increasing:
  "R ⊑⇕ R ∩∩ R"
  by (simp add: ii_down lower_reflexive upper_ii_decreasing)

lemma iu_lower_left_isotone:
  "R ⊑↓ S ⟹ R ∪∪ T ⊑↓ S ∪∪ T"
  by (simp add: down_dist_iu iu_isotone lower_reflexive)

lemma iu_upper_left_isotone:
  "R ⊑↑ S ⟹ R ∪∪ T ⊑↑ S ∪∪ T"
  by (metis (no_types, lifting) iu_assoc iu_commute iu_left_isotone)

lemma iu_convex_left_isotone:
  "R ⊑↕ S ⟹ R ∪∪ T ⊑↕ S ∪∪ T"
  by (simp add: iu_lower_left_isotone iu_upper_left_isotone)

lemma iu_Convex_left_isotone:
  "R ⊑⇕ S ⟹ R ∪∪ T ⊑⇕ S ∪∪ T"
  by (simp add: iu_lower_left_isotone iu_upper_left_isotone)

lemma iu_lower_right_isotone:
  "R ⊑↓ S ⟹ T ∪∪ R ⊑↓ T ∪∪ S"
  by (simp add: iu_commute iu_lower_left_isotone)

lemma iu_upper_right_isotone:
  "R ⊑↑ S ⟹ T ∪∪ R ⊑↑ T ∪∪ S"
  by (simp add: iu_assoc iu_right_isotone)

lemma iu_convex_right_isotone:
  "R ⊑↕ S ⟹ T ∪∪ R ⊑↕ T ∪∪ S"
  by (simp add: iu_lower_right_isotone iu_upper_right_isotone)

lemma iu_Convex_right_isotone:
  "R ⊑⇕ S ⟹ T ∪∪ R ⊑⇕ T ∪∪ S"
  by (simp add: iu_lower_right_isotone iu_upper_right_isotone)

lemma iu_lower_isotone:
  "R ⊑↓ S ⟹ P ⊑↓ Q ⟹ R ∪∪ P ⊑↓ S ∪∪ Q"
  by (simp add: down_dist_iu iu_isotone)

lemma iu_upper_isotone:
  "R ⊑↑ S ⟹ P ⊑↑ Q ⟹ R ∪∪ P ⊑↑ S ∪∪ Q"
  by (simp add: iu_isotone up_dist_iu)

lemma iu_convex_isotone:
  "R ⊑↕ S ⟹ P ⊑↕ Q ⟹ R ∪∪ P ⊑↕ S ∪∪ Q"
  by (simp add: iu_lower_isotone iu_upper_isotone)

lemma iu_Convex_isotone:
  "R ⊑⇕ S ⟹ P ⊑⇕ Q ⟹ R ∪∪ P ⊑⇕ S ∪∪ Q"
  by (simp add: down_dist_iu iu_isotone up_dist_iu)

lemma ii_lower_left_isotone:
  "R ⊑↓ S ⟹ R ∩∩ T ⊑↓ S ∩∩ T"
  by (simp add: down_dist_ii ii_isotone lower_reflexive)

lemma ii_upper_left_isotone:
  "R ⊑↑ S ⟹ R ∩∩ T ⊑↑ S ∩∩ T"
  by (simp add: ii_isotone up_dist_ii upper_reflexive)

lemma ii_convex_left_isotone:
  "R ⊑↕ S ⟹ R ∩∩ T ⊑↕ S ∩∩ T"
  by (simp add: ii_lower_left_isotone ii_upper_left_isotone)

lemma ii_Convex_left_isotone:
  "R ⊑⇕ S ⟹ R ∩∩ T ⊑⇕ S ∩∩ T"
  by (simp add: ii_lower_left_isotone ii_upper_left_isotone)

lemma ii_lower_right_isotone:
  "R ⊑↓ S ⟹ T ∩∩ R ⊑↓ T ∩∩ S"
  by (simp add: ii_assoc ii_right_isotone)

lemma ii_upper_right_isotone:
  "R ⊑↑ S ⟹ T ∩∩ R ⊑↑ T ∩∩ S"
  by (simp add: ii_commute ii_upper_left_isotone)

lemma ii_convex_right_isotone:
  "R ⊑↕ S ⟹ T ∩∩ R ⊑↕ T ∩∩ S"
  by (simp add: ii_lower_right_isotone ii_upper_right_isotone)

lemma ii_Convex_right_isotone:
  "R ⊑⇕ S ⟹ T ∩∩ R ⊑⇕ T ∩∩ S"
  by (simp add: ii_lower_right_isotone ii_upper_right_isotone)

lemma ii_lower_isotone:
  "R ⊑↓ S ⟹ P ⊑↓ Q ⟹ R ∩∩ P ⊑↓ S ∩∩ Q"
  by (simp add: down_dist_ii ii_isotone)

lemma ii_upper_isotone:
  "R ⊑↑ S ⟹ P ⊑↑ Q ⟹ R ∩∩ P ⊑↑ S ∩∩ Q"
  by (simp add: ii_isotone up_dist_ii)

lemma ii_convex_isotone:
  "R ⊑↕ S ⟹ P ⊑↕ Q ⟹ R ∩∩ P ⊑↕ S ∩∩ Q"
  by (simp add: ii_lower_isotone ii_upper_isotone)

lemma ii_Convex_isotone:
  "R ⊑⇕ S ⟹ P ⊑⇕ Q ⟹ R ∩∩ P ⊑⇕ S ∩∩ Q"
  by (simp add: ii_lower_isotone ii_upper_isotone)

lemma ou_lower_left_isotone:
  "R ⊑↓ S ⟹ R ∪ T ⊑↓ S ∪ T"
  by (meson le_sup_iff lower_reflexive lower_transitive)

lemma ou_upper_left_isotone:
  "R ⊑↑ S ⟹ R ∪ T ⊑↑ S ∪ T"
  by (metis Un_subset_iff sup.coboundedI1 up_dist_ou upclosed_ext)

lemma ou_convex_left_isotone:
  "R ⊑↕ S ⟹ R ∪ T ⊑↕ S ∪ T"
  by (meson ou_lower_left_isotone ou_upper_left_isotone)

lemma ou_Convex_left_isotone:
  "R ⊑⇕ S ⟹ R ∪ T ⊑⇕ S ∪ T"
  by (meson le_inf_iff ou_lower_left_isotone ou_upper_left_isotone)

lemma ou_lower_right_isotone:
  "R ⊑↓ S ⟹ T ∪ R ⊑↓ T ∪ S"
  by (metis Un_commute ou_lower_left_isotone)

lemma ou_upper_right_isotone:
  "R ⊑↑ S ⟹ T ∪ R ⊑↑ T ∪ S"
  by (metis Un_commute ou_upper_left_isotone)

lemma ou_convex_right_isotone:
  "R ⊑↕ S ⟹ T ∪ R ⊑↕ T ∪ S"
  by (meson ou_lower_right_isotone ou_upper_right_isotone)

lemma ou_Convex_right_isotone:
  "R ⊑⇕ S ⟹ T ∪ R ⊑⇕ T ∪ S"
  by (metis Un_commute ou_Convex_left_isotone)

lemma ou_lower_isotone:
  "R ⊑↓ S ⟹ P ⊑↓ Q ⟹ R ∪ P ⊑↓ S ∪ Q"
  using down_dist_ou by blast

lemma ou_upper_isotone:
  "R ⊑↑ S ⟹ P ⊑↑ Q ⟹ R ∪ P ⊑↑ S ∪ Q"
  by (simp add: iu_right_dist_ou sup.coboundedI1 sup.coboundedI2)

lemma ou_convex_isotone:
  "R ⊑↕ S ⟹ P ⊑↕ Q ⟹ R ∪ P ⊑↕ S ∪ Q"
  by (meson ou_lower_isotone ou_upper_isotone)

lemma ou_Convex_isotone:
  "R ⊑⇕ S ⟹ P ⊑⇕ Q ⟹ R ∪ P ⊑⇕ S ∪ Q"
  by (metis le_inf_iff ou_lower_isotone ou_upper_isotone)

lemma sp_lower_left_isotone:
  "R ⊑↓ S ⟹ T * R ⊑↓ T * S"
  by (simp add: down_dist_sp s_prod_isor)

lemma sp_upper_left_isotone:
  "R ⊑↑ S ⟹ T * R ⊑↑ T * S"
  by (meson cl3 dual_order.trans s_prod_isor upper_iu_increasing)

lemma sp_convex_left_isotone:
  "R ⊑↕ S ⟹ T * R ⊑↕ T * S"
  by (simp add: sp_lower_left_isotone sp_upper_left_isotone)

lemma sp_Convex_left_isotone:
  "R ⊑⇕ S ⟹ T * R ⊑⇕ T * S"
  by (simp add: sp_lower_left_isotone sp_upper_left_isotone)

lemma cp_lower_left_isotone:
  "R ⊑↓ S ⟹ T ⊙ R ⊑↓ T ⊙ S"
  by (smt (verit) co_prod ic_antidist_ii ic_antidist_iu ic_isotone ic_top sp_upper_left_isotone)

lemma cp_upper_left_isotone:
  "R ⊑↑ S ⟹ T ⊙ R ⊑↑ T ⊙ S"
  by (simp add: cp_right_isotone up_dist_cp)

lemma cp_convex_left_isotone:
  "R ⊑↕ S ⟹ T ⊙ R ⊑↕ T ⊙ S"
  by (simp add: cp_lower_left_isotone cp_upper_left_isotone)

lemma cp_Convex_left_isotone:
  "R ⊑⇕ S ⟹ T ⊙ R ⊑⇕ T ⊙ S"
  by (simp add: cp_lower_left_isotone cp_upper_left_isotone)

lemma lower_ic_upper:
  "R ⊑↓ S ⟷ ∼S ⊑↑ ∼R"
  by (metis ic_down ic_involutive ic_isotone)

lemma upper_ic_lower:
  "R ⊑↑ S ⟷ ∼S ⊑↓ ∼R"
  by (simp add: lower_ic_upper)

lemma convex_ic:
  "R ⊑↕ S ⟷ ∼S ⊑↕ ∼R"
  by (meson lower_ic_upper upper_ic_lower)

lemma Convex_ic:
  "R ⊑⇕ S ⟷ ∼R ⊑⇕ ∼S"
  by (metis le_inf_iff lower_ic_upper upper_ic_lower)

lemma up_lower_isotone:
  "R ⊑↓ S ⟹ R↑ ⊑↓ S↑"
  by (fact iu_lower_left_isotone)

lemma up_upper_isotone:
  "R ⊑↑ S ⟹ R↑ ⊑↑ S↑"
  by (fact iu_left_isotone)

lemma up_convex_isotone:
  "R ⊑↕ S ⟹ R↑ ⊑↕ S↑"
  by (fact iu_convex_left_isotone)

lemma up_Convex_isotone:
  "R ⊑⇕ S ⟹ R↑ ⊑⇕ S↑"
  by (fact iu_Convex_left_isotone)

lemma down_lower_isotone:
  "R ⊑↓ S ⟹ R↓ ⊑↓ S↓"
  by (fact down_isotone)

lemma down_upper_isotone:
  "R ⊑↑ S ⟹ R↓ ⊑↑ S↓"
  by (fact ii_upper_left_isotone)

lemma down_convex_isotone:
  "R ⊑↕ S ⟹ R↓ ⊑↕ S↓"
  by (fact ii_convex_left_isotone)

lemma down_Convex_isotone:
  "R ⊑⇕ S ⟹ R↓ ⊑⇕ S↓"
  by (fact ii_Convex_left_isotone)

lemma convex_lower_isotone:
  "R ⊑↓ S ⟹ R↕ ⊑↓ S↕"
  by (metis convex_idempotent convex_increasing le_inf_iff lower_transitive)

lemma convex_upper_isotone:
  "R ⊑↑ S ⟹ R↕ ⊑↑ S↕"
  by (simp add: convex_lower_isotone ic_convex upper_ic_lower)

lemma convex_convex_isotone:
  "R ⊑↕ S ⟹ R↕ ⊑↕ S↕"
  by (simp add: convex_lower_isotone convex_upper_isotone)

lemma convex_Convex_isotone:
  "R ⊑⇕ S ⟹ R↕ ⊑⇕ S↕"
  by (fact convex_isotone)

lemma subset_lower:
  "R ⊆ S ⟹ R ⊑↓ S"
  using lower_reflexive by auto

lemma subset_upper:
  "R ⊆ S ⟹ S ⊑↑ R"
  using upper_reflexive by blast

lemma subset_Convex:
  "R ⊆ S ⟹ R ⊑⇕ S"
  by (simp add: subset_lower subset_upper)

lemma oi_subset_lower_left_isotone:
  "R ⊆ S ⟹ R ∩ T ⊑↓ S ∩ T"
  using lower_reflexive by fastforce

lemma oi_subset_upper_left_antitone:
  "R ⊆ S ⟹ S ∩ T ⊑↑ R ∩ T"
  using upper_reflexive by force

lemma oi_subset_Convex_left_isotone:
  "R ⊆ S ⟹ R ∩ T ⊑⇕ S ∩ T"
  by (simp add: oi_subset_lower_left_isotone oi_subset_upper_left_antitone)

lemma oi_subset_lower_right_isotone:
  "R ⊆ S ⟹ T ∩ R ⊑↓ T ∩ S"
  by (simp add: oi_subset_lower_left_isotone semilattice_inf_class.inf_commute)

lemma oi_subset_upper_right_antitone:
  "R ⊆ S ⟹ T ∩ S ⊑↑ T ∩ R"
  by (simp add: oi_subset_upper_left_antitone semilattice_inf_class.inf_commute)

lemma oi_subset_Convex_right_isotone:
  "R ⊆ S ⟹ T ∩ R ⊑⇕ T ∩ S"
  using oi_subset_Convex_left_isotone by blast

lemma oi_subset_lower_isotone:
  "R ⊆ S ⟹ P ⊆ Q ⟹ R ∩ P ⊑↓ S ∩ Q"
  by (meson Int_mono subset_lower)

lemma oi_subset_upper_antitone:
  "R ⊆ S ⟹ P ⊆ Q ⟹ S ∩ Q ⊑↑ R ∩ P"
  by (meson Int_mono subset_upper)

lemma oi_subset_Convex_isotone:
  "R ⊆ S ⟹ P ⊆ Q ⟹ R ∩ P ⊑⇕ S ∩ Q"
  by (simp add: oi_subset_lower_isotone oi_subset_upper_antitone)

lemma sp_iu_unit_lower:
  "R * 1⇩∪⇩∪ ⊑↓ R"
  using lower_ii_decreasing sp_right_iu_unit by blast

lemma cp_ii_unit_upper:
  "R ⊑↑ R ⊙ 1⇩∩⇩∩"
  by (meson cp_right_ii_unit in_mono subsetI upper_iu_increasing)

lemma lower_ii_down:
  "R ⊑↓ S ⟷ R↓ = (R ∩∩ S)↓"
  apply safe
    apply (metis down_dist_ii_oi inf.orderE lower_ii_decreasing lower_transitive)
  using ii_assoc lower_ii_decreasing apply blast
  by (metis IntE down_dist_ii_oi lower_reflexive subset_eq)

lemma lower_ii_lower_bound:
  "R ⊑↓ S ⟷ R ⊆ R ∩∩ S"
  by (clarsimp simp: mr_simp) blast

lemma upper_ii_up:
  "R ⊑↑ S ⟷ S↑ = (R ∪∪ S)↑"
  by (metis inf.absorb_iff2 up_dist_iu_oi upclosed_ext upper_iu_increasing upper_transitive)

lemma upper_ii_upper_bound:
  "R ⊑↑ S ⟷ S ⊆ R ∪∪ S"
  by (clarsimp simp: mr_simp) blast

lemma
  "R ⊑↓ S ⟷ R = R ∩∩ S"
  nitpick[expect=genuine,card=1]
  oops

lemma
  "R ⊑↑ S ⟷ S = R ∪∪ S"
  nitpick[expect=genuine,card=1]
  oops

lemma convex_oi_Convex_iu:
  "R↕ ∩ S↕ ⊑⇕ R ∪∪ S"
  by (meson inf_le1 inf_le2 iu_Convex_isotone order_trans subidem_par)

lemma convex_oi_Convex_ii:
  "R↕ ∩ S↕ ⊑⇕ R ∩∩ S"
  by (meson ii_Convex_isotone ii_sub_idempotent inf_le1 inf_le2 order_trans)

lemma convex_oi_iu_ii:
  "R↕ ∩ S↕ = (R ∪∪ S)↑ ∩ (R ∩∩ S)↓"
  by (metis down_dist_ii_oi inf_assoc inf_left_commute up_dist_iu_oi)

lemma ii_lower_iu:
  "R ∩∩ S ⊑↓ R ∪∪ S"
  apply (clarsimp simp: mr_simp)
  by (metis Un_Int_eq(2) inf_left_commute)

lemma ii_upper_iu:
  "R ∩∩ S ⊑↑ R ∪∪ S"
  by (simp add: ic_antidist_ii ic_antidist_iu ii_lower_iu upper_ic_lower)

lemma ii_convex_iu:
  "R ∩∩ S ⊑↕ R ∪∪ S"
  by (simp add: ii_lower_iu ii_upper_iu)

lemma convex_oi_iu_ii_convex:
  "R↕ ∩ S↕ = (R ∪∪ S)↕ ∩ (R ∩∩ S)↕"
  by (metis convex_oi_iu_ii ii_lower_iu ii_upper_iu inf.commute lower_ii_down upper_ii_up)

subsection ‹Functional properties of multirelations›

lemma id_one_converse:
  "Id = 1 ; 1⇧⌣"
  unfolding Id_def converse_def relcomp_unfold s_id_def by force

lemma dom_explicit:
  "Dom R = R ; U ∩ 1"
  by (clarsimp simp: mr_simp Dom_def) blast

lemma dom_explicit_2:
  "Dom R = R ; top ∩ 1"
  apply (clarsimp simp: mr_simp Dom_def)
  apply safe
    apply (simp add: relcomp.relcompI top_def)
   apply blast
  by blast

lemma total_dom:
  "total R ⟷ Dom R = 1"
  unfolding total_def dom_explicit_2
  apply (rule iffI)
  using top_def apply fastforce
  by (metis Int_subset_iff dom_def dom_gla_top dom_top id_one_converse inf.idem inf_le1)

lemma total_eq:
  "total R ⟷ 1⇩∪⇩∪ = R * 1⇩∪⇩∪"
  by (metis total_dom U_c cd_iso dc dc_prop2)

lemma domain_pointwise:
  "x ∈ R * 1⇩∪⇩∪ ⟷ (∃a B . (a,B) ∈ R ∧ x = (a,{}))"
  by (smt (z3) mem_Collect_eq p_id_st)

text ‹‹card› only works for finite sets›

lemma univalent_2:
  "univalent R ⟷ (∀a . finite { B . (a,B) ∈ R } ∧ card { B . (a,B) ∈ R } ≤ one_class.one)"
proof
  assume 1: "univalent R"
  show "∀a . finite { B . (a,B) ∈ R } ∧ card { B . (a,B) ∈ R } ≤ one_class.one"
  proof
    fix a
    let ?B = "{ B . (a,B) ∈ R }"
    show "finite ?B ∧ card ?B ≤ one_class.one"
    proof (rule conjI)
      show 2: "finite ?B"
      proof (rule ccontr)
        assume 3: "infinite ?B"
        from this obtain B where 4: "(a,B) ∈ R"
          using not_finite_existsD by auto
        have "?B = {B}"
        proof
          show "?B ⊆ {B}"
            using 1 4 by (metis (no_types, lifting) univalent_set insertCI mem_Collect_eq subsetI)
        next
          show "{B} ⊆ ?B"
            using 4 by simp
        qed
        thus False
          using 3 by auto
      qed
      show "card ?B ≤ one_class.one"
      proof (rule ccontr)
        assume 5: "¬ card ?B ≤ one_class.one"
        from this obtain B where 6: "(a,B) ∈ R"
          by fastforce
        hence "card (?B - {B}) ≥ one_class.one"
          using 2 5 by auto
        from this obtain C where "(a,C) ∈ R ∧ B ≠ C"
          using 5 by (metis (no_types, lifting) CollectD One_nat_def card.insert_remove card_Diff_singleton_if card.empty card_mono empty_iff finite.emptyI finite.insertI insert_iff subsetI)
        thus False
          using 1 6 by (meson univalent_set)
      qed
    qed
  qed
next
  assume 5: "∀a . finite { B . (a,B) ∈ R } ∧ card { B . (a,B) ∈ R } ≤ one_class.one"
  have "∀a B C . (a,B) ∈ R ∧ (a,C) ∈ R ⟶ B = C"
  proof (intro allI, rule impI)
    fix a B C
    let ?B = "{ B . (a,B) ∈ R }"
    have 6: "finite ?B"
      using 5 by simp
    assume "(a,B) ∈ R ∧ (a,C) ∈ R"
    hence "{B,C} ⊆ ?B"
      by simp
    hence "card {B,C} ≤ one_class.one"
      using 5 6 by (meson card_mono le_trans)
    thus "B = C"
      by (metis One_nat_def card.empty card_insert_disjoint empty_iff finite.emptyI finite.insertI insert_absorb lessI not_le singleton_insert_inj_eq)
  qed
  thus "univalent R"
    by (simp add: univalent_set)
qed

lemma univalent_3:
  "univalent R ⟷ (∀S . R * 1⇩∪⇩∪ = S * 1⇩∪⇩∪ ∧ S ⊆ R ⟶ S = R)"
proof
  assume 1: "∀S . R * 1⇩∪⇩∪ = S * 1⇩∪⇩∪ ∧ S ⊆ R ⟶ S = R"
  have "∀a B C . (a,B) ∈ R ∧ (a,C) ∈ R ⟶ B = C"
  proof (intro allI, rule impI)
    fix a B C
    assume 2: "(a,B) ∈ R ∧ (a,C) ∈ R"
    show "B = C"
    proof (rule ccontr)
      assume 3: "B ≠ C"
      let ?S = "R - { (a,C) }"
      have 4: "R * 1⇩∪⇩∪ = ?S * 1⇩∪⇩∪"
      proof
        show "R * 1⇩∪⇩∪ ⊆ ?S * 1⇩∪⇩∪"
        proof
          fix x::"'a × 'f set"
          assume "x ∈ R * 1⇩∪⇩∪"
          from this obtain b D where "(b,D) ∈ R ∧ x = (b,{})"
            by (meson domain_pointwise)
          thus "x ∈ ?S * 1⇩∪⇩∪"
            using 2 3 by (metis domain_pointwise Pair_inject insertE insert_Diff)
        qed
      next
        show "?S * 1⇩∪⇩∪ ⊆ R * 1⇩∪⇩∪"
          by (simp add: s_prod_isol)
      qed
      have "?S ≠ R"
        using 2 by blast
      thus False
        using 1 4 by blast
    qed
  qed
  thus "univalent R"
    by (simp add: univalent_set)
next
  assume 5: "univalent R"
  show "∀S . R * 1⇩∪⇩∪ = S * 1⇩∪⇩∪ ∧ S ⊆ R ⟶ S = R"
  proof
    fix S
    show "R * 1⇩∪⇩∪ = S * 1⇩∪⇩∪ ∧ S ⊆ R ⟶ S = R"
    proof
      assume 6: "R * 1⇩∪⇩∪ = S * 1⇩∪⇩∪ ∧ S ⊆ R"
      have "R ⊆ S"
      proof
        fix x
        assume 7: "x ∈ R"
        from this obtain a B where 8: "x = (a,B)"
          by fastforce
        show "x ∈ S"
        proof (cases "∃C . C ≠ B ∧ (a,C) ∈ S")
          case True
          thus ?thesis
            using 5 6 7 8 by (metis subsetD univalent_set)
        next
          case False
          thus ?thesis
            using 6 7 8 by (metis (no_types, lifting) domain_pointwise prod.inject)
        qed
      qed
      thus "S = R"
        using 6 by simp
    qed
  qed
qed

lemma total_2:
  "total R ⟷ (∀a . { B . (a,B) ∈ R } ≠ {})"
  by (simp add: total_set)

lemma total_3:
  "total R ⟷ (∀a . finite { B . (a,B) ∈ R } ⟶ card { B . (a,B) ∈ R } ≥ one_class.one)"
  by (metis finite.emptyI nonempty_set_card total_2)

lemma total_4: "total R ⟷ 1⇩∪⇩∪ ⊆ R * 1⇩∪⇩∪"
  by (simp add: c6 order_antisym_conv total_eq)

lemma deterministic_2:
  "deterministic R ⟷ (∀a . card { B . (a,B) ∈ R } = one_class.one)"
  apply (rule iffI)
   apply (metis One_nat_def bot_nat_0.extremum_unique deterministic_def le_simps(2) less_Suc_eq nonempty_set_card total_2 univalent_2)
  by (metis card_1_singletonE deterministic_def finite.emptyI finite_insert order.refl total_3 univalent_2)

lemma univalent_convex:
  assumes "univalent S"
  shows "S = S↕"
  apply (rule antisym)
   apply (simp add: lower_reflexive upper_reflexive)
  apply (clarsimp simp: mr_simp)
  by (metis assms lattice_class.sup_inf_absorb sup_left_idem univalent_set)

lemma univalent_iu_idempotent:
  assumes "univalent S"
  shows "S = S ∪∪ S"
  apply (rule antisym)
   apply (meson convex_reflexive upper_ii_upper_bound)
  apply (clarsimp simp: mr_simp)
  by (metis assms sup.idem univalent_set)

lemma univalent_ii_idempotent:
  assumes "univalent S"
  shows "S = S ∩∩ S"
  apply (rule antisym)
   apply (simp add: ii_sub_idempotent)
  apply (clarsimp simp: mr_simp)
  by (metis assms semilattice_inf_class.inf.idem univalent_set)

lemma univalent_down_iu_idempotent:
  assumes "univalent S"
  shows "S = S↓ ∪∪ S"
  apply (rule antisym)
   apply (meson convex_reflexive subset_upper upper_ii_upper_bound)
  apply (clarsimp simp: mr_simp)
  by (metis assms lattice_class.sup_inf_absorb sup_commute univalent_set)

lemma univalent_up_ii_idempotent:
  assumes "univalent S"
  shows "S = S↑ ∩∩ S"
  apply (rule antisym)
   apply (metis assms ii_left_isotone univalent_ii_idempotent upclosed_ext)
  apply (clarsimp simp: mr_simp)
  by (metis Int_commute assms lattice_class.inf_sup_absorb univalent_set)

lemma univalent_convex_iu_idempotent:
  assumes "univalent S"
  shows "S = S↕ ∪∪ S"
  by (metis assms univalent_convex univalent_iu_idempotent)

lemma univalent_convex_ii_idempotent:
  assumes "univalent S"
  shows "S = S↕ ∩∩ S"
  by (metis assms univalent_convex univalent_ii_idempotent)

lemma univalent_iu_closed:
  "univalent R ⟹ univalent S ⟹ univalent (R ∪∪ S)"
  by (smt (verit, best) case_prodD mem_Collect_eq p_prod_def univalent_set)

lemma univalent_ii_closed:
  "univalent R ⟹ univalent S ⟹ univalent (R ∩∩ S)"
  by (smt (verit, ccfv_SIG) CollectD Pair_inject case_prodE inner_intersection_def univalent_set)

lemma total_lower:
  "total R ⟷ 1⇩∪⇩∪ ⊑↓ R"
  unfolding lower_less_eq
  by (simp add: p_id_def total_set)

lemma total_upper:
  "total R ⟷ R ⊑↑ 1⇩∩⇩∩"
  unfolding upper_less_eq
  by (simp add: ii_unit_def total_set)

lemma total_lower_iu:
  assumes "total T"
  shows "R ⊑↓ R ∪∪ T"
  by (metis assms iu_lower_right_isotone iu_unit total_lower)

lemma total_upper_ii:
  assumes "total T"
  shows "R ∩∩ T ⊑↑ R"
  by (smt (verit, ccfv_threshold) U_par_idem assms iu_assoc iu_commute lower_ii_lower_bound total_lower_iu up_dist_ii upper_ii_up)

lemma total_univalent_lower_iu:
  assumes "total T"
    and "univalent S"
    and "T ⊑↓ S"
  shows "T ∪∪ S = S"
proof -
  have 1: "∀a. ∃B. (a, B) ∈ T"
    by (meson assms(1) total_set)
  have 2: "∀a B C. (a, B) ∈ S ∧ (a, C) ∈ S ⟶ B = C"
    by (meson assms(2) univalent_set)
  hence 3: "T ∪∪ S ⊆ S"
    by (metis assms(2,3) iu_left_isotone univalent_down_iu_idempotent)
  hence "S ⊆ T ∪∪ S"
    apply (clarsimp simp: mr_simp)
    using 1 2 by (metis (mono_tags, lifting) CollectI Un_iff case_prodI subset_Un_eq)
  thus ?thesis
    using 3 by (simp add: subset_antisym)
qed

lemma total_iu_closed:
  "total R ⟹ total S ⟹ total (R ∪∪ S)"
  by (meson lower_transitive total_lower total_lower_iu)

lemma total_ii_closed:
  "total R ⟹ total S ⟹ total (R ∩∩ S)"
  by (metis down_dist_ii_oi le_inf_iff total_lower)

lemma deterministic_lower:
  assumes "deterministic V"
  shows "R ⊑↓ V ⟷ (∀a B C . (a,B) ∈ R ∧ (a,C) ∈ V ⟶ B ⊆ C)"
proof -
  have "R ⊑↓ V ⟷ (∀a B . (a,B) ∈ R ⟶ (∃C . (a,C) ∈ V ∧ B ⊆ C))"
    by (simp add: lower_less_eq)
  also have "... ⟷ (∀a B . (a,B) ∈ R ⟶ (∀C . (a,C) ∈ V ⟶ B ⊆ C))"
    by (metis assms deterministic_set)
  finally show ?thesis
    by blast
qed

lemma deterministic_upper:
  assumes "deterministic V"
  shows "V ⊑↑ R ⟷ (∀a B C . (a,B) ∈ R ∧ (a,C) ∈ V ⟶ C ⊆ B)"
proof -
  have "V ⊑↑ R ⟷ (∀a C . (a,C) ∈ R ⟶ (∃B . (a,B) ∈ V ∧ B ⊆ C))"
    by (simp add: upper_less_eq)
  also have "... ⟷ (∀a C . (a,C) ∈ R ⟶ (∀B . (a,B) ∈ V ⟶ B ⊆ C))"
    by (metis assms deterministic_set)
  finally show ?thesis
    by blast
qed

lemma deterministic_iu_closed:
  "deterministic R ⟹ deterministic S ⟹ deterministic (R ∪∪ S)"
  by (simp add: deterministic_def univalent_iu_closed total_iu_closed)

lemma deterministic_ii_closed:
  "deterministic R ⟹ deterministic S ⟹ deterministic (R ∩∩ S)"
  by (simp add: deterministic_def univalent_ii_closed total_ii_closed)

