Theory UpperPD

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theory UpperPD
imports CompactBasis

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

header {* Upper powerdomain *}

theory UpperPD
imports CompactBasis
begin

subsection {* Basis preorder *}

definition
  upper_le :: "'a pd_basis => 'a pd_basis => bool" (infix "≤\<sharp>" 50) where
  "upper_le = (λu v. ∀y∈Rep_pd_basis v. ∃x∈Rep_pd_basis u. x \<sqsubseteq> y)"

lemma upper_le_refl [simp]: "t ≤\<sharp> t"
unfolding upper_le_def by fast

lemma upper_le_trans: "[|t ≤\<sharp> u; u ≤\<sharp> v|] ==> t ≤\<sharp> 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) trans_less)
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 ≤\<sharp> t"
unfolding upper_le_def Rep_PDUnit by simp

lemma PDUnit_upper_mono: "x \<sqsubseteq> y ==> PDUnit x ≤\<sharp> PDUnit y"
unfolding upper_le_def Rep_PDUnit by simp

lemma PDPlus_upper_mono: "[|s ≤\<sharp> t; u ≤\<sharp> v|] ==> PDPlus s u ≤\<sharp> PDPlus t v"
unfolding upper_le_def Rep_PDPlus by fast

lemma PDPlus_upper_less: "PDPlus t u ≤\<sharp> t"
unfolding upper_le_def Rep_PDPlus by fast

lemma upper_le_PDUnit_PDUnit_iff [simp]:
  "(PDUnit a ≤\<sharp> PDUnit b) = a \<sqsubseteq> b"
unfolding upper_le_def Rep_PDUnit by fast

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

lemma upper_le_PDPlus_iff: "(t ≤\<sharp> PDPlus u v) = (t ≤\<sharp> u ∧ t ≤\<sharp> v)"
unfolding upper_le_def Rep_PDPlus by fast

lemma upper_le_induct [induct set: upper_le]:
  assumes le: "t ≤\<sharp> u"
  assumes 1: "!!a b. a \<sqsubseteq> 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 apply (induct u arbitrary: t rule: pd_basis_induct)
apply (erule rev_mp)
apply (induct_tac t rule: pd_basis_induct)
apply (simp add: 1)
apply (simp add: upper_le_PDPlus_PDUnit_iff)
apply (simp add: 2)
apply (subst PDPlus_commute)
apply (simp add: 2)
apply (simp add: upper_le_PDPlus_iff 3)
done

lemma pd_take_upper_chain:
  "pd_take n t ≤\<sharp> pd_take (Suc n) t"
apply (induct t rule: pd_basis_induct)
apply (simp add: compact_basis.take_chain)
apply (simp add: PDPlus_upper_mono)
done

lemma pd_take_upper_le: "pd_take i t ≤\<sharp> t"
apply (induct t rule: pd_basis_induct)
apply (simp add: compact_basis.take_less)
apply (simp add: PDPlus_upper_mono)
done

lemma pd_take_upper_mono:
  "t ≤\<sharp> u ==> pd_take n t ≤\<sharp> pd_take n u"
apply (erule upper_le_induct)
apply (simp add: compact_basis.take_mono)
apply (simp add: upper_le_PDPlus_PDUnit_iff)
apply (simp add: upper_le_PDPlus_iff)
done


subsection {* Type definition *}

typedef (open) 'a upper_pd =
  "{S::'a pd_basis set. upper_le.ideal S}"
by (fast intro: upper_le.ideal_principal)

instantiation upper_pd :: (profinite) sq_ord
begin

definition
  "x \<sqsubseteq> y <-> Rep_upper_pd x ⊆ Rep_upper_pd y"

instance ..
end

instance upper_pd :: (profinite) po
by (rule upper_le.typedef_ideal_po
    [OF type_definition_upper_pd sq_le_upper_pd_def])

instance upper_pd :: (profinite) cpo
by (rule upper_le.typedef_ideal_cpo
    [OF type_definition_upper_pd sq_le_upper_pd_def])

lemma Rep_upper_pd_lub:
  "chain Y ==> Rep_upper_pd (\<Squnion>i. Y i) = (\<Union>i. Rep_upper_pd (Y i))"
by (rule upper_le.typedef_ideal_rep_contlub
    [OF type_definition_upper_pd sq_le_upper_pd_def])

lemma ideal_Rep_upper_pd: "upper_le.ideal (Rep_upper_pd xs)"
by (rule Rep_upper_pd [unfolded mem_Collect_eq])

definition
  upper_principal :: "'a pd_basis => 'a upper_pd" where
  "upper_principal t = Abs_upper_pd {u. u ≤\<sharp> t}"

