header {* Convex powerdomain *}
theory ConvexPD
imports UpperPD LowerPD
begin
subsection {* Basis preorder *}
definition
convex_le :: "'a pd_basis => 'a pd_basis => bool" (infix "≤\<natural>" 50) where
"convex_le = (λu v. u ≤\<sharp> v ∧ u ≤\<flat> v)"
lemma convex_le_refl [simp]: "t ≤\<natural> t"
unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl)
lemma convex_le_trans: "[|t ≤\<natural> u; u ≤\<natural> v|] ==> t ≤\<natural> v"
unfolding convex_le_def by (fast intro: upper_le_trans lower_le_trans)
interpretation convex_le: preorder convex_le
by (rule preorder.intro, rule convex_le_refl, rule convex_le_trans)
lemma upper_le_minimal [simp]: "PDUnit compact_bot ≤\<natural> t"
unfolding convex_le_def Rep_PDUnit by simp
lemma PDUnit_convex_mono: "x \<sqsubseteq> y ==> PDUnit x ≤\<natural> PDUnit y"
unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono)
lemma PDPlus_convex_mono: "[|s ≤\<natural> t; u ≤\<natural> v|] ==> PDPlus s u ≤\<natural> PDPlus t v"
unfolding convex_le_def by (fast intro: PDPlus_upper_mono PDPlus_lower_mono)
lemma convex_le_PDUnit_PDUnit_iff [simp]:
"(PDUnit a ≤\<natural> PDUnit b) = a \<sqsubseteq> b"
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit by fast
lemma convex_le_PDUnit_lemma1:
"(PDUnit a ≤\<natural> t) = (∀b∈Rep_pd_basis t. a \<sqsubseteq> b)"
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit
using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast
lemma convex_le_PDUnit_PDPlus_iff [simp]:
"(PDUnit a ≤\<natural> PDPlus t u) = (PDUnit a ≤\<natural> t ∧ PDUnit a ≤\<natural> u)"
unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast
lemma convex_le_PDUnit_lemma2:
"(t ≤\<natural> PDUnit b) = (∀a∈Rep_pd_basis t. a \<sqsubseteq> b)"
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit
using Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fast
lemma convex_le_PDPlus_PDUnit_iff [simp]:
"(PDPlus t u ≤\<natural> PDUnit a) = (t ≤\<natural> PDUnit a ∧ u ≤\<natural> PDUnit a)"
unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast
lemma convex_le_PDPlus_lemma:
assumes z: "PDPlus t u ≤\<natural> z"
shows "∃v w. z = PDPlus v w ∧ t ≤\<natural> v ∧ u ≤\<natural> w"
proof (intro exI conjI)
let ?A = "{b∈Rep_pd_basis z. ∃a∈Rep_pd_basis t. a \<sqsubseteq> b}"
let ?B = "{b∈Rep_pd_basis z. ∃a∈Rep_pd_basis u. a \<sqsubseteq> b}"
let ?v = "Abs_pd_basis ?A"
let ?w = "Abs_pd_basis ?B"
have Rep_v: "Rep_pd_basis ?v = ?A"
apply (rule Abs_pd_basis_inverse)
apply (rule Rep_pd_basis_nonempty [of t, folded ex_in_conv, THEN exE])
apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify)
apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE)
apply (simp add: pd_basis_def)
apply fast
done
have Rep_w: "Rep_pd_basis ?w = ?B"
apply (rule Abs_pd_basis_inverse)
apply (rule Rep_pd_basis_nonempty [of u, folded ex_in_conv, THEN exE])
apply (cut_tac z, simp only: convex_le_def lower_le_def, clarify)
apply (drule_tac x=x in bspec, simp add: Rep_PDPlus, erule bexE)
apply (simp add: pd_basis_def)
apply fast
done
show "z = PDPlus ?v ?w"
apply (insert z)
apply (simp add: convex_le_def, erule conjE)
