Theory UpperPD

(*  Title:      HOL/HOLCF/UpperPD.thy
    Author:     Brian Huffman
*)

section ‹Upper powerdomain›

theory UpperPD
imports Compact_Basis
begin

subsection ‹Basis preorder›

definition
  upper_le :: "'a::bifinite pd_basis ⇒ 'a pd_basis ⇒ bool" (infix ‹≤♯› 50) where
  "upper_le = (λu v. ∀y∈Rep_pd_basis v. ∃x∈Rep_pd_basis u. x ⊑ y)"

lemma upper_le_refl [simp]: "t ≤♯ t"
unfolding upper_le_def by fast

lemma upper_le_trans: "⟦t ≤♯ u; u ≤♯ v⟧ ⟹ t ≤♯ v"
unfolding upper_le_def
apply (rule ballI)
apply (drule (1) bspec, erule bexE)
apply (drule (1) bspec, erule bexE)
apply (erule rev_bexI)
apply (erule (1) below_trans)
done

interpretation upper_le: preorder upper_le
by (rule preorder.intro, rule upper_le_refl, rule upper_le_trans)

lemma upper_le_minimal [simp]: "PDUnit compact_bot ≤♯ t"
unfolding upper_le_def Rep_PDUnit by simp

lemma PDUnit_upper_mono: "x ⊑ y ⟹ PDUnit x ≤♯ PDUnit y"
unfolding upper_le_def Rep_PDUnit by simp

lemma PDPlus_upper_mono: "⟦s ≤♯ t; u ≤♯ v⟧ ⟹ PDPlus s u ≤♯ PDPlus t v"
unfolding upper_le_def Rep_PDPlus by fast

lemma PDPlus_upper_le: "PDPlus t u ≤♯ t"
unfolding upper_le_def Rep_PDPlus by fast

lemma upper_le_PDUnit_PDUnit_iff [simp]:
  "(PDUnit a ≤♯ PDUnit b) = (a ⊑ b)"
unfolding upper_le_def Rep_PDUnit by fast

lemma upper_le_PDPlus_PDUnit_iff:
  "(PDPlus t u ≤♯ PDUnit a) = (t ≤♯ PDUnit a ∨ u ≤♯ PDUnit a)"
unfolding upper_le_def Rep_PDPlus Rep_PDUnit by fast

lemma upper_le_PDPlus_iff: "(t ≤♯ PDPlus u v) = (t ≤♯ u ∧ t ≤♯ v)"
unfolding upper_le_def Rep_PDPlus by fast

lemma upper_le_induct [induct set: upper_le]:
  assumes le: "t ≤♯ u"
  assumes 1: "⋀a b. a ⊑ b ⟹ P (PDUnit a) (PDUnit b)"
  assumes 2: "⋀t u a. P t (PDUnit a) ⟹ P (PDPlus t u) (PDUnit a)"
  assumes 3: "⋀t u v. ⟦P t u; P t v⟧ ⟹ P t (PDPlus u v)"
  shows "P t u"
  using le
proof (induct u arbitrary: t rule: pd_basis_induct)
  case (PDUnit a)
  then show ?case
  proof (induct t rule: pd_basis_induct)
    case PDUnit
    then show ?case by (simp add: 1)
  next
    case (PDPlus t u)
    from PDPlus(3) consider (t) "t ≤♯ PDUnit a" | (u) "u ≤♯ PDUnit a"
      by (auto simp: upper_le_PDPlus_PDUnit_iff)
    then show ?case
    proof cases
      case t
      then have "P t (PDUnit a)" by (rule PDPlus(1))
      then show ?thesis by (rule 2)
    next
      case u
      then have "P u (PDUnit a)" by (rule PDPlus(2))
      then have "P (PDPlus u t) (PDUnit a)" by (rule 2)
      then show ?thesis by (simp only: PDPlus_commute)
    qed
  qed
next
  case (PDPlus t t' u)
  then show ?case by (simp add: upper_le_PDPlus_iff 3)
qed