lemma Rep_upper_principal:
  "Rep_upper_pd (upper_principal t) = {u. u ≤\<sharp> t}"
unfolding upper_principal_def
by (simp add: Abs_upper_pd_inverse upper_le.ideal_principal)

interpretation upper_pd:
  ideal_completion upper_le pd_take upper_principal Rep_upper_pd
apply unfold_locales
apply (rule pd_take_upper_le)
apply (rule pd_take_idem)
apply (erule pd_take_upper_mono)
apply (rule pd_take_upper_chain)
apply (rule finite_range_pd_take)
apply (rule pd_take_covers)
apply (rule ideal_Rep_upper_pd)
apply (erule Rep_upper_pd_lub)
apply (rule Rep_upper_principal)
apply (simp only: sq_le_upper_pd_def)
done

text {* Upper powerdomain is pointed *}

lemma upper_pd_minimal: "upper_principal (PDUnit compact_bot) \<sqsubseteq> 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 UU_I, symmetric])

text {* Upper powerdomain is profinite *}

instantiation upper_pd :: (profinite) profinite
begin

definition
  approx_upper_pd_def: "approx = upper_pd.completion_approx"

instance
apply (intro_classes, unfold approx_upper_pd_def)
apply (rule upper_pd.chain_completion_approx)
apply (rule upper_pd.lub_completion_approx)
apply (rule upper_pd.completion_approx_idem)
apply (rule upper_pd.finite_fixes_completion_approx)
done

end

instance upper_pd :: (bifinite) bifinite ..

lemma approx_upper_principal [simp]:
  "approx n·(upper_principal t) = upper_principal (pd_take n t)"
unfolding approx_upper_pd_def
by (rule upper_pd.completion_approx_principal)

lemma approx_eq_upper_principal:
  "∃t∈Rep_upper_pd xs. approx n·xs = upper_principal (pd_take n t)"
unfolding approx_upper_pd_def
by (rule upper_pd.completion_approx_eq_principal)


subsection {* Monadic unit and plus *}

definition
  upper_unit :: "'a -> 'a upper_pd" where
  "upper_unit = compact_basis.basis_fun (λa. upper_principal (PDUnit a))"

definition
  upper_plus :: "'a upper_pd -> 'a upper_pd -> 'a upper_pd" where
  "upper_plus = upper_pd.basis_fun (λt. upper_pd.basis_fun (λu.
      upper_principal (PDPlus t u)))"

abbreviation
  upper_add :: "'a upper_pd => 'a upper_pd => 'a upper_pd"
    (infixl "+\<sharp>" 65) where
  "xs +\<sharp> ys == upper_plus·xs·ys"

syntax
  "_upper_pd" :: "args => 'a upper_pd" ("{_}\<sharp>")

translations
  "{x,xs}\<sharp>" == "{x}\<sharp> +\<sharp> {xs}\<sharp>"
  "{x}\<sharp>" == "CONST upper_unit·x"

lemma upper_unit_Rep_compact_basis [simp]:
  "{Rep_compact_basis a}\<sharp> = upper_principal (PDUnit a)"
unfolding upper_unit_def
by (simp add: compact_basis.basis_fun_principal PDUnit_upper_mono)

lemma upper_plus_principal [simp]:
  "upper_principal t +\<sharp> upper_principal u = upper_principal (PDPlus t u)"
unfolding upper_plus_def
by (simp add: upper_pd.basis_fun_principal
    upper_pd.basis_fun_mono PDPlus_upper_mono)

lemma approx_upper_unit [simp]:
  "approx n·{x}\<sharp> = {approx n·x}\<sharp>"
apply (induct x rule: compact_basis.principal_induct, simp)
apply (simp add: approx_Rep_compact_basis)
done

lemma approx_upper_plus [simp]:
  "approx n·(xs +\<sharp> ys) = (approx n·xs) +\<sharp> (approx n·ys)"
by (induct xs ys rule: upper_pd.principal_induct2, simp, simp, simp)

lemma upper_plus_assoc: "(xs +\<sharp> ys) +\<sharp> zs = xs +\<sharp> (ys +\<sharp> zs)"
apply (induct xs ys arbitrary: zs rule: upper_pd.principal_induct2, simp, simp)
apply (rule_tac x=zs in upper_pd.principal_induct, simp)
apply (simp add: PDPlus_assoc)
done

lemma upper_plus_commute: "xs +\<sharp> ys = ys +\<sharp> xs"
apply (induct xs ys rule: upper_pd.principal_induct2, simp, simp)
apply (simp add: PDPlus_commute)
done