lemma total_univalent_lower_implies_upper:
  assumes "total T"
    and "univalent S"
    and "T ⊑↓ S"
  shows "T ⊑↑ S"
  by (simp add: assms total_univalent_lower_iu upper_ii_upper_bound)

lemma total_univalent_lower_implies_convex:
  assumes "total T"
    and "univalent S"
    and "T ⊑↓ S"
  shows "T ⊑↕ S"
  by (simp add: assms total_univalent_lower_implies_upper)

lemma total_univalent_upper_implies_lower:
  assumes "total T"
    and "univalent S"
    and "S ⊑↑ T"
  shows "S ⊑↓ T"
proof (clarsimp simp: mr_simp)
  fix a B
  assume 1: "(a,B) ∈ S"
  from this obtain C where 2: "(a,C) ∈ T"
    by (meson assms(1) total_set)
  hence "(a,C) ∈ S↑"
    using assms(3) by auto
  from this obtain D where 3: "(a,D) ∈ S ∧ D ⊆ C"
    using 2 by (meson assms(3) upper_less_eq)
  hence "D = B"
    using 1 by (meson assms(2) univalent_set)
  thus "∃C . (∃D . B = C ∩ D) ∧ (a,C) ∈ T"
    using 2 3 by (metis Int_absorb1)
qed

lemma total_univalent_upper_implies_convex:
  assumes "total T"
    and "univalent S"
    and "S ⊑↑ T"
  shows "S ⊑↕ T"
  by (simp add: assms total_univalent_upper_implies_lower)

lemma deterministic_lower_upper:
  assumes "deterministic T"
    and "deterministic S"
  shows "S ⊑↓ T ⟷ S ⊑↑ T"
  by (meson assms deterministic_def total_univalent_lower_implies_convex total_univalent_upper_implies_lower)

lemma deterministic_lower_convex:
  assumes "deterministic T"
    and "deterministic S"
  shows "S ⊑↓ T ⟷ S ⊑↕ T"
  by (simp add: assms deterministic_lower_upper)

lemma deterministic_upper_convex:
  assumes "deterministic T"
    and "deterministic S"
  shows "S ⊑↑ T ⟷ S ⊑↕ T"
  by (simp add: assms deterministic_lower_upper)

lemma total_down_sp_sp_down:
  assumes "total T"
  shows "R↓ * T ⊆ R * T↓"
proof -
  have "R↓ * T ⊆ R * ((1⇩∪⇩∪ ∪ 1) * T)"
    by (simp add: down_sp s_prod_assoc1)
  also have "... = R * (1⇩∪⇩∪ ∪ T * 1)"
    by (simp add: s_prod_distr)
  also have "... = R * (T * 1⇩∪⇩∪ ∪ T * 1)"
    by (metis assms c6 order_antisym_conv total_4)
  also have "... ⊆ R * (T * (1⇩∪⇩∪ ∪ 1))"
    by (metis down_sp le_supI s_prod_isor sp_iu_unit_lower sup_ge2)
  also have "... = R * T↓"
    by (simp add: down_sp)
  finally show ?thesis
    by simp
qed

lemma total_down_sp_semi_commute:
  "total T ⟹ R↓ * T ⊆ (R * T)↓"
  by (simp add: down_dist_sp total_down_sp_sp_down)

lemma total_down_dist_sp:
  "total T ⟹ (R * T)↓ = R↓ * T↓"
  by (smt (verit, best) down_dist_sp equalityI ii_assoc ii_isotone lower_reflexive s_prod_isol top_down total_down_sp_semi_commute)

lemma univalent_ic_closed:
  "univalent R ⟷ univalent (∼R)"
  apply (unfold univalent_set)
  apply (clarsimp simp: mr_simp)
  by (metis double_compl)

lemma total_ic_closed:
  "total R ⟷ total (∼R)"
  by (metis total_dom d_def_expl domain_up_down_conjugate equalityI ic_down ic_top ic_up ii_commute inf.orderE lower_ic_upper top_down top_lower_greatest total_lower total_upper_ii)

lemma deterministic_ic_closed:
  "deterministic R ⟷ deterministic (∼R)"
  by (meson deterministic_def total_ic_closed univalent_ic_closed)

lemma iu_unit_deterministic:
  "deterministic (1⇩∪⇩∪)"
  by (metis Lambda_empty det_lambda)

lemma ii_unit_deterministic:
  "deterministic (1⇩∩⇩∩)"
  using deterministic_ic_closed iu_unit_deterministic by force

lemma univalent_upper_iu:
  assumes "univalent R"
  shows "(R ⊑↑ S) ⟷ (R ∪∪ S = S)"
proof -
  have 1: "R ∪∪ S = S ⟹ R ⊑↑ S"
    using upper_iu_increasing by blast
  have 2: "R ⊑↑ S ⟹ S ⊆ R ∪∪ S"
    by (simp add: upper_ii_upper_bound)
  have "R ⊑↑ S ⟹ R ∪∪ S ⊆ S"
    apply (clarsimp simp: mr_simp)
    by (smt (verit) Ball_Collect assms case_prodD le_iff_sup subset_refl sup.bounded_iff univalent_set)
  thus ?thesis
    using 1 2 by blast
qed

lemma univalent_lower_ii:
  assumes "univalent S"
  shows "(R ⊑↓ S) = (R ∩∩ S = R)"
  apply (clarsimp simp: mr_simp)
  apply safe
    apply (smt (z3) CollectD CollectI Collect_cong Int_iff assms case_prodD inf_set_def subsetD univalent_set)
   apply blast
  by (smt (verit, ccfv_threshold) CollectD Pair_inject case_prodE inf_commute)

subsection ‹Equivalences induced by powerdomain preorders›

abbreviation lower_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹=↓› 50) where
  "R =↓ S ≡ R ⊑↓ S ∧ S ⊑↓ R"

abbreviation upper_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹=↑› 50) where
  "R =↑ S ≡ R ⊑↑ S ∧ S ⊑↑ R"

abbreviation convex_eq :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ bool" (infixl ‹=↕› 50) where
  "R =↕ S ≡ R ⊑↕ S ∧ S ⊑↕ R"

lemma Convex_eq:
  "R =↕ S ≡ R ⊑⇕ S ∧ S ⊑⇕ R"
  by (smt (z3) semilattice_inf_class.le_inf_iff)

lemma convex_lower_upper:
  "R =↕ S ⟷ R =↓ S ∧ R =↑ S"
  by auto

lemma lower_eq_down:
  "R =↓ S ⟷ R↓ = S↓"
  using down_idempotent down_lower_isotone lower_reflexive by blast

lemma upper_eq_up:
  "R =↑ S ⟷ R↑ = S↑"
  by (metis p_prod_comm upclosed_ext upper_ii_up)

lemma convex_eq_convex:
  "R =↕ S ⟷ R↕ = S↕"
  by (metis Convex_lower_upper lower_eq_down upper_eq_up)

lemma lower_eq:
  "R =↓ S ⟷ (∀a B . (∃C . (a,C) ∈ R ∧ B ⊆ C) ⟷ (∃C . (a,C) ∈ S ∧ B ⊆ C))"
  by (meson lower_less_eq order_refl order_trans)

lemma upper_eq:
  "R =↑ S ⟷ (∀a C . (∃B . (a,B) ∈ R ∧ B ⊆ C) ⟷ (∃B . (a,B) ∈ S ∧ B ⊆ C))"
  by (meson order_refl order_trans upper_less_eq)

lemma lower_eq_reflexive:
  "R =↓ R"
  by (simp add: lower_reflexive)

lemma upper_eq_reflexive:
  "R =↑ R"
  by (simp add: upper_reflexive)

lemma convex_eq_reflexive:
  "R =↕ R"
  by (simp add: lower_reflexive upper_reflexive)

lemma lower_eq_symmetric:
  "R =↓ S ⟹ S =↓ R"
  by simp

lemma upper_eq_symmetric:
  "R =↑ S ⟹ S =↑ R"
  by simp

lemma convex_eq_symmetric:
  "R =↕ S ⟹ S =↕ R"
  by simp

lemma lower_eq_transitive:
  "R =↓ S ⟹ S =↓ T ⟹ R =↓ T"
  using lower_transitive by auto

lemma upper_eq_transitive:
  "R =↑ S ⟹ S =↑ T ⟹ R =↑ T"
  using upper_transitive by auto

lemma convex_eq_transitive:
  "R =↕ S ⟹ S =↕ T ⟹ R =↕ T"
  by (meson lower_transitive upper_transitive)

lemma ou_lower_eq_left_congruence:
  "R =↓ S ⟹ R ∪ T =↓ S ∪ T"
  using ou_lower_left_isotone by blast

lemma ou_upper_eq_left_congruence:
  "R =↑ S ⟹ R ∪ T =↑ S ∪ T"
  using ou_upper_left_isotone by blast

lemma ou_convex_eq_left_congruence:
  "R =↕ S ⟹ R ∪ T =↕ S ∪ T"
  by (meson ou_lower_left_isotone ou_upper_left_isotone)

lemma ou_lower_eq_right_congruence:
  "R =↓ S ⟹ T ∪ R =↓ T ∪ S"
  using ou_lower_right_isotone by blast

lemma ou_upper_eq_right_congruence:
  "R =↑ S ⟹ T ∪ R =↑ T ∪ S"
  using ou_upper_right_isotone by blast

lemma ou_convex_eq_right_congruence:
  "R =↕ S ⟹ T ∪ R =↕ T ∪ S"
  by (meson ou_lower_right_isotone ou_upper_right_isotone)

lemma ou_lower_eq_congruence:
  "R =↓ S ⟹ P =↓ Q ⟹ R ∪ P =↓ S ∪ Q"
  using ou_lower_isotone by blast

lemma ou_upper_eq_congruence:
  "R =↑ S ⟹ P =↑ Q ⟹ R ∪ P =↑ S ∪ Q"
  using ou_upper_isotone by blast

lemma ou_convex_eq_congruence:
  "R =↕ S ⟹ P =↕ Q ⟹ R ∪ P =↕ S ∪ Q"
  by (meson ou_lower_isotone ou_upper_isotone)

lemma iu_lower_eq_left_congruence:
  "R =↓ S ⟹ R ∪∪ T =↓ S ∪∪ T"
  using iu_lower_left_isotone by blast

lemma iu_upper_eq_left_congruence:
  "R =↑ S ⟹ R ∪∪ T =↑ S ∪∪ T"
  using iu_upper_left_isotone by blast

lemma iu_convex_eq_left_congruence:
  "R =↕ S ⟹ R ∪∪ T =↕ S ∪∪ T"
  by (simp add: iu_lower_left_isotone iu_upper_left_isotone)

lemma iu_lower_eq_right_congruence:
  "R =↓ S ⟹ T ∪∪ R =↓ T ∪∪ S"
  using iu_lower_right_isotone by blast

lemma iu_upper_eq_right_congruence:
  "R =↑ S ⟹ T ∪∪ R =↑ T ∪∪ S"
  using iu_upper_right_isotone by blast

lemma iu_convex_eq_right_congruence:
  "R =↕ S ⟹ T ∪∪ R =↕ T ∪∪ S"
  by (simp add: iu_lower_right_isotone iu_upper_right_isotone)

lemma iu_lower_eq_congruence:
  "R =↓ S ⟹ P =↓ Q ⟹ R ∪∪ P =↓ S ∪∪ Q"
  using iu_lower_isotone by blast

lemma iu_upper_eq_congruence:
  "R =↑ S ⟹ P =↑ Q ⟹ R ∪∪ P =↑ S ∪∪ Q"
  using iu_upper_isotone by blast

lemma iu_convex_eq_congruence:
  "R =↕ S ⟹ P =↕ Q ⟹ R ∪∪ P =↕ S ∪∪ Q"
  by (simp add: iu_lower_isotone iu_upper_isotone)

lemma ii_lower_eq_left_congruence:
  "R =↓ S ⟹ R ∩∩ T =↓ S ∩∩ T"
  using ii_lower_left_isotone by blast

lemma ii_upper_eq_left_congruence:
  "R =↑ S ⟹ R ∩∩ T =↑ S ∩∩ T"
  using ii_upper_left_isotone by blast

lemma ii_convex_eq_left_congruence:
  "R =↕ S ⟹ R ∩∩ T =↕ S ∩∩ T"
  by (simp add: ii_lower_left_isotone ii_upper_left_isotone)

lemma ii_lower_eq_right_congruence:
  "R =↓ S ⟹ T ∩∩ R =↓ T ∩∩ S"
  using ii_lower_right_isotone by blast

lemma ii_upper_eq_right_congruence:
  "R =↑ S ⟹ T ∩∩ R =↑ T ∩∩ S"
  using ii_upper_right_isotone by blast

lemma ii_convex_eq_right_congruence:
  "R =↕ S ⟹ T ∩∩ R =↕ T ∩∩ S"
  by (simp add: ii_lower_right_isotone ii_upper_right_isotone)

lemma ii_lower_eq_congruence:
  "R =↓ S ⟹ P =↓ Q ⟹ R ∩∩ P =↓ S ∩∩ Q"
  using ii_lower_isotone by blast

lemma ii_upper_eq_congruence:
  "R =↑ S ⟹ P =↑ Q ⟹ R ∩∩ P =↑ S ∩∩ Q"
  using ii_upper_isotone by blast

lemma ii_convex_eq_congruence:
  "R =↕ S ⟹ P =↕ Q ⟹ R ∩∩ P =↕ S ∩∩ Q"
  by (simp add: ii_lower_isotone ii_upper_isotone)

lemma sp_lower_eq_left_congruence:
  "R =↓ S ⟹ T * R =↓ T * S"
  by (simp add: sp_lower_left_isotone)

lemma sp_upper_eq_left_congruence:
  "R =↑ S ⟹ T * R =↑ T * S"
  by (simp add: sp_upper_left_isotone)

lemma sp_convex_eq_left_congruence:
  "R =↕ S ⟹ T * R =↕ T * S"
  by (simp add: sp_lower_left_isotone sp_upper_left_isotone)

lemma cp_lower_eq_left_congruence:
  "R =↓ S ⟹ T ⊙ R =↓ T ⊙ S"
  by (simp add: cp_lower_left_isotone)

lemma cp_upper_eq_left_congruence:
  "R =↑ S ⟹ T ⊙ R =↑ T ⊙ S"
  by (simp add: cp_upper_left_isotone)

lemma cp_convex_eq_left_congruence:
  "R =↕ S ⟹ T ⊙ R =↕ T ⊙ S"
  by (simp add: cp_lower_left_isotone cp_upper_left_isotone)

lemma lower_eq_ic_upper:
  "R =↓ S ⟷ ∼R =↑ ∼S"
  using lower_ic_upper by auto

lemma upper_eq_ic_lower:
  "R =↑ S ⟷ ∼R =↓ ∼S"
  using upper_ic_lower by auto

lemma convex_eq_ic_lower:
  "R =↕ S ⟷ ∼R =↕ ∼S"
  by (meson lower_ic_upper upper_ic_lower)

lemma up_lower_eq_congruence:
  "R =↓ S ⟹ R↑ =↓ S↑"
  by (fact iu_lower_eq_left_congruence)

lemma up_upper_eq_congruence:
  "R =↑ S ⟹ R↑ =↑ S↑"
  by (fact iu_upper_eq_left_congruence)

lemma up_convex_eq_congruence:
  "R =↕ S ⟹ R↑ =↕ S↑"
  by (fact iu_convex_eq_left_congruence)

lemma down_lower_eq_congruence:
  "R =↓ S ⟹ R↓ =↓ S↓"
  by (fact ii_lower_eq_left_congruence)

lemma down_upper_eq_congruence:
  "R =↑ S ⟹ R↓ =↑ S↓"
  by (fact ii_upper_eq_left_congruence)

lemma down_convex_eq_congruence:
  "R =↕ S ⟹ R↓ =↕ S↓"
  by (fact ii_convex_eq_left_congruence)

lemma convex_lower_eq_congruence:
  "R =↓ S ⟹ R↕ =↓ S↕"
  by (simp add: convex_lower_isotone)

lemma convex_upper_eq_congruence:
  "R =↑ S ⟹ R↕ =↑ S↕"
  by (simp add: convex_upper_isotone)

lemma convex_convex_eq_congruence:
  "R =↕ S ⟹ R↕ =↕ S↕"
  by (simp add: convex_lower_isotone convex_upper_isotone)

lemma univalent_lower_eq_subset:
  assumes "univalent S"
    and "S =↓ R"
  shows "S ⊆ R"
proof -
  have 1: "∀a B C. (a, B) ∈ S ∧ (a, C) ∈ S ⟶ B = C"
    using assms(1) by (simp add: univalent_set)
  have "∀a B. (∃A. (a,A) ∈ S ∧ B ⊆ A) = (∃A. (a,A) ∈ R ∧ B ⊆ A)"
    by (meson assms(2) lower_eq)
  hence "∀a B. (a,B) ∈ S ⟶ (a,B) ∈ R"
    using 1 by (smt (verit, del_insts) assms(2) lower_less_eq subset_antisym)
  thus ?thesis
    by (simp add: subset_iff)
qed

lemma univalent_lower_eq:
  assumes "univalent R"
    and "univalent S"
    and "R =↓ S"
  shows "R = S"
  by (meson assms subset_antisym univalent_lower_eq_subset)

lemma univalent_lower_eq_iff:
  assumes "univalent R"
    and "univalent S"
  shows "(R =↓ S) ⟷ (R = S)"
  using assms lower_reflexive univalent_lower_eq by auto

lemma univalent_upper_eq_subset:
  assumes "univalent S"
    and "S =↑ R"
  shows "S ⊆ R"
proof -
  have 1: "∀a B C. (a, B) ∈ S ∧ (a, C) ∈ S ⟶ B = C"
    using assms(1) by (simp add: univalent_set)
  have "∀a B. (∃A. (a,A) ∈ S ∧ A ⊆ B) = (∃A. (a,A) ∈ R ∧ A ⊆ B)"
    by (meson assms(2) upper_eq)
  hence "∀a B. (a,B) ∈ S ⟶ (a,B) ∈ R"
    using 1 by (smt (verit) order_refl subset_antisym)
  thus ?thesis
    by (simp add: subset_iff)
qed

lemma univalent_upper_eq:
  assumes "univalent R"
    and "univalent S"
    and "R =↑ S"
  shows "R = S"
  by (meson assms subset_antisym univalent_upper_eq_subset)

lemma univalent_upper_eq_iff:
  assumes "univalent R"
    and "univalent S"
  shows "(R =↑ S) ⟷ (R = S)"
  using assms univalent_upper_eq upclosed_ext by blast

lemma univalent_convex_eq_iff:
  assumes "univalent R"
    and "univalent S"
  shows "(R =↕ S) ⟷ (R = S)"
  by (metis assms univalent_lower_eq_iff univalent_upper_eq_iff)

lemma total_univalent_upper_ii:
  assumes "total T"
    and "univalent S"
    and "S ⊑↑ T"
  shows "T ∩∩ S = S"
  apply (rule antisym)
   apply (metis assms(2,3) ii_left_isotone univalent_up_ii_idempotent)
  by (metis assms ii_commute lower_ii_lower_bound total_univalent_upper_implies_lower)

lemma lower_eq_down_closed:
  "R =↓ R↓"
  by (simp add: subset_lower)

lemma upper_eq_up_closed:
  "R =↑ R↑"
  by (simp add: subset_upper)

lemma convex_eq_up_closed:
  "R =↕ R↕"
  by (simp add: subset_lower subset_upper)

lemma lower_join:
  "(∀P . Q ⊑↓ P ⟷ R ⊑↓ P ∧ S ⊑↓ P) ⟷ Q =↓ R ∪ S"
  by (meson Un_subset_iff lower_reflexive lower_transitive)

lemma lower_meet:
  "(∀P . P ⊑↓ Q ⟷ P ⊑↓ R ∧ P ⊑↓ S) ⟷ Q =↓ R ∩∩ S"
  by (metis (no_types, lifting) down_dist_ii_oi le_inf_iff lower_eq_down lower_reflexive)

lemma upper_join:
  "(∀P . Q ⊑↑ P ⟷ R ⊑↑ P ∧ S ⊑↑ P) ⟷ Q =↑ R ∪∪ S"
  by (metis (no_types, lifting) convex_increasing le_inf_iff up_dist_iu_oi upper_eq_up)

lemma upper_meet:
  "(∀P . P ⊑↑ Q ⟷ P ⊑↑ R ∧ P ⊑↑ S) ⟷ Q =↑ R ∪ S"
  by (meson Un_subset_iff upper_reflexive upper_transitive)

lemma lower_ii_idempotent:
  "R ∩∩ R =↓ R"
  using ii_down lower_reflexive by blast

lemma upper_iu_idempotent:
  "R ∪∪ R =↑ R"
  using iu_up upper_reflexive by auto

lemma lower_iI_idempotent:
  "I ≠ {} ⟹ (⋂⋂(λj . R)|I) =↓ R"
  by (metis iI_down lower_eq_down)

lemma upper_iU_idempotent:
  "I ≠ {} ⟹ (⋃⋃(λj . R)|I) =↑ R"
  by (metis iU_up upper_eq_up)

lemma down_closed_intersection_closed:
  "R = R↓ ⟹ ∀I . I ≠ {} ⟶ (⋂⋂(λj . R)|I) ⊆ R"
  by (metis lower_iI_idempotent)

lemma up_closed_union_closed:
  "R = R↑ ⟹ ∀I . I ≠ {} ⟶ (⋃⋃(λj . R)|I) ⊆ R"
  by (metis upper_iU_idempotent)

lemma ou_down_lower_eq_ou:
  "R↓ ∪ S↓ =↓ R ∪ S"
  using down_dist_ou lower_eq_down_closed by blast

lemma oi_down_lower_eq_ii:
  "R↓ ∩ S↓ =↓ R ∩∩ S"
  by (simp add: down_dist_ii_oi lower_reflexive)

lemma ou_up_upper_eq_ou:
  "R↑ ∪ S↑ =↑ R ∪ S"
  by (metis ou_upper_isotone up_idempotent upper_reflexive)

lemma oi_up_upper_eq_iu:
  "R↑ ∩ S↑ =↑ R ∪∪ S"
  by (simp add: up_dist_iu_oi upper_reflexive)

lemma oU_down_lower_eq_oU:
  "(⋃R∈X . R↓) =↓ ⋃X"
  by (metis down_dist_oU lower_eq_down_closed)

lemma oI_down_lower_eq_iI:
  "(⋂i∈I . X i↓) =↓ ⋂⋂X|I"
  apply safe
  using down_dist_iI_oI apply fastforce
  by (metis (no_types, lifting) down_dist_iI_oI image_cong image_image lower_eq_down subsetD)

lemma oU_up_upper_eq_oU:
  "(⋃R∈X . R↑) =↑ ⋃X"
  by (metis up_dist_oU upper_eq_up_closed)

lemma oI_up_upper_eq_iI:
  "(⋂i∈I . X i↑) =↑ ⋃⋃X|I"
  by (smt (z3) INT_extend_simps(10) Sup.SUP_cong U_par_idem p_prod_assoc p_prod_comm top_upper_least up_dist_iU_oI upper_ii_upper_bound)

lemma down_order_lower:
  "R↓ ⊆ S↓ ⟷ R ⊑↓ S"
  by (meson lower_eq_down_closed lower_transitive)

lemma up_order_upper:
  "R↑ ⊆ S↑ ⟷ S ⊑↑ R"
  by (meson upper_eq_up_closed upper_transitive)

lemma convex_order_lower_upper:
  "R↕ ⊆ S↕ ⟷ R ⊑↓ S ∧ S ⊑↑ R"
  by (meson convex_eq_up_closed le_inf_iff lower_transitive upper_transitive)

lemma convex_order_Convex:
  "R↕ ⊆ S↕ ⟷ R ⊑⇕ S"
  by (meson Convex_lower_upper convex_order_lower_upper)

subsection ‹Further results for convex-closure›

lemma convex_down:
  "R↕↓ = R↓"
  by (meson convex_eq_up_closed lower_eq_down)

lemma convex_up:
  "R↕↑ = R↑"
  by (meson convex_eq_up_closed upper_eq_up)

lemma iu_dist_oi_convex:
  assumes "R = R↕"
      and "S = S↕"
      and "T = T↕"
    shows "(R ∩ S) ∪∪ T = (R ∪∪ T) ∩ (S ∪∪ T)"
  nitpick[expect=genuine,card=1]
  oops

lemma ii_dist_oi_convex:
  assumes "R = R↕"
      and "S = S↕"
      and "T = T↕"
    shows "(R ∩ S) ∩∩ T = (R ∩∩ T) ∩ (S ∩∩ T)"
  nitpick[expect=genuine,card=1]
  oops

lemma oI_up_closed:
  assumes "∀R∈X . R↑ = R"
    shows "(⋂X)↑ = ⋂X"
  by (meson Inter_iff assms ucl_iff)

lemma oI_down_closed:
  assumes "∀R∈X . R↓ = R"
    shows "(⋂X)↓ = ⋂X"
  by (metis (no_types, opaque_lifting) antisym assms down_isotone dual_order.refl le_Inf_iff lower_reflexive)

lemma oI_convex_closed:
  assumes "∀R∈X . R↕ = R"
    shows "(⋂X)↕ = ⋂X"
  by (metis (no_types, lifting) Inf_lower Inter_greatest antisym assms convex_increasing convex_isotone)

lemma up_dist_Union:
  "(⋃X)↑ = ⋃{ R↑ | R . R ∈ X }"
  by (simp add: Setcompr_eq_image up_dist_oU)

lemma down_dist_Union:
  "(⋃X)↓ = ⋃{ R↓ | R . R ∈ X }"
  by (simp add: Setcompr_eq_image down_dist_oU)

lemma convex_dist_Union:
  "(⋃X)↕ = ⋃{ R↕ | R . R ∈ X }"
  nitpick[expect=genuine,card=1,2]
  oops

lemma up_dist_Inter:
  "(⋂X)↑ = ⋂{ R↑ | R . R ∈ X }"
  nitpick[expect=genuine,card=1]
  oops

lemma down_dist_Inter:
  "(⋂X)↓ = ⋂{ R↓ | R . R ∈ X }"
  nitpick[expect=genuine,card=1]
  oops

lemma convex_dist_Inter:
  "(⋂X)↕ = ⋂{ R↕ | R . R ∈ X }"
  nitpick[expect=genuine,card=1,2]
  oops

lemma Inter_convex_closed:
  "(⋂X)↕ = ⋂X"
  nitpick[expect=genuine,card=1,2]
  oops

abbreviation "convex_iu" (infixl ‹∪∪↕› 70)
  where "R ∪∪↕ S ≡ (R ∪∪ S)↕"

lemma convex_iu:
  "R ∪∪↕ S = (R↓ ∪∪ S↓) ∩ R↑ ∩ S↑"
  by (metis Int_assoc Int_commute down_dist_iu up_dist_iu_oi)

lemma convex_iu_sub:
  "R↕ ∪∪ S ⊆ R ∪∪↕ S"
  by (meson equalityE iu_Convex_left_isotone)

lemma convex_iu_convex_left:
  "R ∪∪↕ S = R↕ ∪∪↕ S"
  by (simp add: convex_down convex_iu convex_up)

lemma convex_iu_convex_right:
  "R ∪∪↕ S = R ∪∪↕ S↕"
  by (simp add: convex_down convex_iu convex_up)

lemma convex_iu_convex:
  "R ∪∪↕ S = R↕ ∪∪↕ S↕"
  by (simp add: convex_down convex_iu convex_up)

lemma convex_iu_assoc:
  "(R ∪∪↕ S) ∪∪↕ T = R ∪∪↕ (S ∪∪↕ T)"
  by (smt (z3) convex_iu_convex_left convex_iu_convex_right p_prod_assoc)

lemma convex_iu_comm:
  "R ∪∪↕ S = S ∪∪↕ R"
  by (simp add: p_prod_comm)

lemma convex_iu_unit:
  "R = R↕ ⟹ R ∪∪↕ 1⇩∪⇩∪ = R"
  by simp

abbreviation "convex_ii" (infixl ‹∩∩↕› 70)
  where "R ∩∩↕ S ≡ (R ∩∩ S)↕"

lemma convex_ii:
  "R ∩∩↕ S = (R↑ ∩∩ S↑) ∩ R↓ ∩ S↓"
  by (simp add: down_dist_ii_oi inf_assoc up_dist_ii)

lemma convex_ii_sub:
  "R↕ ∩∩ S ⊆ R ∩∩↕ S"
  by (meson equalityE ii_Convex_left_isotone)

lemma convex_ii_convex_left:
  "R ∩∩↕ S = R↕ ∩∩↕ S"
  by (simp add: convex_down convex_ii convex_up)

lemma convex_ii_convex_right:
  "R ∩∩↕ S = R ∩∩↕ S↕"
  by (simp add: convex_down convex_ii convex_up)

lemma convex_ii_convex:
  "R ∩∩↕ S = R↕ ∩∩↕ S↕"
  by (simp add: convex_down convex_ii convex_up)

lemma convex_ii_assoc:
  "(R ∩∩↕ S) ∩∩↕ T = R ∩∩↕ (S ∩∩↕ T)"
  by (smt (z3) convex_ii_convex_left convex_ii_convex_right ii_assoc)

lemma convex_ii_comm:
  "R ∩∩↕ S = S ∩∩↕ R"
  by (simp add: ii_commute)

lemma convex_ii_unit:
  "R = R↕ ⟹ R ∩∩↕ 1⇩∩⇩∩ = R"
  by simp

lemma convex_iu_ic:
  "∼(R ∪∪↕ S) = ∼R ∩∩↕ ∼S"
  by (simp add: ic_antidist_iu ic_convex)

lemma convex_ii_ic:
  "∼(R ∩∩↕ S) = ∼R ∪∪↕ ∼S"
  by (simp add: ic_antidist_ii ic_convex)

abbreviation convex_sup :: "('a,'b) mrel set ⇒ ('a,'b) mrel" (‹⋃↕›) where
  "⋃↕X ≡ (⋃X)↕"

lemma convex_sup_convex:
  "⋃↕X = (⋃↕X)↕"
  by simp

lemma convex_sup_inter:
  "⋃↕X = ⋂{ Y . Y = Y↕ ∧ ⋃X ⊆ Y }"
  apply (rule antisym)
  apply (smt (verit) Inf_greatest convex_isotone mem_Collect_eq)
  by (smt (verit, del_insts) Inter_lower convex_down convex_increasing convex_up mem_Collect_eq)

