# Theory ConvexPD

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

section ‹Convex powerdomain›

theory ConvexPD
imports UpperPD LowerPD
begin

subsection ‹Basis preorder›

definition
convex_le :: "'a pd_basis ⇒ 'a pd_basis ⇒ bool" (infix "≤♮" 50) where
"convex_le = (λu v. u ≤♯ v ∧ u ≤♭ v)"

lemma convex_le_refl [simp]: "t ≤♮ t"
unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl)

lemma convex_le_trans: "⟦t ≤♮ u; u ≤♮ v⟧ ⟹ t ≤♮ 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 ≤♮ t"
unfolding convex_le_def Rep_PDUnit by simp

lemma PDUnit_convex_mono: "x ⊑ y ⟹ PDUnit x ≤♮ PDUnit y"
unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono)

lemma PDPlus_convex_mono: "⟦s ≤♮ t; u ≤♮ v⟧ ⟹ PDPlus s u ≤♮ 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 ≤♮ PDUnit b) = (a ⊑ b)"
unfolding convex_le_def upper_le_def lower_le_def Rep_PDUnit by fast

lemma convex_le_PDUnit_lemma1:
"(PDUnit a ≤♮ t) = (∀b∈Rep_pd_basis t. a ⊑ 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 ≤♮ PDPlus t u) = (PDUnit a ≤♮ t ∧ PDUnit a ≤♮ u)"
unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast

lemma convex_le_PDUnit_lemma2:
"(t ≤♮ PDUnit b) = (∀a∈Rep_pd_basis t. a ⊑ 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 ≤♮ PDUnit a) = (t ≤♮ PDUnit a ∧ u ≤♮ PDUnit a)"
unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast

lemma convex_le_PDPlus_lemma:
assumes z: "PDPlus t u ≤♮ z"
shows "∃v w. z = PDPlus v w ∧ t ≤♮ v ∧ u ≤♮ w"
proof (intro exI conjI)
let ?A = "{b∈Rep_pd_basis z. ∃a∈Rep_pd_basis t. a ⊑ b}"
let ?B = "{b∈Rep_pd_basis z. ∃a∈Rep_pd_basis u. a ⊑ 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 ≤♮ ?v" "u ≤♮ ?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 ≤♮ u"
assumes 2: "⋀t u v. ⟦P t u; P u v⟧ ⟹ P t v"
assumes 3: "⋀a b. a ⊑ 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

subsection ‹Type definition›

typedef 'a convex_pd  ("('(_')♮)") =
"{S::'a pd_basis set. convex_le.ideal S}"
by (rule convex_le.ex_ideal)

instantiation convex_pd :: (bifinite) below
begin

definition
"x ⊑ y ⟷ Rep_convex_pd x ⊆ Rep_convex_pd y"

instance ..
end

instance convex_pd :: (bifinite) po
using type_definition_convex_pd below_convex_pd_def
by (rule convex_le.typedef_ideal_po)

instance convex_pd :: (bifinite) cpo
using type_definition_convex_pd below_convex_pd_def
by (rule convex_le.typedef_ideal_cpo)

definition
convex_principal :: "'a pd_basis ⇒ 'a convex_pd" where
"convex_principal t = Abs_convex_pd {u. u ≤♮ t}"

interpretation convex_pd:
ideal_completion convex_le convex_principal Rep_convex_pd
using type_definition_convex_pd below_convex_pd_def
using convex_principal_def pd_basis_countable
by (rule convex_le.typedef_ideal_completion)

text ‹Convex powerdomain is pointed›

lemma convex_pd_minimal: "convex_principal (PDUnit compact_bot) ⊑ 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 bottomI, symmetric])

subsection ‹Monadic unit and plus›

definition
convex_unit :: "'a → 'a convex_pd" where
"convex_unit = compact_basis.extension (λa. convex_principal (PDUnit a))"

definition
convex_plus :: "'a convex_pd → 'a convex_pd → 'a convex_pd" where
"convex_plus = convex_pd.extension (λt. convex_pd.extension (λu.
convex_principal (PDPlus t u)))"

abbreviation
convex_add :: "'a convex_pd ⇒ 'a convex_pd ⇒ 'a convex_pd"
(infixl "∪♮" 65) where
"xs ∪♮ ys == convex_plus⋅xs⋅ys"

syntax
"_convex_pd" :: "args ⇒ logic" ("{_}♮")

translations
"{x,xs}♮" == "{x}♮ ∪♮ {xs}♮"
"{x}♮" == "CONST convex_unit⋅x"