subsection ‹Type definition›

typedef 'a::bifinite upper_pd  (‹(‹notation=‹postfix upper_pd››'(_')♯)›) =
  "{S::'a pd_basis set. upper_le.ideal S}"
by (rule upper_le.ex_ideal)

instantiation upper_pd :: (bifinite) below
begin

definition
  "x ⊑ y ⟷ Rep_upper_pd x ⊆ Rep_upper_pd y"

instance ..
end

instance upper_pd :: (bifinite) po
using type_definition_upper_pd below_upper_pd_def
by (rule upper_le.typedef_ideal_po)

instance upper_pd :: (bifinite) cpo
using type_definition_upper_pd below_upper_pd_def
by (rule upper_le.typedef_ideal_cpo)

definition
  upper_principal :: "'a::bifinite pd_basis ⇒ 'a upper_pd" where
  "upper_principal t = Abs_upper_pd {u. u ≤♯ t}"

interpretation upper_pd:
  ideal_completion upper_le upper_principal Rep_upper_pd
using type_definition_upper_pd below_upper_pd_def
using upper_principal_def pd_basis_countable
by (rule upper_le.typedef_ideal_completion)

text ‹Upper powerdomain is pointed›

lemma upper_pd_minimal: "upper_principal (PDUnit compact_bot) ⊑ ys"
by (induct ys rule: upper_pd.principal_induct, simp, simp)

instance upper_pd :: (bifinite) pcpo
by intro_classes (fast intro: upper_pd_minimal)

lemma inst_upper_pd_pcpo: "⊥ = upper_principal (PDUnit compact_bot)"
by (rule upper_pd_minimal [THEN bottomI, symmetric])


subsection ‹Monadic unit and plus›

definition
  upper_unit :: "'a::bifinite → 'a upper_pd" where
  "upper_unit = compact_basis.extension (λa. upper_principal (PDUnit a))"

definition
  upper_plus :: "'a::bifinite upper_pd → 'a upper_pd → 'a upper_pd" where
  "upper_plus = upper_pd.extension (λt. upper_pd.extension (λu.
      upper_principal (PDPlus t u)))"

abbreviation
  upper_add :: "'a::bifinite upper_pd ⇒ 'a upper_pd ⇒ 'a upper_pd"
    (infixl ‹∪♯› 65) where
  "xs ∪♯ ys == upper_plus⋅xs⋅ys"

syntax
  "_upper_pd" :: "args ⇒ logic"  (‹(‹indent=1 notation=‹mixfix upper_pd enumeration››{_}♯)›)
translations
  "{x,xs}♯" == "{x}♯ ∪♯ {xs}♯"
  "{x}♯" == "CONST upper_unit⋅x"

lemma upper_unit_Rep_compact_basis [simp]:
  "{Rep_compact_basis a}♯ = upper_principal (PDUnit a)"
unfolding upper_unit_def
by (simp add: compact_basis.extension_principal PDUnit_upper_mono)

lemma upper_plus_principal [simp]:
  "upper_principal t ∪♯ upper_principal u = upper_principal (PDPlus t u)"
unfolding upper_plus_def
by (simp add: upper_pd.extension_principal
    upper_pd.extension_mono PDPlus_upper_mono)

interpretation upper_add: semilattice upper_add proof
  fix xs ys zs :: "'a upper_pd"
  show "(xs ∪♯ ys) ∪♯ zs = xs ∪♯ (ys ∪♯ zs)"
    apply (induct xs rule: upper_pd.principal_induct, simp)
    apply (induct ys rule: upper_pd.principal_induct, simp)
    apply (induct zs rule: upper_pd.principal_induct, simp)
    apply (simp add: PDPlus_assoc)
    done
  show "xs ∪♯ ys = ys ∪♯ xs"
    apply (induct xs rule: upper_pd.principal_induct, simp)
    apply (induct ys rule: upper_pd.principal_induct, simp)
    apply (simp add: PDPlus_commute)
    done
  show "xs ∪♯ xs = xs"
    apply (induct xs rule: upper_pd.principal_induct, simp)
    apply (simp add: PDPlus_absorb)
    done
qed

