(*  Title:      HOLCF/UpperPD.thy

    ID:         $Id$

    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 approx_pd_upper_mono1:

  "i \<le> j \<Longrightarrow> approx_pd i t \<le>\<sharp> approx_pd j t"

apply (induct t rule: pd_basis_induct)

apply (simp add: compact_approx_mono1)

apply (simp add: PDPlus_upper_mono)

done

lemma approx_pd_upper_le: "approx_pd i t \<le>\<sharp> t"

apply (induct t rule: pd_basis_induct)

apply (simp add: compact_approx_le)

apply (simp add: PDPlus_upper_mono)

done

lemma approx_pd_upper_mono:

  "t \<le>\<sharp> u \<Longrightarrow> approx_pd n t \<le>\<sharp> approx_pd n u"

apply (erule upper_le_induct)

apply (simp add: compact_approx_mono)

apply (simp add: upper_le_PDPlus_PDUnit_iff)

apply (simp add: upper_le_PDPlus_iff)

done

subsection {* Type definition *}

cpodef (open) 'a upper_pd =

  "{S::'a::profinite pd_basis set. upper_le.ideal S}"

apply (simp add: upper_le.adm_ideal)

apply (fast intro: upper_le.ideal_principal)

done

lemma ideal_Rep_upper_pd: "upper_le.ideal (Rep_upper_pd x)"

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

apply (rule Abs_upper_pd_inverse [unfolded mem_Collect_eq])

apply (rule upper_le.ideal_principal)

done

interpretation upper_pd:

  ideal_completion [upper_le approx_pd upper_principal Rep_upper_pd]

apply unfold_locales

apply (rule approx_pd_upper_le)

apply (rule approx_pd_idem)

apply (erule approx_pd_upper_mono)

apply (rule approx_pd_upper_mono1, simp)

apply (rule finite_range_approx_pd)

apply (rule ex_approx_pd_eq)

apply (rule ideal_Rep_upper_pd)

apply (rule cont_Rep_upper_pd)

apply (rule Rep_upper_principal)

apply (simp only: less_upper_pd_def less_set_eq)

done

lemma upper_principal_less_iff [simp]:

  "upper_principal t \<sqsubseteq> upper_principal u \<longleftrightarrow> t \<le>\<sharp> u"

by (rule upper_pd.principal_less_iff)

lemma upper_principal_eq_iff:

  "upper_principal t = upper_principal u \<longleftrightarrow> t \<le>\<sharp> u \<and> u \<le>\<sharp> t"

by (rule upper_pd.principal_eq_iff)

lemma upper_principal_mono:

  "t \<le>\<sharp> u \<Longrightarrow> upper_principal t \<sqsubseteq> upper_principal u"

by (rule upper_pd.principal_mono)

lemma compact_upper_principal: "compact (upper_principal t)"

by (rule upper_pd.compact_principal)

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])

subsection {* Approximation *}

instance upper_pd :: (profinite) approx ..

defs (overloaded)

  approx_upper_pd_def: "approx \<equiv> upper_pd.completion_approx"

instance upper_pd :: (profinite) profinite

apply (intro_classes, unfold approx_upper_pd_def)

apply (simp add: 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

instance upper_pd :: (bifinite) bifinite ..

lemma approx_upper_principal [simp]:

  "approx n\<cdot>(upper_principal t) = upper_principal (approx_pd 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 (approx_pd n t)"

unfolding approx_upper_pd_def

by (rule upper_pd.completion_approx_eq_principal)

lemma compact_imp_upper_principal:

  "compact xs \<Longrightarrow> \<exists>t. xs = upper_principal t"

apply (drule bifinite_compact_eq_approx)

apply (erule exE)

apply (erule subst)

apply (cut_tac n=i and xs=xs in approx_eq_upper_principal)

apply fast

done

lemma upper_principal_induct:

  "\<lbrakk>adm P; \<And>t. P (upper_principal t)\<rbrakk> \<Longrightarrow> P xs"

by (rule upper_pd.principal_induct)

lemma upper_principal_induct2:

  "\<lbrakk>\<And>ys. adm (\<lambda>xs. P xs ys); \<And>xs. adm (\<lambda>ys. P xs ys);

    \<And>t u. P (upper_principal t) (upper_principal u)\<rbrakk> \<Longrightarrow> P xs ys"

apply (rule_tac x=ys in spec)

apply (rule_tac xs=xs in upper_principal_induct, simp)

apply (rule allI, rename_tac ys)

apply (rule_tac xs=ys in upper_principal_induct, simp)

apply simp

done

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

    upper_principal_mono 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_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_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_principal_induct2, simp, simp)

apply (rule_tac xs=zs in upper_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_principal_induct2, simp, simp)

apply (simp add: PDPlus_commute)

done

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

apply (induct xs rule: upper_principal_induct, simp)

apply (simp add: PDPlus_absorb)

done

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_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 xs="approx i\<cdot>xs" in compact_imp_upper_principal, simp)

    apply (cut_tac xs="approx i\<cdot>ys" in compact_imp_upper_principal, simp)

    apply (cut_tac x="approx i\<cdot>z" in compact_imp_Rep_compact_basis, 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 bifinite_less_ext)

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

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

  apply (cut_tac x="approx i\<cdot>y" in compact_imp_Rep_compact_basis, simp)

  apply (clarify, simp add: compact_le_def)

 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_principal_induct2 [where xs=xs and ys=ys], simp, simp)

apply (simp add: inst_upper_pd_pcpo upper_principal_eq_iff

                 upper_le_PDPlus_PDUnit_iff)

apply auto

done

lemma compact_upper_unit_iff [simp]: "compact {x}\<sharp> \<longleftrightarrow> compact x"

unfolding bifinite_compact_iff by simp

lemma compact_upper_plus [simp]:

  "\<lbrakk>compact xs; compact ys\<rbrakk> \<Longrightarrow> compact (xs +\<sharp> ys)"

apply (drule compact_imp_upper_principal)+

apply (auto simp add: compact_upper_principal)

done

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_principal_induct, rule P)

apply (induct_tac t 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_principal_induct, rule P)

apply (induct_tac t 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: compact_le_def 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_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_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