apply (simp add: Rep_pd_basis_inject [symmetric] Rep_PDPlus)
apply (simp add: Rep_v Rep_w)
apply (rule equalityI)
apply (rule subsetI)
apply (simp only: upper_le_def)
apply (drule (1) bspec, erule bexE)
apply (simp add: Rep_PDPlus)
apply fast
apply fast
done
show "t ≤\<natural> ?v" "u ≤\<natural> ?w"
apply (insert z)
apply (simp_all add: convex_le_def upper_le_def lower_le_def Rep_PDPlus Rep_v Rep_w)
apply fast+
done
qed
lemma convex_le_induct [induct set: convex_le]:
assumes le: "t ≤\<natural> u"
assumes 2: "!!t u v. [|P t u; P u v|] ==> P t v"
assumes 3: "!!a b. a \<sqsubseteq> b ==> P (PDUnit a) (PDUnit b)"
assumes 4: "!!t u v w. [|P t v; P u w|] ==> P (PDPlus t u) (PDPlus v w)"
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_induct1)
apply (simp add: 3)
apply (simp, clarify, rename_tac a b t)
apply (subgoal_tac "P (PDPlus (PDUnit a) (PDUnit a)) (PDPlus (PDUnit b) t)")
apply (simp add: PDPlus_absorb)
apply (erule (1) 4 [OF 3])
apply (drule convex_le_PDPlus_lemma, clarify)
apply (simp add: 4)
done
lemma pd_take_convex_chain:
"pd_take n t ≤\<natural> pd_take (Suc n) t"
apply (induct t rule: pd_basis_induct)
apply (simp add: compact_basis.take_chain)
apply (simp add: PDPlus_convex_mono)
done
lemma pd_take_convex_le: "pd_take i t ≤\<natural> t"
apply (induct t rule: pd_basis_induct)
apply (simp add: compact_basis.take_less)
apply (simp add: PDPlus_convex_mono)
done
lemma pd_take_convex_mono:
"t ≤\<natural> u ==> pd_take n t ≤\<natural> pd_take n u"
apply (erule convex_le_induct)
apply (erule (1) convex_le_trans)
apply (simp add: compact_basis.take_mono)
apply (simp add: PDPlus_convex_mono)
done
subsection {* Type definition *}
typedef (open) 'a convex_pd =
"{S::'a pd_basis set. convex_le.ideal S}"
by (fast intro: convex_le.ideal_principal)
instantiation convex_pd :: (profinite) below
begin
definition
"x \<sqsubseteq> y <-> Rep_convex_pd x ⊆ Rep_convex_pd y"
instance ..
end
instance convex_pd :: (profinite) po
by (rule convex_le.typedef_ideal_po
[OF type_definition_convex_pd below_convex_pd_def])
instance convex_pd :: (profinite) cpo
by (rule convex_le.typedef_ideal_cpo
[OF type_definition_convex_pd below_convex_pd_def])
lemma Rep_convex_pd_lub:
"chain Y ==> Rep_convex_pd (\<Squnion>i. Y i) = (\<Union>i. Rep_convex_pd (Y i))"
by (rule convex_le.typedef_ideal_rep_contlub
[OF type_definition_convex_pd below_convex_pd_def])
lemma ideal_Rep_convex_pd: "convex_le.ideal (Rep_convex_pd xs)"
by (rule Rep_convex_pd [unfolded mem_Collect_eq])
definition
convex_principal :: "'a pd_basis => 'a convex_pd" where
"convex_principal t = Abs_convex_pd {u. u ≤\<natural> t}"
lemma Rep_convex_principal:
"Rep_convex_pd (convex_principal t) = {u. u ≤\<natural> t}"
unfolding convex_principal_def
by (simp add: Abs_convex_pd_inverse convex_le.ideal_principal)
interpretation convex_pd:
ideal_completion convex_le pd_take convex_principal Rep_convex_pd
apply unfold_locales
apply (rule pd_take_convex_le)
apply (rule pd_take_idem)
apply (erule pd_take_convex_mono)
apply (rule pd_take_convex_chain)
apply (rule finite_range_pd_take)
apply (rule pd_take_covers)