lemmas upper_plus_assoc = upper_add.assoc
lemmas upper_plus_commute = upper_add.commute
lemmas upper_plus_absorb = upper_add.idem
lemmas upper_plus_left_commute = upper_add.left_commute
lemmas upper_plus_left_absorb = upper_add.left_idem

text ‹Useful for ‹simp add: upper_plus_ac››
lemmas upper_plus_ac =
  upper_plus_assoc upper_plus_commute upper_plus_left_commute

text ‹Useful for ‹simp only: upper_plus_aci››
lemmas upper_plus_aci =
  upper_plus_ac upper_plus_absorb upper_plus_left_absorb

lemma upper_plus_below1: "xs ∪♯ ys ⊑ xs"
apply (induct xs rule: upper_pd.principal_induct, simp)
apply (induct ys rule: upper_pd.principal_induct, simp)
apply (simp add: PDPlus_upper_le)
done

lemma upper_plus_below2: "xs ∪♯ ys ⊑ ys"
by (subst upper_plus_commute, rule upper_plus_below1)

lemma upper_plus_greatest: "⟦xs ⊑ ys; xs ⊑ zs⟧ ⟹ xs ⊑ ys ∪♯ zs"
apply (subst upper_plus_absorb [of xs, symmetric])
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done

lemma upper_below_plus_iff [simp]:
  "xs ⊑ ys ∪♯ zs ⟷ xs ⊑ ys ∧ xs ⊑ zs"
apply safe
apply (erule below_trans [OF _ upper_plus_below1])
apply (erule below_trans [OF _ upper_plus_below2])
apply (erule (1) upper_plus_greatest)
done

lemma upper_plus_below_unit_iff [simp]:
  "xs ∪♯ ys ⊑ {z}♯ ⟷ xs ⊑ {z}♯ ∨ ys ⊑ {z}♯"
apply (induct xs rule: upper_pd.principal_induct, simp)
apply (induct ys rule: upper_pd.principal_induct, simp)
apply (induct z rule: compact_basis.principal_induct, simp)
apply (simp add: upper_le_PDPlus_PDUnit_iff)
done

lemma upper_unit_below_iff [simp]: "{x}♯ ⊑ {y}♯ ⟷ x ⊑ y"
apply (induct x rule: compact_basis.principal_induct, simp)
apply (induct y rule: compact_basis.principal_induct, simp)
apply simp
done

lemmas upper_pd_below_simps =
  upper_unit_below_iff
  upper_below_plus_iff
  upper_plus_below_unit_iff

lemma upper_unit_eq_iff [simp]: "{x}♯ = {y}♯ ⟷ x = y"
unfolding po_eq_conv by simp

lemma upper_unit_strict [simp]: "{⊥}♯ = ⊥"
using upper_unit_Rep_compact_basis [of compact_bot]
by (simp add: inst_upper_pd_pcpo)

lemma upper_plus_strict1 [simp]: "⊥ ∪♯ ys = ⊥"
by (rule bottomI, rule upper_plus_below1)

lemma upper_plus_strict2 [simp]: "xs ∪♯ ⊥ = ⊥"
by (rule bottomI, rule upper_plus_below2)

lemma upper_unit_bottom_iff [simp]: "{x}♯ = ⊥ ⟷ x = ⊥"
unfolding upper_unit_strict [symmetric] by (rule upper_unit_eq_iff)

lemma upper_plus_bottom_iff [simp]:
  "xs ∪♯ ys = ⊥ ⟷ xs = ⊥ ∨ ys = ⊥"
apply (induct xs rule: upper_pd.principal_induct, simp)
apply (induct ys rule: upper_pd.principal_induct, simp)
apply (simp add: inst_upper_pd_pcpo upper_pd.principal_eq_iff
                 upper_le_PDPlus_PDUnit_iff)
done

lemma compact_upper_unit: "compact x ⟹ compact {x}♯"
by (auto dest!: compact_basis.compact_imp_principal)

