(*  Title:      HOLCF/UpperPD.thy

     Author:     Brian Huffman

 *)

 header {* Upper powerdomain *}

 theory UpperPD

 imports CompactBasis

 begin

 subsection {* Basis preorder *}

 definition

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

   "upper_le = (\<lambda>u v. \<forall>y\<in>Rep_pd_basis v. \<exists>x\<in>Rep_pd_basis u. x \<sqsubseteq> y)"

 lemma upper_le_refl [simp]: "t \<le>\<sharp> t"

 unfolding upper_le_def by fast

 lemma upper_le_trans: "\<lbrakk>t \<le>\<sharp> u; u \<le>\<sharp> v\<rbrakk> \<Longrightarrow> t \<le>\<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) trans_less)

 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 \<le>\<sharp> t"

 unfolding upper_le_def Rep_PDUnit by simp

 lemma PDUnit_upper_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<sharp> PDUnit y"

 unfolding upper_le_def Rep_PDUnit by simp

 lemma PDPlus_upper_mono: "\<lbrakk>s \<le>\<sharp> t; u \<le>\<sharp> v\<rbrakk> \<Longrightarrow> PDPlus s u \<le>\<sharp> PDPlus t v"

 unfolding upper_le_def Rep_PDPlus by fast

 lemma PDPlus_upper_less: "PDPlus t u \<le>\<sharp> t"

 unfolding upper_le_def Rep_PDPlus by fast

 lemma upper_le_PDUnit_PDUnit_iff [simp]:

   "(PDUnit a \<le>\<sharp> PDUnit b) = a \<sqsubseteq> b"

 unfolding upper_le_def Rep_PDUnit by fast

 lemma upper_le_PDPlus_PDUnit_iff:

   "(PDPlus t u \<le>\<sharp> PDUnit a) = (t \<le>\<sharp> PDUnit a \<or> u \<le>\<sharp> PDUnit a)"

 unfolding upper_le_def Rep_PDPlus Rep_PDUnit by fast

 lemma upper_le_PDPlus_iff: "(t \<le>\<sharp> PDPlus u v) = (t \<le>\<sharp> u \<and> t \<le>\<sharp> v)"

 unfolding upper_le_def Rep_PDPlus by fast

 lemma upper_le_induct [induct set: upper_le]:

   assumes le: "t \<le>\<sharp> u"

   assumes 1: "\<And>a b. a \<sqsubseteq> b \<Longrightarrow> P (PDUnit a) (PDUnit b)"

   assumes 2: "\<And>t u a. P t (PDUnit a) \<Longrightarrow> P (PDPlus t u) (PDUnit a)"

   assumes 3: "\<And>t u v. \<lbrakk>P t u; P t v\<rbrakk> \<Longrightarrow> 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 \<le>\<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 \<le>\<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 \<le>\<sharp> u \<Longrightarrow> pd_take n t \<le>\<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) sq_ord

 begin

 definition

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

 instance ..

 end

 instance upper_pd :: (profinite) po

 by (rule upper_le.typedef_ideal_po

     [OF type_definition_upper_pd sq_le_upper_pd_def])

 instance upper_pd :: (profinite) cpo

 by (rule upper_le.typedef_ideal_cpo

     [OF type_definition_upper_pd sq_le_upper_pd_def])

 lemma Rep_upper_pd_lub:

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

   "upper_principal t = Abs_upper_pd {u. u \<le>\<sharp> t}"

 lemma Rep_upper_principal:

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

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

   "upper_unit = compact_basis.basis_fun (\<lambda>a. upper_principal (PDUnit a))"

 definition

   upper_plus :: "'a upper_pd \<rightarrow> 'a upper_pd \<rightarrow> 'a upper_pd" where

   "upper_plus = upper_pd.basis_fun (\<lambda>t. upper_pd.basis_fun (\<lambda>u.

       upper_principal (PDPlus t u)))"

 abbreviation

   upper_add :: "'a upper_pd \<Rightarrow> 'a upper_pd \<Rightarrow> 'a upper_pd"