apply (rule ideal_Rep_convex_pd)
apply (erule Rep_convex_pd_lub)
apply (rule Rep_convex_principal)
apply (simp only: below_convex_pd_def)
done
text {* Convex powerdomain is pointed *}
lemma convex_pd_minimal: "convex_principal (PDUnit compact_bot) \<sqsubseteq> ys"
by (induct ys rule: convex_pd.principal_induct, simp, simp)
instance convex_pd :: (bifinite) pcpo
by intro_classes (fast intro: convex_pd_minimal)
lemma inst_convex_pd_pcpo: "⊥ = convex_principal (PDUnit compact_bot)"
by (rule convex_pd_minimal [THEN UU_I, symmetric])
text {* Convex powerdomain is profinite *}
instantiation convex_pd :: (profinite) profinite
begin
definition
approx_convex_pd_def: "approx = convex_pd.completion_approx"
instance
apply (intro_classes, unfold approx_convex_pd_def)
apply (rule convex_pd.chain_completion_approx)
apply (rule convex_pd.lub_completion_approx)
apply (rule convex_pd.completion_approx_idem)
apply (rule convex_pd.finite_fixes_completion_approx)
done
end
instance convex_pd :: (bifinite) bifinite ..
lemma approx_convex_principal [simp]:
"approx n·(convex_principal t) = convex_principal (pd_take n t)"
unfolding approx_convex_pd_def
by (rule convex_pd.completion_approx_principal)
lemma approx_eq_convex_principal:
"∃t∈Rep_convex_pd xs. approx n·xs = convex_principal (pd_take n t)"
unfolding approx_convex_pd_def
by (rule convex_pd.completion_approx_eq_principal)
subsection {* Monadic unit and plus *}
definition
convex_unit :: "'a -> 'a convex_pd" where
"convex_unit = compact_basis.basis_fun (λa. convex_principal (PDUnit a))"
definition
convex_plus :: "'a convex_pd -> 'a convex_pd -> 'a convex_pd" where
"convex_plus = convex_pd.basis_fun (λt. convex_pd.basis_fun (λu.
convex_principal (PDPlus t u)))"
abbreviation
convex_add :: "'a convex_pd => 'a convex_pd => 'a convex_pd"
(infixl "+\<natural>" 65) where
"xs +\<natural> ys == convex_plus·xs·ys"
syntax
"_convex_pd" :: "args => 'a convex_pd" ("{_}\<natural>")
translations
"{x,xs}\<natural>" == "{x}\<natural> +\<natural> {xs}\<natural>"
"{x}\<natural>" == "CONST convex_unit·x"
lemma convex_unit_Rep_compact_basis [simp]:
"{Rep_compact_basis a}\<natural> = convex_principal (PDUnit a)"
unfolding convex_unit_def
by (simp add: compact_basis.basis_fun_principal PDUnit_convex_mono)
lemma convex_plus_principal [simp]:
"convex_principal t +\<natural> convex_principal u = convex_principal (PDPlus t u)"
unfolding convex_plus_def
by (simp add: convex_pd.basis_fun_principal
convex_pd.basis_fun_mono PDPlus_convex_mono)
lemma approx_convex_unit [simp]:
"approx n·{x}\<natural> = {approx n·x}\<natural>"
apply (induct x rule: compact_basis.principal_induct, simp)
apply (simp add: approx_Rep_compact_basis)
done
lemma approx_convex_plus [simp]:
"approx n·(xs +\<natural> ys) = approx n·xs +\<natural> approx n·ys"
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
interpretation convex_add!: semilattice convex_add proof
fix xs ys zs :: "'a convex_pd"
show "(xs +\<natural> ys) +\<natural> zs = xs +\<natural> (ys +\<natural> zs)"
apply (induct xs ys arbitrary: zs rule: convex_pd.principal_induct2, simp, simp)