lemma compact_upper_unit_iff [simp]: "compact {x}♯ ⟷ compact x"
apply (safe elim!: compact_upper_unit)
apply (simp only: compact_def upper_unit_below_iff [symmetric])
apply (erule adm_subst [OF cont_Rep_cfun2])
done

lemma compact_upper_plus [simp]:
  "⟦compact xs; compact ys⟧ ⟹ compact (xs ∪♯ ys)"
by (auto dest!: upper_pd.compact_imp_principal)


subsection ‹Induction rules›

lemma upper_pd_induct1:
  assumes P: "adm P"
  assumes unit: "⋀x. P {x}♯"
  assumes insert: "⋀x ys. ⟦P {x}♯; P ys⟧ ⟹ P ({x}♯ ∪♯ ys)"
  shows "P (xs::'a::bifinite upper_pd)"
proof (induct xs rule: upper_pd.principal_induct)
  have *: "P {Rep_compact_basis a}♯" for a
    by (rule unit)
  show "P (upper_principal a)" for a
  proof (induct a rule: pd_basis_induct1)
    case (PDUnit a)
    with * show ?case
      by (simp only: upper_unit_Rep_compact_basis [symmetric])
  next
    case (PDPlus a t)
    with * have "P ({Rep_compact_basis a}♯ ∪♯ upper_principal t)"
      by (rule insert)
    then show ?case
      by (simp only: upper_unit_Rep_compact_basis [symmetric]
          upper_plus_principal [symmetric])
  qed
qed (rule P)

lemma upper_pd_induct [case_names adm upper_unit upper_plus, induct type: upper_pd]:
  assumes P: "adm P"
  assumes unit: "⋀x. P {x}♯"
  assumes plus: "⋀xs ys. ⟦P xs; P ys⟧ ⟹ P (xs ∪♯ ys)"
  shows "P (xs::'a::bifinite upper_pd)"
proof (induct xs rule: upper_pd.principal_induct)
  show "P (upper_principal a)" for a
  proof (induct a rule: pd_basis_induct)
    case PDUnit
    then show ?case
      by (simp only: upper_unit_Rep_compact_basis [symmetric] unit)
  next
    case PDPlus
    then show ?case
      by (simp only: upper_plus_principal [symmetric] plus)
  qed
qed (rule P)


subsection ‹Monadic bind›

definition
  upper_bind_basis ::
  "'a::bifinite pd_basis ⇒ ('a → 'b upper_pd) → 'b::bifinite upper_pd" where
  "upper_bind_basis = fold_pd
    (λa. Λ f. f⋅(Rep_compact_basis a))
    (λx y. Λ f. x⋅f ∪♯ y⋅f)"

lemma ACI_upper_bind:
  "semilattice (λx y. Λ f. x⋅f ∪♯ y⋅f)"
apply unfold_locales
apply (simp add: upper_plus_assoc)
apply (simp add: upper_plus_commute)
apply (simp add: eta_cfun)
done

lemma upper_bind_basis_simps [simp]:
  "upper_bind_basis (PDUnit a) =
    (Λ f. f⋅(Rep_compact_basis a))"
  "upper_bind_basis (PDPlus t u) =
    (Λ f. upper_bind_basis t⋅f ∪♯ upper_bind_basis u⋅f)"
unfolding upper_bind_basis_def
apply -
apply (rule fold_pd_PDUnit [OF ACI_upper_bind])
apply (rule fold_pd_PDPlus [OF ACI_upper_bind])
done

lemma upper_bind_basis_mono:
  "t ≤♯ u ⟹ upper_bind_basis t ⊑ upper_bind_basis u"
unfolding cfun_below_iff
apply (erule upper_le_induct, safe)
apply (simp add: monofun_cfun)
apply (simp add: below_trans [OF upper_plus_below1])
apply simp
done

definition
  upper_bind :: "'a::bifinite upper_pd → ('a → 'b upper_pd) → 'b::bifinite upper_pd" where
  "upper_bind = upper_pd.extension upper_bind_basis"

