# Theory LowerPD

theory LowerPD
imports Compact_Basis
```(*  Title:      HOL/HOLCF/LowerPD.thy
Author:     Brian Huffman
*)

section ‹Lower powerdomain›

theory LowerPD
imports Compact_Basis
begin

subsection ‹Basis preorder›

definition
lower_le :: "'a pd_basis ⇒ 'a pd_basis ⇒ bool" (infix "≤♭" 50) where
"lower_le = (λu v. ∀x∈Rep_pd_basis u. ∃y∈Rep_pd_basis v. x ⊑ y)"

lemma lower_le_refl [simp]: "t ≤♭ t"
unfolding lower_le_def by fast

lemma lower_le_trans: "⟦t ≤♭ u; u ≤♭ v⟧ ⟹ t ≤♭ v"
unfolding lower_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 lower_le: preorder lower_le
by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans)

lemma lower_le_minimal [simp]: "PDUnit compact_bot ≤♭ t"
unfolding lower_le_def Rep_PDUnit
by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv])

lemma PDUnit_lower_mono: "x ⊑ y ⟹ PDUnit x ≤♭ PDUnit y"
unfolding lower_le_def Rep_PDUnit by fast

lemma PDPlus_lower_mono: "⟦s ≤♭ t; u ≤♭ v⟧ ⟹ PDPlus s u ≤♭ PDPlus t v"
unfolding lower_le_def Rep_PDPlus by fast

lemma PDPlus_lower_le: "t ≤♭ PDPlus t u"
unfolding lower_le_def Rep_PDPlus by fast

lemma lower_le_PDUnit_PDUnit_iff [simp]:
"(PDUnit a ≤♭ PDUnit b) = (a ⊑ b)"
unfolding lower_le_def Rep_PDUnit by fast

lemma lower_le_PDUnit_PDPlus_iff:
"(PDUnit a ≤♭ PDPlus t u) = (PDUnit a ≤♭ t ∨ PDUnit a ≤♭ u)"
unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast

lemma lower_le_PDPlus_iff: "(PDPlus t u ≤♭ v) = (t ≤♭ v ∧ u ≤♭ v)"
unfolding lower_le_def Rep_PDPlus by fast

lemma lower_le_induct [induct set: lower_le]:
assumes le: "t ≤♭ u"
assumes 1: "⋀a b. a ⊑ b ⟹ P (PDUnit a) (PDUnit b)"
assumes 2: "⋀t u a. P (PDUnit a) t ⟹ P (PDUnit a) (PDPlus t u)"
assumes 3: "⋀t u v. ⟦P t v; P u v⟧ ⟹ P (PDPlus t u) v"
shows "P t u"
using le
apply (induct t arbitrary: u rule: pd_basis_induct)
apply (erule rev_mp)
apply (induct_tac u rule: pd_basis_induct)
apply (simp add: 1)
apply (simp add: lower_le_PDUnit_PDPlus_iff)
apply (simp add: 2)
apply (subst PDPlus_commute)
apply (simp add: 2)
apply (simp add: lower_le_PDPlus_iff 3)
done

subsection ‹Type definition›

typedef 'a lower_pd  ("('(_')♭)") =
"{S::'a pd_basis set. lower_le.ideal S}"
by (rule lower_le.ex_ideal)

instantiation lower_pd :: (bifinite) below
begin

definition
"x ⊑ y ⟷ Rep_lower_pd x ⊆ Rep_lower_pd y"

instance ..
end

instance lower_pd :: (bifinite) po
using type_definition_lower_pd below_lower_pd_def
by (rule lower_le.typedef_ideal_po)

instance lower_pd :: (bifinite) cpo
using type_definition_lower_pd below_lower_pd_def
by (rule lower_le.typedef_ideal_cpo)

definition
lower_principal :: "'a pd_basis ⇒ 'a lower_pd" where
"lower_principal t = Abs_lower_pd {u. u ≤♭ t}"

interpretation lower_pd:
ideal_completion lower_le lower_principal Rep_lower_pd
using type_definition_lower_pd below_lower_pd_def
using lower_principal_def pd_basis_countable
by (rule lower_le.typedef_ideal_completion)

text ‹Lower powerdomain is pointed›

lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) ⊑ ys"
by (induct ys rule: lower_pd.principal_induct, simp, simp)

instance lower_pd :: (bifinite) pcpo
by intro_classes (fast intro: lower_pd_minimal)

lemma inst_lower_pd_pcpo: "⊥ = lower_principal (PDUnit compact_bot)"
by (rule lower_pd_minimal [THEN bottomI, symmetric])

subsection ‹Monadic unit and plus›

definition
lower_unit :: "'a → 'a lower_pd" where
"lower_unit = compact_basis.extension (λa. lower_principal (PDUnit a))"

definition
lower_plus :: "'a lower_pd → 'a lower_pd → 'a lower_pd" where
"lower_plus = lower_pd.extension (λt. lower_pd.extension (λu.