lemma convex_iu_dist_convex_sup:
  "⋃↕X ∪∪↕ S = ⋃↕{ R ∪∪↕ S | R . R ∈ X }"
proof -
  have "⋃↕X ∪∪↕ S = ⋃X ∪∪↕ S"
    using convex_iu_convex_left by blast
  also have "... = ⋃↕{ R ∪∪ S | R . R ∈ X }"
    by (simp add: Setcompr_eq_image iu_right_dist_oU)
  also have "... = ⋃↕{ R ∪∪↕ S | R . R ∈ X }"
  proof (rule antisym)
    show "⋃↕{ R ∪∪ S | R . R ∈ X } ⊆ ⋃↕{ R ∪∪↕ S | R . R ∈ X }"
      by (smt (verit, ccfv_SIG) Convex_transitive Sup_mono dual_order.refl mem_Collect_eq subset_Convex)
    have "∀R∈X . R ∪∪↕ S ⊆ ⋃↕{ R ∪∪ S | R . R ∈ X }"
      by (smt (verit, ccfv_SIG) Union_upper convex_isotone mem_Collect_eq)
    thus "⋃↕{ R ∪∪↕ S | R . R ∈ X } ⊆ ⋃↕{ R ∪∪ S | R . R ∈ X }"
      apply (subst convex_order_Convex)
      apply (rule Union_least)
      by blast
  qed
  finally show ?thesis
    .
qed

lemma convex_ii_dist_convex_sup:
  "⋃↕X ∩∩↕ S = ⋃↕{ R ∩∩↕ S | R . R ∈ X }"
proof -
  have "⋃↕X ∩∩↕ S = ⋃X ∩∩↕ S"
    using convex_ii_convex_left by blast
  also have "... = ⋃↕{ R ∩∩ S | R . R ∈ X }"
    by (simp add: Setcompr_eq_image ii_right_dist_oU)
  also have "... = ⋃↕{ R ∩∩↕ S | R . R ∈ X }"
  proof (rule antisym)
    show "⋃↕{ R ∩∩ S | R . R ∈ X } ⊆ ⋃↕{ R ∩∩↕ S | R . R ∈ X }"
      by (smt (verit, ccfv_SIG) Convex_transitive Sup_mono dual_order.refl mem_Collect_eq subset_Convex)
    have "∀R∈X . R ∩∩↕ S ⊆ ⋃↕{ R ∩∩ S | R . R ∈ X }"
      by (smt (verit, ccfv_SIG) Union_upper convex_isotone mem_Collect_eq)
    thus "⋃↕{ R ∩∩↕ S | R . R ∈ X } ⊆ ⋃↕{ R ∩∩ S | R . R ∈ X }"
      apply (subst convex_order_Convex)
      apply (rule Union_least)
      by blast
  qed
  finally show ?thesis
    .
qed

lemma convex_dist_sup:
  "(⋃X)↕ = ⋃↕{ R↕ | R . R ∈ X }"
proof (rule antisym)
  have "∀R∈X . R ⊆ ⋃↕{ R↕ | R . R ∈ X }"
    by (metis (mono_tags, lifting) Sup_upper2 convex_increasing mem_Collect_eq subset_Convex)
  thus "(⋃X)↕ ⊆ ⋃↕{ R↕ | R . R ∈ X }"
    by (meson Sup_least convex_order_Convex)
  have "∀R∈X . R↕ ⊆ (⋃X)↕"
    by (meson Union_upper convex_isotone)
  thus "⋃↕{ R↕ | R . R ∈ X } ⊆ (⋃X)↕"
    apply (subst convex_order_Convex)
    apply (rule Union_least)
    by blast
qed

section ‹Fusion and Fission›

subsection ‹Atoms and co-atoms›

definition atoms :: "('a,'b) mrel" (‹A⇩∪⇩∪›)
  where "A⇩∪⇩∪ ≡ { (a,{b}) | a b . True }"

definition co_atoms :: "('a,'b) mrel" (‹A⇩∩⇩∩›)
  where "A⇩∩⇩∩ ≡ { (a,UNIV - {b}) | a b . True }"

declare atoms_def [mr_simp] co_atoms_def [mr_simp]

lemma atoms_solution:
  "A⇩∪⇩∪↑ = -1⇩∪⇩∪"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  apply (clarsimp simp: mr_simp)
  by (metis equals0I insert_is_Un mk_disjoint_insert)

lemma atoms_least_solution:
  assumes "R↑ = -1⇩∪⇩∪"
  shows "A⇩∪⇩∪ ⊆ R"
proof
  fix x :: "'a × 'b set"
  assume 1: "x ∈ A⇩∪⇩∪"
  from this obtain a b where 2: "x = (a,{b})"
    by (smt (z3) CollectD atoms_def)
  have 3: "x ∈ R↑"
    using 1 assms atoms_solution upper_reflexive by fastforce
  have "(a,{}) ∉ R↑"
    by (metis ComplD IntE U_c U_par_idem assms domain_pointwise p_prod_assoc up_dist_iu_oi)
  thus "x ∈ R"
    using 2 3 by (smt (verit) U_par_st subsetD subset_singletonD upclosed_ext)
qed

lemma ic_atoms:
  "∼A⇩∪⇩∪ = A⇩∩⇩∩"
  apply (clarsimp simp: mr_simp)
  by fastforce

lemma ic_co_atoms:
  "∼A⇩∩⇩∩ = A⇩∪⇩∪"
  by (metis ic_atoms ic_involutive)

lemma co_atoms_solution:
  "A⇩∩⇩∩↓ = -1⇩∩⇩∩"
  by (metis atoms_solution ic_atoms ic_dist_oc ic_iu_unit ic_up)

lemma co_atoms_least_solution:
  assumes "R↓ = -1⇩∩⇩∩"
  shows "A⇩∩⇩∩ ⊆ R"
  by (metis assms atoms_least_solution ic_atoms ic_dist_oc ic_down ic_ii_unit ic_involutive ic_isotone)

lemma iu_unit_atoms_disjoint:
  "1⇩∪⇩∪ ∩ A⇩∪⇩∪ = {}"
  by (metis Compl_disjoint atoms_solution iu_unit_down oi_down_up_iff)

lemma ii_unit_co_atoms_disjoint:
  "1⇩∩⇩∩ ∩ A⇩∩⇩∩ = {}"
  using co_atoms_solution lower_reflexive by fastforce

lemma atoms_sp_idempotent:
  "A⇩∪⇩∪ * A⇩∪⇩∪ = A⇩∪⇩∪"
  by (auto simp: mr_simp)

lemma atoms_sp_cp:
  "(R ∩ A⇩∪⇩∪) * S = (R ∩ A⇩∪⇩∪) ⊙ S"
  by (auto simp: mr_simp)

subsection ‹Inner-functional properties›

abbreviation inner_univalent :: "('a,'b) mrel ⇒ bool" where
  "inner_univalent R ≡ R ⊆ 1⇩∪⇩∪ ∪ A⇩∪⇩∪"

abbreviation inner_total :: "('a,'b) mrel ⇒ bool" where
  "inner_total R ≡ R ⊆ -1⇩∪⇩∪"

abbreviation inner_deterministic :: "('a,'b) mrel ⇒ bool" where
  "inner_deterministic R ≡ inner_total R ∧ inner_univalent R"

lemma inner_deterministic_atoms:
  "inner_deterministic R ⟷ R ⊆ A⇩∪⇩∪"
  using atoms_solution upper_reflexive by fastforce

lemma inner_univalent:
  "inner_univalent R ⟷ (∀a b c B . (a,B) ∈ R ∧ b ∈ B ∧ c ∈ B ⟶ b = c)"
  apply (clarsimp simp: mr_simp, safe)
   apply blast
  by (smt (z3) UNIV_I UN_iff equals0I insertI1 insert_absorb singleton_insert_inj_eq' subsetI)

lemma inner_univalent_2:
  "inner_univalent R ⟷ (∀a B . (a,B) ∈ R ⟶ finite B ∧ card B ≤ one_class.one)"
  apply (rule iffI)
   apply (metis card_eq_0_iff finite.emptyI inner_univalent is_singletonI' is_singleton_altdef linear nonempty_set_card)
  by (metis all_not_in_conv card_1_singletonE eq_iff inner_univalent nonempty_set_card singletonD)

lemma inner_total:
  "inner_total R ⟷ (∀a B . (a,B) ∈ R ⟶ (∃b . b ∈ B))"
  apply (rule iffI)
   apply (smt (verit, del_insts) Collect_empty_eq all_not_in_conv disjoint_eq_subset_Compl p_id_zero_st)
  by (smt (verit) Collect_empty_eq disjoint_eq_subset_Compl ex_in_conv p_id_zero_st)

lemma inner_total_2:
  "inner_total R ⟷ (∀a B . (a,B) ∈ R ⟶ B ≠ {})"
  by (meson all_not_in_conv inner_total)

lemma inner_total_3:
  "inner_total R ⟷ (∀a B . (a,B) ∈ R ∧ finite B ⟶ card B ≥ one_class.one)"
  by (metis finite.emptyI inner_total_2 nonempty_set_card)

lemma inner_deterministic:
  "inner_deterministic R ⟷ (∀a B . (a,B) ∈ R ⟶ (∃!b . b ∈ B))"
  by (metis (no_types, lifting) inner_total inner_univalent)

lemma inner_deterministic_2:
  "inner_deterministic R ⟷ (∀a B . (a,B) ∈ R ⟶ card B = one_class.one)"
  by (metis card_1_singletonE eq_iff finite.emptyI finite_insert inner_total_3 inner_univalent_2)

lemma inner_deterministic_sp_unit:
  "inner_deterministic 1"
  by (simp add: inner_deterministic s_id_def)

lemma inner_univalent_down:
  assumes "inner_univalent S"
  shows "S↓ ⊆ S ∪ 1⇩∪⇩∪"
  using assms by (auto simp: mr_simp)

lemma inner_deterministic_lower_eq:
  assumes "inner_deterministic V"
    and "inner_deterministic W"
    and "V =↓ W"
  shows "V = W"
  using assms inner_univalent_down by blast

lemma inner_total_down_closed:
  "inner_total T ⟹ R ⊆ T ⟹ inner_total R"
  by simp

lemma inner_univalent_down_closed:
  "inner_univalent T ⟹ R ⊆ T ⟹ inner_univalent R"
  by simp

lemma inner_deterministic_down_closed:
  "inner_deterministic T ⟹ R ⊆ T ⟹ inner_deterministic R"
  by blast

lemma inner_univalent_convex:
  assumes "inner_univalent R"
  shows "R = R↕"
  apply (rule antisym)
  using convex_increasing apply blast
  apply (clarsimp simp: mr_simp)
  by (smt (verit) Un_Int_eq(2) Un_Int_eq(3) assms boolean_algebra_cancel.sup0 disjoint_iff inner_univalent semilattice_inf_class.inf.orderE subsetI)

lemma inner_deterministic_alt_closure:
  "inner_deterministic R = (R O converse 1 O 1 = R)"
  apply (clarsimp simp: mr_simp)
  apply safe
    apply force
  by blast+

lemma inner_deterministic_s_id_conv_epsiloff:
  "inner_deterministic R ⟹ R O converse s_id = R O epsiloff"
  apply (clarsimp simp: mr_simp)
  unfolding epsiloff_def
  by blast

lemma inner_deterministic_lower_iff:
  assumes "inner_deterministic R"
    and "inner_deterministic S"
  shows "(R ⊑↓ S) ⟷ (R ⊆ S)"
  apply standard
   apply (smt (verit, ccfv_threshold) Un_commute assms disjoint_eq_subset_Compl inf.orderE inf.orderI inf_commute inf_sup_distrib2 inner_univalent_down sup.orderE sup_bot_left)
  by (simp add: subset_lower)

lemma inner_deterministic_upper_iff:
  assumes "inner_deterministic R"
    and "inner_deterministic S"
  shows "(R ⊑↑ S) ⟷ (S ⊆ R)"
  apply standard
   apply (clarsimp simp: mr_simp)
  using inner_deterministic apply (smt (verit, del_insts) Ball_Collect Un_subset_iff assms case_prodD subsetD subsetI subset_antisym)
  by (simp add: subset_upper)

lemma inner_deterministic_lower_eq_iff:
  assumes "inner_deterministic R"
    and "inner_deterministic S"
  shows "(R =↓ S) ⟷ (R = S)"
  by (meson assms inner_deterministic_lower_eq lower_reflexive)

lemma inner_deterministic_upper_eq_iff:
  assumes "inner_deterministic R"
    and "inner_deterministic S"
  shows "(R =↑ S) ⟷ (R = S)"
  by (simp add: antisym assms inner_deterministic_upper_iff)

lemma inner_deterministic_convex_eq_iff:
  assumes "inner_deterministic R"
    and "inner_deterministic S"
  shows "(R =↕ S) ⟷ (R = S)"
  by (metis assms inner_deterministic_lower_eq_iff inner_deterministic_upper_eq_iff)

lemma
  "inner_univalent R ⟹ inner_univalent S ⟹ inner_univalent (R ∪∪ S)"
  nitpick[expect=genuine,card=1,2]
  oops

lemma inner_univalent_ii_closed:
  "inner_univalent R ⟹ inner_univalent S ⟹ inner_univalent (R ∩∩ S)"
  by (metis (no_types, lifting) Un_subset_iff convex_reflexive down_dist_ii_oi inner_univalent_down inner_univalent_down_closed le_inf_iff subsetI)

lemma inner_total_iu_closed:
  "inner_total R ⟹ inner_total S ⟹ inner_total (R ∪∪ S)"
  by (metis U_par_idem U_par_p_id atoms_solution c_prod_idr iu_upper_isotone s_prod_p_idl top_upper_least)

lemma
  "inner_total R ⟹ inner_total S ⟹ inner_total (R ∩∩ S)"
  nitpick[expect=genuine,card=1,2]
  oops

  subsection ‹Fusion›

lemma fusion_set:
  "fus R ≡ { (a,B) . B = ⋃{ C . (a,C) ∈ R } }"
  unfolding fus_set Image_singleton
  by (smt (verit) Collect_cong Pair_inject case_prodE case_prodI2)

declare fusion_set [mr_simp]

lemma fusion_lower_increasing:
  "R ⊑↓ fus R"
  apply (clarsimp simp: mr_simp)
  by blast

lemma fusion_deterministic:
  "deterministic (fus R)"
  by (simp add: deterministic_set fusion_set)

lemma fusion_least:
  assumes "R ⊑↓ S"
    and "deterministic S"
  shows "fus R ⊑↓ S"
proof (clarsimp simp: mr_simp)
  fix a::'a
  from assms(2) obtain C::"'b set" where 1: "(a,C) ∈ S"
    by (meson deterministic_set)
  hence "⋃{ B . (a,B) ∈ R } ⊆ C"
    using assms deterministic_lower by (smt (verit, del_insts) Sup_le_iff mem_Collect_eq)
  thus "∃C . (∃D . ⋃{ B . (a,B) ∈ R } = C ∩ D) ∧ (a,C) ∈ S"
    using 1 by blast
qed

lemma fusion_unique:
  assumes "∀R . R ⊑↓ f R"
    and "∀R . deterministic (f R)"
    and "∀R S . R ⊑↓ S ∧ deterministic S ⟶ f R ⊑↓ S"
  shows "f T = fus T"
  apply (rule univalent_lower_eq)
  using assms(2) deterministic_def apply blast
  using deterministic_def fusion_deterministic apply blast
  by (simp add: assms fusion_deterministic fusion_least fusion_lower_increasing)

lemma fusion_down_char:
  "(fus R)↓ = -((-(R↓) ∩ A⇩∪⇩∪)↑)"
proof
  show "(fus R)↓ ⊆ -((-(R↓) ∩ A⇩∪⇩∪)↑)"
    apply (clarsimp simp: mr_simp)
    by blast
next
  show "-((-(R↓) ∩ A⇩∪⇩∪)↑) ⊆ (fus R)↓"
  proof (clarsimp simp: mr_simp)
    fix a A
    assume 1: "∀B . (∀C . A ≠ B ∪ C) ∨ (∃C . (∃D . B = C ∩ D) ∧ (a,C) ∈ R) ∨ (∀b . B ≠ {b})"
    have "A ⊆ ⋃{ C . (a,C) ∈ R }"
    proof
      fix x
      assume "x ∈ A"
      from this obtain C where "x ∈ C ∧ (a,C) ∈ R"
        using 1 by (metis IntD1 insert_Diff insert_is_Un singletonI)
      thus "x ∈ ⋃{ C . (a,C) ∈ R }"
        by blast
    qed
    thus "∃D . A = ⋃{ C . (a,C) ∈ R } ∩ D"
      by auto
  qed
qed

lemma fusion_up_char:
  "(fus R)↑ = -((∼(R↓) ∩ A⇩∩⇩∩)↓)"
proof
  show "(fus R)↑ ⊆ -((∼(R↓) ∩ A⇩∩⇩∩)↓)"
    apply (clarsimp simp: mr_simp)
    by blast
next
  show "-((∼(R↓) ∩ A⇩∩⇩∩)↓) ⊆ (fus R)↑"
  proof (clarsimp simp: mr_simp)
    fix a A
    assume 1: "∀C . (∀D . A ≠ C ∩ D) ∨ (∀B . (∀D . - C ≠ B ∩ D) ∨ (a,B) ∉ R) ∨ (∀b . C ≠ UNIV - {b})"
    have "⋃{ C . (a,C) ∈ R } ⊆ A"
    proof
      fix x
      assume "x ∈ ⋃{ C . (a,C) ∈ R }"
      from this obtain C where "x ∈ C ∧ (a,C) ∈ R"
        by blast
      thus "x ∈ A"
        using 1 by (metis Compl_eq_Diff_UNIV Diff_Diff_Int Diff_cancel Diff_eq Diff_insert_absorb Int_commute double_complement insert_Diff insert_inter_insert)
    qed
    thus "∃C . A = ⋃{ C . (a,C) ∈ R } ∪ C"
      by auto
  qed
qed

lemma fusion_up_char_2:
  "(fus R)↑ = -(((R↓ ∩ A⇩∪⇩∪) * ∼1)↓)"
  by (simp add: atoms_sp_cp co_prod fusion_up_char ic_atoms ic_dist_oi)

lemma fusion_char:
  "fus R = -((-(R↓) ∩ A⇩∪⇩∪)↑) ∩ -((∼(R↓) ∩ A⇩∩⇩∩)↓)"
  by (metis deterministic_def fusion_deterministic fusion_down_char fusion_up_char inf_commute univalent_convex)

lemma fusion_char_2:
  "fus R = -((-(R↓) ∩ A⇩∪⇩∪)↑) ∩ -(((R↓ ∩ A⇩∪⇩∪) * ∼1)↓)"
  using fusion_char fusion_up_char fusion_up_char_2 by blast

lemma fusion_lower_isotone:
  "R ⊑↓ S ⟹ fus R ⊑↓ fus S"
  by (meson fusion_deterministic fusion_least fusion_lower_increasing lower_transitive)

lemma fusion_iu_idempotent:
  "fus R ∪∪ fus R = fus R"
  using deterministic_def fusion_deterministic univalent_iu_idempotent by blast

lemma fusion_down:
  "fus R = fus (R↓)"
  by (simp add: fusion_char)

lemma fusion_iu_total:
  "total T ⟹ T ∪∪ fus T = fus T"
  by (meson deterministic_def fusion_deterministic fusion_lower_increasing total_univalent_lower_iu)

lemma fusion_deterministic_fixpoint:
  "deterministic R ⟷ R = fus R"
  by (metis deterministic_def fusion_deterministic fusion_iu_total fusion_least lower_reflexive p_prod_comm total_univalent_lower_iu)

abbreviation non_empty :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹ne _› [100] 100)
  where "ne R ≡ R ∩ -1⇩∪⇩∪"

lemma non_empty:
  "ne R = { (a,B) | a B . (a,B) ∈ R ∧ B ≠ {} }"
  by (auto simp: mr_simp)

lemma ne_equality:
  "ne R = R ⟷ R ⊆ -1⇩∪⇩∪"
  by blast

lemma ne_dist_ou:
  "ne (R ∪ S) = ne R ∪ ne S"
  by (fact inf_sup_distrib2)

lemma ne_down_idempotent:
  "ne ((ne (R↓))↓) = ne (R↓)"
  by (auto simp: mr_simp)

lemma ne_up:
  "(ne R)↑ = ne R * 1↑"
proof
  show "(ne R)↑ ⊆ ne R * 1↑"
    apply (clarsimp simp: mr_simp)
    by (metis UN_constant Un_insert_left insert_absorb)
next
  show "ne R * 1↑ ⊆ (ne R)↑"
  proof (clarsimp simp: mr_simp)
    fix a B f
    assume 1: "(a,B) ∈ R" and 2: "B ≠ {}" and "∀b∈B . ∃C . f b = insert b C"
    hence "B ⊆ (⋃x∈B . f x)"
      by blast
    thus "∃D . (∃C . (⋃x∈B . f x) = D ∪ C) ∧ (a,D) ∈ R ∧ D ≠ {}"
      using 1 2 by blast
  qed
qed

lemma ne_dist_down_sp:
  "ne (R↓ * S) = ne (R↓) * ne S"
proof (rule antisym)
  show "ne (R↓ * S) ⊆ ne (R↓) * ne S"
  proof (clarsimp simp: mr_simp)
    fix a C f D x
    assume 1: "(a,C) ∈ R" and 2: "∀b∈C∩D . (b,f b) ∈ S" and 3: "f x ≠ {}" and 4: "x ∈ C" and 5: "x ∈ D"
    let ?B = "{ b∈C∩D . f b ≠ {} }"
    have 6: "∃C . (∃D . ?B = C ∩ D) ∧ (a,C) ∈ R"
      using 1 by auto
    have 7: "?B ≠ {}"
      using 3 4 5 by auto
    have 8: "∀b∈?B . (b,f b) ∈ S ∧ f b ≠ {}"
      using 2 by auto
    have "(⋃x∈C∩D . f x) = (⋃x∈?B . f x)"
      by auto
    thus "∃B . (∃C . (∃D . B = C ∩ D) ∧ (a,C) ∈ R) ∧ B ≠ {} ∧ (∃g . (∀b∈B . (b,g b) ∈ S ∧ g b ≠ {}) ∧ (⋃x∈C∩D . f x) = (⋃x∈B . g x))"
      using 6 7 8 by blast
  qed
next
  show "ne (R↓) * ne S ⊆ ne (R↓ * S)"
    by (clarsimp simp: mr_simp) blast
qed

lemma total_ne_down_dist_sp:
  "total T ⟹ ne ((R * T)↓) = ne (R↓) * ne (T↓)"
  by (simp add: ne_dist_down_sp total_down_dist_sp)

lemma inner_univalent_char:
  "inner_univalent S ⟷ (∀R . fus R = fus S ∧ R ⊑↓ S ⟶ ne R = ne S)"
proof
  assume 1: "inner_univalent S"
  show "∀R . fus R = fus S ∧ R ⊑↓ S ⟶ ne R = ne S"
  proof (rule allI, rule impI)
    fix R
    assume 2: "fus R = fus S ∧ R ⊑↓ S"
    show "ne R = ne S"
    proof (rule antisym)
      show "ne R ⊆ ne S"
      proof
        fix x
        assume 3: "x ∈ ne R"
        from this obtain a B where 4: "x = (a,B) ∧ x ∈ R ∧ B ≠ {}"
          by (metis Int_iff Int_lower2 inner_total_2 surj_pair)
        from this obtain C where 5: "B ⊆ C ∧ (a,C) ∈ S"
          using 2 by (meson lower_less_eq)
        from this obtain b where "C = {b}"
          using 1 4 by (metis Un_empty inner_univalent is_singletonI' is_singleton_the_elem subset_Un_eq)
        hence "B = C"
          using 4 5 by blast
        thus "x ∈ ne S"
          using 3 4 5 by blast
      qed
    next
      show "ne S ⊆ ne R"
      proof
        fix x
        assume 6: "x ∈ ne S"
        from this obtain a B where 7: "x = (a,B) ∧ x ∈ S ∧ B ≠ {}"
          by (metis Int_absorb Int_iff inner_total_2 semilattice_inf_class.inf.absorb_iff2 surj_pair)
        from this obtain b where 8: "B = {b}"
          using 1 by (meson antisym card_1_singletonE inner_univalent_2 nonempty_set_card)
        from this obtain C where 9: "b ∈ C ∧ (a,C) ∈ fus S"
          using 7 by (meson fusion_lower_increasing insert_subset lower_less_eq)
        hence "(a,C) ∈ fus R"
          using 2 by simp
        hence "C = ⋃{ D . (a,D) ∈ R }"
          by (clarsimp simp: mr_simp)
        from this obtain D where 10: "b ∈ D ∧ (a,D) ∈ R"
          using 9 by blast
        from this obtain E where 11: "D ⊆ E ∧ (a,E) ∈ S"
          using 2 by (meson lower_less_eq)
        from this obtain c where "E = {c}"
          using 1 10 by (metis antisym card_1_singletonE empty_iff inner_univalent_2 nonempty_set_card subsetI)
        hence "D = {b}"
          using 10 11 by blast
        thus "x ∈ ne R"
          using 6 7 8 10 by blast
      qed
    qed
  qed
next
  assume 12: "∀R . fus R = fus S ∧ R ⊑↓ S ⟶ ne R = ne S"
  show "inner_univalent S"
  proof (unfold inner_univalent, intro allI, rule impI)
    fix a b c B
    assume 13: "(a,B) ∈ S ∧ b ∈ B ∧ c ∈ B"
    let ?R = "{ (f,{e}) | f e . ∃C . (f,C) ∈ S ∧ e ∈ C }"
    have 14: "fus ?R = fus S"
      apply (clarsimp simp: mr_simp)
      by blast
    have "?R ⊑↓ S"
      using lower_less_eq by fastforce
    hence "ne ?R = ne S"
      using 12 14 by simp
    hence "(a,B) ∈ ?R"
      using 13 by (smt (verit, del_insts) empty_iff mem_Collect_eq non_empty)
    thus "b = c"
      using 13 by blast
  qed
qed

lemma ne_dist_oU:
  "ne (⋃X) = ⋃(non_empty ` X)"
  by blast

subsection ‹Fission›

lemma fission_set:
  "fis R = { (a,{b}) | a b . ∃B . (a,B) ∈ R ∧ b ∈ B }"
  unfolding fis_set Image_singleton
  by simp

declare fission_set [mr_simp]

lemma fission_var:
  "fis R = R↓ ∩ A⇩∪⇩∪"
  apply (clarsimp simp: mr_simp)
  by blast

lemma fission_lower_decreasing:
  "fis R ⊑↓ R"
  by (simp add: fission_var)

lemma fission_inner_deterministic:
  "inner_deterministic (fis R)"
  by (simp add: fission_var inner_deterministic_atoms)

lemma fission_greatest:
  assumes "S ⊑↓ R"
    and "inner_deterministic S"
  shows "S ⊑↓ fis R"
proof (clarsimp simp: mr_simp)
  fix a B
  assume 1: "(a,B) ∈ S"
  from this obtain b where 2: "B = {b}"
    using assms(2) by (meson card_1_singletonE inner_deterministic_2)
  from 1 obtain C where "B ⊆ C ∧ (a,C) ∈ R"
    using assms(1) by (meson lower_less_eq)
  thus "∃C . (∃D . B = C ∩ D) ∧ (∃b . C = {b} ∧ (∃E . (a,E) ∈ R ∧ b ∈ E))"
    using 2 by auto
qed

lemma fission_unique:
  assumes "∀R . f R ⊑↓ R"
    and "∀R . inner_deterministic (f R)"
    and "∀R S . S ⊑↓ R ∧ inner_deterministic S ⟶ S ⊑↓ f R"
  shows "f T = fis T"
  apply (rule inner_deterministic_lower_eq)
    apply (simp add: assms(2))
   apply (simp add: fission_inner_deterministic)
  by (simp add: assms fission_greatest fission_inner_deterministic fission_lower_decreasing)

lemma fission_lower_isotone:
  "R ⊑↓ S ⟹ fis R ⊑↓ fis S"
  by (meson fission_greatest fission_inner_deterministic fission_lower_decreasing lower_transitive)

lemma fission_idempotent:
  "fis (fis R) = fis R"
  by (metis comp_apply fis_fis)

lemma fission_top:
  "fis U = A⇩∪⇩∪"
  using fission_var top_down top_upper_least by fastforce

lemma fission_down:
  "fis R = fis (R↓)"
  by (simp add: fission_var)

lemma fission_ne_fixpoint:
  "fis R = ne (fis R)"
  using fission_inner_deterministic by blast

lemma fission_down_ne_fixpoint:
  "fis R = ne ((fis R)↓)"
  by (metis fission_inner_deterministic fission_ne_fixpoint fusion_down inner_univalent_char lower_ii_decreasing)

lemma fission_inner_deterministic_fixpoint:
  "inner_deterministic R ⟷ R = fis R"
  apply (rule iffI)
   apply (metis comp_eq_dest_lhs fission_lower_decreasing fission_ne_fixpoint fus_fis inner_univalent_char le_iff_inf)
  using fission_inner_deterministic by auto

lemma fission_sp_subdist:
  "fis (R * S) ⊆ fis R * fis S"
proof
  fix x
  assume "x ∈ fis (R * S)"
  from this obtain a b B where 1: "x = (a,{b}) ∧ (a,B) ∈ R * S ∧ b ∈ B"
    by (smt (z3) CollectD fission_set)
  from this obtain C f where 2: "(a,C) ∈ R ∧ (∀c ∈ C . (c,f c) ∈ S) ∧ B = ⋃{ f c | c . c ∈ C }"
    by (simp add: mr_simp) blast
  from this obtain c where 3: "b ∈ f c ∧ c ∈ C"
    using 1 by blast
  let ?B = "{c}"
  let ?f = "λx . {b}"
  have 4: "(a,?B) ∈ fis R"
    using 2 3 fission_set by blast
  have 5: "∀b∈?B . (b,?f b) ∈ fis S"
    using 2 3 fission_set by blast
  have "{b} = ⋃{ ?f b | b . b ∈ ?B }"
    by simp
  hence "∃f . (∀b∈?B . (b,f b) ∈ fis S) ∧ {b} = ⋃{ f b | b . b ∈ ?B }"
    using 5 by auto
  hence "(a,{b}) ∈ fis R * fis S"
    apply (unfold s_prod_def)
    using 4 by auto
  thus "x ∈ fis R * fis S"
    using 1 by simp
qed

lemma fission_sp_total_dist:
  assumes "total T"
  shows "fis (R * T) = fis R * fis T"
  by (metis assms atoms_sp_idempotent fis_lax fission_var sp_oi_subdist_2 subset_antisym total_down_dist_sp)

lemma fission_dist_ou:
  "fis (R ∪ S) = fis R ∪ fis S"
  by (simp add: down_dist_ou fission_var inf_sup_distrib2)

lemma fission_sp_iu_unit:
  "fis (R * 1⇩∪⇩∪) = {}"
  by (metis c_nc down_sp fission_lower_decreasing nu_def nu_fis nu_fis_var s_prod_zerol subset_empty)

lemma fission_fusion_lower_decreasing:
  "fis (fus R) ⊑↓ R"
  apply (clarsimp simp: mr_simp)
  by blast

