Quelle LowerPD.thy Sprache: Isabelle

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

section \<open>Lower powerdomain\<close>

theory LowerPD
imports Compact_Basis
begin

subsection \<open>Basis preorder\<close>

definition
  lower_le :: "'a::bifinite pd_basis \ 'a pd_basis \ bool" (infix \\\\ 50) where
  "lower_le = (\u 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
proof (induct t arbitrary: u rule: pd_basis_induct)
  case (PDUnit a)
  then show ?case
  proof (induct u rule: pd_basis_induct)
    case PDUnit
    then show ?case by (simp add: 1)
  next
    case (PDPlus t u)
    from PDPlus(3) consider (t) "PDUnit a \\ t" | (u) "PDUnit a \\ u"
      by (auto simp: lower_le_PDUnit_PDPlus_iff)
    then show ?case
    proof cases
      case t
      then have "P (PDUnit a) t" by (rule PDPlus(1))
      then show ?thesis by (rule 2)
    next
      case u
      then have "P (PDUnit a) u" by (rule PDPlus(2))
      then have "P (PDUnit a) (PDPlus u t)" by (rule 2)
      then show ?thesis by (simp only: PDPlus_commute)
    qed
  qed
next
  case (PDPlus t t')
  then show ?case by (simp add: lower_le_PDPlus_iff 3)
qed

subsection \<open>Type definition\<close>

typedef 'a::bifinite lower_pd (\(\notation=\postfix lower_pd\\'(_')\)\) =
  "{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::bifinite 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 \<open>Lower powerdomain is pointed\<close>

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 \<open>Monadic unit and plus\<close>

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

definition
  lower_plus :: "'a::bifinite lower_pd \ 'a lower_pd \ 'a lower_pd" where
  "lower_plus = lower_pd.extension (\t. lower_pd.extension (\u.
      lower_principal (PDPlus t u)))"

abbreviation
  lower_add :: "'a::bifinite lower_pd \ 'a lower_pd \ 'a lower_pd"
    (infixl \<open>\<union>\<flat>\<close> 65) where
  "xs \\ ys == lower_plus\xs\ys"

syntax
  "_lower_pd" :: "args \ logic" (\(\indent=1 notation=\mixfix lower_pd enumeration\\{_}\)\)
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::bifinite 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 \<open>Useful for \<open>simp add: lower_plus_ac\<close>\<close>
lemmas lower_plus_ac =
  lower_plus_assoc lower_plus_commute lower_plus_left_commute

text \<open>Useful for \<open>simp only: lower_plus_aci\<close>\<close>
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 \<open>Induction rules\<close>

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::bifinite lower_pd)"
proof (induct xs rule: lower_pd.principal_induct)
  have *: "P {Rep_compact_basis a}\" for a
    by (rule unit)
  show "P (lower_principal a)" for a
  proof (induct a rule: pd_basis_induct1)
    case PDUnit
    from * show ?case
      by (simp only: lower_unit_Rep_compact_basis [symmetric])
  next
    case (PDPlus a t)
    with * have "P ({Rep_compact_basis a}\ \\ lower_principal t)"
      by (rule insert)
    then show ?case
      by (simp only: lower_unit_Rep_compact_basis [symmetric] lower_plus_principal [symmetric])
  qed
qed (rule P)

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::bifinite lower_pd)"
proof (induct xs rule: lower_pd.principal_induct)
  show "P (lower_principal a)" for a
  proof (induct a rule: pd_basis_induct)
    case PDUnit
    then show ?case
      by (simp only: lower_unit_Rep_compact_basis [symmetric] unit)
  next
    case PDPlus
    then show ?case
      by (simp only: lower_plus_principal [symmetric] plus)
  qed
qed (rule P)

subsection \<open>Monadic bind\<close>

definition
  lower_bind_basis ::
  "'a::bifinite pd_basis \ ('a \ 'b lower_pd) \ 'b::bifinite lower_pd" where
  "lower_bind_basis = fold_pd
    (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))
    (\<lambda>x y. \<Lambda> f. x\<cdot>f \<union>\<flat> y\<cdot>f)"

lemma ACI_lower_bind:
  "semilattice (\x y. \ 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) =
    (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"
  "lower_bind_basis (PDPlus t u) =
    (\<Lambda> f. lower_bind_basis t\<cdot>f \<union>\<flat> lower_bind_basis u\<cdot>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::bifinite lower_pd \ ('a \ 'b lower_pd) \ 'b::bifinite lower_pd" where
  "lower_bind = lower_pd.extension lower_bind_basis"

syntax
  "_lower_bind" :: "[logic, logic, logic] \ logic"
    (\<open>(\<open>indent=3 notation=\<open>binder lower_bind\<close>\<close>\<Union>\<flat>_\<in>_./ _)\<close> [0, 0, 10] 10)

translations
  "\\x\xs. e" == "CONST lower_bind\xs\(\ 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\(\ x. lower_bind\(f\x)\g)"
by (induct xs, simp_all)

subsection \<open>Map\<close>

definition
  lower_map :: "('a::bifinite \ 'b::bifinite) \ 'a lower_pd \ 'b lower_pd" where
  "lower_map = (\ f xs. lower_bind\xs\(\ x. {f\x}\))"

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\(\ 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\(\ 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\(\ 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\(\ 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
  from d.deflation_axioms show "deflation (lower_map\d)"
    by (rule deflation_lower_map)
  have "finite (range (\x. d\x))" by (rule d.finite_range)
  hence "finite (Rep_compact_basis -` range (\x. d\x))"
    by (rule finite_vimageI, simp add: inj_on_def Rep_compact_basis_inject)
  hence "finite (Pow (Rep_compact_basis -` range (\x. d\x)))" by simp
  hence "finite (Rep_pd_basis -` (Pow (Rep_compact_basis -` range (\x. 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 (\x. d\x))))" by simp
  hence "finite (range (\xs. 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 \<open>Lower powerdomain is bifinite\<close>

lemma approx_chain_lower_map:
  assumes "approx_chain a"
  shows "approx_chain (\i. 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 \<open>Join\<close>

definition
  lower_join :: "'a::bifinite lower_pd lower_pd \ 'a lower_pd" where
  "lower_join = (\ xss. lower_bind\xss\(\ 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\<cdot>f\<cdot>(lower_join\<cdot>xss)"
by (induct xss rule: lower_pd_induct, simp_all)

end

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