lower_principal (PDPlus t u)))"

abbreviation
lower_add :: "'a lower_pd ⇒ 'a lower_pd ⇒ 'a lower_pd"
(infixl "∪♭" 65) where
"xs ∪♭ ys == lower_plus⋅xs⋅ys"

syntax
"_lower_pd" :: "args ⇒ logic" ("{_}♭")

translations
"{x,xs}♭" == "{x}♭ ∪♭ {xs}♭"
"{x}♭" == "CONST lower_unit⋅x"

lemma lower_unit_Rep_compact_basis [simp]:
"{Rep_compact_basis a}♭ = lower_principal (PDUnit a)"
unfolding lower_unit_def
by (simp add: compact_basis.extension_principal PDUnit_lower_mono)

lemma lower_plus_principal [simp]:
"lower_principal t ∪♭ lower_principal u = lower_principal (PDPlus t u)"
unfolding lower_plus_def
by (simp add: lower_pd.extension_principal
lower_pd.extension_mono PDPlus_lower_mono)

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

lemmas lower_plus_assoc = lower_add.assoc
lemmas lower_plus_commute = lower_add.commute
lemmas lower_plus_absorb = lower_add.idem
lemmas lower_plus_left_commute = lower_add.left_commute
lemmas lower_plus_left_absorb = lower_add.left_idem

text ‹Useful for ‹simp add: lower_plus_ac››
lemmas lower_plus_ac =
lower_plus_assoc lower_plus_commute lower_plus_left_commute

text ‹Useful for ‹simp only: lower_plus_aci››
lemmas lower_plus_aci =
lower_plus_ac lower_plus_absorb lower_plus_left_absorb

lemma lower_plus_below1: "xs ⊑ xs ∪♭ ys"
apply (induct xs rule: lower_pd.principal_induct, simp)
apply (induct ys rule: lower_pd.principal_induct, simp)
apply (simp add: PDPlus_lower_le)
done

lemma lower_plus_below2: "ys ⊑ xs ∪♭ ys"
by (subst lower_plus_commute, rule lower_plus_below1)

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

lemma lower_plus_below_iff [simp]:
"xs ∪♭ ys ⊑ zs ⟷ xs ⊑ zs ∧ ys ⊑ zs"
apply safe
apply (erule below_trans [OF lower_plus_below1])
apply (erule below_trans [OF lower_plus_below2])
apply (erule (1) lower_plus_least)
done

lemma lower_unit_below_plus_iff [simp]:
"{x}♭ ⊑ ys ∪♭ zs ⟷ {x}♭ ⊑ ys ∨ {x}♭ ⊑ zs"
apply (induct x rule: compact_basis.principal_induct, simp)
apply (induct ys rule: lower_pd.principal_induct, simp)
apply (induct zs rule: lower_pd.principal_induct, simp)
apply (simp add: lower_le_PDUnit_PDPlus_iff)
done

lemma lower_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 lower_pd_below_simps =
lower_unit_below_iff
lower_plus_below_iff
lower_unit_below_plus_iff

lemma lower_unit_eq_iff [simp]: "{x}♭ = {y}♭ ⟷ x = y"
by (simp add: po_eq_conv)

lemma lower_unit_strict [simp]: "{⊥}♭ = ⊥"
using lower_unit_Rep_compact_basis [of compact_bot]
by (simp add: inst_lower_pd_pcpo)

lemma lower_unit_bottom_iff [simp]: "{x}♭ = ⊥ ⟷ x = ⊥"
unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff)

lemma lower_plus_bottom_iff [simp]:
"xs ∪♭ ys = ⊥ ⟷ xs = ⊥ ∧ ys = ⊥"
apply safe
apply (rule bottomI, erule subst, rule lower_plus_below1)
apply (rule bottomI, erule subst, rule lower_plus_below2)
apply (rule lower_plus_absorb)
done

lemma lower_plus_strict1 [simp]: "⊥ ∪♭ ys = ys"
apply (rule below_antisym [OF _ lower_plus_below2])
apply (simp add: lower_plus_least)
done

lemma lower_plus_strict2 [simp]: "xs ∪♭ ⊥ = xs"
apply (rule below_antisym [OF _ lower_plus_below1])
apply (simp add: lower_plus_least)
done

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

lemma compact_lower_unit_iff [simp]: "compact {x}♭ ⟷ compact x"
apply (safe elim!: compact_lower_unit)
apply (simp only: compact_def lower_unit_below_iff [symmetric])