apply (rule_tac x=zs in convex_pd.principal_induct, simp)
apply (simp add: PDPlus_assoc)
done
show "xs +\<natural> ys = ys +\<natural> xs"
apply (induct xs ys rule: convex_pd.principal_induct2, simp, simp)
apply (simp add: PDPlus_commute)
done
show "xs +\<natural> xs = xs"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (simp add: PDPlus_absorb)
done
qed
lemmas convex_plus_assoc = convex_add.assoc
lemmas convex_plus_commute = convex_add.commute
lemmas convex_plus_absorb = convex_add.idem
lemmas convex_plus_left_commute = convex_add.left_commute
lemmas convex_plus_left_absorb = convex_add.left_idem
text {* Useful for @{text "simp add: convex_plus_ac"} *}
lemmas convex_plus_ac =
convex_plus_assoc convex_plus_commute convex_plus_left_commute
text {* Useful for @{text "simp only: convex_plus_aci"} *}
lemmas convex_plus_aci =
convex_plus_ac convex_plus_absorb convex_plus_left_absorb
lemma convex_unit_below_plus_iff [simp]:
"{x}\<natural> \<sqsubseteq> ys +\<natural> zs <-> {x}\<natural> \<sqsubseteq> ys ∧ {x}\<natural> \<sqsubseteq> zs"
apply (rule iffI)
apply (subgoal_tac
"adm (λf. f·{x}\<natural> \<sqsubseteq> f·ys ∧ f·{x}\<natural> \<sqsubseteq> f·zs)")
apply (drule admD, rule chain_approx)
apply (drule_tac f="approx i" in monofun_cfun_arg)
apply (cut_tac x="approx i·x" in compact_basis.compact_imp_principal, simp)
apply (cut_tac x="approx i·ys" in convex_pd.compact_imp_principal, simp)
apply (cut_tac x="approx i·zs" in convex_pd.compact_imp_principal, simp)
apply (clarify, simp)
apply simp
apply simp
apply (erule conjE)
apply (subst convex_plus_absorb [of "{x}\<natural>", symmetric])
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done
lemma convex_plus_below_unit_iff [simp]:
"xs +\<natural> ys \<sqsubseteq> {z}\<natural> <-> xs \<sqsubseteq> {z}\<natural> ∧ ys \<sqsubseteq> {z}\<natural>"
apply (rule iffI)
apply (subgoal_tac
"adm (λf. f·xs \<sqsubseteq> f·{z}\<natural> ∧ f·ys \<sqsubseteq> f·{z}\<natural>)")
apply (drule admD, rule chain_approx)
apply (drule_tac f="approx i" in monofun_cfun_arg)
apply (cut_tac x="approx i·xs" in convex_pd.compact_imp_principal, simp)
apply (cut_tac x="approx i·ys" in convex_pd.compact_imp_principal, simp)
apply (cut_tac x="approx i·z" in compact_basis.compact_imp_principal, simp)
apply (clarify, simp)
apply simp
apply simp
apply (erule conjE)
apply (subst convex_plus_absorb [of "{z}\<natural>", symmetric])
apply (erule (1) monofun_cfun [OF monofun_cfun_arg])
done
lemma convex_unit_below_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> <-> 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
lemma convex_unit_eq_iff [simp]: "{x}\<natural> = {y}\<natural> <-> x = y"
unfolding po_eq_conv by simp
lemma convex_unit_strict [simp]: "{⊥}\<natural> = ⊥"
unfolding inst_convex_pd_pcpo Rep_compact_bot [symmetric] by simp
lemma convex_unit_strict_iff [simp]: "{x}\<natural> = ⊥ <-> x = ⊥"
unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)
lemma compact_convex_unit_iff [simp]:
"compact {x}\<natural> <-> compact x"
unfolding profinite_compact_iff by simp
lemma compact_convex_plus [simp]:
"[|compact xs; compact ys|] ==> compact (xs +\<natural> ys)"
by (auto dest!: convex_pd.compact_imp_principal)
subsection {* Induction rules *}
lemma convex_pd_induct1:
assumes P: "adm P"
assumes unit: "!!x. P {x}\<natural>"
assumes insert: "!!x ys. [|P {x}\<natural>; P ys|] ==> P ({x}\<natural> +\<natural> ys)"
shows "P (xs::'a convex_pd)"
apply (induct xs rule: convex_pd.principal_induct, rule P)
apply (induct_tac a rule: pd_basis_induct1)
apply (simp only: convex_unit_Rep_compact_basis [symmetric])
apply (rule unit)
apply (simp only: convex_unit_Rep_compact_basis [symmetric]
convex_plus_principal [symmetric])
apply (erule insert [OF unit])
done
lemma convex_pd_induct:
assumes P: "adm P"
assumes unit: "!!x. P {x}\<natural>"
assumes plus: "!!xs ys. [|P xs; P ys|] ==> P (xs +\<natural> ys)"
shows "P (xs::'a convex_pd)"
apply (induct xs rule: convex_pd.principal_induct, rule P)
apply (induct_tac a rule: pd_basis_induct)
apply (simp only: convex_unit_Rep_compact_basis [symmetric] unit)
apply (simp only: convex_plus_principal [symmetric] plus)
done
subsection {* Monadic bind *}
definition
convex_bind_basis ::
"'a pd_basis => ('a -> 'b convex_pd) -> 'b convex_pd" where
"convex_bind_basis = fold_pd
(λa. Λ f. f·(Rep_compact_basis a))
(λx y. Λ f. x·f +\<natural> y·f)"
lemma ACI_convex_bind:
"class.ab_semigroup_idem_mult (λx y. Λ f. x·f +\<natural> y·f)"
apply unfold_locales
apply (simp add: convex_plus_assoc)
apply (simp add: convex_plus_commute)
apply (simp add: eta_cfun)
done
lemma convex_bind_basis_simps [simp]:
"convex_bind_basis (PDUnit a) =
(Λ f. f·(Rep_compact_basis a))"
"convex_bind_basis (PDPlus t u) =
(Λ f. convex_bind_basis t·f +\<natural> convex_bind_basis u·f)"
unfolding convex_bind_basis_def
apply -
apply (rule fold_pd_PDUnit [OF ACI_convex_bind])
apply (rule fold_pd_PDPlus [OF ACI_convex_bind])
done
lemma monofun_LAM:
"[|cont f; cont g; !!x. f x \<sqsubseteq> g x|] ==> (Λ x. f x) \<sqsubseteq> (Λ x. g x)"
by (simp add: expand_cfun_below)
lemma convex_bind_basis_mono:
"t ≤\<natural> u ==> convex_bind_basis t \<sqsubseteq> convex_bind_basis u"
apply (erule convex_le_induct)
apply (erule (1) below_trans)
apply (simp add: monofun_LAM monofun_cfun)
apply (simp add: monofun_LAM monofun_cfun)
done
definition
convex_bind :: "'a convex_pd -> ('a -> 'b convex_pd) -> 'b convex_pd" where
"convex_bind = convex_pd.basis_fun convex_bind_basis"
lemma convex_bind_principal [simp]:
"convex_bind·(convex_principal t) = convex_bind_basis t"
unfolding convex_bind_def
apply (rule convex_pd.basis_fun_principal)
apply (erule convex_bind_basis_mono)
done
lemma convex_bind_unit [simp]:
"convex_bind·{x}\<natural>·f = f·x"
by (induct x rule: compact_basis.principal_induct, simp, simp)
lemma convex_bind_plus [simp]:
"convex_bind·(xs +\<natural> ys)·f = convex_bind·xs·f +\<natural> convex_bind·ys·f"
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
lemma convex_bind_strict [simp]: "convex_bind·⊥·f = f·⊥"
unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)
subsection {* Map and join *}
definition
convex_map :: "('a -> 'b) -> 'a convex_pd -> 'b convex_pd" where
