(*  Title:      HOLCF/ConvexPD.thy

    Author:     Brian Huffman

*)

header {* Convex powerdomain *}

theory ConvexPD

imports UpperPD LowerPD

begin

subsection {* Basis preorder *}

definition

  convex_le :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> bool" (infix "\<le>\<natural>" 50) where

  "convex_le = (\<lambda>u v. u \<le>\<sharp> v \<and> u \<le>\<flat> v)"

lemma convex_le_refl [simp]: "t \<le>\<natural> t"

unfolding convex_le_def by (fast intro: upper_le_refl lower_le_refl)

lemma convex_le_trans: "\<lbrakk>t \<le>\<natural> u; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> t \<le>\<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 \<le>\<natural> t"

unfolding convex_le_def Rep_PDUnit by simp

lemma PDUnit_convex_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<natural> PDUnit y"

unfolding convex_le_def by (fast intro: PDUnit_upper_mono PDUnit_lower_mono)

lemma PDPlus_convex_mono: "\<lbrakk>s \<le>\<natural> t; u \<le>\<natural> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<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 \<le>\<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 \<le>\<natural> t) = (\<forall>b\<in>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 \<le>\<natural> PDPlus t u) = (PDUnit a \<le>\<natural> t \<and> PDUnit a \<le>\<natural> u)"

unfolding convex_le_PDUnit_lemma1 Rep_PDPlus by fast

lemma convex_le_PDUnit_lemma2:

  "(t \<le>\<natural> PDUnit b) = (\<forall>a\<in>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 \<le>\<natural> PDUnit a) = (t \<le>\<natural> PDUnit a \<and> u \<le>\<natural> PDUnit a)"

unfolding convex_le_PDUnit_lemma2 Rep_PDPlus by fast

lemma convex_le_PDPlus_lemma:

  assumes z: "PDPlus t u \<le>\<natural> z"

  shows "\<exists>v w. z = PDPlus v w \<and> t \<le>\<natural> v \<and> u \<le>\<natural> w"

proof (intro exI conjI)

  let ?A = "{b\<in>Rep_pd_basis z. \<exists>a\<in>Rep_pd_basis t. a \<sqsubseteq> b}"

  let ?B = "{b\<in>Rep_pd_basis z. \<exists>a\<in>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 \<le>\<natural> ?v" "u \<le>\<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 \<le>\<natural> u"

  assumes 2: "\<And>t u v. \<lbrakk>P t u; P u v\<rbrakk> \<Longrightarrow> P t v"

  assumes 3: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)"

  assumes 4: "\<And>t u v w. \<lbrakk>P t v; P u w\<rbrakk> \<Longrightarrow> 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 \<le>\<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 \<le>\<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 \<le>\<natural> u \<Longrightarrow> pd_take n t \<le>\<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) sq_ord

begin

definition

  "x \<sqsubseteq> y \<longleftrightarrow> Rep_convex_pd x \<subseteq> Rep_convex_pd y"

instance ..

end

instance convex_pd :: (profinite) po

by (rule convex_le.typedef_ideal_po

    [OF type_definition_convex_pd sq_le_convex_pd_def])

instance convex_pd :: (profinite) cpo

by (rule convex_le.typedef_ideal_cpo

    [OF type_definition_convex_pd sq_le_convex_pd_def])

lemma Rep_convex_pd_lub:

  "chain Y \<Longrightarrow> 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 sq_le_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 \<Rightarrow> 'a convex_pd" where

  "convex_principal t = Abs_convex_pd {u. u \<le>\<natural> t}"

lemma Rep_convex_principal:

  "Rep_convex_pd (convex_principal t) = {u. u \<le>\<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: sq_le_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: "\<bottom> = 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\<cdot>(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:

  "\<exists>t\<in>Rep_convex_pd xs. approx n\<cdot>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 \<rightarrow> 'a convex_pd" where

  "convex_unit = compact_basis.basis_fun (\<lambda>a. convex_principal (PDUnit a))"

definition

  convex_plus :: "'a convex_pd \<rightarrow> 'a convex_pd \<rightarrow> 'a convex_pd" where

  "convex_plus = convex_pd.basis_fun (\<lambda>t. convex_pd.basis_fun (\<lambda>u.

      convex_principal (PDPlus t u)))"

abbreviation

  convex_add :: "'a convex_pd \<Rightarrow> 'a convex_pd \<Rightarrow> 'a convex_pd"

    (infixl "+\<natural>" 65) where

  "xs +\<natural> ys == convex_plus\<cdot>xs\<cdot>ys"

syntax

  "_convex_pd" :: "args \<Rightarrow> 'a convex_pd" ("{_}\<natural>")

translations

  "{x,xs}\<natural>" == "{x}\<natural> +\<natural> {xs}\<natural>"