apply (erule adm_subst [OF cont_Rep_cfun2])
done

lemma compact_lower_plus [simp]:
"⟦compact xs; compact ys⟧ ⟹ compact (xs ∪♭ ys)"
by (auto dest!: lower_pd.compact_imp_principal)

subsection ‹Induction rules›

lemma lower_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 lower_pd)"
apply (induct xs rule: lower_pd.principal_induct, rule P)
apply (induct_tac a rule: pd_basis_induct1)
apply (simp only: lower_unit_Rep_compact_basis [symmetric])
apply (rule unit)
apply (simp only: lower_unit_Rep_compact_basis [symmetric]
lower_plus_principal [symmetric])
apply (erule insert [OF unit])
done

lemma lower_pd_induct
[case_names adm lower_unit lower_plus, induct type: lower_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 lower_pd)"
apply (induct xs rule: lower_pd.principal_induct, rule P)
apply (induct_tac a rule: pd_basis_induct)
apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit)
apply (simp only: lower_plus_principal [symmetric] plus)
done

subsection ‹Monadic bind›

definition
lower_bind_basis ::
"'a pd_basis ⇒ ('a → 'b lower_pd) → 'b lower_pd" where
"lower_bind_basis = fold_pd
(λa. Λ f. f⋅(Rep_compact_basis a))
(λx y. Λ f. x⋅f ∪♭ y⋅f)"

lemma ACI_lower_bind:
"semilattice (λx y. Λ f. x⋅f ∪♭ y⋅f)"
apply unfold_locales
apply (simp add: lower_plus_assoc)
apply (simp add: lower_plus_commute)
apply (simp add: eta_cfun)
done

lemma lower_bind_basis_simps [simp]:
"lower_bind_basis (PDUnit a) =
(Λ f. f⋅(Rep_compact_basis a))"
"lower_bind_basis (PDPlus t u) =
(Λ f. lower_bind_basis t⋅f ∪♭ lower_bind_basis u⋅f)"
unfolding lower_bind_basis_def
apply -
apply (rule fold_pd_PDUnit [OF ACI_lower_bind])
apply (rule fold_pd_PDPlus [OF ACI_lower_bind])
done

lemma lower_bind_basis_mono:
"t ≤♭ u ⟹ lower_bind_basis t ⊑ lower_bind_basis u"
unfolding cfun_below_iff
apply (erule lower_le_induct, safe)
apply (simp add: monofun_cfun)
apply (simp add: rev_below_trans [OF lower_plus_below1])
apply simp
done

definition
lower_bind :: "'a lower_pd → ('a → 'b lower_pd) → 'b lower_pd" where
"lower_bind = lower_pd.extension lower_bind_basis"

syntax
"_lower_bind" :: "[logic, logic, logic] ⇒ logic"
("(3⋃♭_∈_./ _)" [0, 0, 10] 10)

translations
"⋃♭x∈xs. e" == "CONST lower_bind⋅xs⋅(Λ x. e)"

lemma lower_bind_principal [simp]:
"lower_bind⋅(lower_principal t) = lower_bind_basis t"
unfolding lower_bind_def
apply (rule lower_pd.extension_principal)
apply (erule lower_bind_basis_mono)
done

lemma lower_bind_unit [simp]:
"lower_bind⋅{x}♭⋅f = f⋅x"
by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma lower_bind_plus [simp]:
"lower_bind⋅(xs ∪♭ ys)⋅f = lower_bind⋅xs⋅f ∪♭ lower_bind⋅ys⋅f"
by (induct xs rule: lower_pd.principal_induct, simp,
induct ys rule: lower_pd.principal_induct, simp, simp)

lemma lower_bind_strict [simp]: "lower_bind⋅⊥⋅f = f⋅⊥"
unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit)

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

subsection ‹Map›

definition
lower_map :: "('a → 'b) → 'a lower_pd → 'b lower_pd" where
"lower_map = (Λ f xs. lower_bind⋅xs⋅(Λ x. {f⋅x}♭))"

lemma lower_map_unit [simp]:
"lower_map⋅f⋅{x}♭ = {f⋅x}♭"
unfolding lower_map_def by simp

lemma lower_map_plus [simp]:
"lower_map⋅f⋅(xs ∪♭ ys) = lower_map⋅f⋅xs ∪♭ lower_map⋅f⋅ys"
unfolding lower_map_def by simp

lemma lower_map_bottom [simp]: "lower_map⋅f⋅⊥ = {f⋅⊥}♭"
unfolding lower_map_def by simp