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

   "xs +\<sharp> ys == upper_plus\<cdot>xs\<cdot>ys"

 syntax

   "_upper_pd" :: "args \<Rightarrow> 'a upper_pd" ("{_}\<sharp>")

 translations

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

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

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

 lemma upper_plus_assoc: "(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

 lemma upper_plus_commute: "xs +\<sharp> ys = ys +\<sharp> xs"

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

 apply (simp add: PDPlus_commute)

 done

 lemma upper_plus_absorb: "xs +\<sharp> xs = xs"

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

 apply (simp add: PDPlus_absorb)

 done

 class_interpretation aci_upper_plus: ab_semigroup_idem_mult ["op +\<sharp>"]

   by unfold_locales

     (rule upper_plus_assoc upper_plus_commute upper_plus_absorb)+

 lemma upper_plus_left_commute: "xs +\<sharp> (ys +\<sharp> zs) = ys +\<sharp> (xs +\<sharp> zs)"

 by (rule aci_upper_plus.mult_left_commute)

 lemma upper_plus_left_absorb: "xs +\<sharp> (xs +\<sharp> ys) = xs +\<sharp> ys"

 by (rule aci_upper_plus.mult_left_idem)

 lemmas upper_plus_aci = aci_upper_plus.mult_ac_idem

 lemma upper_plus_less1: "xs +\<sharp> ys \<sqsubseteq> xs"

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

 apply (simp add: PDPlus_upper_less)

 done

 lemma upper_plus_less2: "xs +\<sharp> ys \<sqsubseteq> ys"

 by (subst upper_plus_commute, rule upper_plus_less1)

 lemma upper_plus_greatest: "\<lbrakk>xs \<sqsubseteq> ys; xs \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> 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_less_plus_iff:

   "xs \<sqsubseteq> ys +\<sharp> zs \<longleftrightarrow> xs \<sqsubseteq> ys \<and> xs \<sqsubseteq> zs"

 apply safe

 apply (erule trans_less [OF _ upper_plus_less1])

 apply (erule trans_less [OF _ upper_plus_less2])

 apply (erule (1) upper_plus_greatest)

 done

 lemma upper_plus_less_unit_iff:

   "xs +\<sharp> ys \<sqsubseteq> {z}\<sharp> \<longleftrightarrow> xs \<sqsubseteq> {z}\<sharp> \<or> ys \<sqsubseteq> {z}\<sharp>"

  apply (rule iffI)

   apply (subgoal_tac

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

    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 upper_pd.compact_imp_principal, simp)

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

     apply (cut_tac x="approx i\<cdot>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 trans_less [OF upper_plus_less1])

  apply (erule trans_less [OF upper_plus_less2])

 done

 lemma upper_unit_less_iff [simp]: "{x}\<sharp> \<sqsubseteq> {y}\<sharp> \<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

 lemmas upper_pd_less_simps =

   upper_unit_less_iff

   upper_less_plus_iff

   upper_plus_less_unit_iff

 lemma upper_unit_eq_iff [simp]: "{x}\<sharp> = {y}\<sharp> \<longleftrightarrow> x = y"

 unfolding po_eq_conv by simp

 lemma upper_unit_strict [simp]: "{\<bottom>}\<sharp> = \<bottom>"

 unfolding inst_upper_pd_pcpo Rep_compact_bot [symmetric] by simp

 lemma upper_plus_strict1 [simp]: "\<bottom> +\<sharp> ys = \<bottom>"

 by (rule UU_I, rule upper_plus_less1)

 lemma upper_plus_strict2 [simp]: "xs +\<sharp> \<bottom> = \<bottom>"

 by (rule UU_I, rule upper_plus_less2)

 lemma upper_unit_strict_iff [simp]: "{x}\<sharp> = \<bottom> \<longleftrightarrow> x = \<bottom>"

 unfolding upper_unit_strict [symmetric] by (rule upper_unit_eq_iff)

 lemma upper_plus_strict_iff [simp]:

   "xs +\<sharp> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<or> ys = \<bottom>"

 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> \<longleftrightarrow> compact x"

 unfolding profinite_compact_iff by simp

 lemma compact_upper_plus [simp]:

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

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

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

   "upper_bind_basis = fold_pd

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

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

 lemma ACI_upper_bind:

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

 apply unfold_locales

 apply (simp add: upper_plus_assoc)

 apply (simp add: upper_plus_commute)

 apply (simp add: upper_plus_absorb eta_cfun)

 done

 lemma upper_bind_basis_simps [simp]:

   "upper_bind_basis (PDUnit a) =

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

   "upper_bind_basis (PDPlus t u) =

     (\<Lambda> f. upper_bind_basis t\<cdot>f +\<sharp> upper_bind_basis u\<cdot>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 \<le>\<sharp> u \<Longrightarrow> upper_bind_basis t \<sqsubseteq> upper_bind_basis u"

 unfolding expand_cfun_less

 apply (erule upper_le_induct, safe)

 apply (simp add: monofun_cfun)

 apply (simp add: trans_less [OF upper_plus_less1])

 apply (simp add: upper_less_plus_iff)

 done

 definition

   upper_bind :: "'a upper_pd \<rightarrow> ('a \<rightarrow> 'b upper_pd) \<rightarrow> 'b upper_pd" where

   "upper_bind = upper_pd.basis_fun upper_bind_basis"

 lemma upper_bind_principal [simp]:

   "upper_bind\<cdot>(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\<cdot>{x}\<sharp>\<cdot>f = f\<cdot>x"

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

 lemma upper_bind_plus [simp]:

   "upper_bind\<cdot>(xs +\<sharp> ys)\<cdot>f = upper_bind\<cdot>xs\<cdot>f +\<sharp> upper_bind\<cdot>ys\<cdot>f"

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

 lemma upper_bind_strict [simp]: "upper_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"

 unfolding upper_unit_strict [symmetric] by (rule upper_bind_unit)

 subsection {* Map and join *}

 definition

   upper_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a upper_pd \<rightarrow> 'b upper_pd" where

   "upper_map = (\<Lambda> f xs. upper_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<sharp>))"

 definition

   upper_join :: "'a upper_pd upper_pd \<rightarrow> 'a upper_pd" where

   "upper_join = (\<Lambda> xss. upper_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"

 lemma upper_map_unit [simp]:

   "upper_map\<cdot>f\<cdot>{x}\<sharp> = {f\<cdot>x}\<sharp>"

 unfolding upper_map_def by simp

 lemma upper_map_plus [simp]:

   "upper_map\<cdot>f\<cdot>(xs +\<sharp> ys) = upper_map\<cdot>f\<cdot>xs +\<sharp> upper_map\<cdot>f\<cdot>ys"

 unfolding upper_map_def by simp

 lemma upper_join_unit [simp]:

   "upper_join\<cdot>{xs}\<sharp> = xs"

 unfolding upper_join_def by simp

 lemma upper_join_plus [simp]:

   "upper_join\<cdot>(xss +\<sharp> yss) = upper_join\<cdot>xss +\<sharp> upper_join\<cdot>yss"

 unfolding upper_join_def by simp

 lemma upper_map_ident: "upper_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"

 by (induct xs rule: upper_pd_induct, simp_all)

 lemma upper_map_map:

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

 by (induct xs rule: upper_pd_induct, simp_all)

 lemma upper_join_map_unit:

   "upper_join\<cdot>(upper_map\<cdot>upper_unit\<cdot>xs) = xs"

 by (induct xs rule: upper_pd_induct, simp_all)

 lemma upper_join_map_join:

   "upper_join\<cdot>(upper_map\<cdot>upper_join\<cdot>xsss) = upper_join\<cdot>(upper_join\<cdot>xsss)"

 by (induct xsss rule: upper_pd_induct, simp_all)

 lemma upper_join_map_map:

   "upper_join\<cdot>(upper_map\<cdot>(upper_map\<cdot>f)\<cdot>xss) =

    upper_map\<cdot>f\<cdot>(upper_join\<cdot>xss)"

 by (induct xss rule: upper_pd_induct, simp_all)

 lemma upper_map_approx: "upper_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"

 by (induct xs rule: upper_pd_induct, simp_all)

 end