  "{x}\<natural>" == "CONST convex_unit\<cdot>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\<cdot>{x}\<natural> = {approx n\<cdot>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\<cdot>(xs +\<natural> ys) = approx n\<cdot>xs +\<natural> approx n\<cdot>ys"

by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)

lemma convex_plus_assoc:

  "(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

lemma convex_plus_commute: "xs +\<natural> ys = ys +\<natural> xs"

apply (induct xs ys rule: convex_pd.principal_induct2, simp, simp)

apply (simp add: PDPlus_commute)

done

lemma convex_plus_absorb: "xs +\<natural> xs = xs"

apply (induct xs rule: convex_pd.principal_induct, simp)

apply (simp add: PDPlus_absorb)

done

class_interpretation aci_convex_plus: ab_semigroup_idem_mult ["op +\<natural>"]

  by unfold_locales

    (rule convex_plus_assoc convex_plus_commute convex_plus_absorb)+

lemma convex_plus_left_commute: "xs +\<natural> (ys +\<natural> zs) = ys +\<natural> (xs +\<natural> zs)"

by (rule aci_convex_plus.mult_left_commute)

lemma convex_plus_left_absorb: "xs +\<natural> (xs +\<natural> ys) = xs +\<natural> ys"

by (rule aci_convex_plus.mult_left_idem)

lemmas convex_plus_aci = aci_convex_plus.mult_ac_idem

lemma convex_unit_less_plus_iff [simp]:

  "{x}\<natural> \<sqsubseteq> ys +\<natural> zs \<longleftrightarrow> {x}\<natural> \<sqsubseteq> ys \<and> {x}\<natural> \<sqsubseteq> zs"

 apply (rule iffI)

  apply (subgoal_tac

    "adm (\<lambda>f. f\<cdot>{x}\<natural> \<sqsubseteq> f\<cdot>ys \<and> f\<cdot>{x}\<natural> \<sqsubseteq> f\<cdot>zs)")

   apply (drule admD, rule chain_approx)

    apply (drule_tac f="approx i" in monofun_cfun_arg)

    apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp)

    apply (cut_tac x="approx i\<cdot>ys" in convex_pd.compact_imp_principal, simp)

    apply (cut_tac x="approx i\<cdot>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_less_unit_iff [simp]:

  "xs +\<natural> ys \<sqsubseteq> {z}\<natural> \<longleftrightarrow> xs \<sqsubseteq> {z}\<natural> \<and> ys \<sqsubseteq> {z}\<natural>"

 apply (rule iffI)

  apply (subgoal_tac

    "adm (\<lambda>f. f\<cdot>xs \<sqsubseteq> f\<cdot>{z}\<natural> \<and> f\<cdot>ys \<sqsubseteq> f\<cdot>{z}\<natural>)")

   apply (drule admD, rule chain_approx)

    apply (drule_tac f="approx i" in monofun_cfun_arg)

    apply (cut_tac x="approx i\<cdot>xs" in convex_pd.compact_imp_principal, simp)

    apply (cut_tac x="approx i\<cdot>ys" in convex_pd.compact_imp_principal, simp)

    apply (cut_tac x="approx i\<cdot>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_less_iff [simp]: "{x}\<natural> \<sqsubseteq> {y}\<natural> \<longleftrightarrow> x \<sqsubseteq> y"

 apply (rule iffI)

  apply (rule profinite_less_ext)

  apply (drule_tac f="approx i" in monofun_cfun_arg, simp)

  apply (cut_tac x="approx i\<cdot>x" in compact_basis.compact_imp_principal, simp)

  apply (cut_tac x="approx i\<cdot>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> \<longleftrightarrow> x = y"

unfolding po_eq_conv by simp

lemma convex_unit_strict [simp]: "{\<bottom>}\<natural> = \<bottom>"

unfolding inst_convex_pd_pcpo Rep_compact_bot [symmetric] by simp

lemma convex_unit_strict_iff [simp]: "{x}\<natural> = \<bottom> \<longleftrightarrow> x = \<bottom>"

unfolding convex_unit_strict [symmetric] by (rule convex_unit_eq_iff)

lemma compact_convex_unit_iff [simp]:

  "compact {x}\<natural> \<longleftrightarrow> compact x"

unfolding profinite_compact_iff by simp

lemma compact_convex_plus [simp]:

  "\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> 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: "\<And>x. P {x}\<natural>"

  assumes insert: "\<And>x ys. \<lbrakk>P {x}\<natural>; P ys\<rbrakk> \<Longrightarrow> 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: "\<And>x. P {x}\<natural>"

  assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> 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 \<Rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where

  "convex_bind_basis = fold_pd

    (\<lambda>a. \<Lambda> f. f\<cdot>(Rep_compact_basis a))

