# Theory ConvexPD

Up to index of Isabelle/HOL/HOLCF

theory ConvexPD
imports UpperPD LowerPD
`(*  Title:      HOL/HOLCF/ConvexPD.thy    Author:     Brian Huffman*)header {* Convex powerdomain *}theory ConvexPDimports UpperPD LowerPDbeginsubsection {* 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_leby (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 simplemma 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 fastlemma 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_PDUnitusing Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fastlemma 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 fastlemma 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_PDUnitusing Rep_pd_basis_nonempty [of t, folded ex_in_conv] by fastlemma 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 fastlemma 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+   doneqedlemma 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)donesubsection {* Type definition *}typedef 'a convex_pd =  "{S::'a pd_basis set. convex_le.ideal S}"by (rule convex_le.ex_ideal)type_notation (xsymbols) convex_pd ("('(_')\<natural>)")instantiation convex_pd :: (bifinite) belowbegindefinition  "x \<sqsubseteq> y <-> Rep_convex_pd x ⊆ Rep_convex_pd y"instance ..endinstance convex_pd :: (bifinite) pousing type_definition_convex_pd below_convex_pd_defby (rule convex_le.typedef_ideal_po)instance convex_pd :: (bifinite) cpousing type_definition_convex_pd below_convex_pd_defby (rule convex_le.typedef_ideal_cpo)definition  convex_principal :: "'a pd_basis => 'a convex_pd" where  "convex_principal t = Abs_convex_pd {u. u ≤\<natural> t}"interpretation convex_pd:  ideal_completion convex_le convex_principal Rep_convex_pdusing type_definition_convex_pd below_convex_pd_defusing convex_principal_def pd_basis_countableby (rule convex_le.typedef_ideal_completion)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) pcpoby 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 "∪\<natural>" 65) where  "xs ∪\<natural> ys == convex_plus·xs·ys"syntax  "_convex_pd" :: "args => logic" ("{_}\<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_defby (simp add: compact_basis.extension_principal PDUnit_convex_mono)lemma convex_plus_principal [simp]:  "convex_principal t ∪\<natural> convex_principal u = convex_principal (PDPlus t u)"unfolding convex_plus_defby (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 ∪\<natural> ys) ∪\<natural> zs = xs ∪\<natural> (ys ∪\<natural> 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 ∪\<natural> ys = ys ∪\<natural> 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 ∪\<natural> xs = xs"    apply (induct xs rule: convex_pd.principal_induct, simp)    apply (simp add: PDPlus_absorb)    doneqedlemmas convex_plus_assoc = convex_add.assoclemmas convex_plus_commute = convex_add.commutelemmas convex_plus_absorb = convex_add.idemlemmas convex_plus_left_commute = convex_add.left_commutelemmas convex_plus_left_absorb = convex_add.left_idemtext {* Useful for @{text "simp add: convex_plus_ac"} *}lemmas convex_plus_ac =  convex_plus_assoc convex_plus_commute convex_plus_left_commutetext {* Useful for @{text "simp only: convex_plus_aci"} *}lemmas convex_plus_aci =  convex_plus_ac convex_plus_absorb convex_plus_left_absorblemma convex_unit_below_plus_iff [simp]:  "{x}\<natural> \<sqsubseteq> ys ∪\<natural> zs <-> {x}\<natural> \<sqsubseteq> ys ∧ {x}\<natural> \<sqsubseteq> 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 simpdonelemma convex_plus_below_unit_iff [simp]:  "xs ∪\<natural> ys \<sqsubseteq> {z}\<natural> <-> xs \<sqsubseteq> {z}\<natural> ∧ ys \<sqsubseteq> {z}\<natural>"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 simpdonelemma convex_unit_below_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> <-> x \<sqsubseteq> y"apply (induct x rule: compact_basis.principal_induct, simp)apply (induct y rule: compact_basis.principal_induct, simp)apply simpdonelemma convex_unit_eq_iff [simp]: "{x}\<natural> = {y}\<natural> <-> x = y"unfolding po_eq_conv by simplemma convex_unit_strict [simp]: "{⊥}\<natural> = ⊥"using convex_unit_Rep_compact_basis [of compact_bot]by (simp add: inst_convex_pd_pcpo)lemma convex_unit_bottom_iff [simp]: "{x}\<natural> = ⊥ <-> x = ⊥"unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)lemma compact_convex_unit: "compact x ==> compact {x}\<natural>"by (auto dest!: compact_basis.compact_imp_principal)lemma compact_convex_unit_iff [simp]: "compact {x}\<natural> <-> 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])donelemma 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])donelemma convex_pd_induct  [case_names adm convex_unit convex_plus, induct type: convex_pd]:  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)donesubsection {* 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_localesapply (simp add: convex_plus_assoc)apply (simp add: convex_plus_commute)apply (simp add: eta_cfun)donelemma 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_defapply -apply (rule fold_pd_PDUnit [OF ACI_convex_bind])apply (rule fold_pd_PDPlus [OF ACI_convex_bind])donelemma 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)donedefinition  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\<Union>\<natural>_∈_./ _)" [0, 0, 10] 10)translations  "\<Union>\<natural>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_defapply (rule convex_pd.extension_principal)apply (erule convex_bind_basis_mono)donelemma 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 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}\<natural>))"lemma convex_map_unit [simp]:  "convex_map·f·{x}\<natural> = {f·x}\<natural>"unfolding convex_map_def by simplemma convex_map_plus [simp]:  "convex_map·f·(xs ∪\<natural> ys) = convex_map·f·xs ∪\<natural> convex_map·f·ys"unfolding convex_map_def by simplemma convex_map_bottom [simp]: "convex_map·f·⊥ = {f·⊥}\<natural>"unfolding convex_map_def by simplemma 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 defaultapply (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)donelemma deflation_convex_map: "deflation d ==> deflation (convex_map·d)"apply defaultapply (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  have "deflation d" by fact  thus "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)qedsubsection {* 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) bifiniteproof  show "∃(a::nat => 'a convex_pd -> 'a convex_pd). approx_chain a"    using bifinite [where 'a='a]    by (fast intro!: approx_chain_convex_map)qedsubsection {* 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}\<natural> = xs"unfolding convex_join_def by simplemma convex_join_plus [simp]:  "convex_join·(xss ∪\<natural> yss) = convex_join·xss ∪\<natural> convex_join·yss"unfolding convex_join_def by simplemma convex_join_bottom [simp]: "convex_join·⊥ = ⊥"unfolding convex_join_def by simplemma 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 ≤\<natural> u ==> t ≤\<sharp> u"unfolding convex_le_def by simpdefinition  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_defapply (rule convex_pd.extension_principal)apply (rule upper_pd.principal_mono)apply (erule convex_le_imp_upper_le)donelemma 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 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 ≤\<natural> u ==> t ≤\<flat> u"unfolding convex_le_def by simpdefinition  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_defapply (rule convex_pd.extension_principal)apply (rule lower_pd.principal_mono)apply (erule convex_le_imp_lower_le)donelemma 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 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 \<sqsubseteq> ys) =    (convex_to_upper·xs \<sqsubseteq> convex_to_upper·ys ∧     convex_to_lower·xs \<sqsubseteq> 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)donelemmas convex_plus_below_plus_iff =  convex_pd_below_iff [where xs="xs ∪\<natural> ys" and ys="zs ∪\<natural> ws"]  for xs ys zs wslemmas 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_simpsend`