syntax
  "_upper_bind" :: "[logic, logic, logic] ⇒ logic"
    (‹(‹indent=3 notation=‹binder upper_bind››⋃♯_∈_./ _)› [0, 0, 10] 10)

translations
  "⋃♯x∈xs. e" == "CONST upper_bind⋅xs⋅(Λ x. e)"

lemma upper_bind_principal [simp]:
  "upper_bind⋅(upper_principal t) = upper_bind_basis t"
unfolding upper_bind_def
apply (rule upper_pd.extension_principal)
apply (erule upper_bind_basis_mono)
done

lemma upper_bind_unit [simp]:
  "upper_bind⋅{x}♯⋅f = f⋅x"
by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma upper_bind_plus [simp]:
  "upper_bind⋅(xs ∪♯ ys)⋅f = upper_bind⋅xs⋅f ∪♯ upper_bind⋅ys⋅f"
by (induct xs rule: upper_pd.principal_induct, simp,
    induct ys rule: upper_pd.principal_induct, simp, simp)

lemma upper_bind_strict [simp]: "upper_bind⋅⊥⋅f = f⋅⊥"
unfolding upper_unit_strict [symmetric] by (rule upper_bind_unit)

lemma upper_bind_bind:
  "upper_bind⋅(upper_bind⋅xs⋅f)⋅g = upper_bind⋅xs⋅(Λ x. upper_bind⋅(f⋅x)⋅g)"
by (induct xs, simp_all)


subsection ‹Map›

definition
  upper_map :: "('a::bifinite → 'b::bifinite) → 'a upper_pd → 'b upper_pd" where
  "upper_map = (Λ f xs. upper_bind⋅xs⋅(Λ x. {f⋅x}♯))"

lemma upper_map_unit [simp]:
  "upper_map⋅f⋅{x}♯ = {f⋅x}♯"
unfolding upper_map_def by simp

lemma upper_map_plus [simp]:
  "upper_map⋅f⋅(xs ∪♯ ys) = upper_map⋅f⋅xs ∪♯ upper_map⋅f⋅ys"
unfolding upper_map_def by simp

lemma upper_map_bottom [simp]: "upper_map⋅f⋅⊥ = {f⋅⊥}♯"
unfolding upper_map_def by simp

lemma upper_map_ident: "upper_map⋅(Λ x. x)⋅xs = xs"
by (induct xs rule: upper_pd_induct, simp_all)

lemma upper_map_ID: "upper_map⋅ID = ID"
by (simp add: cfun_eq_iff ID_def upper_map_ident)

lemma upper_map_map:
  "upper_map⋅f⋅(upper_map⋅g⋅xs) = upper_map⋅(Λ x. f⋅(g⋅x))⋅xs"
by (induct xs rule: upper_pd_induct, simp_all)

lemma upper_bind_map:
  "upper_bind⋅(upper_map⋅f⋅xs)⋅g = upper_bind⋅xs⋅(Λ x. g⋅(f⋅x))"
by (simp add: upper_map_def upper_bind_bind)

lemma upper_map_bind:
  "upper_map⋅f⋅(upper_bind⋅xs⋅g) = upper_bind⋅xs⋅(Λ x. upper_map⋅f⋅(g⋅x))"
by (simp add: upper_map_def upper_bind_bind)

lemma ep_pair_upper_map: "ep_pair e p ⟹ ep_pair (upper_map⋅e) (upper_map⋅p)"
apply standard
apply (induct_tac x rule: upper_pd_induct, simp_all add: ep_pair.e_inverse)
apply (induct_tac y rule: upper_pd_induct)
apply (simp_all add: ep_pair.e_p_below monofun_cfun del: upper_below_plus_iff)
done

lemma deflation_upper_map: "deflation d ⟹ deflation (upper_map⋅d)"
apply standard
apply (induct_tac x rule: upper_pd_induct, simp_all add: deflation.idem)
apply (induct_tac x rule: upper_pd_induct)
apply (simp_all add: deflation.below monofun_cfun del: upper_below_plus_iff)
done