lemma fusion_fission_lower_increasing:
  "R ⊑↓ fus (fis R)"
  apply (clarsimp simp: mr_simp)
  by blast

lemma fission_fusion_galois:
  "fis R ⊑↓ S ⟷ R ⊑↓ fus S"
  apply (rule iffI)
   apply (meson fusion_fission_lower_increasing fusion_lower_isotone lower_transitive)
  by (meson fission_fusion_lower_decreasing fission_lower_isotone lower_transitive)

lemma fission_fusion:
  "fis (fus R) = fis R"
  by (metis fission_fusion_lower_decreasing fission_idempotent fission_inner_deterministic fission_lower_isotone fusion_lower_increasing inner_deterministic_lower_eq)

lemma fusion_fission:
  "fus (fis R) = fus R"
  by (metis comp_def fus_fis)

lemma same_fusion_fission_lower:
  "fus R = fus S ⟹ fis R ⊑↓ S"
  by (metis fission_fusion_galois fusion_lower_increasing)

lemma fission_below_ne_down_fusion:
  "fis R ⊆ ne ((fus R)↓)"
  using fission_fusion fission_inner_deterministic fission_lower_decreasing by blast

lemma ne_fusion_fission:
  "(ne ((fus R)↓))↑ = (fis R)↑"
  by (metis (mono_tags, lifting) atoms_solution fission_below_ne_down_fusion fission_fusion oi_down_sub_up subset_trans upper_eq_up upper_reflexive fission_var)

lemma fission_up_ne_down_up:
  "(fis R)↑ = (ne (R↓))↑"
  by (metis (mono_tags, lifting) atoms_solution fission_ne_fixpoint fission_top oi_down_sub_up semilattice_inf_class.inf_le2 semilattice_inf_class.inf_left_commute subset_trans upper_eq_up fission_var)

lemma fusion_idempotent:
  "fus (fus R) = fus R"
  by (metis fission_fusion fusion_fission)

lemma fission_dist_oU:
  "fis (⋃X) = ⋃(fis ` X)"
  by (metis (no_types, lifting) SUP_cong UN_simps(4) fission_var ii_right_dist_oU)

subsection ‹Co-fusion and co-fission›

definition co_fusion :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹⨅⨅ _› [80] 80) where
  "⨅⨅R ≡ { (a,B) . B = ⋂{ C . (a,C) ∈ R } }"

declare co_fusion_def [mr_simp]

lemma co_fusion_upper_decreasing:
  "⨅⨅R ⊑↑ R"
  apply (clarsimp simp: mr_simp)
  by blast

lemma co_fusion_deterministic:
  "deterministic (⨅⨅R)"
  by (simp add: deterministic_set co_fusion_def)

lemma co_fusion_greatest:
  assumes "S ⊑↑ R"
    and "deterministic S"
  shows "S ⊑↑ ⨅⨅R"
proof (clarsimp simp: mr_simp)
  fix a
  from assms(2) obtain B where 1: "(a,B) ∈ S"
    by (meson deterministic_set)
  hence "B ⊆ ⋂{ C . (a,C) ∈ R }"
    using assms deterministic_upper by (smt (verit, ccfv_threshold) Inter_greatest mem_Collect_eq)
  thus "∃B . (∃D . ⋂{ C . (a,C) ∈ R } = B ∪ D) ∧ (a,B) ∈ S"
    using 1 by blast
qed

lemma co_fusion_unique:
  assumes "∀R . f R ⊑↑ R"
    and "∀R . deterministic (f R)"
    and "∀R S . S ⊑↑ R ∧ deterministic S ⟶ S ⊑↑ f R"
  shows "f T = ⨅⨅T"
  apply (rule univalent_upper_eq)
  using assms(2) deterministic_def apply blast
  using co_fusion_deterministic deterministic_def apply blast
  by (simp add: assms co_fusion_deterministic co_fusion_greatest co_fusion_upper_decreasing)

lemma co_fusion_up_char:
  "(⨅⨅R)↑ = -((-(R↑) ∩ A⇩∩⇩∩)↓)"
proof
  show "(⨅⨅R)↑ ⊆ -((-(R↑) ∩ A⇩∩⇩∩)↓)"
    apply (clarsimp simp: mr_simp)
    by blast
next
  show "-((-(R↑) ∩ A⇩∩⇩∩)↓) ⊆ (⨅⨅R)↑"
  proof (clarsimp simp: mr_simp)
    fix a A
    assume 1: "∀B . (∀C . A ≠ B ∩ C) ∨ (∃C . (∃D . B = C ∪ D) ∧ (a,C) ∈ R) ∨ (∀b . B ≠ UNIV - {b})"
    have "⋂{ C . (a,C) ∈ R } ⊆ A"
    proof
      fix x
      assume "x ∈ ⋂{ C . (a,C) ∈ R }"
      hence "∀C . A ≠ (UNIV - {x}) ∩ C"
        using 1 by blast
      thus "x ∈ A"
        by blast
    qed
    thus "∃D . A = ⋂{ C . (a,C) ∈ R } ∪ D"
      by auto
  qed
qed

lemma co_fusion_down_char:
  "(⨅⨅R)↓ = -((∼(R↑) ∩ A⇩∪⇩∪)↑)"
proof
  show "(⨅⨅R)↓ ⊆ -((∼(R↑) ∩ A⇩∪⇩∪)↑)"
    apply (clarsimp simp: mr_simp)
    by blast
next
  show "-((∼(R↑) ∩ A⇩∪⇩∪)↑) ⊆ (⨅⨅R)↓"
  proof (clarsimp simp: mr_simp)
    fix a A
    assume 1: "∀C . (∀D . A ≠ C ∪ D) ∨ (∀B . (∀D . - C ≠ B ∪ D) ∨ (a,B) ∉ R) ∨ (∀b . C ≠ {b})"
    have "A ⊆ ⋂{ C . (a,C) ∈ R }"
    proof
      fix x
      assume "x ∈ A"
      hence "∀B . (∀D . - {x} ≠ B ∪ D) ∨ (a,B) ∉ R"
        using 1 by blast
      thus "x ∈ ⋂{ C . (a,C) ∈ R }"
        by blast
    qed
    thus "∃C . A = ⋂{ C . (a,C) ∈ R } ∩ C"
      by auto
  qed
qed

lemma co_fusion_down_char_2:
  "(⨅⨅R)↓ = -(((R↑ ∩ A⇩∩⇩∩) ⊙ ∼1)↑)"
  by (metis co_fusion_down_char ic_co_atoms ic_cp_ic_unit ic_dist_oi)

lemma co_fusion_char:
  "⨅⨅R = -((-(R↑) ∩ A⇩∩⇩∩)↓) ∩ -((∼(R↑) ∩ A⇩∪⇩∪)↑)"
  by (metis deterministic_def co_fusion_deterministic co_fusion_down_char co_fusion_up_char univalent_convex)

lemma co_fusion_char_2:
  "⨅⨅R = -((-(R↑) ∩ A⇩∩⇩∩)↓) ∩ -(((R↑ ∩ A⇩∩⇩∩) ⊙ ∼1)↑)"
  using co_fusion_char co_fusion_down_char co_fusion_down_char_2 by blast

lemma co_fusion_upper_isotone:
  "R ⊑↑ S ⟹ ⨅⨅R ⊑↑ ⨅⨅S"
  by (meson co_fusion_deterministic co_fusion_greatest co_fusion_upper_decreasing upper_transitive)

lemma co_fusion_ii_idempotent:
  "⨅⨅R ∩∩ ⨅⨅R = ⨅⨅R"
  by (metis deterministic_def co_fusion_deterministic univalent_ii_idempotent)

lemma co_fusion_up:
  "⨅⨅R = ⨅⨅(R↑)"
  by (simp add: co_fusion_char)

lemma co_fusion_ii_total:
  "total T ⟹ T ∩∩ ⨅⨅T = ⨅⨅T"
  by (meson co_fusion_deterministic co_fusion_upper_decreasing deterministic_def total_univalent_upper_ii)

lemma co_fusion_deterministic_fixpoint:
  "deterministic R ⟷ R = ⨅⨅R"
  apply (rule iffI)
   apply (metis deterministic_def co_fusion_deterministic co_fusion_greatest co_fusion_ii_total ii_commute total_univalent_upper_ii upper_reflexive)
  by (metis co_fusion_deterministic)

abbreviation co_fission :: "('a,'b) mrel ⇒ ('a,'b) mrel" (‹at⇩∩⇩∩ _› [80] 80) where
  "at⇩∩⇩∩ R ≡ R↑ ∩ A⇩∩⇩∩"

lemma co_fission:
  "at⇩∩⇩∩ R = { (a,B) | a B . (∃b . -B = {b}) ∧ (∃C . (a,C) ∈ R ∧ C ⊆ B) }"
  apply (clarsimp simp: mr_simp)
  by blast

declare co_fission [mr_simp]

lemma co_fission_upper_increasing:
  "R ⊑↑ at⇩∩⇩∩ R"
  by (fact semilattice_inf_class.inf_le1)

lemma co_fission_ic_inner_deterministic:
  "inner_deterministic (∼at⇩∩⇩∩ R)"
  by (simp add: ic_co_atoms ic_dist_oi inner_deterministic_atoms)

lemma co_fission_least:
  assumes "R ⊑↑ S"
    and "inner_deterministic (∼S)"
  shows "at⇩∩⇩∩ R ⊑↑ S"
proof (clarsimp simp: mr_simp)
  fix a B
  assume 1: "(a,B) ∈ S"
  hence "(a,-B) ∈ ∼S"
    by (simp add: inner_complement_def)
  from this obtain b where 2: "-B = {b}"
    using assms(2) by (meson card_1_singletonE inner_deterministic_2)
  from 1 obtain C where "C ⊆ B ∧ (a,C) ∈ R"
    using assms(1) by (meson upper_less_eq)
  thus "∃C . (∃D . B = C ∪ D) ∧ (∃E . (∃D . C = E ∪ D) ∧ (a,E) ∈ R) ∧ (∃b . C = UNIV - {b})"
    using 2 by (metis Compl_eq_Diff_UNIV double_compl subset_Un_eq sup.idem)
qed

lemma co_fission_unique:
  assumes "∀R . R ⊑↑ f R"
    and "∀R . inner_deterministic (∼f R)"
    and "∀R S . R ⊑↑ S ∧ inner_deterministic (∼S) ⟶ f R ⊑↑ S"
  shows "f T = at⇩∩⇩∩ T"
  apply (rule ic_injective)
  apply (rule inner_deterministic_lower_eq)
    apply (simp add: assms(2))
   apply (simp add: co_fission_ic_inner_deterministic)
  by (meson assms co_fission_ic_inner_deterministic co_fission_least semilattice_inf_class.inf_le1 upper_ic_lower)

lemma co_fission_upper_isotone:
  "R ⊑↑ S ⟹ at⇩∩⇩∩ R ⊑↑ at⇩∩⇩∩ S"
  by (simp add: oi_subset_upper_left_antitone upper_transitive)

lemma co_fission_idempotent:
  "at⇩∩⇩∩ (at⇩∩⇩∩ R) = at⇩∩⇩∩ R"
  by (meson equalityI semilattice_inf_class.inf_le1 semilattice_inf_class.inf_le2 semilattice_inf_class.le_inf_iff upper_reflexive upper_transitive)

lemma co_fission_top:
  "at⇩∩⇩∩ U = A⇩∩⇩∩"
  using top_lower_greatest U_par_idem top_down by blast

lemma co_fission_up:
  "at⇩∩⇩∩ R = at⇩∩⇩∩ (R↑)"
  by simp

lemma co_fission_ic_inner_deterministic_fixpoint:
  "inner_deterministic (∼R) ⟷ R = at⇩∩⇩∩ R"
  apply (rule iffI)
   apply (simp add: fission_var fission_inner_deterministic_fixpoint ic_antidist_iu ic_co_atoms ic_dist_oi ic_injective ic_up)
  by (metis co_fission_ic_inner_deterministic)

lemma co_fusion_co_fission_upper_decreasing:
  "⨅⨅(at⇩∩⇩∩ R) ⊑↑ R"
proof (clarsimp simp: mr_simp)
  fix a B
  assume 1: "(a,B) ∈ R"
  have "⋂{ D . (∃E . (∃F . D = E ∪ F) ∧ (a,E) ∈ R) ∧ (∃b . D = UNIV - {b}) } ⊆ B"
  proof
    fix x
    assume 2: "x ∈ ⋂{ D . (∃E . (∃F . D = E ∪ F) ∧ (a,E) ∈ R) ∧ (∃b . D = UNIV - {b}) }"
    show "x ∈ B"
    proof (rule ccontr)
      let ?D = "-{x}"
      assume 3: "x ∉ B"
      hence "B ⊆ ?D"
        by simp
      hence "⋂{ D . (∃E . (∃F . D = E ∪ F) ∧ (a,E) ∈ R) ∧ (∃b . D = UNIV - {b}) } ⊆ ?D"
        using 1 by (smt (z3) CollectI Compl_eq_Diff_UNIV Inf_lower subset_Un_eq)
      thus False
        using 2 by auto
    qed
  qed
  thus "∃C . B = ⋂{ D . (∃E . (∃F . D = E ∪ F) ∧ (a,E) ∈ R) ∧ (∃b . D = UNIV - {b}) } ∪ C"
    by auto
qed

lemma co_fission_co_fusion_upper_increasing:
  "R ⊑↑ at⇩∩⇩∩ (⨅⨅R)"
proof (clarsimp simp: mr_simp)
  fix a b B
  assume "⋂{ C . (a,C) ∈ R } ∪ B = UNIV - {b}"
  hence "b ∉ ⋂{ C . (a,C) ∈ R }"
    by blast
  hence "∃C . b ∉ C ∧ (a,C) ∈ R"
    by blast
  thus "∃C . (∃D . UNIV - {b} = C ∪ D) ∧ (a,C) ∈ R"
    by blast
qed

lemma co_fusion_co_fission_galois:
  "⨅⨅R ⊑↑ S ⟷ R ⊑↑ at⇩∩⇩∩ S"
  apply (rule iffI)
   apply (meson co_fission_co_fusion_upper_increasing co_fission_upper_isotone upper_transitive)
  by (meson co_fusion_co_fission_upper_decreasing co_fusion_upper_isotone upper_transitive)

lemma co_fission_co_fusion:
  "at⇩∩⇩∩ (⨅⨅R) = at⇩∩⇩∩ R"
  using co_fission_co_fusion_upper_increasing co_fission_idempotent co_fission_upper_isotone co_fusion_upper_decreasing by blast

lemma co_fusion_co_fission:
  "⨅⨅(at⇩∩⇩∩ R) = ⨅⨅R"
  apply (rule antisym)
   apply (metis deterministic_def co_fission_co_fusion co_fission_co_fusion_upper_increasing co_fusion_co_fission_upper_decreasing co_fusion_deterministic co_fusion_upper_isotone univalent_upper_eq_subset)
  by (metis deterministic_def co_fission_co_fusion co_fission_co_fusion_upper_increasing co_fusion_co_fission_upper_decreasing co_fusion_deterministic co_fusion_upper_isotone univalent_upper_eq_subset)

lemma same_co_fusion_co_fission_upper:
  "⨅⨅R = ⨅⨅S ⟹ S ⊑↑ at⇩∩⇩∩ R"
  by (metis co_fusion_co_fission_galois co_fusion_upper_decreasing)

lemma co_fusion_idempotent:
  "⨅⨅(⨅⨅R) = ⨅⨅R"
  by (metis co_fission_co_fusion co_fusion_co_fission)

section ‹Modalities›

subsection ‹Tests›

abbreviation test :: "('a,'a) mrel ⇒ bool" where
  "test R ≡ R ⊆ 1"

lemma test:
  "test R ⟷ (∀a B . (a,B) ∈ R ⟶ B = {a})"
  by (force simp: s_id_def)

lemma test_fix: "test R ≡ R ∩ 1⇩σ = R"
  by (simp add: le_iff_inf)

lemma test_ou_closed:
  "test p ⟹ test q ⟹ test (p ∪ q)"
  by (fact sup_least)

lemma test_oi_closed:
  "test p ⟹ test (p ∩ q)"
  by blast

abbreviation test_complement :: "('a,'a) mrel ⇒ ('a,'a) mrel" (‹≀ _› [80] 80) where
  "≀ R ≡ -R ∩ 1"

lemma test_complement_closed:
  "test (≀ p)"
  by simp

lemma test_double_complement:
  "test p ⟷ p = ≀ ≀ p"
  by blast

lemma test_complement:
  "(a,{a}) ∈ ≀ p ⟷ ¬ (a,{a}) ∈ p"
  by (simp add: s_id_def)

declare test_complement [mr_simp]

lemma test_complement_antitone:
  assumes "test p"
  shows "p ⊆ q ⟷ ≀ q ⊆ ≀ p"
  using assms(1) by blast

lemma test_complement_huntington:
  "test p ⟹ p = ≀ (≀ p ∪ ≀ q) ∪ ≀ (≀ p ∪ q)"
  by blast

abbreviation test_implication :: "('a,'a) mrel ⇒ ('a,'a) mrel ⇒ ('a,'a) mrel" (infixl ‹→› 65) where
  "p → q ≡ ≀ p ∪ q"

lemma test_implication_closed:
  "test q ⟹ test (p → q)"
  by simp

lemma test_implication:
  "(a,{a}) ∈ p → q ⟷ ((a,{a}) ∈ p ⟶ (a,{a}) ∈ q)"
  by (simp add: s_id_def)

declare test_implication [mr_simp]

lemma test_implication_left_antitone:
  assumes "test p"
  shows "p ⊆ r ⟹ r → q ⊆ p → q"
  by blast

lemma test_implication_right_isotone:
  assumes "test p"
  shows "q ⊆ r ⟹ p → q ⊆ p → r"
  by blast

lemma test_sp_idempotent:
  "test p ⟹ p * p = p"
  by (metis d_rest_ax inf.order_iff s_subid_iff2)

lemma test_sp:
  assumes "test p"
  shows "p * R = (p * U) ∩ R"
  apply (clarsimp simp: mr_simp)
  apply safe
    apply blast
  using assms subid_aux2 by fastforce+

lemma sp_test:
  "test p ⟹ R * p = R ∩ (U * p)"
  apply (rule antisym)
   apply (metis (no_types, lifting) U_par_idem inf.absorb_iff2 inf.idem le_inf_iff s_prod_idr sp_oi_subdist top_upper_least)
  using test_fix by (smt (z3) IntE s_prod_test_aux1 s_prod_test_aux2 subrelI)

lemma sp_test_dist_oi:
  "test p ⟹ (R ∩ S) * p = (R * p) ∩ (S * p)"
  by (smt (z3) Int_left_commute semilattice_inf_class.inf.assoc semilattice_inf_class.inf.right_idem sp_test)

lemma sp_test_dist_oi_left:
  "test p ⟹ (R ∩ S) * p = (R * p) ∩ S"
  by (smt (z3) Int_commute semilattice_inf_class.inf.left_commute sp_test)

lemma sp_test_dist_oi_right:
  "test p ⟹ (R ∩ S) * p = R ∩ (S * p)"
  by (metis semilattice_inf_class.inf.commute sp_test_dist_oi_left)

lemma sp_test_sp_oi_left:
  "test p ⟹ (R ∩ (U * p)) * T = R * p * T"
  by (metis sp_test)

lemma sp_test_sp_oi_right:
  "test p ⟹ R * ((p * U) ∩ T) = R * p * T"
  by (metis inf.orderE test_assoc1 test_sp)

lemma test_sp_ne:
  "test p ⟹ p * ne R = ne (p * R)"
  by (smt (z3) lattice_class.inf_sup_aci(1) lattice_class.inf_sup_aci(3) test_sp)

lemma ne_sp_test:
  "test p ⟹ ne R * p = ne (R * p)"
  by (fact sp_test_dist_oi_left)

lemma top_sp_test_down_closed:
  assumes "test p"
  shows "U * p = (U * p)↓"
proof -
  have 1: "p ∩ 1⇩σ = p"
    using assms by blast
  hence "(U * p)↓ = {(a,A). (a,A) ∈ U ∧ (∀a ∈ A. (a,{a}) ∈ p)} ∩∩ U"
    by (smt (verit) Collect_cong case_prodI2 case_prod_conv s_prod_test_var)
  also have "… = {(a,A). ∀a ∈ A. (a,{a}) ∈ p} ∩∩ U"
    by (smt (verit) Collect_cong ii_assoc lower_ii_down mem_Collect_eq split_cong subsetD top_down)
  also have "… = {(a,A). (a,A) ∈ U ∧ (∀a ∈ A. (a,{a}) ∈ p)}"
    by (auto simp: mr_simp)
  also have "… = U * p"
    using 1 by (smt (verit) Collect_cong case_prodI2 case_prod_conv s_prod_test_var)
  finally show ?thesis
    by blast
qed

lemma oc_top_sp_test_up_closed:
  "test p ⟹ -(U * p) = (-(U * p))↑"
  by (metis antisym convex_reflexive disjoint_eq_subset_Compl inf_compl_bot oi_down_up_iff semilattice_inf_class.inf.commute top_sp_test_down_closed)

lemma top_sp_test:
  "test p ⟹ (a,B) ∈ U * p ⟷ (∀b∈B . (b,{b}) ∈ p)"
  using test_fix by (metis IntE UNIV_I s_prod_test sp_test)

lemma oc_top_sp_test:
  "test p ⟹ (a,B) ∈ -(U * p) ⟷ (∃b∈B . (b,{b}) ∉ p)"
  by (simp add: top_sp_test)

declare top_sp_test [mr_simp] oc_top_sp_test [mr_simp]

lemma oc_top_sp_test_0:
  "-1⇩∪⇩∪ * ≀ p = ne (U * ≀ p)"
  by (metis Int_lower1 semilattice_inf_class.inf.commute sp_test)

lemma oc_top_sp_test_1:
  assumes "test p"
  shows "-(U * p) = (ne (U * ≀ p))↑"
proof (rule antisym)
  show "-(U * p) ⊆ (ne (U * ≀ p))↑"
  proof
    fix x::"'c × 'a set"
    assume 1: "x ∈ -(U * p)"
    from this obtain a B where 2: "x = (a,B)"
      by force
    from this obtain c where 3: "c ∈ B ∧ (c,{c}) ∉ p"
      using 1 by (meson assms oc_top_sp_test)
    hence 4: "(a,{c}) ∈ U * ≀ p"
      by (metis singletonD test_complement test_complement_closed top_sp_test)
    have "(a,{c}) ∈ -1⇩∪⇩∪"
      using oc_top_sp_test by (smt (verit, del_insts) ComplI Int_iff assms boolean_algebra.conj_cancel_left inf.coboundedI2 p_id_zero s_prod_test_aux1 singleton_iff)
    hence "(a,{c}) ∈ ne (U * ≀ p)"
      using 4 by simp
    thus "x ∈ (ne (U * ≀ p))↑"
      using 2 3 by (metis (no_types, lifting) U_par_st singletonD subset_eq)
  qed
next
  have "(U * p)↓ = U * p"
    using assms top_sp_test_down_closed by auto
  also have "... ⊆ -(-1⇩∪⇩∪ * ≀ p)"
    by (smt (verit) Compl_disjoint assms disjoint_eq_subset_Compl inf_commute oc_top_sp_test_0 p_id_zero s_prod_idl sp_test_dist_oi_right test_assoc1 test_double_complement)
  also have "... = -ne (U * ≀ p)"
    by (simp add: oc_top_sp_test_0)
  finally have "U * p ⊆ -((ne (U * ≀ p))↑)"
    by (simp add: down_double_complement_up)
  thus "(ne (U * ≀ p))↑ ⊆ -(U * p)"
    by auto
qed

lemma oc_top_sp_test_2:
  "test p ⟹ -(U * p) = (-1⇩∪⇩∪ * ≀ p)↑"
  by (simp add: oc_top_sp_test_1 oc_top_sp_test_0)

lemma split_sp_test:
  assumes "test p"
  shows "R = (R * p) ∪ (ne R ∩ (ne (R↓ * ≀ p))↑)"
proof (rule antisym)
  show "R ⊆ (R * p) ∪ (ne R ∩ (ne (R↓ * ≀ p))↑)"
  proof
    fix x
    assume 1: "x ∈ R"
    from this obtain a B where 2: "x = (a,B)"
      by force
    show "x ∈ (R * p) ∪ (ne R ∩ (ne (R↓ * ≀ p))↑)"
    proof (cases "∀b∈B . (b,{b}) ∈ p")
      case True
      hence "(a,B) ∈ U * p"
        by (simp add: assms top_sp_test)
      thus ?thesis
        using 1 2 by (metis Int_iff UnCI assms sp_test)
    next
      case False
      from this obtain b where 3: "{b} ⊆ B ∧ (b,{b}) ∉ p"
        by auto
      hence "(a,{b}) ∈ R↓"
        using 1 2 down by fastforce
      hence "(a,{b}) ∈ R↓ * ≀ p"
        using 3 by (metis s_prod_test_aux2 singletonD test_complement)
      hence "(a,{b}) ∈ ne (R↓ * ≀ p)"
        by (simp add: non_empty)
      hence "(a,B) ∈ (ne (R↓ * ≀ p))↑"
        using 3 by (meson U_par_st)
      thus ?thesis
        using 1 2 3 non_empty by auto
    qed
  qed
next
  show "(R * p) ∪ (ne R ∩ (ne (R↓ * ≀ p))↑) ⊆ R"
    using assms sp_test by auto
qed

lemma top_sp_test_down_iff_1:
  assumes "test p"
  shows "R ⊆ U * p ⟷ R↓ ⊆ U * p"
  by (smt (verit, del_insts) assms down_order_lower top_sp_test_down_closed)

lemma test_ne:
  "test p ⟹ ne p = p"
  using inner_deterministic_sp_unit by blast

lemma ne_test_up:
  "test p ⟹ ne (p↑) = p↑"
  by (metis atoms_solution ne_equality test_ne up_idempotent up_isotone)

lemma ne_sp_test_up:
  "test p ⟹ (ne (R * p))↑ = ne R * p↑"
  using test_fix by (smt (z3) ne_up sp_test_dist_oi_left test_assoc1 test_ne)

lemma ne_down_sp_test_up:
  "test p ⟹ ne (R↓ * p↑) = ne (R↓) * p↑"
  by (simp add: ne_dist_down_sp ne_test_up)

lemma test_up_sp:
  "test p ⟹ p↑ = p * 1↑"
  by (metis ne_up test_ne)

lemma top_test_oi_top_complement:
  "test p ⟹ (U * p) ∩ (U * ≀ p) = 1⇩∪⇩∪"
  by (smt (verit) Compl_disjoint U_par_idem inf.absorb_iff2 inf_commute p_id_zero s_prod_idl sp_test_dist_oi_right test_assoc1 top_upper_least)

lemma sp_test_oi_complement:
  "test p ⟹ (R * p) ∩ (R * ≀ p) = R ∩ 1⇩∪⇩∪"
  by (smt (z3) semilattice_inf_class.inf_idem sp_test sp_test_dist_oi_left sp_test_dist_oi_right test_complement_closed top_test_oi_top_complement)

lemma ne_top_sp_test_complement:
  assumes "test p"
  shows "ne (U * p) * ≀ p = {}"
  by (metis Compl_disjoint Int_assoc assms oc_top_sp_test_0 semilattice_inf_class.inf_le2 sp_test_dist_oi_right top_test_oi_top_complement)

lemma complement_test_sp_top:
  assumes "test p"
  shows "-(p * U) = ≀ p * U"
proof -
  have "-(p * U) = -{(a,A). (a,{a}) ∈ p ∧ (a,A) ∈ U}"
    by (metis (no_types, lifting) Collect_cong assms inf.orderE split_cong test_s_prod_var)
  also have "… = -{(a,A). (a,{a}) ∈ p}"
    using top_upper_least by auto
  also have "… = {(a,A). (a,{a}) ∉ p}"
    by force
  also have "… = {(a,A). (a,{a}) ∈ ≀ p}"
    by (meson test_complement)
  also have "… = {(a,A). (a,{a}) ∈ ≀ p ∧ (a,A) ∈ U}"
    using U_par_idem top_upper_least by auto
  also have "… =  ≀ p * U"
    by (simp add: test_s_prod_var)
  finally show ?thesis
    .
qed

lemma top_sp_test_shunt:
  assumes "test p"
  shows "R ⊆ U * p ⟶ R * ≀ p ⊆ 1⇩∪⇩∪"
  by (metis assms inf.absorb_iff1 sp_test sp_test_dist_oi test_complement_closed top_test_oi_top_complement)

lemma top_sp_test_down_iff_2:
  assumes "test p"
  shows "R↓ ⊆ U * p ⟷ R↓ * ≀ p ⊆ 1⇩∪⇩∪"
proof
  assume "R↓ ⊆ U * p"
  thus "R↓ * ≀ p ⊆ 1⇩∪⇩∪"
    using assms top_sp_test_shunt by blast
next
  assume 1: "R↓ * ≀ p ⊆ 1⇩∪⇩∪"
  have "R ⊆ U * p"
  proof
    fix x
    assume "x ∈ R"
    from this obtain a B where 2: "x = (a,B) ∧ x ∈ R"
      by force
    have "∀b∈B . (b,{b}) ∈ p"
    proof
      fix b
      assume 3: "b ∈ B"
      have "(b,{b}) ∉ ≀ p"
      proof
        assume 4: "(b,{b}) ∈ ≀ p"
        have "(a,{b}) ∈ R↓"
          using 2 3 down by fastforce
        hence "(a,{b}) ∈ R↓ * ≀ p"
          using 4 by (simp add: s_prod_test)
        thus False
          using 1 by (metis Pair_inject domain_pointwise insert_not_empty p_subid_iff)
      qed
      thus "(b,{b}) ∈ p"
        by (meson test_complement)
    qed
    thus "x ∈ U * p"
      using 2 by (simp add: assms top_sp_test)
  qed
  thus "R↓ ⊆ U * p"
    using assms top_sp_test_down_iff_1 by blast
qed

lemma top_sp_test_down_iff_3:
  "R↓ * ≀ p ⊆ 1⇩∪⇩∪ ⟷ ne (R↓) * ≀ p ⊆ {}"
  by (simp add: disjoint_eq_subset_Compl ne_sp_test)

lemma top_sp_test_down_iff_4:
  assumes "test p"
  shows "R↓ ∩ (U * ≀ p) ⊆ 1⇩∪⇩∪ ⟷ R↓ ⊆ 1⇩∪⇩∪ ∪ (U * p)"
  by (metis assms lattice_class.sup_inf_absorb semilattice_inf_class.inf_le2 sp_test sup_commute top_sp_test_down_iff_2 top_test_oi_top_complement)

lemma top_sp_test_down_iff_5:
  assumes "test p"
  shows "R↓ ⊆ U * p ⟷ R↓ ⊆ 1⇩∪⇩∪ ∪ (U * p)"
  by (metis assms semilattice_inf_class.inf_le1 sup.absorb2 top_test_oi_top_complement)