lemma upper_plus_absorb [simp]: "xs +\<sharp> xs = xs"
apply (induct xs rule: upper_pd.principal_induct, simp)
apply (simp add: PDPlus_absorb)
done

lemma upper_plus_left_commute: "xs +\<sharp> (ys +\<sharp> zs) = ys +\<sharp> (xs +\<sharp> zs)"
by (rule mk_left_commute [of "op +\<sharp>", OF upper_plus_assoc upper_plus_commute])

lemma upper_plus_left_absorb [simp]: "xs +\<sharp> (xs +\<sharp> ys) = xs +\<sharp> ys"
by (simp only: upper_plus_assoc [symmetric] upper_plus_absorb)

text {* Useful for @{text "simp add: upper_plus_ac"} *}
lemmas upper_plus_ac =
  upper_plus_assoc upper_plus_commute upper_plus_left_commute

text {* Useful for @{text "simp only: upper_plus_aci"} *}
lemmas upper_plus_aci =
  upper_plus_ac upper_plus_absorb upper_plus_left_absorb

lemma upper_plus_less1: "xs +\<sharp> ys \<sqsubseteq> xs"
apply (induct xs ys rule: upper_pd.principal_induct2, simp, simp)
apply (simp add: PDPlus_upper_less)
done

lemma upper_plus_less2: "xs +\<sharp> ys \<sqsubseteq> ys"
by (subst upper_plus_commute, rule upper_plus_less1)

lemma upper_plus_greatest: "[|xs \<sqsubseteq> ys; xs \<sqsubseteq> zs|] ==> xs \<sqsubseteq> ys +\<sharp> zs"
apply (subst upper_plus_absorb [of xs, symmetric])
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done

lemma upper_less_plus_iff:
  "xs \<sqsubseteq> ys +\<sharp> zs <-> xs \<sqsubseteq> ys ∧ xs \<sqsubseteq> zs"
apply safe
apply (erule trans_less [OF _ upper_plus_less1])
apply (erule trans_less [OF _ upper_plus_less2])
apply (erule (1) upper_plus_greatest)
done

lemma upper_plus_less_unit_iff:
  "xs +\<sharp> ys \<sqsubseteq> {z}\<sharp> <-> xs \<sqsubseteq> {z}\<sharp> ∨ ys \<sqsubseteq> {z}\<sharp>"
 apply (rule iffI)
  apply (subgoal_tac
    "adm (λf. f·xs \<sqsubseteq> f·{z}\<sharp> ∨ f·ys \<sqsubseteq> f·{z}\<sharp>)")
   apply (drule admD, rule chain_approx)
    apply (drule_tac f="approx i" in monofun_cfun_arg)
    apply (cut_tac x="approx i·xs" in upper_pd.compact_imp_principal, simp)
    apply (cut_tac x="approx i·ys" in upper_pd.compact_imp_principal, simp)
    apply (cut_tac x="approx i·z" in compact_basis.compact_imp_principal, simp)
    apply (clarify, simp add: upper_le_PDPlus_PDUnit_iff)
   apply simp
  apply simp
 apply (erule disjE)
  apply (erule trans_less [OF upper_plus_less1])
 apply (erule trans_less [OF upper_plus_less2])
done

lemma upper_unit_less_iff [simp]: "{x}\<sharp> \<sqsubseteq> {y}\<sharp> <-> x \<sqsubseteq> y"
 apply (rule iffI)
  apply (rule profinite_less_ext)
  apply (drule_tac f="approx i" in monofun_cfun_arg, simp)
  apply (cut_tac x="approx i·x" in compact_basis.compact_imp_principal, simp)
  apply (cut_tac x="approx i·y" in compact_basis.compact_imp_principal, simp)
  apply clarsimp
 apply (erule monofun_cfun_arg)
done

lemmas upper_pd_less_simps =
  upper_unit_less_iff
  upper_less_plus_iff
  upper_plus_less_unit_iff

lemma upper_unit_eq_iff [simp]: "{x}\<sharp> = {y}\<sharp> <-> x = y"
unfolding po_eq_conv by simp

lemma upper_unit_strict [simp]: "{⊥}\<sharp> = ⊥"
unfolding inst_upper_pd_pcpo Rep_compact_bot [symmetric] by simp

lemma upper_plus_strict1 [simp]: "⊥ +\<sharp> ys = ⊥"
by (rule UU_I, rule upper_plus_less1)

lemma upper_plus_strict2 [simp]: "xs +\<sharp> ⊥ = ⊥"
by (rule UU_I, rule upper_plus_less2)

lemma upper_unit_strict_iff [simp]: "{x}\<sharp> = ⊥ <-> x = ⊥"
unfolding upper_unit_strict [symmetric] by (rule upper_unit_eq_iff)