lemma convex_unit_Rep_compact_basis [simp]:
"{Rep_compact_basis a}♮ = convex_principal (PDUnit a)"
unfolding convex_unit_def
by (simp add: compact_basis.extension_principal PDUnit_convex_mono)

lemma convex_plus_principal [simp]:
"convex_principal t ∪♮ convex_principal u = convex_principal (PDPlus t u)"
unfolding convex_plus_def
by (simp add: convex_pd.extension_principal
convex_pd.extension_mono PDPlus_convex_mono)

interpretation convex_add: semilattice convex_add proof
fix xs ys zs :: "'a convex_pd"
show "(xs ∪♮ ys) ∪♮ zs = xs ∪♮ (ys ∪♮ zs)"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (induct zs rule: convex_pd.principal_induct, simp)
apply (simp add: PDPlus_assoc)
done
show "xs ∪♮ ys = ys ∪♮ xs"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (simp add: PDPlus_commute)
done
show "xs ∪♮ 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 ‹simp add: convex_plus_ac››
lemmas convex_plus_ac =
convex_plus_assoc convex_plus_commute convex_plus_left_commute

text ‹Useful for ‹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}♮ ⊑ ys ∪♮ zs ⟷ {x}♮ ⊑ ys ∧ {x}♮ ⊑ zs"
apply (induct x rule: compact_basis.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (induct zs rule: convex_pd.principal_induct, simp)
apply simp
done

lemma convex_plus_below_unit_iff [simp]:
"xs ∪♮ ys ⊑ {z}♮ ⟷ xs ⊑ {z}♮ ∧ ys ⊑ {z}♮"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (induct z rule: compact_basis.principal_induct, simp)
apply simp
done

lemma convex_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

lemma convex_unit_eq_iff [simp]: "{x}♮ = {y}♮ ⟷ x = y"
unfolding po_eq_conv by simp

lemma convex_unit_strict [simp]: "{⊥}♮ = ⊥"
using convex_unit_Rep_compact_basis [of compact_bot]
by (simp add: inst_convex_pd_pcpo)

lemma convex_unit_bottom_iff [simp]: "{x}♮ = ⊥ ⟷ x = ⊥"
unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)

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

lemma compact_convex_unit_iff [simp]: "compact {x}♮ ⟷ compact x"
apply (safe elim!: compact_convex_unit)
apply (simp only: compact_def convex_unit_below_iff [symmetric])
apply (erule adm_subst [OF cont_Rep_cfun2])
done

lemma compact_convex_plus [simp]:
"⟦compact xs; compact ys⟧ ⟹ compact (xs ∪♮ 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}♮"
assumes insert: "⋀x ys. ⟦P {x}♮; P ys⟧ ⟹ P ({x}♮ ∪♮ 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
[case_names adm convex_unit convex_plus, induct type: convex_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 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 ∪♮ y⋅f)"

lemma ACI_convex_bind:
"semilattice (λx y. Λ f. x⋅f ∪♮ 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 ∪♮ 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 convex_bind_basis_mono:
"t ≤♮ u ⟹ convex_bind_basis t ⊑ 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.extension convex_bind_basis"

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

translations
"⋃♮x∈xs. e" == "CONST convex_bind⋅xs⋅(Λ x. e)"

lemma convex_bind_principal [simp]:
"convex_bind⋅(convex_principal t) = convex_bind_basis t"
unfolding convex_bind_def
apply (rule convex_pd.extension_principal)
apply (erule convex_bind_basis_mono)
done

lemma convex_bind_unit [simp]:
"convex_bind⋅{x}♮⋅f = f⋅x"
by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma convex_bind_plus [simp]:
"convex_bind⋅(xs ∪♮ ys)⋅f = convex_bind⋅xs⋅f ∪♮ convex_bind⋅ys⋅f"
by (induct xs rule: convex_pd.principal_induct, simp,
induct ys rule: convex_pd.principal_induct, simp, simp)

lemma convex_bind_strict [simp]: "convex_bind⋅⊥⋅f = f⋅⊥"
unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)

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

subsection ‹Map›

definition
convex_map :: "('a → 'b) → 'a convex_pd → 'b convex_pd" where
"convex_map = (Λ f xs. convex_bind⋅xs⋅(Λ x. {f⋅x}♮))"

lemma convex_map_unit [simp]:
"convex_map⋅f⋅{x}♮ = {f⋅x}♮"
unfolding convex_map_def by simp

lemma convex_map_plus [simp]:
"convex_map⋅f⋅(xs ∪♮ ys) = convex_map⋅f⋅xs ∪♮ convex_map⋅f⋅ys"
unfolding convex_map_def by simp

lemma convex_map_bottom [simp]: "convex_map⋅f⋅⊥ = {f⋅⊥}♮"
unfolding convex_map_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: cfun_eq_iff 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_bind_map:
"convex_bind⋅(convex_map⋅f⋅xs)⋅g = convex_bind⋅xs⋅(Λ x. g⋅(f⋅x))"
by (simp add: convex_map_def convex_bind_bind)

lemma convex_map_bind:
"convex_map⋅f⋅(convex_bind⋅xs⋅g) = convex_bind⋅xs⋅(Λ x. convex_map⋅f⋅(g⋅x))"
by (simp add: convex_map_def convex_bind_bind)

lemma ep_pair_convex_map: "ep_pair e p ⟹ ep_pair (convex_map⋅e) (convex_map⋅p)"
apply standard
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 standard
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