lemma iu_test_sp_left_zero:
  assumes "q ⊆ 1⇩∪⇩∪"
  shows "q * R = q"
  by (metis assms p_id_assoc2 p_subid_iff s_prod_p_idl)

lemma test_iu_test_split:
  "t ⊆ 1 ∪ 1⇩∪⇩∪ ⟷ (∃p q . p ⊆ 1 ∧ q ⊆ 1⇩∪⇩∪ ∧ t = p ∪ q)"
  by (meson subset_UnE sup.mono)

lemma test_iu_test_sp_assoc_1:
  "t ⊆ 1 ∪ 1⇩∪⇩∪ ⟹ t * (R * S) = (t * R) * S"
  unfolding test_iu_test_split
  by (smt (verit, ccfv_threshold) inf.orderE p_id_assoc2 p_subid_iff s_prod_distr s_prod_p_idl test_assoc2)

lemma test_iu_test_sp_assoc_2:
  "t ⊆ 1⇩∪⇩∪ ⟹ R * (t * S) = (R * t) * S"
proof -
  assume 1: "t ⊆ 1⇩∪⇩∪"
  have "R * (t * S) = R * (t * {})"
    using 1 by (metis iu_test_sp_left_zero p_id_assoc2 s_prod_p_idl)
  also have "… = (R * t) * {}"
    by (metis cl5 s_prod_idl)
  also have "… ⊆ (R * t) * S"
    by (simp add: s_prod_isor)
  finally have "R * (t * S) ⊆ (R * t) * S"
    .
  thus ?thesis
    by (simp add: s_prod_assoc1 set_eq_subset)
qed

lemma test_iu_test_sp_assoc_3:
  assumes "t ⊆ 1 ∪ 1⇩∪⇩∪"
  shows "R * (t * S) = (R * t) * S"
proof
  let ?g = "λb . if (b,{b}) ∈ t ∧ (b,{}) ∉ t then {b} else {}"
  show "R * (t * S) ⊆ (R * t) * S"
  proof
    fix x
    assume "x ∈ R * (t * S)"
    from this obtain a B C f where 1: "x = (a,C) ∧ (a,B) ∈ R ∧ (∀b∈B . (b,f b) ∈ t * S) ∧ C = ⋃{ f b | b . b∈B }"
      by (simp add: mr_simp) blast
    hence "∀b∈B . ∃D . (b,D) ∈ t ∧ (∃g . (∀e∈D . (e,g e) ∈ S) ∧ f b = ⋃{ g e | e . e ∈ D })"
      by (simp add: mr_simp Setcompr_eq_image)
    hence "∃Db . ∀b∈B . (b,Db b) ∈ t ∧ (∃g . (∀e∈Db b . (e,g e) ∈ S) ∧ f b = ⋃{ g e | e . e ∈ Db b })"
      by (rule bchoice)
    from this obtain Db where 2: "∀b∈B . (b,Db b) ∈ t ∧ (∃g . (∀e∈Db b . (e,g e) ∈ S) ∧ f b = ⋃{ g e | e . e ∈ Db b })"
      by auto
    let ?D = "⋃{ Db b | b . b ∈ B }"
    have "∀b∈B . (b,Db b) ∈ t"
      using 2 by auto
    hence 3: "∀b∈B . Db b = {b} ∨ Db b = {}"
      using assms by (metis Pair_inject Un_iff domain_pointwise inf.orderE p_subid_iff subid_aux2 test_iu_test_split)
    have 4: "(a,?D) ∈ R * t"
      apply (simp add: mr_simp)
      apply (rule exI[where ?x="B"])
      apply (rule conjI)
      using 1 apply simp
      apply (rule exI[where ?x="Db"])
      using 2 by auto
    have 5: "∀b∈?D . (b,f b) ∈ S"
    proof
      fix b
      assume "b ∈ ?D"
      hence "b ∈ B ∧ Db b = {b}"
        using 3 by auto
      thus "(b,f b) ∈ S"
        using 2 by force
    qed
    have 6: "C = ⋃{ f b | b . b ∈ ?D }"
    proof
      show "C ⊆ ⋃{ f b | b . b ∈ ?D }"
      proof
        fix y
        assume "y ∈ C"
        from this 1 obtain b where 7: "b ∈ B ∧ y ∈ f b"
          by auto
        hence "Db b = {b}"
          using 2 3 by blast
        thus "y ∈ ⋃{ f b | b . b ∈ ?D }"
          using 7 by blast
      qed
    next
      show "⋃{ f b | b . b ∈ ?D } ⊆ C"
      proof
        fix y
        assume "y ∈ ⋃{ f b | b . b ∈ ?D }"
        from this obtain b where 8: "b ∈ ?D ∧ y ∈ f b"
          by auto
        hence "b ∈ B"
          using 3 by auto
        thus "y ∈ C"
          using 1 8 by auto
      qed
    qed
    have "(a,C) ∈ (R * t) * S"
      using 4 5 6 apply (clarsimp simp: mr_simp) by blast
    thus "x ∈ (R * t) * S"
      using 1 by simp
  qed
next
  show "(R * t) * S ⊆ R * (t * S)"
    using s_prod_assoc1 by blast
qed

lemma test_iu_test_sp_assoc_4:
  "t ⊆ 1⇩∪⇩∪ ⟹ R * (S * t) = (R * S) * t"
  by (metis cl5 iu_test_sp_left_zero)

lemma test_iu_test_sp_assoc_5:
  assumes "t ⊆ 1 ∪ 1⇩∪⇩∪"
  shows "R * (S * t) = (R * S) * t"
proof
  show "R * (S * t) ⊆ (R * S) * t"
  proof
    fix x
    assume "x ∈ R * (S * t)"
    from this obtain a B C f where 1: "x = (a,C) ∧ (a,B) ∈ R ∧ (∀b∈B . (b,f b) ∈ S * t) ∧ C = ⋃{ f b | b . b∈B }"
      by (clarsimp simp: mr_simp) blast
    hence "∀b∈B . ∃D . (b,D) ∈ S ∧ (∃g . (∀e∈D . (e,g e) ∈ t) ∧ f b = ⋃{ g e | e . e ∈ D })"
      by (clarsimp simp: mr_simp Setcompr_eq_image)
    hence "∃Db . ∀b∈B . (b,Db b) ∈ S ∧ (∃g . (∀e∈Db b . (e,g e) ∈ t) ∧ f b = ⋃{ g e | e . e ∈ Db b })"
      by (rule bchoice)
    from this obtain Db where 2: "∀b∈B . (b,Db b) ∈ S ∧ (∃g . (∀e∈Db b . (e,g e) ∈ t) ∧ f b = ⋃{ g e | e . e ∈ Db b })"
      by auto
    hence "∃gb . ∀b∈B . (∀e∈Db b . (e,gb b e) ∈ t) ∧ f b = ⋃{ gb b e | e . e ∈ Db b }"
      by (auto intro: bchoice)
    from this obtain gb where 3: "∀b∈B . (∀e∈Db b . (e,gb b e) ∈ t) ∧ f b = ⋃{ gb b e | e . e ∈ Db b }"
      by auto
    let ?g = "λx . ⋃{ gb b x | b . b ∈ B ∧ x ∈ Db b }"
    have 4: "∀b∈B . ∀e∈Db b . gb b e = {} ∨ gb b e = {e}"
    proof (intro ballI)
      fix b e
      assume "b ∈ B" "e ∈ Db b"
      hence "(e,gb b e) ∈ t"
        using 3 by simp
      thus "gb b e = {} ∨ gb b e = {e}"
        by (metis Un_iff assms domain_pointwise inf.orderE p_subid_iff prod.inject subid_aux2 test_iu_test_split)
    qed
    hence "∀e . ?g e ⊆ {e}"
      by auto
    hence 5: "∀e . ?g e = {} ∨ ?g e = {e}"
      by auto
    let ?D = "⋃{ Db b | b . b ∈ B }"
    have 6: "(a,?D) ∈ R * S"
      apply (unfold s_prod_def)
      using 1 2 by blast
    have 7: "∀b∈?D . (b,?g b) ∈ t"
    proof
      fix e
      assume "e ∈ ?D"
      from this obtain b where 8: "b ∈ B ∧ e ∈ Db b"
        by auto
      show "(e,?g e) ∈ t"
      proof (cases "?g e = {}")
        case True
        hence "gb b e = {}"
          using 8 by auto
        thus ?thesis
          using 3 8 True by metis
      next
        case False
        hence 9: "?g e = {e}"
          using 5 by auto
        from this obtain bb where 10: "bb ∈ B ∧ e ∈ Db bb ∧ e ∈ gb bb e"
          by auto
        hence "gb bb e = {e}"
          using 4 by auto
        thus ?thesis
          using 3 9 10 by force
      qed
    qed
    have 11: "C = ⋃{ ?g e | e . e ∈ ?D }"
    proof
      show "C ⊆ ⋃{ ?g e | e . e ∈ ?D }"
      proof
        fix y
        assume "y ∈ C"
        from this obtain b where 12: "b ∈ B ∧ y ∈ f b"
          using 1 by auto
        from this 3 obtain e where "e ∈ Db b ∧ y ∈ gb b e"
          by auto
        thus "y ∈ ⋃{ ?g e | e . e ∈ ?D }"
          using 4 12 by auto
      qed
    next
      show "⋃{ ?g e | e . e ∈ ?D } ⊆ C"
      proof
        fix y
        assume "y ∈ ⋃{ ?g e | e . e ∈ ?D }"
        from this obtain b e where 13: "b ∈ B ∧ e ∈ Db b ∧ y ∈ gb b e"
          by auto
        hence "y ∈ f b"
          using 3 by auto
        thus "y ∈ C"
          using 1 13 by auto
      qed
    qed
    have "(a,C) ∈ (R * S) * t"
      apply (simp add: mr_simp)
      apply (rule exI[where ?x="?D"])
      apply (rule conjI)
      using 6 apply (simp add: mr_simp)
      apply (rule exI[where ?x="?g"])
      using 7 11 by auto
    thus "x ∈ (R * S) * t"
      using 1 by simp
  qed
next
  show "(R * S) * t ⊆ R * (S * t)"
    using s_prod_assoc1 by blast
qed

lemma inner_deterministic_sp_assoc:
  assumes "inner_univalent t"
  shows "t * (R * S) = (t * R) * S"
proof (rule antisym)
  show "t * (R * S) ⊆ (t * R) * S"
  proof
    fix x
    assume "x ∈ t * (R * S)"
    from this obtain a B D f where 1: "x = (a,D) ∧ (a,B) ∈ t ∧ (∀b∈B . (b,f b) ∈ R * S) ∧ D = ⋃{ f b | b . b∈B }"
      by (simp add: mr_simp) blast
    have "(a,D) ∈ (t * R) * S"
    proof (cases "(a,B) ∈ 1⇩∪⇩∪")
      case True
      hence "B = {}"
        by (metis Pair_inject domain_pointwise iu_test_sp_left_zero order_refl)
      hence "D = {}"
        using 1 by fastforce
      hence "(a,D) ∈ t * (R * {})"
        using 1 ‹(B::'b::type set) = {}› s_prod_def by fastforce
      hence "(a,D) ∈ (t * R) * {}"
        by (metis cl5 s_prod_idl)
      thus ?thesis
        using s_prod_isor by auto
    next
      case False
      hence "(a,B) ∈ A⇩∪⇩∪"
        using 1 assms by blast
      from this obtain b where 2: "B = {b}"
        by (smt (z3) atoms_def Pair_inject mem_Collect_eq)
      hence "D = f b ∧ (b,f b) ∈ R * S"
        using 1 by auto
      from this obtain C g where 3: "(b,C) ∈ R ∧ (∀c∈C . (c,g c) ∈ S) ∧ D = ⋃{ g c | c . c∈C }"
        by (clarsimp simp: mr_simp) blast
      have "(a,C) ∈ t * R"
        apply (simp add: mr_simp)
        apply (rule exI[where ?x="B"])
        using 1 2 3 by auto
      thus ?thesis
        using 3 s_prod_def by blast
    qed
    thus "x ∈ (t * R) * S"
      using 1 by auto
  qed
next
  show "(t * R) * S ⊆ t * (R * S)"
    using s_prod_assoc1 by blast
qed

lemma iu_unit_below_top_sp_test:
  "1⇩∪⇩∪ ⊆ U * R"
  by (clarsimp simp: mr_simp) force

lemma ne_oi_complement_top_sp_test_1:
  "ne (R ∩ -(U * S)) = R ∩ -(U * S)"
  using iu_unit_below_top_sp_test by auto

lemma ne_oi_complement_top_sp_test_2:
  "ne R ∩ -(U * S) = R ∩ -(U * S)"
  using iu_unit_below_top_sp_test by auto

lemma schroeder_test:
  assumes "test p"
  shows "R * p ⊆ S ⟷ -S * p ⊆ -R"
  by (metis assms disjoint_eq_subset_Compl double_compl semilattice_inf_class.inf_commute sp_test_dist_oi_right)

lemma complement_test_sp_test:
  "test p ⟹ -p * p ⊆ -1"
  by (simp add: schroeder_test)

lemma test_sp_commute:
  "test p ⟹ test q ⟹ p * q = q * p"
  by (metis s_prod_idl sp_test_dist_oi_left sp_test_dist_oi_right test_fix)

lemma test_shunting:
  assumes "test p"
    and "test q"
    and "test r"
  shows "p * q ⊆ r ⟷ p ⊆ r ∪ ≀ q"
proof
  assume 1: "p * q ⊆ r"
  have "p = p * q ∪ p * ≀ q"
    by (smt (verit, del_insts) Int_Un_distrib assms(1,2) inf_sup_aci(1) inf_sup_ord(2) le_iff_inf s_distl_test s_prod_idr sup_neg_inf)
  also have "... ⊆ r ∪ ≀ q"
    using 1 by (metis assms(1) inf_le2 sup.mono test_sp)
  finally show "p ⊆ r ∪ ≀ q"
    .
next
  assume "p ⊆ r ∪ ≀ q"
  hence "p * q ⊆ p * r"
    by (smt (verit) assms boolean_algebra_class.boolean_algebra.double_compl inf.orderE inf_le1 le_inf_iff schroeder_test sup_neg_inf test_double_complement test_s_prod_is_meet test_sp_commute)
  also have "... ⊆ r"
    using assms(1) test_sp by auto
  finally show "p * q ⊆ r"
    .
qed

lemma test_sp_is_iu:
  "test p ⟹ test q ⟹ p * q = p ∪∪ q"
  by (metis Int_assoc inf.right_idem p_prod_comm s_prod_idr test_p_prod_is_meet test_sp)

lemma test_set:
  "test p ⟹ p = { (a,{a}) | a . (a,{a}) ∈ p }"
  using s_id_st by blast

lemma test_sp_is_ii:
  assumes "test p"
    and "test q"
  shows "p * q = p ∩∩ q"
proof -
  have "p ∩ q = { (a,{a}) | a . (a,{a}) ∈ p } ∩ { (a,{a}) | a . (a,{a}) ∈ q }"
    using assms test_set by blast
  also have "... = { (a,{a}) | a . (a,{a}) ∈ p } ∩∩ { (a,{a}) | a . (a,{a}) ∈ q }"
    apply (rule antisym)
     apply (clarsimp simp: mr_simp)
    apply (rule Int_greatest)
     apply (clarsimp simp: mr_simp)
    by (clarsimp simp: mr_simp)
  also have "... = p ∩∩ q"
    using test_set[symmetric, OF assms(1)] test_set[symmetric, OF assms(2)] by simp
  finally show "p * q = p ∩∩ q"
    by (metis assms inf.orderE test_oi_closed test_s_prod_is_meet)
qed

lemma test_galois_1:
  assumes "test p"
    and "test q"
  shows "p * q ⊆ r ⟷ q ⊆ p → r"
  by (metis (no_types, lifting) Int_Un_eq(2) assms inf.orderE sup_neg_inf test_double_complement test_s_prod_is_meet test_sp_commute)

lemma test_sp_shunting:
  assumes "test p"
  shows "≀ p * R ⊆ {} ⟷ R ⊆ p * R"
  apply (rule iffI)
   apply (metis (no_types, opaque_lifting) assms complement_test_sp_top disjoint_eq_subset_Compl double_compl semilattice_inf_class.inf.absorb_iff2 semilattice_inf_class.inf_commute semilattice_inf_class.inf_right_idem test_sp)
  by (metis (no_types, opaque_lifting) assms complement_test_sp_top disjoint_eq_subset_Compl double_compl semilattice_inf_class.inf_commute semilattice_inf_class.inf_le1 subset_antisym test_sp)

lemma test_oU_closed:
  "∀p∈X . test p ⟹ test (⋃X)"
  by blast

lemma test_oI_closed:
  "∃p∈X . test p ⟹ test (⋂X)"
  by blast

lemma sp_test_dist_oI:
  assumes "test p"
    and "X ≠ {}"
  shows "(⋂X) * p = (⋂R∈X . R * p)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply blast
  apply (clarsimp simp: mr_simp)
  subgoal for a C proof -
    assume 1: "∀R∈X . ∃B . (a,B) ∈ R ∧ (∃f . (∀b∈B . (b,f b) ∈ p) ∧ C = ⋃(f ` B))"
    have 2: "(∀R∈X . (a,C) ∈ R ∧ (∀b∈C . (b,{b}) ∈ p))"
    proof
      fix R
      assume "R ∈ X"
      from this obtain B where 3: "(a,B) ∈ R ∧ (∃f . (∀b∈B . (b,f b) ∈ p) ∧ C = ⋃(f ` B))"
        using 1 by force
      from this obtain f where 4: "∀b∈B . (b,f b) ∈ p" and 5: "C = ⋃(f ` B)"
        by auto
      hence 6: "∀b∈B . f b = {b}"
        using assms(1) test by blast
      hence 7: "C = B"
        using 5 by auto
      hence "(a,C) ∈ R"
        using 3 by auto
      thus "(a,C) ∈ R ∧ (∀b∈C . (b,{b}) ∈ p)"
        using 4 6 7 by fastforce
    qed
    show ?thesis
      apply (rule exI[of _ C])
      apply (rule conjI)
      using 2 apply simp
      apply (rule exI[of _ "λx . {x}"])
      apply (rule conjI)
      using 2 assms(2) apply blast
      by simp
  qed
  done

lemma test_iU_is_iI:
  assumes "∀i∈I . test (X i)"
    and "I ≠ {}"
  shows "⋃⋃X|I = ⋂⋂X|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  subgoal for a f proof -
    assume 1: "∀i∈I . (a,f i) ∈ X i"
    hence "∀i∈I . f i = {a}"
      using assms(1) by (meson test)
    hence "⋃(f ` I) = ⋂(f ` I) ∧ (∀i∈I . (a,f i) ∈ X i)"
      using 1 assms(2) by auto
    thus ?thesis
      by meson
  qed
  apply (clarsimp simp: mr_simp)
  by (metis (no_types, opaque_lifting) INF_eq_const SUP_eq_const assms test)

lemma test_iU_is_oI:
  assumes "∀i∈I . test (X i)"
    and "I ≠ {}"
  shows "⋃⋃X|I = ⋂(X ` I)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply (metis (no_types, opaque_lifting) SUP_eq_const assms test)
  apply (clarsimp simp: mr_simp)
  by (metis UN_constant assms(2))

subsection ‹Domain and antidomain›

declare Dom_def [mr_simp]

abbreviation aDom :: "('a,'b) mrel ⇒ ('a,'a) mrel" where
  "aDom R ≡ ≀ Dom R"

lemma ad_set: "aDom R = {(a,{a})| a. ¬(∃A. (a,A) ∈ R)}"
  by (clarsimp simp: mr_simp) force

lemma d_test:
  "test (Dom R)"
  unfolding Dom_def using s_id_def by fastforce

lemma ad_test:
  "test (aDom R)"
  by simp

lemma ad_expl:
  "aDom R = -((R * 1⇩∪⇩∪) ∪∪ 1) ∩ 1"
  by (simp add: d_def_expl)

lemma ad_expl_2:
  "aDom (R::('a,'b) mrel) = -((R * (1⇩∪⇩∪::('b,'a) mrel))↑) ∩ (1::('a,'a) mrel)"
proof
  have "-((R * 1⇩∪⇩∪)↑) ∩ 1 = -((R * 1⇩∪⇩∪) ∪∪ U) ∩ 1"
    by simp
  also have "… ⊆ -((R * 1⇩∪⇩∪) ∪∪ 1) ∩ 1"
    by (metis c6 convex_increasing iu_commute iu_isotone iu_unit sp_unit_convex test_complement_antitone upper_iu_increasing)
  also have "… = aDom R"
    by (simp add: d_def_expl)
  finally show "-((R * 1⇩∪⇩∪)↑) ∩ 1 ⊆ aDom R"
    using ‹≀ ((R::('a::type × 'b::type set) set) * 1⇩∪⇩∪)↑ ⊆ ≀ (R * 1⇩∪⇩∪ ∪∪ 1)› by blast
  have "aDom R = {(a,{a}) |a. ¬(∃B. (a,B) ∈ R)}"
    by (simp add: ad_set)
  also have "… ⊆ {(a,{a}) |a. (a,{a}) ∉ (R * (p_id::('b,'a) mrel))↑}"
    by (simp add: U_par_st domain_pointwise)
  also have "… = {(a,{a}) |a. (a,{a}) ∈ -(R * (p_id::('b,'a) mrel))↑ ∩ 1}"
    using test_complement by fastforce
  finally show "aDom R ⊆ -((R * (1⇩∪⇩∪::('b,'a) mrel))↑) ∩ (1::('a,'a) mrel)"
    by blast
qed

lemma aDom:
  "aDom R = { (a,{a}) | a . ¬(∃B . (a,B) ∈ R) }"
  by (simp add: ad_set)

declare aDom [mr_simp]

lemma d_down_oi_up_1:
  "Dom (R↓ ∩ S) = Dom (R ∩ S↑)"
  by (metis Int_commute d_def_expl domain_up_down_conjugate)

lemma d_down_oi_up_2:
  "Dom (R↓ ∩ S) = Dom (R↓ ∩ S↑)"
  by (simp add: d_down_oi_up_1)

lemma d_ne_down_dp_complement_test:
  assumes "test p"
  shows "Dom (R ∩ -(U * p)) = Dom (ne (R↓) * ≀ p)"
  by (simp add: assms d_down_oi_up_1 oc_top_sp_test_0 oc_top_sp_test_1 sp_test_dist_oi_right)

lemma d_strict:
  "R = {} ⟷ Dom R = {}"
  using Dom_def by fastforce

lemma d_sp_strict:
  "R * S = {} ⟷ R * Dom S = {}"
  apply (clarsimp simp: mr_simp)
  apply safe
   apply metis
  by (metis UN_singleton)

lemma d_complement_ad:
  "Dom R = ≀ aDom R"
  using d_test by blast

lemma down_sp_below_iu_unit:
  "R↓ * S ⊆ 1⇩∪⇩∪ ⟷ R ⊆ U * aDom (ne S)"
proof -
  have "R↓ * S ⊆ 1⇩∪⇩∪ ⟷ ne (R↓ * S) = {}"
    by (simp add: disjoint_eq_subset_Compl)
  also have "... ⟷ ne (R↓) * ne S = {}"
    by (simp add: ne_dist_down_sp)
  also have "... ⟷ ne (R↓) * Dom (ne S) = {}"
    using d_sp_strict by auto
  also have "... ⟷ ne (R↓) * ≀ aDom (ne S) = {}"
    by (metis d_complement_ad)
  also have "... ⟷ R ⊆ U * aDom (ne S)"
    by (metis top_sp_test_down_iff_1 top_sp_test_down_iff_2 top_sp_test_down_iff_3 ad_test subset_empty)
  finally show ?thesis
    .
qed

lemma ad_sp_bot:
  "aDom R * R = {}"
  by (simp add: d_s_id_ax d_sp_strict inf_sup_aci(1) sp_test_dist_oi_left)

lemma sp_top_d:
  "R * U ⊆ Dom R * U"
  by (simp add: cl8_var iu_unit_up sp_upper_left_isotone)

lemma d_sp_top:
  "Dom (R * U) = Dom R"
  by (clarsimp simp: mr_simp) blast

lemma d_down:
  "Dom (R↓) = Dom R"
  by (metis U_par_idem d_down_oi_up_1 inf.orderE top_down top_lower_greatest)

lemma d_up:
  "Dom (R↑) = Dom R"
  by (metis Int_absorb1 U_par_idem d_down_oi_up_1 top_down top_upper_least)

lemma d_isotone:
  "R ⊆ S ⟹ Dom R ⊆ Dom S"
  unfolding Dom_def by blast

lemma ad_antitone:
  "R ⊆ S ⟹ aDom S ⊆ aDom R"
  by (simp add: Int_commute d_isotone semilattice_inf_class.le_infI2)

lemma d_dist_ou:
  "Dom (R ∪ S) = Dom R ∪ Dom S"
  unfolding Dom_def by blast

lemma d_dist_iu:
  "Dom (R ∪∪ S) = Dom R * Dom S"
  by (clarsimp simp: mr_simp) auto

lemma d_dist_ii:
  "Dom (R ∩∩ S) = Dom R * Dom S"
  by (metis antisym_conv d_U d_dist_iu d_down d_isotone ii_convex_iu s_prod_idr)

lemma d_loc:
  "Dom (R * Dom S) = Dom (R * S)"
  apply (clarsimp simp: mr_simp)
  apply safe
   apply metis
  by (metis UN_singleton)

lemma ad_loc:
  "aDom (R * Dom S) = aDom (R * S)"
  by simp

lemma d_ne_down:
  "Dom (ne (R↓)) = Dom (ne R)"
  by (metis atoms_solution d_down_oi_up_1 d_down_oi_up_2)

lemma ne_sp_iu_unit_up:
  "ne R = R ⟹ (R * 1⇩∪⇩∪)↑ = R * U"
  apply (clarsimp simp: mr_simp)
  apply safe
   apply (metis (no_types, lifting) Compl_iff IntE Inter_iff UNIV_I UN_simps(2) image_eqI singletonI)
  apply clarsimp
  by (metis SUP_bot sup_bot_left)

lemma ne_d_expl:
  "ne R = R ⟹ Dom R = R * U ∩ 1"
  by (metis cl8_var d_def_expl d_test ne_sp_iu_unit_up test_sp)

lemma ne_a_expl:
  "ne R = R ⟹ aDom R = -(R * U) ∩ 1"
  by (simp add: ad_expl_2 ne_sp_iu_unit_up)

lemma d_dist_oU:
  "Dom (⋃X) = ⋃(Dom ` X)"
  apply (clarsimp simp: mr_simp)
  by blast

lemma d_dist_iU_iI:
  "Dom (⋃⋃X|I) = Dom (⋂⋂X|I)"
  by (clarsimp simp: mr_simp)

lemma d_dist_iU_oI:
  assumes "I ≠ {}"
  shows "Dom (⋃⋃X|I) = ⋂(Dom ` X ` I)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply blast
  apply (clarsimp simp: mr_simp)
  by (meson all_not_in_conv assms)

subsection ‹Left residual›

definition sp_lres :: "('a,'c) mrel ⇒ ('b,'c) mrel ⇒ ('a,'b) mrel" (infixl ‹⊘› 65) where
  "Q ⊘ R ≡ { (a,B) . ∀f . (∀b ∈ B . (b,f b) ∈ R) ⟶ (a,⋃{ f b | b . b ∈ B }) ∈ Q }"

declare sp_lres_def [mr_simp]

lemma sp_lres_galois:
  "S * R ⊆ Q ⟷ S ⊆ Q ⊘ R"
proof
  assume 1: "S * R ⊆ Q"
  show "S ⊆ Q ⊘ R"
  proof
    fix x
    assume "x ∈ S"
    from this obtain a B where 2: "x = (a,B) ∧ (a,B) ∈ S"
      by (metis surj_pair)
    have "∀f . (∀b ∈ B . (b,f b) ∈ R) ⟶ (a,⋃{ f b | b . b ∈ B }) ∈ Q"
    proof (rule allI, rule impI)
      fix f
      assume "∀b ∈ B . (b,f b) ∈ R"
      hence "(a,⋃{ f b | b . b ∈ B }) ∈ S * R"
        apply (unfold s_prod_def)
        using 2 by auto
      thus "(a,⋃{ f b | b . b ∈ B }) ∈ Q"
        using 1 by auto
    qed
    thus "x ∈ Q ⊘ R"
      using 2 sp_lres_def by auto
  qed
next
  assume 3: "S ⊆ Q ⊘ R"
  have "(Q ⊘ R) * R ⊆ Q"
  proof
    fix x
    assume "x ∈ (Q ⊘ R) * R"
    from this obtain a D where 4: "x = (a,D) ∧ (a,D) ∈ (Q ⊘ R) * R"
      by (metis surj_pair)
    from this obtain C where "(a,C) ∈ Q ⊘ R ∧ (∃g . (∀c∈C . (c,g c) ∈ R) ∧ D = ⋃{ g c | c . c ∈ C })"
      by (simp add: mr_simp) blast
    thus "x ∈ Q"
      apply (unfold sp_lres_def)
      using 4 by auto
  qed
  thus "S * R ⊆ Q"
    using 3 by (meson dual_order.trans s_prod_isol)
qed

lemma sp_lres_expl:
  "Q ⊘ R = ⋃{ S . S * R ⊆ Q }"
  using sp_lres_galois by blast

lemma bot_sp_lres_d:
  "{} ⊘ R = {} ⊘ Dom R"
  by (metis d_sp_strict order_refl sp_lres_galois subset_antisym subset_empty)

lemma bot_sp_lres_expl:
  "{} ⊘ R = -(U * Dom R)"
  apply (rule antisym)
   apply (metis d_sp_strict d_test disjoint_eq_subset_Compl order_refl sp_lres_galois sp_test subset_empty)
  by (metis Compl_disjoint2 bot_sp_lres_d d_test sp_lres_galois sp_test subset_empty)

lemma sp_lres_sp_below:
  "(Q ⊘ R) * R ⊆ Q"
  by (simp add: sp_lres_galois)

lemma sp_lres_left_isotone:
  "Q ⊆ S ⟹ Q ⊘ R ⊆ S ⊘ R"
  by (meson dual_order.refl sp_lres_galois subset_trans)

lemma sp_lres_right_antitone:
  "S ⊆ R ⟹ Q ⊘ R ⊆ Q ⊘ S"
  by (meson dual_order.trans s_prod_isor sp_lres_galois sp_lres_sp_below)

lemma sp_lres_down_closed_1:
  "Q↓ ⊘ R = Q↓ ⊘ R↓"
proof
  show "Q↓ ⊘ R ⊆ Q↓ ⊘ R↓"
    by (metis down_dist_sp down_idempotent down_isotone sp_lres_galois sp_lres_sp_below)
next
  show "Q↓ ⊘ R↓ ⊆ Q↓ ⊘ R"
    by (simp add: lower_reflexive sp_lres_right_antitone)
qed