lemma lower_map_ident: "lower_map⋅(Λ x. x)⋅xs = xs"
by (induct xs rule: lower_pd_induct, simp_all)

lemma lower_map_ID: "lower_map⋅ID = ID"
by (simp add: cfun_eq_iff ID_def lower_map_ident)

lemma lower_map_map:
"lower_map⋅f⋅(lower_map⋅g⋅xs) = lower_map⋅(Λ x. f⋅(g⋅x))⋅xs"
by (induct xs rule: lower_pd_induct, simp_all)

lemma lower_bind_map:
"lower_bind⋅(lower_map⋅f⋅xs)⋅g = lower_bind⋅xs⋅(Λ x. g⋅(f⋅x))"
by (simp add: lower_map_def lower_bind_bind)

lemma lower_map_bind:
"lower_map⋅f⋅(lower_bind⋅xs⋅g) = lower_bind⋅xs⋅(Λ x. lower_map⋅f⋅(g⋅x))"
by (simp add: lower_map_def lower_bind_bind)

lemma ep_pair_lower_map: "ep_pair e p ⟹ ep_pair (lower_map⋅e) (lower_map⋅p)"
apply standard
apply (induct_tac x rule: lower_pd_induct, simp_all add: ep_pair.e_inverse)
apply (induct_tac y rule: lower_pd_induct)
apply (simp_all add: ep_pair.e_p_below monofun_cfun del: lower_plus_below_iff)
done

lemma deflation_lower_map: "deflation d ⟹ deflation (lower_map⋅d)"
apply standard
apply (induct_tac x rule: lower_pd_induct, simp_all add: deflation.idem)
apply (induct_tac x rule: lower_pd_induct)
apply (simp_all add: deflation.below monofun_cfun del: lower_plus_below_iff)
done

(* FIXME: long proof! *)
lemma finite_deflation_lower_map:
assumes "finite_deflation d" shows "finite_deflation (lower_map⋅d)"
proof (rule finite_deflation_intro)
interpret d: finite_deflation d by fact
have "deflation d" by fact
thus "deflation (lower_map⋅d)" by (rule deflation_lower_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 (lower_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (λx. d⋅x))))" by simp
hence "finite (range (λxs. lower_map⋅d⋅xs))"
apply (rule rev_finite_subset)
apply clarsimp
apply (induct_tac xs rule: lower_pd.principal_induct)
apply (simp add: adm_mem_finite *)
apply (rename_tac t, induct_tac t rule: pd_basis_induct)
apply (simp only: lower_unit_Rep_compact_basis [symmetric] lower_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: lower_plus_principal [symmetric] lower_map_plus)
apply clarsimp
apply (rule imageI)
apply (rule vimageI2)
apply (simp add: Rep_PDPlus)
done
thus "finite {xs. lower_map⋅d⋅xs = xs}"
by (rule finite_range_imp_finite_fixes)
qed

subsection ‹Lower powerdomain is bifinite›

lemma approx_chain_lower_map:
assumes "approx_chain a"
shows "approx_chain (λi. lower_map⋅(a i))"
using assms unfolding approx_chain_def
by (simp add: lub_APP lower_map_ID finite_deflation_lower_map)

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

subsection ‹Join›

definition
lower_join :: "'a lower_pd lower_pd → 'a lower_pd" where
"lower_join = (Λ xss. lower_bind⋅xss⋅(Λ xs. xs))"

lemma lower_join_unit [simp]:
"lower_join⋅{xs}♭ = xs"
unfolding lower_join_def by simp

lemma lower_join_plus [simp]:
"lower_join⋅(xss ∪♭ yss) = lower_join⋅xss ∪♭ lower_join⋅yss"
unfolding lower_join_def by simp

lemma lower_join_bottom [simp]: "lower_join⋅⊥ = ⊥"
unfolding lower_join_def by simp

lemma lower_join_map_unit:
"lower_join⋅(lower_map⋅lower_unit⋅xs) = xs"
by (induct xs rule: lower_pd_induct, simp_all)

lemma lower_join_map_join:
"lower_join⋅(lower_map⋅lower_join⋅xsss) = lower_join⋅(lower_join⋅xsss)"
by (induct xsss rule: lower_pd_induct, simp_all)

lemma lower_join_map_map:
"lower_join⋅(lower_map⋅(lower_map⋅f)⋅xss) =
lower_map⋅f⋅(lower_join⋅xss)"
by (induct xss rule: lower_pd_induct, simp_all)

end
```