"convex_map = (Λ f xs. convex_bind·xs·(Λ x. {f·x}\<natural>))"
definition
convex_join :: "'a convex_pd convex_pd -> 'a convex_pd" where
"convex_join = (Λ xss. convex_bind·xss·(Λ xs. xs))"
lemma convex_map_unit [simp]:
"convex_map·f·(convex_unit·x) = convex_unit·(f·x)"
unfolding convex_map_def by simp
lemma convex_map_plus [simp]:
"convex_map·f·(xs +\<natural> ys) = convex_map·f·xs +\<natural> convex_map·f·ys"
unfolding convex_map_def by simp
lemma convex_join_unit [simp]:
"convex_join·{xs}\<natural> = xs"
unfolding convex_join_def by simp
lemma convex_join_plus [simp]:
"convex_join·(xss +\<natural> yss) = convex_join·xss +\<natural> convex_join·yss"
unfolding convex_join_def by simp
lemma convex_map_ident: "convex_map·(Λ x. x)·xs = xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma convex_map_ID: "convex_map·ID = ID"
by (simp add: expand_cfun_eq ID_def convex_map_ident)
lemma convex_map_map:
"convex_map·f·(convex_map·g·xs) = convex_map·(Λ x. f·(g·x))·xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma convex_join_map_unit:
"convex_join·(convex_map·convex_unit·xs) = xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma convex_join_map_join:
"convex_join·(convex_map·convex_join·xsss) = convex_join·(convex_join·xsss)"
by (induct xsss rule: convex_pd_induct, simp_all)
lemma convex_join_map_map:
"convex_join·(convex_map·(convex_map·f)·xss) =
convex_map·f·(convex_join·xss)"
by (induct xss rule: convex_pd_induct, simp_all)
lemma convex_map_approx: "convex_map·(approx n)·xs = approx n·xs"
by (induct xs rule: convex_pd_induct, simp_all)
lemma ep_pair_convex_map:
"ep_pair e p ==> ep_pair (convex_map·e) (convex_map·p)"
apply default
apply (induct_tac x rule: convex_pd_induct, simp_all add: ep_pair.e_inverse)
apply (induct_tac y rule: convex_pd_induct)
apply (simp_all add: ep_pair.e_p_below monofun_cfun)
done
lemma deflation_convex_map: "deflation d ==> deflation (convex_map·d)"
apply default
apply (induct_tac x rule: convex_pd_induct, simp_all add: deflation.idem)
apply (induct_tac x rule: convex_pd_induct)
apply (simp_all add: deflation.below monofun_cfun)
done
subsection {* Conversions to other powerdomains *}
text {* Convex to upper *}
lemma convex_le_imp_upper_le: "t ≤\<natural> u ==> t ≤\<sharp> u"
unfolding convex_le_def by simp
definition
convex_to_upper :: "'a convex_pd -> 'a upper_pd" where
"convex_to_upper = convex_pd.basis_fun upper_principal"
lemma convex_to_upper_principal [simp]:
"convex_to_upper·(convex_principal t) = upper_principal t"
unfolding convex_to_upper_def
apply (rule convex_pd.basis_fun_principal)
apply (rule upper_pd.principal_mono)
apply (erule convex_le_imp_upper_le)
done
lemma convex_to_upper_unit [simp]:
"convex_to_upper·{x}\<natural> = {x}\<sharp>"
by (induct x rule: compact_basis.principal_induct, simp, simp)
lemma convex_to_upper_plus [simp]:
"convex_to_upper·(xs +\<natural> ys) = convex_to_upper·xs +\<sharp> convex_to_upper·ys"
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
lemma approx_convex_to_upper:
"approx i·(convex_to_upper·xs) = convex_to_upper·(approx i·xs)"