(* FIXME: long proof! *)
lemma finite_deflation_convex_map:
assumes "finite_deflation d" shows "finite_deflation (convex_map⋅d)"
proof (rule finite_deflation_intro)
interpret d: finite_deflation d by fact
from d.deflation_axioms show "deflation (convex_map⋅d)"
by (rule deflation_convex_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 (convex_principal ` Rep_pd_basis -` (Pow (Rep_compact_basis -` range (λx. d⋅x))))" by simp
hence "finite (range (λxs. convex_map⋅d⋅xs))"
apply (rule rev_finite_subset)
apply clarsimp
apply (induct_tac xs rule: convex_pd.principal_induct)
apply (simp add: adm_mem_finite *)
apply (rename_tac t, induct_tac t rule: pd_basis_induct)
apply (simp only: convex_unit_Rep_compact_basis [symmetric] convex_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: convex_plus_principal [symmetric] convex_map_plus)
apply clarsimp
apply (rule imageI)
apply (rule vimageI2)
apply (simp add: Rep_PDPlus)
done
thus "finite {xs. convex_map⋅d⋅xs = xs}"
by (rule finite_range_imp_finite_fixes)
qed

subsection ‹Convex powerdomain is bifinite›

lemma approx_chain_convex_map:
assumes "approx_chain a"
shows "approx_chain (λi. convex_map⋅(a i))"
using assms unfolding approx_chain_def
by (simp add: lub_APP convex_map_ID finite_deflation_convex_map)

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

subsection ‹Join›

definition
convex_join :: "'a convex_pd convex_pd → 'a convex_pd" where
"convex_join = (Λ xss. convex_bind⋅xss⋅(Λ xs. xs))"

lemma convex_join_unit [simp]:
"convex_join⋅{xs}♮ = xs"
unfolding convex_join_def by simp

lemma convex_join_plus [simp]:
"convex_join⋅(xss ∪♮ yss) = convex_join⋅xss ∪♮ convex_join⋅yss"
unfolding convex_join_def by simp

lemma convex_join_bottom [simp]: "convex_join⋅⊥ = ⊥"
unfolding convex_join_def by simp

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)

subsection ‹Conversions to other powerdomains›

text ‹Convex to upper›

lemma convex_le_imp_upper_le: "t ≤♮ u ⟹ t ≤♯ u"
unfolding convex_le_def by simp

definition
convex_to_upper :: "'a convex_pd → 'a upper_pd" where
"convex_to_upper = convex_pd.extension 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.extension_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}♮ = {x}♯"
by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma convex_to_upper_plus [simp]:
"convex_to_upper⋅(xs ∪♮ ys) = convex_to_upper⋅xs ∪♯ convex_to_upper⋅ys"
by (induct xs rule: convex_pd.principal_induct, simp,
induct ys rule: convex_pd.principal_induct, 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 ≤♮ u ⟹ t ≤♭ u"
unfolding convex_le_def by simp

definition
convex_to_lower :: "'a convex_pd → 'a lower_pd" where
"convex_to_lower = convex_pd.extension 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.extension_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}♮ = {x}♭"
by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma convex_to_lower_plus [simp]:
"convex_to_lower⋅(xs ∪♮ ys) = convex_to_lower⋅xs ∪♭ convex_to_lower⋅ys"
by (induct xs rule: convex_pd.principal_induct, simp,
induct ys rule: convex_pd.principal_induct, 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 ⊑ ys) =
(convex_to_upper⋅xs ⊑ convex_to_upper⋅ys ∧
convex_to_lower⋅xs ⊑ convex_to_lower⋅ys)"
apply (induct xs rule: convex_pd.principal_induct, simp)
apply (induct ys rule: convex_pd.principal_induct, simp)
apply (simp add: convex_le_def)
done

lemmas convex_plus_below_plus_iff =
convex_pd_below_iff [where xs="xs ∪♮ ys" and ys="zs ∪♮ ws"]
for xs ys zs ws

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
```