lemma sp_lres_down_closed_2:
  assumes "R↓ = R"
    and "total T"
  shows "(R ⊘ T)↓ = R ⊘ T"
proof -
  have "(R ⊘ T)↓ ⊆ R ⊘ T"
    by (metis assms lower_transitive sp_lres_galois sp_lres_sp_below total_down_sp_semi_commute)
  thus ?thesis
    by (simp add: lower_reflexive subset_antisym)
qed

lemma down_sp_sp:
  "R↓ * S = R * (1⇩∪⇩∪ ∪ S)"
proof -
  have "R↓ * S = R * (1⇩∪⇩∪ ∪ 1) * S"
    by (simp add: down_sp)
  also have "... = R * ((1⇩∪⇩∪ ∪ 1) * S)"
    by (simp add: test_iu_test_sp_assoc_3)
  also have "... = R * (1⇩∪⇩∪ ∪ S)"
    apply (clarsimp simp: mr_simp)
    apply safe
     apply (smt (z3) Sup_empty ccpo_Sup_singleton image_empty image_insert singletonI)
    by (smt (verit, del_insts) Sup_empty all_not_in_conv ccpo_Sup_singleton image_insert image_is_empty singletonD)
  finally show ?thesis
    .
qed

lemma iu_unit_sp_lres_iu_unit_ou:
  "U * aDom (ne R) = 1⇩∪⇩∪ ⊘ (1⇩∪⇩∪ ∪ R)"
  apply (rule antisym)
   apply (metis down_sp_sp sp_lres_galois down_sp_below_iu_unit order_refl)
  by (metis down_sp_sp sp_lres_galois down_sp_below_iu_unit order_refl)

lemma bot_sl_below_complement_d:
  "{} ⊘ R ⊆ - Dom R"
  by (metis Compl_anti_mono bot_sp_lres_expl d_test dual_order.refl inf.order_iff sp_test test_s_prod_is_meet)

lemma sp_unit_oi_bot_sp_lres:
  "1 ∩ - Dom R = 1 ∩ ({} ⊘ R)"
  by (smt (verit, ccfv_SIG) ad_sp_bot boolean_algebra_cancel.inf1 bot_sl_below_complement_d inf.orderE inf_bot_right inf_commute inf_le2 sp_lres_galois)

lemma ad_explicit_d:
  "aDom R = -(U * Dom R) ∩ 1"
  by (simp add: bot_sp_lres_expl lattice_class.inf_sup_aci(1) sp_unit_oi_bot_sp_lres)

lemma top_test_sp_lres_total_expl_1:
  assumes "test p"
  shows "∀S . S↓ ⊆ (U * p) ⊘ R ⟷ S ⊆ U * aDom (R ∩ -(U * p))"
proof
  fix S::"('b,'c) mrel"
  have "S ⊆ U * aDom (R ∩ -(U * p)) ⟷ ne (S↓) * Dom (R ∩ -(U * p)) = {}"
    by (metis (no_types, lifting) d_complement_ad inf_le2 subset_empty top_sp_test_down_iff_1 top_sp_test_down_iff_2 top_sp_test_down_iff_3)
  also have "... ⟷ ne (S↓) * Dom (ne (R↓) * ≀ p) = {}"
    by (simp add: assms d_ne_down_dp_complement_test)
  also have "... ⟷ ne (S↓) * (ne (R↓) * ≀ p) = {}"
    using d_sp_strict by auto
  also have "... ⟷ ne (S↓) * ne (R↓) * ≀ p = {}"
    by (simp add: test_assoc3)
  also have "... ⟷ ne ((S↓ * R)↓) * ≀ p = {}"
    by (simp add: down_dist_sp ne_dist_down_sp)
  also have "... ⟷ S↓ * R ⊆ U * p"
    by (metis assms top_sp_test_down_iff_1 top_sp_test_down_iff_2 top_sp_test_down_iff_3 subset_empty)
  also have "... ⟷ S↓ ⊆ (U * p) ⊘ R"
    by (simp add: sp_lres_galois)
  finally show "S↓ ⊆ (U * p) ⊘ R ⟷ S ⊆ U * aDom (R ∩ -(U * p))"
    by simp
qed

lemma top_test_sp_lres_total_expl_2:
  assumes "test p"
    and "total T"
  shows "(U * p) ⊘ T = U * aDom (T ∩ -(U * p))"
proof -
  have "∀S . S ⊆ (U * p) ⊘ T ⟷ S ⊆ U * aDom (T ∩ -(U * p))"
    by (smt (z3) assms lower_reflexive sp_lres_down_closed_2 subset_trans top_sp_test_down_closed top_test_sp_lres_total_expl_1)
  thus ?thesis
    by blast
qed

lemma top_test_sp_lres_total_expl_3:
  assumes "test p"
  shows "((U * p) ⊘ R) ∩ 1 = aDom (R ∩ -(U * p))"
proof
  have "(((U * p) ⊘ R) ∩ 1)↓ * R = (((U * p) ⊘ R) ∩ 1) * (1⇩∪⇩∪ ∪ R)"
    using down_sp_sp by blast
  also have "... = (((U * p) ⊘ R) ∩ 1) * 1⇩∪⇩∪ ∪ (((U * p) ⊘ R) ∩ 1) * R"
    by (simp add: s_distl_test)
  also have "... ⊆ 1⇩∪⇩∪ ∪ (((U * p) ⊘ R) ∩ 1) * R"
    using c6 by auto
  also have "... ⊆ 1⇩∪⇩∪ ∪ ((U * p) ⊘ R) * R"
    by (metis Un_Int_eq(4) inf_le2 iu_unit_convex iu_unit_down sp_oi_subdist sup.mono)
  also have "... ⊆ 1⇩∪⇩∪ ∪ U * p"
    using sp_lres_sp_below by auto
  also have "... = U * p"
    by (simp add: iu_unit_below_top_sp_test sup.absorb2)
  finally have "(((U * p) ⊘ R) ∩ 1)↓ ⊆ (U * p) ⊘ R"
    by (simp add: sp_lres_galois)
  hence "((U * p) ⊘ R) ∩ 1 ⊆ U * aDom (R ∩ -(U * p))"
    by (metis assms top_test_sp_lres_total_expl_1)
  thus "((U * p) ⊘ R) ∩ 1 ⊆ aDom (R ∩ -(U * p))"
    by (metis (no_types, lifting) inf.idem inf.orderE inf_commute sp_oi_subdist sp_test test_s_prod_is_meet)
next
  have "aDom (R ∩ -(U * p)) = U * aDom (R ∩ -(U * p)) ∩ 1"
    by (metis (no_types, lifting) inf.absorb_iff2 inf.idem inf_commute inf_le2 sp_test test_s_prod_is_meet)
  thus "aDom (R ∩ -(U * p)) ⊆ ((U * p) ⊘ R) ∩ 1"
    by (metis Int_mono ad_test assms order_refl top_sp_test_down_closed top_test_sp_lres_total_expl_1)
qed

lemma top_test_sp_lres_total_expl_4:
  assumes "test p"
  shows "aDom (ne (R↓) * ≀ p) = ((U * p) ⊘ R) ∩ 1"
  by (simp add: assms d_ne_down_dp_complement_test top_test_sp_lres_total_expl_3)

lemma oi_complement_top_sp_test_top_1:
  assumes "test p"
  shows "(R ∩ -(U * p)) * U = (R↓ ∩ -(U * p)) * U"
proof (rule antisym)
  show "(R ∩ -(U * p)) * U ⊆ (R↓ ∩ -(U * p)) * U"
    by (metis (no_types, lifting) assms cl8_var d_down_oi_up_1 equalityD2 ne_oi_complement_top_sp_test_1 ne_sp_iu_unit_up oc_top_sp_test_up_closed)
next
  have "R↓ ∩ -(U * p) ⊆ (R ∩ -(U * p))↓"
    by (metis oc_top_sp_test_up_closed down_oi_up_closed assms)
  also have "... ⊆ (R ∩ -(U * p)) * U"
    by (simp add: down_below_sp_top)
  finally show "(R↓ ∩ -(U * p)) * U ⊆ (R ∩ -(U * p)) * U"
    by (metis assms domain_up_down_conjugate inf_commute ne_oi_complement_top_sp_test_1 ne_sp_iu_unit_up oc_top_sp_test_up_closed set_eq_subset)
qed

lemma oi_complement_top_sp_test_top_2:
  assumes "test p"
  shows "(R↓ ∩ -(U * p)) * U = ne (R↓) * ≀ p * U"
proof (rule antisym)
  have "R↓ ∩ -(U * p) * U ⊆ Dom (R↓ ∩ -(U * p)) * U"
    using sp_top_d by blast
  also have "... = Dom (ne (R↓) * ≀ p) * U"
    by (simp add: assms d_ne_down_dp_complement_test)
  also have "... = Dom (ne (R↓ * ≀ p)) * U"
    by (simp add: ne_sp_test)
  also have "... = ne (R↓ * ≀ p) * U"
    by (simp add: cl8_var ne_sp_iu_unit_up)
  also have "... = ne (R↓) * ≀ p * U"
    by (simp add: ne_sp_test)
  finally show "(R↓ ∩ -(U * p)) * U ⊆ ne (R↓) * ≀ p * U"
    .
next
  have "ne (R↓) * ≀ p ⊆ -(U * p)"
    by (metis assms disjoint_eq_subset_Compl double_complement inf_compl_bot_right schroeder_test sp_test test_complement_closed top_test_oi_top_complement)
  thus "ne (R↓) * ≀ p * U ⊆ (R↓ ∩ -(U * p)) * U"
    by (metis (no_types) assms cl8_var d_down_oi_up_1 d_ne_down_dp_complement_test ne_oi_complement_top_sp_test_1 ne_sp_iu_unit_up oc_top_sp_test_up_closed sp_top_d)
qed

lemma oi_complement_top_sp_test_top_3:
  assumes "test p"
  shows "(R↓ ∩ -(U * p)) * U = ne (R↓) * -(p * U)"
  by (simp add: assms complement_test_sp_top oi_complement_top_sp_test_top_2 test_assoc1)

lemma split_sp_test_2:
  "test p ⟹ R ⊆ R * p ∪ ne (R↓) * (≀ p)↑"
proof -
  assume "test p"
  hence "R ⊆ R * p ∪ (ne (R↓ * ≀ p))↑"
    by (smt (verit, best) IntE UnCI UnE split_sp_test subsetI)
  thus "R ⊆ R * p ∪ ne (R↓) * (≀ p)↑"
    by (simp add: ne_sp_test_up)
qed

lemma split_sp_test_3:
  "test p ⟹ R ⊆ R * p ∪ R↓ * (≀ p)↑"
  by (smt (z3) IntE UnCI UnE ne_dist_down_sp ne_sp_test_up ne_test_up split_sp_test subsetI)

lemma split_sp_test_4:
  assumes "test p"
    and "test q"
  shows "R * (p ∪ q) ⊆ R * p ∪ ne (R↓) * q↑"
proof -
  have 1: "(p ∪ q) * p = p"
    by (metis Un_Int_eq(1) assms sp_test_dist_oi_left test_ou_closed test_sp_commute test_sp_idempotent)
  have "(R * (p ∪ q))↓ * ≀ p ⊆ R↓ * q"
  proof -
    have "(R * (p ∪ q))↓ * ≀ p = R * (p ∪ q) * (1⇩∪⇩∪ ∪ ≀ p)"
      using down_sp_sp by blast
    also have "... = R * ((p ∪ q) * 1⇩∪⇩∪ ∪ (p ∪ q) * ≀ p)"
      by (smt (verit) assms inf.orderE s_distl_test test_assoc1 test_ou_closed)
    also have "... ⊆ R * (1⇩∪⇩∪ ∪ (p ∪ q) * ≀ p)"
      by (meson c6 s_prod_isor subset_refl sup.mono)
    also have "... = R↓ * ((p ∪ q) * ≀ p)"
      using down_sp_sp by blast
    also have "... = R↓ * (p * ≀ p ∪ q * ≀ p)"
      by (metis assms ii_right_dist_ou inf_commute inf_le1 test_ou_closed test_sp_is_ii)
    also have "... = R↓ * (q * ≀ p)"
      by (metis Compl_disjoint assms(1) inf_commute inf_le1 s_prod_idl sp_test_dist_oi_left subset_Un_eq test_sp_commute)
    also have "... ⊆ R↓ * q"
      by (metis inf_le1 inf_le2 s_prod_isor sp_test)
    finally show ?thesis
      .
  qed
  hence 2: "ne ((R * (p ∪ q))↓) * ≀ p ⊆ ne (R↓) * q"
    by (metis assms(2) inf_le2 ne_dist_ou ne_sp_test subset_Un_eq)
  have "R * (p ∪ q) ⊆ R * (p ∪ q) * p ∪ ne ((R * (p ∪ q))↓) * (≀ p)↑"
    by (simp add: assms split_sp_test_2)
  also have "... = R * p ∪ ne ((R * (p ∪ q))↓) * (≀ p)↑"
    using 1 by (metis assms(1) inf.orderE test_assoc3)
  also have "... ⊆ R * p ∪ ne (R↓) * q↑"
    using 2 by (metis (no_types, lifting) Un_Int_eq(1) assms(2) inf_le2 iu_isotone ne_sp_test ne_sp_test_up sup.mono)
  finally show ?thesis
    .
qed

lemma split_sp_test_5:
  assumes "test p"
    and "test q"
  shows "R * (p ∪ q) ⊆ R * p ∪ R↓ * q↑"
proof -
  have "R * (p ∪ q) ⊆ R * p ∪ ne (R↓) * q↑"
    by (simp add: assms split_sp_test_4)
  thus ?thesis
    by (metis (no_types, lifting) assms(2) le_inf_iff ne_dist_down_sp ne_test_up sup_neg_inf)
qed

lemma split_sp_test_6:
  assumes "test p"
    and "test q"
  shows "Dom (R * (p ∪ q)) ⊆ Dom (R * p ∪ ne (R↓) * q)"
proof -
  have "Dom (R * (p ∪ q)) ⊆ Dom (R * p ∪ ne (R↓) * q↑)"
    by (simp add: assms d_isotone split_sp_test_4)
  also have "... = Dom (R * p ∪ (ne (R↓) * q)↑)"
    by (simp add: assms(2) ne_sp_test ne_sp_test_up)
  also have "... ⊆ Dom (R * p ∪ ne (R↓) * q)"
    by (metis d_up subsetI up_dist_ou up_idempotent)
  finally show ?thesis
    .
qed

lemma split_sp_test_7:
  assumes "test p"
    and "test q"
  shows "Dom (ne (R↓) * (p ∪ q)) = Dom (ne (R↓) * p ∪ ne (R↓) * q)"
  apply (rule antisym)
   apply (metis assms ne_down_idempotent split_sp_test_6)
  by (smt (verit, ccfv_SIG) Un_Int_eq(1) Un_subset_iff assms(2) d_isotone inf.orderE inf_le1 le_inf_iff lower_eq_down ne_dist_ou sp_oi_subdist_2 subset_Un_eq sup.mono)

lemma test_sp_left_dist_iu_1:
  "test p ⟹ p * (R ∪∪ S) = p * R ∪∪ S"
  by (metis cl8_var inf.orderE p_prod_assoc s_subid_iff2)

lemma test_sp_left_dist_iu_2:
  "test p ⟹ p * (R ∪∪ S) = R ∪∪ p * S"
  by (metis iu_commute test_sp_left_dist_iu_1)

lemma d_sp_below_iu_down:
  "Dom R * S ⊆ (R ∪∪ S)↓"
  by (simp add: cl8_var iu_lower_left_isotone sp_iu_unit_lower)

lemma d_sp_ne_down_below_ne_iu_down:
  "Dom R * ne (S↓) ⊆ ne ((R ∪∪ S)↓)"
proof -
  have "Dom R * S↓ ⊆ (R ∪∪ S)↓"
    by (simp add: cl8_var iu_lower_isotone sp_iu_unit_lower)
  hence "ne (Dom R * S↓) ⊆ ne ((R ∪∪ S)↓)"
    by blast
  thus ?thesis
    by (smt (z3) d_test test_sp_ne)
qed

lemma top_test:
  "test p ⟹ U * p = { (a,B) . (∀b∈B . (b,{b}) ∈ p) }"
  apply (unfold test)
  apply (clarsimp simp: mr_simp)
  by fastforce

lemma iu_oi_complement_top_test_ou_up:
  "test p ⟹ (R ∪∪ S) ∩ -(U * p) ⊆ ((R ∪ S) ∩ -(U * p))↑"
  apply (unfold top_test)
  apply (clarsimp simp: mr_simp)
  by blast

lemma d_ne_iu_down_sp_test_ou:
  assumes "test p"
  shows "Dom (ne ((R ∪∪ S)↓) * p) ⊆ Dom ((ne (R↓) ∪ ne (S↓)) * p)"
proof -
  have "Dom (ne ((R ∪∪ S)↓) * p) = Dom ((R ∪∪ S) ∩ -(U * ≀ p))"
    by (metis assms d_ne_down_dp_complement_test test_double_complement)
  also have "... ⊆ Dom ((R ∪ S) ∩ -(U * ≀ p))"
    by (metis iu_oi_complement_top_test_ou_up d_isotone d_up semilattice_inf_class.inf_le2)
  also have "... = Dom (ne ((R ∪ S)↓) * p)"
    by (metis assms d_ne_down_dp_complement_test test_double_complement)
  finally show ?thesis
    by (simp add: down_dist_ou ne_dist_ou)
qed

lemma test_sp_left_dist_iU:
  assumes "test p"
    and "I ≠ {}"
  shows "p * (⋃⋃X|I) = ⋃⋃(λi . p * X i)|I"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  subgoal for a B f proof -
    assume 1: "(a,B) ∈ p"
    hence 2: "B = {a}"
      by (metis assms(1) test)
    assume "∀b∈B . ∃g . f b = ⋃(g ` I) ∧ (∀i∈I . (b,g i) ∈ X i)"
    from this obtain g where 3: "f a = ⋃(g ` I) ∧ (∀i∈I . (a,g i) ∈ X i)"
      using 2 by auto
    have "⋃(f ` B) = ⋃(g ` I) ∧ (∀i∈I . ∃B . (a,B) ∈ p ∧ (∃f . (∀b∈B . (b,f b) ∈ X i) ∧ g i = ⋃(f ` B)))"
      apply (rule conjI)
      using 2 3 apply blast
      apply (rule ballI)
      apply (rule exI[of _ B])
      apply (rule conjI)
      using 1 apply simp
      subgoal for i
        apply (rule exI[of _ "λb . g i"])
        using 2 3 by blast
      done
    thus ?thesis
      by auto
  qed
  apply (clarsimp simp: mr_simp)
  subgoal for a f proof -
    assume 4: "∀i∈I . ∃B . (a,B) ∈ p ∧ (∃g . (∀b∈B . (b,g b) ∈ X i) ∧ f i = ⋃(g ` B))"
    have "(a,{a}) ∈ p ∧ (∃g . (∀b∈{a} . ∃f . g b = ⋃(f ` I) ∧ (∀i∈I . (b,f i) ∈ X i)) ∧ ⋃(f ` I) = ⋃(g ` {a}))"
      apply (rule conjI)
      using 4 apply (metis assms equals0I test)
      apply (rule exI[of _ "λa . ⋃(f ` I)"])
      apply clarsimp
      apply (rule exI[of _ f])
      using 4 assms(1) test by fastforce
    thus ?thesis
      by auto
  qed
  done

subsection ‹Modal operations›

definition adia :: "('a,'b) mrel ⇒ ('b,'b) mrel ⇒ ('a,'a) mrel" (‹| _ ⟩ _› [50,90] 95) where
  "|R⟩p ≡ { (a,{a}) | a . ∃B . (a,B) ∈ R ∧ (∀b∈B . (b,{b}) ∈ p) }"

definition abox :: "('a,'b) mrel ⇒ ('b,'b) mrel ⇒ ('a,'a) mrel" (‹| _ ] _› [50,90] 95) where
  "|R]p ≡ { (a,{a}) | a . ∀B . (a,B) ∈ R ⟶ (∀b∈B . (b,{b}) ∈ p) }"

definition edia :: "('a,'b) mrel ⇒ ('b,'b) mrel ⇒ ('a,'a) mrel" (‹| _ ⟩⟩ _› [50,90] 95) where
  "|R⟩⟩p ≡ { (a,{a}) | a . ∃B . (a,B) ∈ R ∧ (∃b∈B . (b,{b}) ∈ p) }"

definition ebox :: "('a,'b) mrel ⇒ ('b,'b) mrel ⇒ ('a,'a) mrel" (‹| _ ]] _› [50,90] 95) where
  "|R]]p ≡ { (a,{a}) | a . ∀B . (a,B) ∈ R ⟶ (∃b∈B . (b,{b}) ∈ p) }"

declare adia_def [mr_simp] abox_def [mr_simp] edia_def [mr_simp] ebox_def [mr_simp]

lemma adia:
  assumes "test p"
  shows "|R⟩p = Dom (R * p)"
proof
  show "|R⟩p ⊆ Dom (R * p)"
  proof
    fix x
    assume "x ∈ |R⟩p"
    from this obtain a B where 1: "x = (a,{a}) ∧ (a,B) ∈ R ∧ (∀b∈B . (b,{b}) ∈ p)"
      by (smt (z3) adia_def surj_pair mem_Collect_eq)
    have "(a,B) ∈ R * p"
      apply (clarsimp simp: s_prod_def)
      apply (rule exI[where ?x="B"])
      apply (rule conjI)
      using 1 apply simp
      apply (rule exI[where ?x="λx . {x}"])
      using 1 by auto
    thus "x ∈ Dom (R * p)"
      using 1 Dom_def by auto
  qed
next
  show "Dom (R * p) ⊆ |R⟩p"
  proof
    fix x
    assume "x ∈ Dom (R * p)"
    from this obtain a A where 2: "x = (a,{a}) ∧ (a,A) ∈ R * p"
      by (smt (z3) Dom_def surj_pair mem_Collect_eq)
    from this obtain B f where 3: "(a,B) ∈ R ∧ (∀b∈B . (b,f b) ∈ p) ∧ A = ⋃{ f b | b . b ∈ B }"
      by (simp add: mr_simp) blast
    hence "∀b∈B . (b,{b}) ∈ p"
      using assms subid_aux2 by fastforce
    thus "x ∈ |R⟩p"
      using 2 3 adia_def by blast
  qed
qed

lemma abox_1:
  assumes "test p"
  shows "|R]p = aDom (R ∩ -(U * p))"
proof
  show "|R]p ⊆ aDom (R ∩ -(U * p))"
  proof
    fix x
    assume "x ∈ |R]p"
    from this obtain a where 1: "x = (a,{a}) ∧ (∀B . (a,B) ∈ R ⟶ (∀b∈B . (b,{b}) ∈ p))"
      by (smt (z3) abox_def surj_pair mem_Collect_eq)
    have "¬(∃B . (a,B) ∈ R ∩ -(U * p))"
    proof
      assume "∃B . (a,B) ∈ R ∩ -(U * p)"
      from this obtain B where "(a,B) ∈ R ∧ (a,B) ∉ U * p"
        by auto
      thus False
        using 1 by (metis (no_types, lifting) assms top_sp_test)
    qed
    thus "x ∈ aDom (R ∩ -(U * p))"
      using 1 aDom by blast
  qed
next
  show "aDom (R ∩ -(U * p)) ⊆ |R]p"
  proof
    fix x
    assume "x ∈ aDom (R ∩ -(U * p))"
    from this obtain a where 2: "x = (a,{a}) ∧ ¬(∃B . (a,B) ∈ R ∩ -(U * p))"
      by (smt (z3) aDom surj_pair mem_Collect_eq)
    hence "∀B . (a,B) ∈ R ⟶ (∀b∈B . (b,{b}) ∈ p)"
      using assms by (metis (no_types, lifting) IntI oc_top_sp_test)
    thus "x ∈ |R]p"
      using 2 abox_def by blast
  qed
qed

lemma abox:
  assumes "test p"
  shows "|R]p = aDom (ne (R↓) * ≀ p)"
  by (simp add: abox_1 assms d_ne_down_dp_complement_test)

lemma edia_1:
  assumes "test p"
  shows "|R⟩⟩p = Dom (R ∩ -(U * ≀ p))"
proof
  show "|R⟩⟩p ⊆ Dom (R ∩ -(U * ≀ p))"
  proof
    fix x
    assume "x ∈ |R⟩⟩p"
    from this obtain a b B where 1: "x = (a,{a}) ∧ (a,B) ∈ R ∧ b ∈ B ∧ (b,{b}) ∈ p"
      by (smt (z3) edia_def surj_pair mem_Collect_eq)
    hence "(a,B) ∈ -(U * ≀ p)"
      by (metis (no_types, lifting) lattice_class.inf_sup_ord(2) oc_top_sp_test test_complement)
    thus "x ∈ Dom (R ∩ -(U * ≀ p))"
      using 1 Dom_def by auto
  qed
next
  show "Dom (R ∩ -(U * ≀ p)) ⊆ |R⟩⟩p"
  proof
    fix x
    assume "x ∈ Dom (R ∩ -(U * ≀ p))"
    from this obtain a B where 2: "x = (a,{a}) ∧ (a,B) ∈ R ∧ (a,B) ∈ -(U * ≀ p)"
      by (smt (z3) Dom_def surj_pair mem_Collect_eq IntE)
    hence "∃b∈B . (b,{b}) ∈ p"
      by (meson oc_top_sp_test test_complement test_complement_closed)
    thus "x ∈ |R⟩⟩p"
      using 2 edia_def by blast
  qed
qed

lemma edia:
  assumes "test p"
  shows "|R⟩⟩p = Dom (ne (R↓) * p)"
  by (metis assms d_ne_down_dp_complement_test edia_1 test_double_complement)

lemma ebox:
  assumes "test p"
  shows "|R]]p = aDom (R * ≀ p)"
proof
  show "|R]]p ⊆ aDom (R * ≀ p)"
  proof
    fix x
    assume "x ∈ |R]]p"
    from this obtain a where 1: "x = (a,{a}) ∧ (∀B . (a,B) ∈ R ⟶ (∃b∈B . (b,{b}) ∈ p))"
      by (smt (z3) ebox_def surj_pair mem_Collect_eq)
    hence "¬(∃B . (a,B) ∈ R * ≀ p)"
      by (metis (no_types, lifting) s_prod_test test_complement)
    thus "x ∈ aDom (R * ≀ p)"
      using 1 aDom by blast
  qed
next
  show "aDom (R * ≀ p) ⊆ |R]]p"
  proof
    fix x
    assume "x ∈ aDom (R * ≀ p)"
    from this obtain a where 2: "x = (a,{a}) ∧ ¬(∃B . (a,B) ∈ R * ≀ p)"
      by (smt (z3) aDom surj_pair mem_Collect_eq)
    have "∀B . (a,B) ∈ R ⟶ (∃b∈B . (b,{b}) ∈ p)"
    proof (rule allI, rule impI)
      fix B
      assume "(a,B) ∈ R"
      hence "(a,B) ∉ U * ≀ p"
        using 2 by (metis Int_iff Int_lower2 sp_test)
      thus "∃b∈B . (b,{b}) ∈ p"
        by (meson test_complement test_complement_closed top_sp_test)
    qed
    thus "x ∈ |R]]p"
      using 2 ebox_def by blast
  qed
qed

lemma abox_2:
  assumes "test p"
  shows "|R]p = -((R ∩ -(U * p)) * U) ∩ 1"
  by (simp add: abox_1 assms ne_a_expl ne_oi_complement_top_sp_test_1)

lemma abox_3:
  assumes "test p"
  shows "|R]p = -(ne (R↓) * ≀ p * U) ∩ 1"
  by (simp add: abox assms ne_a_expl ne_sp_test)

lemma abox_4:
  assumes "test p"
  shows "|R]p = ((U * p) ⊘ R) ∩ 1"
  by (simp add: abox_1 assms top_test_sp_lres_total_expl_3)

lemma abox_ebox:
  assumes "test p"
  shows "|R]p = |ne (R↓)]]p"
  by (simp add: abox assms ebox)

lemma abox_edia:
  assumes "test p"
  shows "|R]p = ≀ |R⟩⟩(≀ p)"
  by (simp add: abox assms edia)

lemma abox_adia:
  assumes "test p"
  shows "|R]p = ≀ |ne (R↓)⟩(≀ p)"
  by (simp add: abox adia assms)

lemma edia_adia:
  assumes "test p"
  shows "|R⟩⟩p = |ne (R↓)⟩p"
  by (simp add: adia assms edia)

lemma edia_abox:
  assumes "test p"
  shows "|R⟩⟩p = ≀ |R](≀ p)"
  by (metis abox_1 assms d_complement_ad edia_1 semilattice_inf_class.inf.cobounded2)

lemma edia_ebox:
  assumes "test p"
  shows "|R⟩⟩p = ≀ |ne (R↓)]](≀ p)"
  by (simp add: abox assms ebox edia_abox)

lemma abox_ne_down:
  assumes "test p"
  shows "|R]p = |ne (R↓)]p"
  by (simp add: abox assms ne_down_idempotent)

lemma edia_ne_down:
  assumes "test p"
  shows "|R⟩⟩p = |ne (R↓)⟩⟩p"
  by (simp add: assms edia ne_down_idempotent)

lemma adia_up:
  assumes "test p"
  shows "|R⟩p = |R↑⟩p"
proof -
  have "|R↑⟩p = Dom (R↑ ∩ U * p)"
    by (metis adia assms iu_assoc iu_unit_up up_dist_iu_oi)
  also have "... = Dom (R ∩ U * p)"
    by (metis assms d_def_expl domain_up_down_conjugate sp_test_dist_oi_right top_sp_test_down_closed)
  also have "... = |R⟩p"
    by (metis adia assms inf.absorb_iff2 inf_commute top_down top_lower_greatest)
  finally show ?thesis
    by simp
qed

lemma ebox_up:
  assumes "test p"
  shows "|R]]p = |R↑]]p"
  by (metis Int_commute adia adia_up assms ebox semilattice_inf_class.inf_le1)

lemma adia_ebox:
  assumes "test p"
  shows "|R⟩p = ≀ |R]](≀ p)"
  by (metis (no_types, lifting) adia assms d_complement_ad ebox test_double_complement)

lemma ebox_adia:
  assumes "test p"
  shows "|R]]p = ≀ |R⟩(≀ p)"
  by (simp add: adia assms ebox)

lemma abox_down:
  assumes "test p"
  shows "|R]p = |R↓]p"
  by (simp add: abox assms)

lemma edia_down:
  assumes "test p"
  shows "|R⟩⟩p = |R↓⟩⟩p"
  by (simp add: assms edia)

lemma fusion_oi_complement_top_test_up:
  "test p ⟹ fus R ∩ -(U * p) ⊆ (R ∩ -(U * p))↑"
  apply (unfold top_test)
  apply (clarsimp simp: mr_simp)
  by blast