    (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"

lemma ACI_convex_bind:

  "ab_semigroup_idem_mult (\<lambda>x y. \<Lambda> f. x\<cdot>f +\<natural> y\<cdot>f)"

apply unfold_locales

apply (simp add: convex_plus_assoc)

apply (simp add: convex_plus_commute)

apply (simp add: convex_plus_absorb eta_cfun)

done

lemma convex_bind_basis_simps [simp]:

  "convex_bind_basis (PDUnit a) =

    (\<Lambda> f. f\<cdot>(Rep_compact_basis a))"

  "convex_bind_basis (PDPlus t u) =

    (\<Lambda> f. convex_bind_basis t\<cdot>f +\<natural> convex_bind_basis u\<cdot>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:

  "\<lbrakk>cont f; cont g; \<And>x. f x \<sqsubseteq> g x\<rbrakk> \<Longrightarrow> (\<Lambda> x. f x) \<sqsubseteq> (\<Lambda> x. g x)"

by (simp add: expand_cfun_less)

lemma convex_bind_basis_mono:

  "t \<le>\<natural> u \<Longrightarrow> convex_bind_basis t \<sqsubseteq> convex_bind_basis u"

apply (erule convex_le_induct)

apply (erule (1) trans_less)

apply (simp add: monofun_LAM monofun_cfun)

apply (simp add: monofun_LAM monofun_cfun)

done

definition

  convex_bind :: "'a convex_pd \<rightarrow> ('a \<rightarrow> 'b convex_pd) \<rightarrow> 'b convex_pd" where

  "convex_bind = convex_pd.basis_fun convex_bind_basis"

lemma convex_bind_principal [simp]:

  "convex_bind\<cdot>(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\<cdot>{x}\<natural>\<cdot>f = f\<cdot>x"

by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma convex_bind_plus [simp]:

  "convex_bind\<cdot>(xs +\<natural> ys)\<cdot>f = convex_bind\<cdot>xs\<cdot>f +\<natural> convex_bind\<cdot>ys\<cdot>f"

by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)

lemma convex_bind_strict [simp]: "convex_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"

unfolding convex_unit_strict [symmetric] by (rule convex_bind_unit)

subsection {* Map and join *}

definition

  convex_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a convex_pd \<rightarrow> 'b convex_pd" where

  "convex_map = (\<Lambda> f xs. convex_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<natural>))"

definition

  convex_join :: "'a convex_pd convex_pd \<rightarrow> 'a convex_pd" where

  "convex_join = (\<Lambda> xss. convex_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"

lemma convex_map_unit [simp]:

  "convex_map\<cdot>f\<cdot>(convex_unit\<cdot>x) = convex_unit\<cdot>(f\<cdot>x)"

unfolding convex_map_def by simp

lemma convex_map_plus [simp]:

  "convex_map\<cdot>f\<cdot>(xs +\<natural> ys) = convex_map\<cdot>f\<cdot>xs +\<natural> convex_map\<cdot>f\<cdot>ys"

unfolding convex_map_def by simp

lemma convex_join_unit [simp]:

  "convex_join\<cdot>{xs}\<natural> = xs"

unfolding convex_join_def by simp

lemma convex_join_plus [simp]:

  "convex_join\<cdot>(xss +\<natural> yss) = convex_join\<cdot>xss +\<natural> convex_join\<cdot>yss"

unfolding convex_join_def by simp

lemma convex_map_ident: "convex_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"

by (induct xs rule: convex_pd_induct, simp_all)

lemma convex_map_map:

  "convex_map\<cdot>f\<cdot>(convex_map\<cdot>g\<cdot>xs) = convex_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>xs"

by (induct xs rule: convex_pd_induct, simp_all)

lemma convex_join_map_unit:

  "convex_join\<cdot>(convex_map\<cdot>convex_unit\<cdot>xs) = xs"

by (induct xs rule: convex_pd_induct, simp_all)

lemma convex_join_map_join:

  "convex_join\<cdot>(convex_map\<cdot>convex_join\<cdot>xsss) = convex_join\<cdot>(convex_join\<cdot>xsss)"

by (induct xsss rule: convex_pd_induct, simp_all)

lemma convex_join_map_map:

  "convex_join\<cdot>(convex_map\<cdot>(convex_map\<cdot>f)\<cdot>xss) =

   convex_map\<cdot>f\<cdot>(convex_join\<cdot>xss)"

by (induct xss rule: convex_pd_induct, simp_all)

lemma convex_map_approx: "convex_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"

by (induct xs rule: convex_pd_induct, simp_all)

subsection {* Conversions to other powerdomains *}

text {* Convex to upper *}

lemma convex_le_imp_upper_le: "t \<le>\<natural> u \<Longrightarrow> t \<le>\<sharp> u"

unfolding convex_le_def by simp

definition

  convex_to_upper :: "'a convex_pd \<rightarrow> 'a upper_pd" where

  "convex_to_upper = convex_pd.basis_fun upper_principal"

lemma convex_to_upper_principal [simp]:

  "convex_to_upper\<cdot>(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\<cdot>{x}\<natural> = {x}\<sharp>"

by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma convex_to_upper_plus [simp]:

  "convex_to_upper\<cdot>(xs +\<natural> ys) = convex_to_upper\<cdot>xs +\<sharp> convex_to_upper\<cdot>ys"

by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)

lemma approx_convex_to_upper:

  "approx i\<cdot>(convex_to_upper\<cdot>xs) = convex_to_upper\<cdot>(approx i\<cdot>xs)"

by (induct xs rule: convex_pd_induct, simp, simp, simp)

lemma convex_to_upper_bind [simp]:

  "convex_to_upper\<cdot>(convex_bind\<cdot>xs\<cdot>f) =

    upper_bind\<cdot>(convex_to_upper\<cdot>xs)\<cdot>(convex_to_upper oo f)"

by (induct xs rule: convex_pd_induct, simp, simp, simp)

lemma convex_to_upper_map [simp]:

  "convex_to_upper\<cdot>(convex_map\<cdot>f\<cdot>xs) = upper_map\<cdot>f\<cdot>(convex_to_upper\<cdot>xs)"

by (simp add: convex_map_def upper_map_def cfcomp_LAM)

lemma convex_to_upper_join [simp]:

  "convex_to_upper\<cdot>(convex_join\<cdot>xss) =

    upper_bind\<cdot>(convex_to_upper\<cdot>xss)\<cdot>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 \<le>\<natural> u \<Longrightarrow> t \<le>\<flat> u"

unfolding convex_le_def by simp

definition

  convex_to_lower :: "'a convex_pd \<rightarrow> 'a lower_pd" where

  "convex_to_lower = convex_pd.basis_fun lower_principal"

lemma convex_to_lower_principal [simp]:

  "convex_to_lower\<cdot>(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\<cdot>{x}\<natural> = {x}\<flat>"

by (induct x rule: compact_basis.principal_induct, simp, simp)

lemma convex_to_lower_plus [simp]:

  "convex_to_lower\<cdot>(xs +\<natural> ys) = convex_to_lower\<cdot>xs +\<flat> convex_to_lower\<cdot>ys"

by (induct xs ys rule: convex_pd.principal_induct2, simp, simp, simp)

lemma approx_convex_to_lower:

  "approx i\<cdot>(convex_to_lower\<cdot>xs) = convex_to_lower\<cdot>(approx i\<cdot>xs)"

by (induct xs rule: convex_pd_induct, simp, simp, simp)

lemma convex_to_lower_bind [simp]:

  "convex_to_lower\<cdot>(convex_bind\<cdot>xs\<cdot>f) =

    lower_bind\<cdot>(convex_to_lower\<cdot>xs)\<cdot>(convex_to_lower oo f)"

by (induct xs rule: convex_pd_induct, simp, simp, simp)

lemma convex_to_lower_map [simp]:

  "convex_to_lower\<cdot>(convex_map\<cdot>f\<cdot>xs) = lower_map\<cdot>f\<cdot>(convex_to_lower\<cdot>xs)"

by (simp add: convex_map_def lower_map_def cfcomp_LAM)

lemma convex_to_lower_join [simp]:

  "convex_to_lower\<cdot>(convex_join\<cdot>xss) =

    lower_bind\<cdot>(convex_to_lower\<cdot>xss)\<cdot>convex_to_lower"

by (simp add: convex_join_def lower_join_def cfcomp_LAM eta_cfun)

text {* Ordering property *}

lemma convex_pd_less_iff:

  "(xs \<sqsubseteq> ys) =

    (convex_to_upper\<cdot>xs \<sqsubseteq> convex_to_upper\<cdot>ys \<and>

     convex_to_lower\<cdot>xs \<sqsubseteq> convex_to_lower\<cdot>ys)"

 apply (safe elim!: monofun_cfun_arg)

 apply (rule profinite_less_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\<cdot>xs" in convex_pd.compact_imp_principal, simp)

 apply (cut_tac x="approx i\<cdot>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_less_plus_iff =

  convex_pd_less_iff [where xs="xs +\<natural> ys" and ys="zs +\<natural> ws", standard]

lemmas convex_pd_less_simps =

  convex_unit_less_plus_iff

  convex_plus_less_unit_iff

  convex_plus_less_plus_iff

  convex_unit_less_iff

  convex_to_upper_unit

  convex_to_upper_plus

  convex_to_lower_unit

  convex_to_lower_plus

  upper_pd_less_simps

  lower_pd_less_simps

end