lemma upper_plus_strict_iff [simp]:
  "xs +\<sharp> ys = ⊥ <-> xs = ⊥ ∨ ys = ⊥"
apply (rule iffI)
apply (erule rev_mp)
apply (rule upper_pd.principal_induct2 [where x=xs and y=ys], simp, simp)
apply (simp add: inst_upper_pd_pcpo upper_pd.principal_eq_iff
                 upper_le_PDPlus_PDUnit_iff)
apply auto
done

lemma compact_upper_unit_iff [simp]: "compact {x}\<sharp> <-> compact x"
unfolding profinite_compact_iff by simp

lemma compact_upper_plus [simp]:
  "[|compact xs; compact ys|] ==> compact (xs +\<sharp> 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}\<sharp>"
  assumes insert: "!!x ys. [|P {x}\<sharp>; P ys|] ==> P ({x}\<sharp> +\<sharp> ys)"
  shows "P (xs::'a upper_pd)"
apply (induct xs rule: upper_pd.principal_induct, rule P)
apply (induct_tac a rule: pd_basis_induct1)
apply (simp only: upper_unit_Rep_compact_basis [symmetric])
apply (rule unit)
apply (simp only: upper_unit_Rep_compact_basis [symmetric]
                  upper_plus_principal [symmetric])
apply (erule insert [OF unit])
done

lemma upper_pd_induct:
  assumes P: "adm P"
  assumes unit: "!!x. P {x}\<sharp>"
  assumes plus: "!!xs ys. [|P xs; P ys|] ==> P (xs +\<sharp> ys)"
  shows "P (xs::'a upper_pd)"
apply (induct xs rule: upper_pd.principal_induct, rule P)
apply (induct_tac a rule: pd_basis_induct)
apply (simp only: upper_unit_Rep_compact_basis [symmetric] unit)
apply (simp only: upper_plus_principal [symmetric] plus)
done


subsection {* Monadic bind *}

definition
  upper_bind_basis ::
  "'a pd_basis => ('a -> 'b upper_pd) -> 'b upper_pd" where
  "upper_bind_basis = fold_pd
    (λa. Λ f. f·(Rep_compact_basis a))
    (λx y. Λ f. x·f +\<sharp> y·f)"

lemma ACI_upper_bind:
  "ab_semigroup_idem_mult (λx y. Λ f. x·f +\<sharp> 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 +\<sharp> 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 ≤\<sharp> u ==> upper_bind_basis t \<sqsubseteq> upper_bind_basis u"
unfolding expand_cfun_less
apply (erule upper_le_induct, safe)
apply (simp add: monofun_cfun)
apply (simp add: trans_less [OF upper_plus_less1])
apply (simp add: upper_less_plus_iff)
done

definition
  upper_bind :: "'a upper_pd -> ('a -> 'b upper_pd) -> 'b upper_pd" where
  "upper_bind = upper_pd.basis_fun upper_bind_basis"

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

lemma upper_bind_unit [simp]:
  "upper_bind·{x}\<sharp>·f = f·x"
by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma upper_bind_plus [simp]:
  "upper_bind·(xs +\<sharp> ys)·f = upper_bind·xs·f +\<sharp> upper_bind·ys·f"
by (induct xs ys rule: upper_pd.principal_induct2, simp, simp, simp)

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


subsection {* Map and join *}

definition
  upper_map :: "('a -> 'b) -> 'a upper_pd -> 'b upper_pd" where
  "upper_map = (Λ f xs. upper_bind·xs·(Λ x. {f·x}\<sharp>))"

definition
  upper_join :: "'a upper_pd upper_pd -> 'a upper_pd" where
  "upper_join = (Λ xss. upper_bind·xss·(Λ xs. xs))"

lemma upper_map_unit [simp]:
  "upper_map·f·{x}\<sharp> = {f·x}\<sharp>"
unfolding upper_map_def by simp

lemma upper_map_plus [simp]:
  "upper_map·f·(xs +\<sharp> ys) = upper_map·f·xs +\<sharp> upper_map·f·ys"
unfolding upper_map_def by simp

lemma upper_join_unit [simp]:
  "upper_join·{xs}\<sharp> = xs"
unfolding upper_join_def by simp

lemma upper_join_plus [simp]:
  "upper_join·(xss +\<sharp> yss) = upper_join·xss +\<sharp> upper_join·yss"
unfolding upper_join_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_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_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)

lemma upper_map_approx: "upper_map·(approx n)·xs = approx n·xs"
by (induct xs rule: upper_pd_induct, simp_all)

end