lemma adia_left_isotone:
  "test p ⟹ R ⊆ S ⟹ |R⟩p ⊆ |S⟩p"
  by (metis adia d_isotone inf.absorb_iff1 sp_test_dist_oi)

lemma adia_right_isotone:
  "test p ⟹ test q ⟹ p ⊆ q ⟹ |R⟩p ⊆ |R⟩q"
  by (metis (no_types, opaque_lifting) adia d_isotone inf.orderE inf_commute inf_le1 sp_test test_assoc3 test_s_prod_is_meet)

lemma abox_left_antitone:
  "test p ⟹ R ⊆ S ⟹ |S]p ⊆ |R]p"
  apply (clarsimp simp: mr_simp) by force

lemma abox_right_isotone:
  "test p ⟹ test q ⟹ p ⊆ q ⟹ |R]p ⊆ |R]q"
  by (smt (verit, ccfv_threshold) IntE abox_def inf.orderE mem_Collect_eq subsetI)

lemma edia_left_isotone:
  "test p ⟹ R ⊆ S ⟹ |R⟩⟩p ⊆ |S⟩⟩p"
  by (metis Int_mono adia_left_isotone down_isotone edia_adia order_refl)

lemma edia_right_isotone:
  "test p ⟹ test q ⟹ p ⊆ q ⟹ |R⟩⟩p ⊆ |R⟩⟩q"
  by (simp add: adia_right_isotone edia_adia)

lemma ebox_left_antitone:
  "test p ⟹ R ⊆ S ⟹ |S]]p ⊆ |R]]p"
  by (metis (no_types, lifting) adia_ebox adia_left_isotone ebox_adia test_complement_antitone test_double_complement)

lemma ebox_right_isotone:
  "test p ⟹ test q ⟹ p ⊆ q ⟹ |R]]p ⊆ |R]]q"
  by (smt (verit, ccfv_SIG) adia_ebox adia_right_isotone ebox inf_le2 test_complement_antitone test_double_complement)

lemma edia_fusion:
  assumes "test p"
  shows "|R⟩⟩p = |fus R⟩⟩p"
proof
  have "|fus R⟩⟩p = Dom (fus R ∩ -(U * ≀ p))"
    using assms edia_1 by blast
  also have "... ⊆ Dom (R ∩ -(U * ≀ p))"
    by (metis fusion_oi_complement_top_test_up d_isotone d_up semilattice_inf_class.inf_le2)
  also have "... = |R⟩⟩p"
    using assms edia_1 by blast
  finally show "|fus R⟩⟩p ⊆ |R⟩⟩p"
    .
next
  have "|R⟩⟩p ⊆ |(fus R)↓⟩⟩p"
    by (simp add: assms edia_left_isotone fusion_lower_increasing)
  thus "|R⟩⟩p ⊆ |fus R⟩⟩p"
    using assms edia_down by blast
qed

lemma abox_fusion:
  assumes "test p"
  shows "|R]p = |fus R]p"
  by (metis Int_lower2 abox_edia assms edia_fusion)

lemma abox_fission:
  assumes "test p"
  shows "|R]p = |fis R]p"
  by (metis assms abox_fusion fusion_fission)

lemma edia_fission:
  assumes "test p"
  shows "|R⟩⟩p = |fis R⟩⟩p"
  by (metis assms edia_fusion fusion_fission)

lemma fission_below:
  "fis R ⊆ S ⟷ (∀a b B . (a,B) ∈ R ∧ b ∈ B ⟶ (a,{b}) ∈ S)"
  apply standard
   apply (simp add: basic_trans_rules(31) fission_set)
  apply (clarsimp simp: mr_simp)
  by blast

lemma below_fission_up:
  "S ⊆ (fis R)↑ ⟷ (∀a B . (a,B) ∈ S ⟶ (∃C . (a,C) ∈ R ∧ C ∩ B ≠ {}))"
proof
  assume "S ⊆ (fis R)↑"
  thus "∀a B . (a,B) ∈ S ⟶ (∃C . (a,C) ∈ R ∧ C ∩ B ≠ {})"
    apply (clarsimp simp: mr_simp)
    by fastforce
next
  assume 1: "∀a B . (a,B) ∈ S ⟶ (∃C . (a,C) ∈ R ∧ C ∩ B ≠ {})"
  show "S ⊆ (fis R)↑"
  proof
    fix x
    assume "x ∈ S"
    from this obtain a B where 2: "x = (a,B) ∧ (a,B) ∈ S"
      by (metis surj_pair)
    hence "∃C . (a,C) ∈ R ∧ C ∩ B ≠ {}"
      using 1 by simp
    from this obtain C b where 3: "(a,C) ∈ R ∧ b ∈ C ∧ b ∈ B"
      by auto
    hence "(a,{b}) ∈ fis R"
      using fission_set by blast
    thus "x ∈ (fis R)↑"
      using 2 3 U_par_st by fastforce
  qed
qed

lemma ebox_below_abox:
  assumes "test p"
    and "fis R ⊆ S"
  shows "|S]]p ⊆ |R]p"
  by (metis abox_ebox abox_fission assms ebox_left_antitone fission_down_ne_fixpoint)

lemma abox_below_ebox:
  assumes "test p"
    and "S ⊆ (fis R)↑"
  shows "|R]p ⊆ |S]]p"
  by (metis abox_ebox abox_fission assms ebox_left_antitone ebox_up fission_down_ne_fixpoint)

lemma abox_eq_ebox:
  assumes "test p"
    and "fis R ⊆ S"
    and "S ⊆ (fis R)↑"
  shows "|R]p = |S]]p"
  by (simp add: abox_below_ebox assms ebox_below_abox subset_antisym)

lemma abox_eq_ebox_sufficient:
  "S = fis R ∨ S = ne (R↓) ∨ S = (ne (R↓))↑ ⟶ fis R ⊆ S ∧ S ⊆ (fis R)↑"
  apply (unfold imp_disjL)
  apply (intro conjI)
    apply (simp add: convex_reflexive)
   apply (simp add: fission_inner_deterministic fission_up_ne_down_up oi_subset_upper_right_antitone same_fusion_fission_lower)
  by (metis convex_reflexive fission_up_ne_down_up order_refl)

lemma ebox_fission_abox:
  "test p ⟹ |R]p = |fis R]]p"
  by (metis abox abox_fission ebox fission_down_ne_fixpoint)

lemma ebox_down_ne_up_abox:
  "test p ⟹ |R]p = |(ne (R↓))↑]]p"
  using abox_ebox ebox_up by blast

lemma same_fusion:
  assumes "fis R ⊑↓ S"
    and "S ⊑↓ fus R"
  shows "fis R = fis S"
  by (metis assms fission_down fission_fusion fission_fusion_galois subset_antisym)

lemma same_abox:
  assumes "fis R ⊑↓ S"
    and "S ⊑↓ fus R"
    and "test p"
  shows "|R]p = |S]p"
  by (metis assms ebox_fission_abox same_fusion)

lemma abox_ebox_inner_deterministic:
  assumes "test p"
    and "inner_deterministic R"
  shows "|R]p = |R]]p"
  apply (rule abox_eq_ebox)
    apply (simp add: assms(1))
  using assms(2) fission_inner_deterministic_fixpoint apply blast
  by (metis assms(2) convex_reflexive fission_inner_deterministic_fixpoint)

lemma adia_edia_inner_deterministic:
  assumes "test p"
    and "inner_deterministic R"
  shows "|R⟩p = |R⟩⟩p"
  by (metis assms edia_adia fission_down_ne_fixpoint fission_inner_deterministic_fixpoint)

lemma abox_adia_deterministic:
  assumes "test p"
    and "deterministic R"
  shows "|R]p = |R⟩p"
proof
  show "|R]p ⊆ |R⟩p"
  proof
    fix x
    assume "x ∈ |R]p"
    from this obtain a where 1: "x = (a,{a}) ∧ (∀B . (a,B) ∈ R ⟶ (∀b∈B . (b,{b}) ∈ p))"
      using abox_def by force
    from assms(2) obtain B where "(a,B) ∈ R"
      by (meson deterministic_set)
    thus "x ∈ |R⟩p"
      using 1 adia_def by fastforce
  qed
next
  show "|R⟩p ⊆ |R]p"
  proof
    fix x
    assume "x ∈ |R⟩p"
    from this obtain a B where 2: "x = (a,{a}) ∧ (a,B) ∈ R ∧ (∀b∈B . (b,{b}) ∈ p)"
      by (smt (z3) adia_def mem_Collect_eq)
    have "∀C . (a,C) ∈ R ⟶ (∀b∈C . (b,{b}) ∈ p)"
    proof (rule allI, rule impI)
      fix C
      assume "(a,C) ∈ R"
      hence "B = C"
        using 2 by (metis assms(2) deterministic_set)
      thus "∀b∈C . (b,{b}) ∈ p"
        using 2 by simp
    qed
    thus "x ∈ |R]p"
      using 2 abox_def by blast
  qed
qed

lemma ebox_edia_deterministic:
  assumes "test p"
    and "deterministic R"
  shows "|R]]p = |R⟩⟩p"
  by (simp add: assms abox_adia_deterministic ebox_adia edia_abox)

lemma abox_ebox_fusion:
  assumes "test p"
  shows "|fis R]p = |fis R]]p"
  by (metis abox_fission assms ebox_fission_abox)

lemma abox_fission_edia_fusion:
  assumes "test p"
  shows "|fis R]p = |fus R⟩p"
  by (simp add: abox_adia_deterministic abox_fusion assms fusion_deterministic fusion_fission)

lemma abox_adia_fusion:
  assumes "test p"
  shows "|fus R]p = |fus R⟩p"
  by (simp add: abox_adia_deterministic assms fusion_deterministic)

subsection ‹Goldblatt's axioms without star›

lemma abox_sp_unit:
  "|R]1 = 1"
  apply (clarsimp simp: mr_simp) by force

lemma ou_unit_abox:
  "test p ⟹ |{}]p = 1"
  by (metis abox_1 abox_sp_unit disjoint_eq_subset_Compl empty_subsetI inf.absorb_iff2 test_complement_closed)

lemma ou_unit_test_implication:
  "test p ⟹ {} → p = 1"
  by blast

lemma sp_unit_abox:
  "test p ⟹ |1]p = p"
  by (smt (verit) Int_left_commute abox_1 c1 cl8_var convex_reflexive d_ne_down_dp_complement_test fission_down_ne_fixpoint fission_inner_deterministic_fixpoint inf.absorb_iff2 inf_commute inner_deterministic_sp_unit s_subid_iff2 test_double_complement test_sp)

lemma sp_unit_test_implication:
  "test p ⟹ 1 → p = p"
  by simp

lemma test_abox_ebox:
  "test p ⟹ test q ⟹ |q]p = |q]]p"
  apply (rule antisym)
   apply (metis abox_ebox_inner_deterministic dual_order.trans inner_deterministic_sp_unit subset_refl)
  by (metis abox_ebox_inner_deterministic dual_order.eq_iff inner_deterministic_sp_unit inner_univalent_down_closed ne_equality test_ne)

lemma test_abox:
  "test p ⟹ test q ⟹ |q]p = q → p"
  by (smt (z3) Int_commute Int_lower2 abox cl9_var compl_sup d_complement_ad d_ne_down_dp_complement_test lattice_class.inf_sup_aci(2) sp_unit_abox test_ou_closed)

lemma abox_ou_adia_sp_unit:
  assumes "test p"
  shows "|R]p ∪ |R⟩1 = 1"
  apply (rule antisym)
   apply (simp add: assms abox adia_ebox)
  by (clarsimp simp: mr_simp)

lemma d_test_sp:
  "test p ⟹ Dom (p * R) = p * Dom R"
  by (simp add: c4 d_def_expl test_sp_left_dist_iu_1)

lemma ad_test_sp:
  "test p ⟹ aDom (p * R) = ≀ p ∪ aDom R"
  by (metis (no_types, opaque_lifting) Int_commute boolean_algebra.conj_disj_distrib boolean_algebra.de_Morgan_conj d_s_id_inter d_test_sp s_subid_iff2 test_fix)

lemma adia_test_sp:
  "test p ⟹ test q ⟹ |p * R⟩q = p * |R⟩q"
  by (metis (no_types, lifting) adia d_test_sp test_assoc3 test_double_complement)

lemma ebox_test_sp:
  "test p ⟹ test q ⟹ |p * R]]q = ≀ p ∪ |R]]q"
  by (simp add: ad_test_sp ebox test_assoc3)

lemma abox_test_sp:
  assumes "test p"
    and "test q"
  shows "|p * R]q = ≀ p ∪ |R]q"
proof -
  have "|p * R]q = aDom ((p * R) ∩ -(U * q))"
    by (simp add: abox_1 assms(2))
  also have "... = aDom (p * (R ∩ -(U * q)))"
    by (metis Int_assoc assms(1) test_sp)
  also have "... = ≀ p ∪ |R]q"
    by (simp add: abox_1 ad_test_sp assms)
  finally show ?thesis
    .
qed

lemma abox_test_sp_2:
  "test p ⟹ test q ⟹ p ∪ |R]q = |≀ p * R]q"
  by (simp add: abox_test_sp test_double_complement)

lemma abox_test_sp_3:
  "test p ⟹ test q ⟹ p → |R]q = |p * R]q"
  by (simp add: abox_test_sp)

lemma fission_sp_dist:
  "fis (R * S) = fis (R * Dom S) * fis S"
proof -
  have "S = Dom S * (S ∪ aDom S * 1⇩∪⇩∪)"
    by (auto simp: mr_simp)
  hence "fis (R * S) = fis (R * Dom S * (S ∪ aDom S * 1⇩∪⇩∪))"
    by (metis d_s_id_ax sp_test_sp_oi_right test_sp)
  also have "... = fis (R * Dom S) * fis (S ∪ aDom S * 1⇩∪⇩∪)"
    apply (rule fission_sp_total_dist)
    by (smt (verit) total_dom Compl_disjoint ad_sp_bot ad_test_sp c6 compl_inf_bot d_complement_ad d_dist_ou inf_le2 iu_unit_down subset_Un_eq sup_ge2 sup_inf_absorb total_lower)
  also have "... = fis (R * Dom S) * (fis S ∪ fis (aDom S * 1⇩∪⇩∪))"
    by (simp add: fission_dist_ou)
  also have "... = fis (R * Dom S) * fis S"
    by (simp add: fission_sp_iu_unit)
  finally show ?thesis
    .
qed

lemma abox_test:
  "test p ⟹ test ( |R]p )"
  by (simp add: abox)

lemma adia_test:
  "test p ⟹ test ( |R⟩p )"
  by (simp add: adia d_test)

lemma ebox_test:
  "test p ⟹ test ( |R]]p )"
  by (simp add: ebox)

lemma edia_test:
  "test p ⟹ test ( |R⟩⟩p )"
  by (simp add: edia d_test)

lemma abox_sp:
  assumes "test p"
    and "test q"
  shows "|R](p * q) = |R]p * |R]q"
proof -
  have "|R](p * q) = aDom (ne (R↓) * (≀ p ∪ ≀ q))"
    by (metis (no_types, lifting) abox_1 ad_test_sp assms cl9_var d_ne_down_dp_complement_test sp_test test_double_complement test_oi_closed)
  also have "... = aDom (ne (R↓) * ≀ p) * aDom (ne (R↓) * ≀ q)"
    by (smt (z3) ad_test_sp cl9_var d_complement_ad d_dist_ou d_test_sp semilattice_inf_class.inf_le2 split_sp_test_7)
  also have "... = |R]p * |R]q"
    by (simp add: abox assms)
  finally show ?thesis
    .
qed

lemma adia_ou_below_ne_down:
  assumes "test p"
  shows "|R⟩(p ∪ ≀ q) ⊆ |R⟩p ∪ |ne (R↓)⟩(≀ q)"
  by (metis adia assms d_dist_ou split_sp_test_6 test_complement_closed test_ou_closed)

lemma abox_adia_mp:
  assumes "test p"
    and "test q"
  shows "|R⟩(p → q) * |R]p ⊆ |R⟩q"
  by (smt (z3) adia_ou_below_ne_down test_shunting abox adia assms d_complement_ad sup_commute test_complement_closed test_implication_closed)

lemma adia_abox_mp:
  assumes "test p"
    and "test q"
  shows "|R⟩p * |R](p → q) ⊆ |R⟩q"
proof -
  have "p ⊆ p → q → q"
    using assms(1) by blast
  hence "|R⟩p ⊆ |R⟩((p → q) → q)"
    by (simp add: adia_right_isotone assms)
  thus ?thesis
    by (smt (z3) abox_adia_mp abox_test adia_test assms(2) semilattice_inf_class.inf.orderE semilattice_inf_class.le_infI2 test_implication_closed test_shunting)
qed

lemma abox_implication_adia:
  assumes "test p"
    and "test q"
  shows "|R](p → q) ⊆ |R⟩p → |R⟩q"
  by (metis adia_abox_mp test_shunting test_sp_commute Int_lower2 Un_commute abox_test adia_test assms test_ou_closed)

lemma abox_adia_implication:
  assumes "test p"
    and "test q"
  shows "|R]p ⊆ |R⟩q → |R⟩(p * q)"
proof -
  have "p ⊆ q → p * q"
    by (metis assms subset_refl test_galois_1 test_sp_commute)
  hence "|R]p ⊆ |R](q → p * q)"
    by (simp add: abox_right_isotone assms test_galois_1)
  thus ?thesis
    by (metis (no_types, lifting) Int_Un_eq(2) abox_implication_adia assms le_sup_iff subset_Un_eq test_galois_1)
qed

lemma abox_mp:
  assumes "test p"
    and "test q"
  shows "|R]p * |R](p → q) ⊆ |R]q"
  by (metis (no_types, lifting) abox_right_isotone abox_sp assms semilattice_inf_class.inf.absorb_iff1 sp_test_dist_oi_left subset_refl sup_commute test_implication_closed test_shunting test_sp_commute)

lemma abox_implication:
  assumes "test p"
    and "test q"
  shows "|R](p → q) ⊆ |R]p → |R]q"
  by (metis abox_mp test_shunting test_sp_commute abox_test assms sup_commute test_implication_closed)

lemma ebox_left_dist_ou:
  assumes "test p"
  shows "|R ∪ S]]p = |R]]p * |S]]p"
  by (auto simp: mr_simp)

lemma abox_left_dist_ou:
  assumes "test p"
  shows "|R ∪ S]p = |R]p * |S]p"
  by (simp add: abox_ebox assms ebox_left_dist_ou ii_right_dist_ou ne_dist_ou)

lemma adia_left_dist_ou:
  assumes "test p"
  shows "|R ∪ S⟩p = |R⟩p ∪ |S⟩p"
  by (auto simp: mr_simp)

lemma edia_left_dist_ou:
  assumes "test p"
  shows "|R ∪ S⟩⟩p = |R⟩⟩p ∪ |S⟩⟩p"
  by (simp add: assms boolean_algebra.conj_disj_distrib2 d_dist_ou edia_1)

lemma abox_dist_iu_1:
  assumes "test p"
  shows "|R ∪∪ S]p = |Dom R * ne (S↓)]]p * |Dom S * ne (R↓)]]p"
proof
  have 1: "|R ∪∪ S]p ⊆ |Dom R * ne (S↓)]]p"
    by (metis abox_ebox assms d_sp_ne_down_below_ne_iu_down ebox_left_antitone)
  have "|R ∪∪ S]p ⊆ |Dom S * ne (R↓)]]p"
    by (metis abox_ebox assms d_sp_ne_down_below_ne_iu_down ebox_left_antitone iu_commute)
  thus "|R ∪∪ S]p ⊆ |Dom R * ne (S↓)]]p * |Dom S * ne (R↓)]]p"
    using 1 by (simp add: assms ebox)
next
  have "|Dom R * ne (S↓)]]p * |Dom S * ne (R↓)]]p ⊆ |Dom R * Dom S * ne (S↓)]]p * |Dom S * ne (R↓)]]p"
    apply (clarsimp simp: mr_simp)
    by (metis UN_I singletonI)
  also have "... ⊆ |Dom R * Dom S * ne (S↓)]]p * |Dom R * Dom S * ne (R↓)]]p"
    by (simp add: assms d_lb2 ebox_left_antitone s_prod_isol s_prod_isor)
  also have "... = |Dom R * Dom S * ne (S↓) ∪ Dom R * Dom S * ne (R↓)]]p"
    using assms ebox_left_dist_ou by blast
  also have "... = |ne (Dom R * Dom S * S↓) ∪ ne (Dom R * Dom S * R↓)]]p"
    by (metis d_dist_ii d_test test_sp_ne)
  also have "... = |ne ((Dom R * Dom S * S)↓) ∪ ne ((Dom R * Dom S * R)↓)]]p"
    by (simp add: down_dist_sp)
  also have "... = aDom ((ne ((Dom R * Dom S * S)↓) ∪ ne ((Dom R * Dom S * R)↓)) * ≀ p)"
    using assms ebox by blast
  also have "... ⊆ aDom ((ne ((Dom R * Dom S * S ∪∪ Dom R * Dom S * R)↓)) * ≀ p)"
    using d_ne_iu_down_sp_test_ou by blast
  also have "... = |Dom R * Dom S * S ∪∪ Dom R * Dom S * R]p"
    using abox assms by blast
  also have "... = |Dom R * Dom S * (R ∪∪ S)]p"
    by (metis d_assoc1 d_inter_r p_prod_comm)
  also have "... = |R ∪∪ S]p"
    by (metis c1 cl8_var d_dist_iu)
  finally show "|Dom R * ne (S↓)]]p * |Dom S * ne (R↓)]]p ⊆ |R ∪∪ S]p"
    .
qed

lemma abox_dist_iu_2:
  assumes "test p"
  shows "|R ∪∪ S]p = |Dom R * S]p * |Dom S * R]p"
proof -
  have "|Dom R * ne (S↓)]]p * |Dom S * ne (R↓)]]p = |ne ((Dom R * S)↓)]]p * |ne ((Dom S * R)↓)]]p"
    by (simp add: d_test down_dist_sp test_sp_ne)
  also have "... = |Dom R * S]p * |Dom S * R]p"
    by (simp add: abox_ebox assms)
  finally show ?thesis
    using assms abox_dist_iu_1 by blast
qed

lemma abox_dist_iu_3:
  assumes "test p"
  shows "|R ∪∪ S]p = ( |R⟩1 → |S]p ) * ( |S⟩1 → |R]p )"
  by (metis abox_dist_iu_2 adia assms abox_test_sp d_test s_prod_idr subset_refl)

lemma abox_adia_sp_one_set:
  "|R]|S⟩1 = { (a,{a}) | a . ∀B . (a,B) ∈ R ⟶ (∀b∈B . ∃D . (b,D) ∈ S) }"
  by (auto simp: abox_def Dom_def adia)

lemma abox_abox_set:
  "|R]|S]p = { (a,{a}) | a . ∀B . (a,B) ∈ R ⟶ (∀C . (∃b∈B . (b,C) ∈ S) ⟶ (∀c∈C . (c,{c}) ∈ p)) }"
  by (auto simp: abox_def)

lemma sp_abox_set:
  "|R * S]p = { (a,{a}) | a . ∀B . (a,B) ∈ R ⟶ (∀C . (∃f . (∀b∈B . (b,f b) ∈ S) ∧ C = ⋃{ f b | b . b ∈ B }) ⟶ (∀c∈C . (c,{c}) ∈ p)) }"
  apply (unfold abox_def s_prod_def)
  by blast

lemma abox_sp_1:
  assumes "test p"
  shows "|R]|S⟩1 * |R * S]p ⊆ |R]|S]p"
proof -
  have "|R]|S⟩1 * |R * S]p = |R]|S⟩1 ∩ |R * S]p"
    by (smt (verit, ccfv_SIG) abox_test adia_test assms convex_increasing inf.orderE inf_assoc sp_unit_convex test_s_prod_is_meet)
  also have "... ⊆ |R]|S]p"
  proof
    fix x
    assume "x ∈ |R]|S⟩1 ∩ |R * S]p"
    from this obtain a where 1: "x = (a,{a}) ∧ x ∈ |R]|S⟩1 ∧ x ∈ |R * S]p"
      by (metis Int_iff abox_test adia_test order_refl subid_aux2 subsetD surj_pair)
    hence 2: "∀B . (a,B) ∈ R ⟶ (∀b∈B . ∃D . (b,D) ∈ S)"
      by (smt (z3) abox_adia_sp_one_set mem_Collect_eq old.prod.inject)
    have 3: "∀B . (a,B) ∈ R ⟶ (∀C . (∃f . (∀b∈B . (b,f b) ∈ S) ∧ C = ⋃{ f b | b . b ∈ B }) ⟶ (∀c∈C . (c,{c}) ∈ p))"
      using 1 by (smt (z3) sp_abox_set mem_Collect_eq old.prod.inject)
    have "∀B . (a,B) ∈ R ⟶ (∀C . (∃b∈B . (b,C) ∈ S) ⟶ (∀c∈C . (c,{c}) ∈ p))"
    proof (rule allI, rule impI)
      fix B
      assume 4: "(a,B) ∈ R"
      hence "∃DD . ∀b∈B . (b,DD b) ∈ S"
        using 2 by (auto intro: bchoice)
      from this obtain DD where 5: "∀b∈B . (b,DD b) ∈ S"
        by auto
      show "∀C . (∃b∈B . (b,C) ∈ S) ⟶ (∀c∈C . (c,{c}) ∈ p)"
      proof (rule allI, rule impI)
        fix C
        assume "∃b∈B . (b,C) ∈ S"
        from this obtain b where 6: "b ∈ B ∧ (b,C) ∈ S"
          by auto
        let ?f = "λx . if x = b then C else DD x"
        let ?C = "C ∪ ⋃{ ?f x | x . x ∈ B ∧ x ≠ b }"
        have "∃f . (∀b∈B . (b,f b) ∈ S) ∧ ?C = ⋃{ f b | b . b ∈ B }"
          apply (rule exI[where ?x="?f"])
          using 5 6 by auto
        hence "∀c∈?C . (c,{c}) ∈ p"
          using 3 4 by auto
        thus "∀c∈C . (c,{c}) ∈ p"
          by blast
      qed
    qed
    thus "x ∈ |R]|S]p"
      using 1 abox_abox_set by blast
  qed
  finally show ?thesis
    .
qed

lemma abox_sp_2:
  assumes "test p"
  shows "|R]|S]p = |R↓ * S]p"
proof -
  have "|R]|S]p = aDom (ne (R↓) * Dom (ne (S↓) * ≀ p))"
    by (metis abox abox_test assms d_complement_ad)
  also have "... = aDom (ne (R↓) * ne (S↓) * ≀ p)"
    by (simp add: test_assoc3)
  also have "... = aDom (ne ((R↓ * S)↓) * ≀ p)"
    by (simp add: down_dist_sp ne_dist_down_sp)
  also have "... = |R↓ * S]p"
    by (simp add: abox assms)
  finally show ?thesis
    .
qed

lemma abox_sp_3:
  assumes "test p"
  shows "|R]|S]p ⊆ |R * S]p"
  by (clarsimp simp: mr_simp) auto

lemma abox_sp_4:
  assumes "test p"
  shows "|R * S]p ⊆ |R]|S⟩1 → |R]|S]p"
proof -
  have "|R]|S⟩1 * |R * S]p ⊆ |R]|S]p"
    by (auto simp: assms abox_sp_1)
  hence "|R]|S⟩1 ∩ |R * S]p ⊆ |R]|S]p"
    by (smt (verit) abox_test adia_test assms convex_increasing inf.orderE sp_unit_convex test_oi_closed test_s_prod_is_meet)
  thus ?thesis
    using abox_test assms by blast
qed

lemma abox_sp_5:
  assumes "test p"
  shows "|R]|S⟩1 * |R * S]p = |R]|S⟩1 * |R]|S]p"
proof (rule antisym)
  have "|R * S]p ⊆ |R]|S⟩1 → |R]|S]p"
    by (simp add: abox_sp_4 assms)
  hence "|R]|S⟩1 ∩ |R * S]p ⊆ |R]|S]p"
    by blast
  hence "|R]|S⟩1 ∩ |R * S]p ⊆ |R]|S⟩1 ∩ |R]|S]p"
    by blast
  thus "|R]|S⟩1 * |R * S]p ⊆ |R]|S⟩1 * |R]|S]p"
    by (smt (verit, del_insts) abox_test adia_test assms convex_increasing inf.orderE sp_unit_convex test_oi_closed test_s_prod_is_meet)
  show "|R]|S⟩1 * |R]|S]p ⊆ |R]|S⟩1 * |R * S]p"
    by (simp add: abox_sp_3 assms s_prod_isor)
qed

lemma abox_sp_6:
  assumes "test p"
  shows "|R]|S⟩1 → |R * S]p = |R]|S⟩1 → |R]|S]p"
  by (smt (z3) Int_commute abox_sp_3 abox_sp_4 assms inf_sup_distrib2 lattice_class.inf_sup_absorb semilattice_inf_class.inf.absorb_iff2 sup_commute)

lemma abox_sp_7:
  assumes "test p"
    and "total S"
  shows "|R * S]p = |R]|S]p"
  by (metis (no_types, lifting) abox_ebox abox_sp_2 assms down_dist_sp total_down_dist_sp)

lemma adia_sp_associative:
  assumes "test p"
  shows "|Q * (R * S)⟩p = |(Q * R) * S⟩p"
proof -
  have "|Q * (R * S)⟩p = |Q⟩( |R⟩( |S⟩p))"
    by (metis (no_types, lifting) adia adia_test assms d_loc_ax inf.orderE test_assoc3)
  also have "… = |(Q * R) * S⟩p"
    by (smt (verit, best) adia adia_test assms d_loc test_assoc3 test_double_complement)
  finally show "|Q * (R * S)⟩p = |(Q * R) * S⟩p"
    .
qed

lemma ebox_sp_associative:
  assumes "test p"
  shows "|Q * (R * S)]]p = |(Q * R) * S]]p"
  by (simp add: adia_sp_associative assms ebox_adia)

lemma edia_sp_associative:
  assumes "test p"
  shows "|Q * (R * S)⟩⟩p = |(Q * R) * S⟩⟩p"
proof -
  have "|fis (Q * (R * S))⟩⟩p = |fis (Q * Dom (R * S)) * (fis (R * Dom S) * fis S)⟩⟩p"
    by (metis fission_sp_dist)
  also have "... = |(fis (Q * Dom (R * S)) * fis (R * Dom S)) * fis S⟩⟩p"
    by (simp add: inner_deterministic_sp_assoc semilattice_inf_class.inf_commute semilattice_inf_class.le_infI1 fission_var)
  also have "... = |fis (Q * Dom (R * Dom S)) * fis (R * Dom S) * fis S⟩⟩p"
    by simp
  also have "... = |fis (Q * (R * Dom S)) * fis S⟩⟩p"
    by (metis fission_sp_dist)
  also have "... = |fis ((Q * R) * Dom S) * fis S⟩⟩p"
    by (metis d_complement_ad test_assoc3)
  also have "... = |fis ((Q * R) * S)⟩⟩p"
    by (metis fission_sp_dist)
  finally show ?thesis
    using assms edia_fission by blast
qed