(* FIXME: long proof! *)
lemma finite_deflation_upper_map:
  assumes "finite_deflation d" shows "finite_deflation (upper_map⋅d)"
proof (rule finite_deflation_intro)
  interpret d: finite_deflation d by fact
  from d.deflation_axioms show "deflation (upper_map⋅d)"
    by (rule deflation_upper_map)
  have "finite (range (λx. d⋅x))" by (rule d.finite_range)
  hence "finite (Rep_compact_basis -` range (λx. d⋅x))"
    by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
  hence "finite (Pow (Rep_compact_basis -` range (λx. d⋅x)))" by simp
  hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (λx. d⋅x))))"
    by (rule finite_vimageI, simp add: inj_on_def Rep_pd_basis_inject)
  hence *: "finite (upper_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (λx. d⋅x))))" by simp
  hence "finite (range (λxs. upper_map⋅d⋅xs))"
    apply (rule rev_finite_subset)
    apply clarsimp
    apply (induct_tac xs rule: upper_pd.principal_induct)
    apply (simp add: adm_mem_finite *)
    apply (rename_tac t, induct_tac t rule: pd_basis_induct)
    apply (simp only: upper_unit_Rep_compact_basis [symmetric] upper_map_unit)
    apply simp
    apply (subgoal_tac "∃b. d⋅(Rep_compact_basis a) = Rep_compact_basis b")
    apply clarsimp
    apply (rule imageI)
    apply (rule vimageI2)
    apply (simp add: Rep_PDUnit)
    apply (rule range_eqI)
    apply (erule sym)
    apply (rule exI)
    apply (rule Abs_compact_basis_inverse [symmetric])
    apply (simp add: d.compact)
    apply (simp only: upper_plus_principal [symmetric] upper_map_plus)
    apply clarsimp
    apply (rule imageI)
    apply (rule vimageI2)
    apply (simp add: Rep_PDPlus)
    done
  thus "finite {xs. upper_map⋅d⋅xs = xs}"
    by (rule finite_range_imp_finite_fixes)
qed


subsection ‹Upper powerdomain is bifinite›

lemma approx_chain_upper_map:
  assumes "approx_chain a"
  shows "approx_chain (λi. upper_map⋅(a i))"
  using assms unfolding approx_chain_def
  by (simp add: lub_APP upper_map_ID finite_deflation_upper_map)

instance upper_pd :: (bifinite) bifinite
proof
  show "∃(a::nat ⇒ 'a upper_pd → 'a upper_pd). approx_chain a"
    using bifinite [where 'a='a]
    by (fast intro!: approx_chain_upper_map)
qed


subsection ‹Join›

definition
  upper_join :: "'a::bifinite upper_pd upper_pd → 'a upper_pd" where
  "upper_join = (Λ xss. upper_bind⋅xss⋅(Λ xs. xs))"

lemma upper_join_unit [simp]:
  "upper_join⋅{xs}♯ = xs"
unfolding upper_join_def by simp

lemma upper_join_plus [simp]:
  "upper_join⋅(xss ∪♯ yss) = upper_join⋅xss ∪♯ upper_join⋅yss"
unfolding upper_join_def by simp

lemma upper_join_bottom [simp]: "upper_join⋅⊥ = ⊥"
unfolding upper_join_def by simp

lemma upper_join_map_unit:
  "upper_join⋅(upper_map⋅upper_unit⋅xs) = xs"
by (induct xs rule: upper_pd_induct, simp_all)

lemma upper_join_map_join:
  "upper_join⋅(upper_map⋅upper_join⋅xsss) = upper_join⋅(upper_join⋅xsss)"
by (induct xsss rule: upper_pd_induct, simp_all)

lemma upper_join_map_map:
  "upper_join⋅(upper_map⋅(upper_map⋅f)⋅xss) =
   upper_map⋅f⋅(upper_join⋅xss)"
by (induct xss rule: upper_pd_induct, simp_all)

end