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) 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 ≤\<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_le: "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) below
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 below_upper_pd_def])
instance upper_pd :: (profinite) cpo
by (rule upper_le.typedef_ideal_cpo
[OF type_definition_upper_pd below_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 below_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: below_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)
interpretation upper_add!: semilattice upper_add proof
fix xs ys zs :: "'a upper_pd"
show "(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
show "xs +\<sharp> ys = ys +\<sharp> xs"
apply (induct xs ys rule: upper_pd.principal_induct2, simp, simp)
apply (simp add: PDPlus_commute)
done
show "xs +\<sharp> 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 @{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_below1: "xs +\<sharp> ys \<sqsubseteq> xs"
apply (induct xs ys rule: upper_pd.principal_induct2, simp, simp)
apply (simp add: PDPlus_upper_le)
done
lemma upper_plus_below2: "xs +\<sharp> ys \<sqsubseteq> ys"
by (subst upper_plus_commute, rule upper_plus_below1)
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_below_plus_iff:
"xs \<sqsubseteq> ys +\<sharp> zs <-> xs \<sqsubseteq> ys ∧ xs \<sqsubseteq> 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:
"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 below_trans [OF upper_plus_below1])
apply (erule below_trans [OF upper_plus_below2])
done
lemma upper_unit_below_iff [simp]: "{x}\<sharp> \<sqsubseteq> {y}\<sharp> <-> x \<sqsubseteq> y"
apply (rule iffI)
apply (rule profinite_below_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_below_simps =
upper_unit_below_iff
upper_below_plus_iff
upper_plus_below_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_below1)
lemma upper_plus_strict2 [simp]: "xs +\<sharp> ⊥ = ⊥"
by (rule UU_I, rule upper_plus_below2)
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:
"class.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_below
apply (erule upper_le_induct, safe)
apply (simp add: monofun_cfun)
apply (simp add: below_trans [OF upper_plus_below1])
apply (simp add: upper_below_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_ID: "upper_map·ID = ID"
by (simp add: expand_cfun_eq 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_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)
lemma ep_pair_upper_map: "ep_pair e p ==> ep_pair (upper_map·e) (upper_map·p)"
apply default
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)
done
lemma deflation_upper_map: "deflation d ==> deflation (upper_map·d)"
apply default
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)
done
end