lemma abox_sp_associative:
  assumes "test p"
  shows "|Q * (R * S)]p = |(Q * R) * S]p"
  by (simp add: edia_sp_associative assms abox_edia)

lemma abox_oI:
  assumes "X ≠ {}"
  shows "|R]⋂X = (⋂p∈X . |R]p)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
  apply (clarsimp simp: mr_simp)
  using assms by blast

lemma ebox_left_dist_oU:
  assumes "X ≠ {}"
  shows "|⋃X]]p = (⋂R∈X . |R]]p)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply blast
  apply (clarsimp simp: mr_simp)
  using assms by blast

lemma abox_left_dist_oU:
  assumes "X ≠ {}"
  shows "|⋃X]p = (⋂R∈X . |R]p)"
  apply (rule antisym)
   apply (clarsimp simp: mr_simp)
   apply blast
  apply (clarsimp simp: mr_simp)
  using assms by blast

lemma adia_left_dist_oU:
  "|⋃X⟩p = (⋃R∈X . |R⟩p)"
  apply (clarsimp simp: mr_simp)
  by blast

lemma edia_left_dist_oU:
  "|⋃X⟩⟩p = (⋃R∈X . |R⟩⟩p)"
  apply (clarsimp simp: mr_simp)
  by blast

subsection ‹Goldblatt's axioms with star›

unbundle no rtrancl_syntax
notation star (‹_⇧*› [1000] 999)

lemma star_induct_1:
  assumes "1 ⊆ X"
    and "R * X ⊆ X"
  shows "R⇧* ⊆ X"
  apply (unfold star_def)
  apply (rule lfp_lowerbound)
  by (simp add: assms)

lemma star_induct:
  assumes "S ⊆ 1 ∪ 1⇩∪⇩∪"
    and "S ⊆ X"
    and "R * X ⊆ X"
  shows "R⇧* * S ⊆ X"
proof -
  have "R⇧* ⊆ X ⊘ S"
  proof (rule star_induct_1)
    show "1 ⊆ X ⊘ S"
      by (metis (no_types, opaque_lifting) Int_subset_iff assms(2) dual_order.eq_iff sp_lres_galois test_sp)
  next
    have "(X ⊘ S) * S ⊆ X"
      by (simp add: sp_lres_sp_below)
    hence "R * (X ⊘ S) * S ⊆ R * X"
      by (metis assms(1) s_prod_isor test_iu_test_sp_assoc_5)
    also have "... ⊆ X"
      by (simp add: assms(3))
    finally show "R * (X ⊘ S) ⊆ X ⊘ S"
      by (simp add: sp_lres_galois)
  qed
  thus ?thesis
    by (simp add: sp_lres_galois)
qed

lemma star_total:
  "total (R⇧*)"
  by (metis s_prod_idl s_prod_isol star_refl total_4)

lemma star_down:
  "R⇧*↓ = (R↓)⇧* ∪ 1⇩∪⇩∪"
proof
  have "R⇧* * (1 ∪ 1⇩∪⇩∪) ⊆ (R↓)⇧* ∪ 1⇩∪⇩∪"
  proof (rule star_induct)
    show "1 ∪ 1⇩∪⇩∪ ⊆ 1 ∪ 1⇩∪⇩∪"
      by simp
  next
    show "1 ∪ 1⇩∪⇩∪ ⊆ (R↓)⇧* ∪ 1⇩∪⇩∪"
      using star_refl by auto
  next
    have "ne (R * ((R↓)⇧* ∪ 1⇩∪⇩∪)) ⊆ ne (R↓ * ((R↓)⇧* ∪ 1⇩∪⇩∪))"
      by (simp add: down_sp_sp sup_commute)
    also have "... = ne (R↓) * ne ((R↓)⇧* ∪ 1⇩∪⇩∪)"
      by (simp add: ne_dist_down_sp)
    also have "... = ne (R↓) * ne ((R↓)⇧*)"
      by (metis down_idempotent down_sp_sp ne_dist_down_sp sup_commute)
    also have "... ⊆ R↓ * (R↓)⇧*"
      using sp_oi_subdist by blast
    also have "... ⊆ (R↓)⇧*"
      using star_unfold_eq by blast
    finally show "R * ((R↓)⇧* ∪ 1⇩∪⇩∪) ⊆ (R↓)⇧* ∪ 1⇩∪⇩∪"
      by blast
  qed
  thus "R⇧*↓ ⊆ (R↓)⇧* ∪ 1⇩∪⇩∪"
    by (simp add: down_sp sup_commute)
next
  have "(R↓)⇧* ⊆ R⇧*↓"
  proof (rule star_induct_1)
    show "1 ⊆ R⇧*↓"
      by (simp add: star_refl subset_lower)
  next
    show "R↓ * R⇧*↓ ⊆ R⇧*↓"
      by (metis total_dom Un_Int_eq(1) d_isotone d_test ii_right_dist_ou inf_le2 le_sup_iff s_subid_iff2 star_unfold_eq subset_antisym total_down_dist_sp)
  qed
  thus "(R↓)⇧* ∪ 1⇩∪⇩∪ ⊆ R⇧*↓"
    using star_total total_lower by blast
qed

lemma ne_star_down:
  "ne (R⇧*↓) = ne ((R↓)⇧*)"
  by (simp add: ne_dist_ou star_down)

lemma ne_down_star:
  "ne ((R↓)⇧*) = (ne (R↓))⇧*"
proof
  have "(R↓)⇧* ⊆ (ne (R↓))⇧* ∪ 1⇩∪⇩∪"
  proof (rule star_induct_1)
    show "1 ⊆ (ne (R↓))⇧* ∪ 1⇩∪⇩∪"
      by (simp add: le_supI1 star_refl)
  next
    have "ne (R↓ * ((ne (R↓))⇧* ∪ 1⇩∪⇩∪)) = ne (R↓) * ne ((ne (R↓))⇧*)"
      by (metis down_idempotent down_sp_sp ne_dist_down_sp sup_commute)
    also have "... ⊆ (ne (R↓))⇧*"
      by (metis (no_types, lifting) IntE UnCI inf.absorb_iff2 sp_oi_subdist star_unfold_eq subsetI)
    finally show "R↓ * ((ne (R↓))⇧* ∪ 1⇩∪⇩∪) ⊆ ((ne (R↓))⇧* ∪ 1⇩∪⇩∪)"
      by blast
  qed
  thus "ne ((R↓)⇧*) ⊆ (ne (R↓))⇧*"
    by (smt (z3) Compl_disjoint2 Int_commute Int_left_commute ne_dist_ou semilattice_inf_class.le_iff_inf sup_bot.right_neutral)
next
  show "(ne (R↓))⇧* ⊆ ne ((R↓)⇧*)"
  proof (rule star_induct_1)
    show "1 ⊆ ne ((R↓)⇧*)"
      using star_refl test_ne by auto
  next
    show "ne (R↓) * ne ((R↓)⇧*) ⊆ ne ((R↓)⇧*)"
      by (metis IntE IntI UnCI ne_dist_down_sp star_unfold_eq subsetI)
  qed
qed

lemma abox_star_unfold:
  "test p ⟹ |R⇧*]p = p * |R]|R⇧*]p"
  by (metis abox_left_dist_ou abox_sp_7 sp_unit_abox star_total star_unfold_eq)

lemma star_sp_test_commute:
  assumes "S ⊆ 1 ∪ 1⇩∪⇩∪"
    and "Q * S ⊆ S * R"
  shows "Q⇧* * S ⊆ S * R⇧*"
proof (rule star_induct)
  show "S ⊆ 1 ∪ 1⇩∪⇩∪"
    by (simp add: assms(1))
next
  show "S ⊆ S * R⇧*"
    by (metis s_prod_idr s_prod_isor star_refl)
next
  have "Q * (S * R⇧*) ⊆ S * R * R⇧*"
    by (metis (no_types, lifting) assms s_prod_distr subset_Un_eq test_iu_test_sp_assoc_3)
  thus "Q * (S * R⇧*) ⊆ S * R⇧*"
    by (metis (no_types, lifting) UnCI dual_order.trans s_prod_assoc1 s_prod_isor star_unfold subset_eq)
qed

lemma adia_star_induct:
  assumes "test p"
  shows "|R⟩p ⊆ p ⟷ |R⇧*⟩p ⊆ p"
proof
  assume "|R⟩p ⊆ p"
  hence "≀ p * Dom (R * p) = {}"
    by (metis adia assms d_idem2 s_prod_isol subset_empty test_sp_shunting)
  hence "R * p ⊆ p * (R * p)"
    by (metis assms d_sp_strict subset_refl test_sp_shunting)
  hence "R * p ⊆ p * R"
    by (metis assms inf.absorb_iff2 inf_commute sp_test_dist_oi_right test_assoc3 test_sp_idempotent)
  hence "R⇧* * p ⊆ p * R⇧*"
    by (simp add: assms le_supI1 star_sp_test_commute)
  hence "R⇧* * p ⊆ p * (R⇧* * p)"
    by (metis assms inf.absorb_iff2 inf.orderE sp_oi_subdist test_assoc3 test_sp_idempotent)
  hence "≀ p * (R⇧* * p) = {}"
    by (meson assms subset_empty test_sp_shunting)
  hence "≀ p * Dom (R⇧* * p) = {}"
    using d_sp_strict by blast
  thus "|R⇧*⟩p ⊆ p"
    by (metis adia assms d_test empty_subsetI semilattice_inf_class.le_inf_iff sp_test test_sp_shunting)
next
  assume "|R⇧*⟩p ⊆ p"
  thus "|R⟩p ⊆ p"
    by (metis adia_left_isotone assms dual_order.trans s_prod_idr s_prod_isor star_refl star_unfold sup.coboundedI2)
qed

lemma ebox_star_induct:
  assumes "test p"
  shows "p ⊆ |R]]p ⟷ p ⊆ |R⇧*]]p"
  by (smt (verit, best) adia adia_star_induct assms d_complement_ad ebox_adia test_complement_antitone test_double_complement)

lemma abox_star_induct:
  assumes "test p"
  shows "p ⊆ |R]p ⟷ p ⊆ |R⇧*]p"
proof -
  have "p ⊆ |ne (R↓)]]p ⟷ p ⊆ |ne (R⇧*↓)]]p"
    by (metis assms ebox_star_induct ne_down_star ne_star_down)
  thus ?thesis
    by (metis abox_ebox assms)
qed

lemma edia_star_induct:
  assumes "test p"
  shows "|R⟩⟩p ⊆ p ⟷ |R⇧*⟩⟩p ⊆ p"
  by (metis adia_star_induct assms edia_adia ne_down_star ne_star_down)

lemma abox_star_induct_1:
  assumes "test p"
    and "test q"
    and "q ⊆ p * |R]q"
  shows "q ⊆ |R⇧*]p"
proof -
  have "q ⊆ p ∧ q ⊆ |R⇧*]q"
    by (metis Int_subset_iff abox_star_induct abox_test assms test_sp test_sp_commute)
  thus ?thesis
    using abox_right_isotone assms(1,2) by blast
qed

lemma adia_star_induct_1:
  assumes "test p"
    and "test q"
    and "p ∪ |R⟩q ⊆ q"
  shows "|R⇧*⟩p ⊆ q"
  by (meson adia_right_isotone adia_star_induct assms order.trans sup.bounded_iff)

lemma abox_segerberg:
  assumes "test p"
  shows "|R⇧*](p → |R]p) ⊆ p → |R⇧*]p"
proof -
  have "p * |R⇧*](p → |R]p) ⊆ |R⇧*]p"
  proof (rule abox_star_induct_1)
    show "test p"
      by (simp add: assms)
  next
    show "test (p * |R⇧*](p → |R]p))"
      by (simp add: abox_test assms test_galois_1)
  next
    have "p * |R⇧*](p → |R]p) = p * (p → |R]p) * |R]|R⇧*](p → |R]p)"
      by (metis abox_star_unfold abox_test assms inf_le2 le_infE sp_unit_convex sp_unit_down test_iu_test_sp_assoc_1 test_ou_closed)
    also have "... = p * |R]p * |R]|R⇧*](p → |R]p)"
      by (smt (verit, best) Un_Int_eq(4) abox_left_dist_ou abox_test assms equalityD1 le_infE s_prod_isol sp_unit_abox sp_unit_convex sp_unit_down subset_Un_eq subset_antisym test_galois_1 test_iu_test_sp_assoc_1 test_ou_closed test_sp_commute)
    also have "... = p * |R](p * |R⇧*](p → |R]p))"
      by (metis (no_types, lifting) abox_sp abox_test abox_test_sp_3 assms test_assoc2 test_double_complement)
    finally show "p * |R⇧*](p → |R]p) ⊆ p * |R](p * |R⇧*](p → |R]p))"
      by simp
  qed
  thus ?thesis
    by (meson abox_test assms test_galois_1 test_implication_closed)
qed

lemma abox_segerberg_adia:
  assumes "test p"
  shows "|R⇧*]( |R⟩p → p ) ⊆ |R⇧*⟩p → p"
proof -
  let ?q = "|R⇧*]( |R⟩p → p )"
  have "|R⇧*⟩p ⊆ ?q → p"
  proof (rule adia_star_induct_1)
    show "test p"
      by (simp add: assms)
  next
    show "test ( ?q → p )"
      by (simp add: assms)
  next
    have "|R⟩(?q → p) * |R]?q * ( |R⟩p → p ) ⊆ |R⟩p * ( |R⟩p → p )"
      by (metis (no_types, lifting) abox_adia_mp abox_test assms inf.absorb_iff2 sp_test_dist_oi test_implication_closed)
    also have "... ⊆ p"
      by (meson adia_test assms equalityD2 test_galois_1 test_implication_closed)
    finally have "|R⟩(?q → p) ⊆ ( |R⟩p → p ) * |R]?q → p"
      by (smt (verit) abox_star_unfold abox_test adia assms d_complement_ad test_assoc3 test_double_complement test_galois_1 test_implication_closed test_sp_commute)
    also have "... = ?q → p"
      by (metis abox_star_unfold assms test_implication_closed)
    finally show "p ∪ |R⟩( ?q → p ) ⊆ ?q → p"
      by (metis le_sup_iff order_refl)
  qed
  thus ?thesis
    by (smt (z3) abox_test adia_test assms sup_commute test_galois_1 test_implication_closed test_shunting)
qed

lemma s_p_id_sp:
  "(s_id ∪ p_id) * R = R ∪ p_id"
  by (simp add: s_prod_distr)

subsection ‹Propositional Hoare logic›

abbreviation hoare :: "('a,'a) mrel ⇒ ('a,'b) mrel ⇒ ('b,'b) mrel ⇒ bool" (‹_⦃_⦄_› [50,60,50] 95)
  where "p⦃R⦄q ≡ p ⊆ |R]q"

abbreviation if_then_else :: "('a,'a) mrel ⇒ ('a,'b) mrel ⇒ ('a,'b) mrel ⇒ ('a,'b) mrel"
  where "if_then_else p R S ≡ p * R ∪ ≀ p * S"

abbreviation while_do :: "('a,'a) mrel ⇒ ('a,'a) mrel ⇒ ('a,'a) mrel"
  where "while_do p R ≡ (p * R)⇧* * ≀ p"

lemma hoare_skip:
  assumes "test p"
    shows "p⦃1⦄p"
  by (simp add: assms sp_unit_abox)

lemma hoare_cons:
  assumes "test s"
      and "r ⊆ p"
      and "q ⊆ s"
      and "p⦃R⦄q"
    shows "r⦃R⦄s"
  by (meson abox_right_isotone assms order_trans)

lemma hoare_seq:
  assumes "test q"
      and "test r"
      and "p⦃R⦄q"
      and "q⦃S⦄r"
    shows "p⦃R*S⦄r"
proof -
  have "p ⊆ |R]q"
    by (simp add: assms(3))
  also have "... ⊆ |R]|S]r"
    by (simp add: abox_right_isotone abox_test assms(1,2,4))
  also have "... ⊆ |R*S]r"
    by (simp add: abox_sp_3 assms(2))
  finally show ?thesis
    by simp
qed

lemma hoare_if:
  assumes "test p"
      and "test q"
      and "test r"
      and "(p*q)⦃R⦄r"
      and "((≀ p)*q)⦃S⦄r"
  shows "q⦃if_then_else p R S⦄r"
  by (smt (verit, ccfv_threshold) abox_left_dist_ou abox_test_sp assms inf.absorb_iff2 inf_le2 sp_oi_subdist test_galois_1 test_sp_idempotent)

lemma hoare_while:
  assumes "test p"
      and "test q"
      and "(p*q)⦃R⦄q"
    shows "q⦃while_do p R⦄(q*(≀ p))"
proof -
  have "q ⊆ p → |R]q"
    by (meson assms test_galois_1)
  also have "... = |p]|R]q"
    by (simp add: abox_test assms(1,2) test_abox)
  also have "... ⊆ |p*R]q"
    by (simp add: abox_sp_3 assms(2))
  finally have "q ⊆ |(p*R)⇧*]q"
    by (meson abox_star_induct assms(2))
  also have "... ⊆ |(p*R)⇧*](q ∪ p)"
    by (simp add: abox_right_isotone assms(1,2))
  also have "... = |(p*R)⇧*](≀p → (q*(≀ p)))"
    by (smt (verit, best) Un_assoc Un_upper1 assms(1,2) inf_commute sup.commute sup.order_iff test_double_complement test_galois_1 test_s_prod_is_meet)
  also have "... = |(p*R)⇧*]|≀p](q*(≀ p))"
    by (simp add: assms(2) test_abox test_shunting)
  also have "... ⊆ |while_do p R](q*(≀ p))"
    by (simp add: abox_sp_3 assms(2) test_galois_1)
  finally show ?thesis
    by simp
qed

lemma hoare_par:
  assumes "test q"
      and "p⦃R⦄q"
      and "p⦃S⦄q"
    shows "p⦃R ∪∪ S⦄q"
proof -
  have 1: "p ⊆ |R⟩1 → |S]q"
    by (simp add: assms(3) le_supI2)
  have "p ⊆ |S⟩1 → |R]q"
    by (simp add: assms(2) le_supI2)
  hence "p ⊆ ( |R⟩1 → |S]q) ∩ ( |S⟩1 → |R]q)"
    using 1 by simp
  also have "... = |R ∪∪ S]q"
    by (smt (verit) abox_dist_iu_3 abox_test assms(1) inf.orderE test_implication_closed test_oi_closed test_s_prod_is_meet)
  finally show ?thesis
    by simp
qed

section ‹Counterexamples›

locale counterexamples 
begin

lemma counter_01:
  "¬ ((U::('a,'b) mrel) * -((U::('b,'c) mrel) * (R::('c,'d) mrel)) ⊆ -(U * R))"
  by (metis UNIV_I U_par_zero disjoint_eq_subset_Compl emptyE iu_unit_below_top_sp_test iu_unit_up le_inf_iff s_prod_zerol subset_empty top_upper_least)

abbreviation "a_1 ≡ finite_1.a⇩1"

lemma counter_02:
  "∃R::(Enum.finite_1,Enum.finite_1) mrel . ∃p . ¬ (test p ⟶ (R ∩ -(U * p)) * U = R * -(p * U))"
  apply (rule exI[where ?x="{(a_1,{})}"])
  apply (rule exI[where ?x="{}"])
  apply (clarsimp simp: s_id_def)
  by (smt (verit, ccfv_SIG) Compl_empty_eq Diff_eq Int_insert_left_if0 U_par_p_id cl8_var complement_test_sp_top d_U d_sp_strict dc empty_subsetI inf_le2 inf_top_left inner_total_2 insert_not_empty s_prod_zerol x_split_var x_y_split zero_nc)

lemma counter_03:
  "∃R::(Enum.finite_1,Enum.finite_1) mrel . ∃p . ¬ (test p ⟶ (R ∩ -(U * p)) * 1⇩∪⇩∪ = R * (-(p * U) ∩ 1⇩∪⇩∪))"
  apply (rule exI[where ?x="{(a_1,{})}"])
  apply (rule exI[where ?x="{}"])
  apply (clarsimp simp: s_id_def)
  by (smt (z3) Int_Un_eq(3) Int_absorb2 U_c ad_sp_bot cd_iso dc_prop1 disjoint_eq_subset_Compl inf_compl_bot_right inner_total_2 insertI1 p_id_zero singleton_Un_iff sp_oi_subdist)

abbreviation "b_1 ≡ finite_2.a⇩1"
abbreviation "b_2 ≡ finite_2.a⇩2"
abbreviation "b_1_0 ≡ (b_1,{})"
abbreviation "b_1_1 ≡ (b_1,{b_1})"
abbreviation "b_1_2 ≡ (b_1,{b_2})"
abbreviation "b_1_3 ≡ (b_1,{b_1,b_2})"
abbreviation "b_2_0 ≡ (b_2,{})"
abbreviation "b_2_1 ≡ (b_2,{b_1})"
abbreviation "b_2_2 ≡ (b_2,{b_2})"
abbreviation "b_2_3 ≡ (b_2,{b_1,b_2})"

lemma counter_04:
  "∃R::(Enum.finite_2,Enum.finite_2) mrel . ∃p q . ¬ (test p ⟶ test q ⟶ |R * p]q = |R]|p]q)"
  apply (rule exI[where ?x="{b_1_3}"])
  apply (rule exI[where ?x="{b_1_1}"])
  apply (rule exI[where ?x="{}"])
  apply (subst sp_test)
   apply (clarsimp simp: s_id_def)
  apply (subst top_test)
   apply (clarsimp simp: s_id_def)
  apply (unfold abox_def)
  apply (clarsimp simp: s_id_def)
  by blast

lemma counter_05:
  "¬ (∃f . ∀R p . test p ⟶ |R⟩p = |f R]p)"
  by (smt (verit, ccfv_threshold) Int_lower1 Int_lower2 abox_sp_unit adia_test_sp counter_01 iu_test_sp_left_zero s_prod_idl subset_refl)

lemma counter_06:
  "¬ (∃f . ∀R p . test p ⟶ |R]]p = |f R]p)"
  by (metis abox_adia_fusion abox_fusion abox_sp_unit adia counter_05 d_complement_ad disjoint_eq_subset_Compl ebox_adia empty_subsetI s_prod_idr order_refl)

lemma counter_07:
  "¬ (∃f . mono f ∧ (∀R . fus R = lfp (λX . f R X)))"
proof
  assume "∃f::('a,'b) mrel ⇒ ('a,'b) mrel ⇒ ('a,'b) mrel . mono f ∧ (∀R . fus R = lfp (λX . f R X))"
  from this obtain f :: "('a,'b) mrel ⇒ ('a,'b) mrel ⇒ ('a,'b) mrel" where "mono f ∧ (∀R . fus R = lfp (λX . f R X))"
    by auto
  hence "fus {} ⊆ fus (U::('a,'b) mrel)"
    by (simp add: le_fun_def mono_def lfp_mono)
  thus False
    by (auto simp: mr_simp)
qed

abbreviation "c_1 ≡ finite_3.a⇩1"
abbreviation "c_2 ≡ finite_3.a⇩2"
abbreviation "c_3 ≡ finite_3.a⇩3"

lemma counter_08:
  "¬ (∼(1::(Enum.finite_3,Enum.finite_3) mrel) * ∼1 ∈ {1, ∼1})"
proof -
  let ?c = "(c_1,{c_1,c_2,c_3})"
  have 1: "?c ∈ ∼1 * ∼1"
    apply (clarsimp simp: mr_simp)
    apply (rule exI[where ?x="{c_2,c_3}"])
    using UNIV_finite_3 by auto
  have "?c ∉ 1 ∧ ?c ∉ ∼1"
    by (auto simp: mr_simp)
  thus ?thesis
    using 1 by auto
qed

lemma counter_09:
  "¬ (∼(1::(Enum.finite_3,Enum.finite_3) mrel) ⊙ 1 ∈ {1, ∼1})"
  by (metis counter_08 co_prod empty_iff ic_involutive insert_iff)

lemma ex_2_cases:
  "∃b. b = b_1 ∨ b = b_2"
  by auto

lemma all_2_cases:
  "(∀b. b = b_2 ∧ b = b_1) = False"
  by auto

lemma impl_2_cases:
  "⋃{ X . ∃b. (b = b_1 ⟶ X = Y) ∧ (b = b_2 ⟶ X = Z)} = Y ∪ Z"
  by auto

lemma ex_2_set_cases:
  "(∃B::Enum.finite_2 set . P B) ⟷ P {} ∨ P {b_1} ∨ P {b_2} ∨ P {b_1,b_2}"
proof -
  let ?U = "UNIV::Enum.finite_2 set set"
  have "?U ⊆ {{},{b_1},{b_2},{b_1,b_2}}"
  proof
    fix x
    have "x ⊆ {b_1,b_2}"
      by auto
    thus "x ∈ {{},{b_1},{b_2},{b_1,b_2}}"
      by auto
  qed
  hence "?U = {{},{b_1},{b_2},{b_1,b_2}}"
    by auto
  thus ?thesis
    by (metis UNIV_I empty_iff insertE)
qed

abbreviation "B_0 ≡ {}::Enum.finite_2 set"
abbreviation "B_1 ≡ {b_1}"
abbreviation "B_2 ≡ {b_2}"
abbreviation "B_3 ≡ {b_1,b_2}"
abbreviation "mkf x y ≡ λz . if z = b_1 then x else y"

lemma mkf:
  "f = mkf (f b_1) (f b_2)"
  by auto

lemma mkf2:
  "f b_1 = X ∧ f b_2 = Y ⟹ f = mkf X Y"
  by auto

lemma ex_2_mrel_cases:
  "(∃f::Enum.finite_2 ⇒ Enum.finite_2 set . P f) ⟷
    P (mkf B_0 B_0) ∨ P (mkf B_0 B_1) ∨ P (mkf B_0 B_2) ∨ P (mkf B_0 B_3) ∨
    P (mkf B_1 B_0) ∨ P (mkf B_1 B_1) ∨ P (mkf B_1 B_2) ∨ P (mkf B_1 B_3) ∨
    P (mkf B_2 B_0) ∨ P (mkf B_2 B_1) ∨ P (mkf B_2 B_2) ∨ P (mkf B_2 B_3) ∨
    P (mkf B_3 B_0) ∨ P (mkf B_3 B_1) ∨ P (mkf B_3 B_2) ∨ P (mkf B_3 B_3)"
proof
  assume "∃f::Enum.finite_2 ⇒ Enum.finite_2 set . P f"
  from this obtain f where 1: "P f"
    by auto
  have "⋀x . f x ⊆ B_3"
    by auto
  hence 2: "⋀x . f x = B_0 ∨ f x = B_1 ∨ f x = B_2 ∨ f x = B_3"
    by auto
  have "f = mkf B_0 B_0 ∨ f = mkf B_0 B_1 ∨ f = mkf B_0 B_2 ∨ f = mkf B_0 B_3 ∨
        f = mkf B_1 B_0 ∨ f = mkf B_1 B_1 ∨ f = mkf B_1 B_2 ∨ f = mkf B_1 B_3 ∨
        f = mkf B_2 B_0 ∨ f = mkf B_2 B_1 ∨ f = mkf B_2 B_2 ∨ f = mkf B_2 B_3 ∨
        f = mkf B_3 B_0 ∨ f = mkf B_3 B_1 ∨ f = mkf B_3 B_2 ∨ f = mkf B_3 B_3"
    using 2[of b_1] 2[of b_2] mkf2[of f] by blast
  thus "P (mkf B_0 B_0) ∨ P (mkf B_0 B_1) ∨ P (mkf B_0 B_2) ∨ P (mkf B_0 B_3) ∨
        P (mkf B_1 B_0) ∨ P (mkf B_1 B_1) ∨ P (mkf B_1 B_2) ∨ P (mkf B_1 B_3) ∨
        P (mkf B_2 B_0) ∨ P (mkf B_2 B_1) ∨ P (mkf B_2 B_2) ∨ P (mkf B_2 B_3) ∨
        P (mkf B_3 B_0) ∨ P (mkf B_3 B_1) ∨ P (mkf B_3 B_2) ∨ P (mkf B_3 B_3)"
    using 1 by auto
next
  assume "P (mkf B_0 B_0) ∨ P (mkf B_0 B_1) ∨ P (mkf B_0 B_2) ∨ P (mkf B_0 B_3) ∨
          P (mkf B_1 B_0) ∨ P (mkf B_1 B_1) ∨ P (mkf B_1 B_2) ∨ P (mkf B_1 B_3) ∨
          P (mkf B_2 B_0) ∨ P (mkf B_2 B_1) ∨ P (mkf B_2 B_2) ∨ P (mkf B_2 B_3) ∨
          P (mkf B_3 B_0) ∨ P (mkf B_3 B_1) ∨ P (mkf B_3 B_2) ∨ P (mkf B_3 B_3)"
  thus "∃f::Enum.finite_2 ⇒ Enum.finite_2 set . P f"
    by auto
qed

lemma counter_10:
  "∃R::(Enum.finite_2,Enum.finite_2) mrel . ¬ (U::(Enum.finite_2,Enum.finite_2) mrel) * (U * R) ⊆ U * R"
  apply (rule exI[where ?x="{b_1_1,b_1_2}"])
  apply (unfold s_prod_def)
  apply (unfold ex_2_set_cases)
  apply (unfold ex_2_mrel_cases)
  apply (clarsimp simp: mr_simp ex_2_cases all_2_cases impl_2_cases)
  by auto

lemma counter_11:
  "∃(R::(Enum.finite_2,Enum.finite_2) mrel) (s::(Enum.finite_2,Enum.finite_2) mrel) (t::(Enum.finite_2,Enum.finite_2) mrel) . ¬ (inner_univalent s ∧ inner_univalent t ⟶ R * (s * t) = (R * s) * t)"
  apply (rule exI[where ?x="{b_1_3}"])
  apply (rule exI[where ?x="{b_1_1,b_2_1}"])
  apply (rule exI[where ?x="{b_1_1,b_1_2}"])
  apply (unfold s_prod_def)
  apply (unfold ex_2_set_cases)
  apply (unfold ex_2_mrel_cases)
  apply (clarsimp simp: mr_simp ex_2_cases all_2_cases impl_2_cases)
  by (auto simp: times_eq_iff)

lemma counter_12:
  "¬(∃S . 1⇩∪⇩∪ ⊙ S = 1⇩∪⇩∪)"
  by (metis Int_absorb2 UNIV_I U_U cl9 co_prod cp_ii_unit_upper disjoint_iff_not_equal ic_antidist_ii ic_iu_unit ic_top iu_unit_down p_prod_comm p_prod_ild s_prod_idl s_prod_p_idl)

lemma counter_13:
  "¬(∃S . ∀R . R ⊙ S = R)"
  by (meson counter_12)
end

end