by (induct xs rule: convex_pd_induct, simp, simp, simp)
lemma convex_to_upper_bind [simp]:
"convex_to_upper·(convex_bind·xs·f) =
upper_bind·(convex_to_upper·xs)·(convex_to_upper oo f)"
by (induct xs rule: convex_pd_induct, simp, simp, simp)
lemma convex_to_upper_map [simp]:
"convex_to_upper·(convex_map·f·xs) = upper_map·f·(convex_to_upper·xs)"
by (simp add: convex_map_def upper_map_def cfcomp_LAM)
lemma convex_to_upper_join [simp]:
"convex_to_upper·(convex_join·xss) =
upper_bind·(convex_to_upper·xss)·convex_to_upper"
by (simp add: convex_join_def upper_join_def cfcomp_LAM eta_cfun)
text {* Convex to lower *}
lemma convex_le_imp_lower_le: "t ≤\<natural> u ==> t ≤\<flat> u"
unfolding convex_le_def by simp
definition
convex_to_lower :: "'a convex_pd -> 'a lower_pd" where
"convex_to_lower = convex_pd.basis_fun lower_principal"
lemma convex_to_lower_principal [simp]:
"convex_to_lower·(convex_principal t) = lower_principal t"
unfolding convex_to_lower_def
apply (rule convex_pd.basis_fun_principal)
apply (rule lower_pd.principal_mono)
apply (erule convex_le_imp_lower_le)
done
lemma convex_to_lower_unit [simp]:
"convex_to_lower·{x}\<natural> = {x}\<flat>"
by (induct x rule: compact_basis.principal_induct, simp, simp)
lemma convex_to_lower_plus [simp]:
"convex_to_lower·(xs +\<natural> ys) = convex_to_lower·xs +\<flat> convex_to_lower·ys"
by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)
lemma approx_convex_to_lower:
"approx i·(convex_to_lower·xs) = convex_to_lower·(approx i·xs)"
by (induct xs rule: convex_pd_induct, simp, simp, simp)
lemma convex_to_lower_bind [simp]:
"convex_to_lower·(convex_bind·xs·f) =
lower_bind·(convex_to_lower·xs)·(convex_to_lower oo f)"
by (induct xs rule: convex_pd_induct, simp, simp, simp)
lemma convex_to_lower_map [simp]:
"convex_to_lower·(convex_map·f·xs) = lower_map·f·(convex_to_lower·xs)"
by (simp add: convex_map_def lower_map_def cfcomp_LAM)
lemma convex_to_lower_join [simp]:
"convex_to_lower·(convex_join·xss) =
lower_bind·(convex_to_lower·xss)·convex_to_lower"
by (simp add: convex_join_def lower_join_def cfcomp_LAM eta_cfun)
text {* Ordering property *}
lemma convex_pd_below_iff:
"(xs \<sqsubseteq> ys) =
(convex_to_upper·xs \<sqsubseteq> convex_to_upper·ys ∧
convex_to_lower·xs \<sqsubseteq> convex_to_lower·ys)"
apply (safe elim!: monofun_cfun_arg)
apply (rule profinite_below_ext)
apply (drule_tac f="approx i" in monofun_cfun_arg)
apply (drule_tac f="approx i" in monofun_cfun_arg)
apply (cut_tac x="approx i·xs" in convex_pd.compact_imp_principal, simp)
apply (cut_tac x="approx i·ys" in convex_pd.compact_imp_principal, simp)
apply clarify
apply (simp add: approx_convex_to_upper approx_convex_to_lower convex_le_def)
done
lemmas convex_plus_below_plus_iff =
convex_pd_below_iff [where xs="xs +\<natural> ys" and ys="zs +\<natural> ws", standard]
lemmas convex_pd_below_simps =
convex_unit_below_plus_iff
convex_plus_below_unit_iff
convex_plus_below_plus_iff
convex_unit_below_iff
convex_to_upper_unit
convex_to_upper_plus
convex_to_lower_unit
convex_to_lower_plus
upper_pd_below_simps
lower_pd_below_simps
end