(*  Title:      HOLCF/LowerPD.thy

    ID:         $Id$

    Author:     Brian Huffman

*)

header {* Lower powerdomain *}

theory LowerPD

imports CompactBasis

begin

subsection {* Basis preorder *}

definition

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

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

lemma lower_le_refl [simp]: "t \<le>\<flat> t"

unfolding lower_le_def by fast

lemma lower_le_trans: "\<lbrakk>t \<le>\<flat> u; u \<le>\<flat> v\<rbrakk> \<Longrightarrow> t \<le>\<flat> v"

unfolding lower_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 lower_le: preorder [lower_le]

by (rule preorder.intro, rule lower_le_refl, rule lower_le_trans)

lemma lower_le_minimal [simp]: "PDUnit compact_bot \<le>\<flat> t"

unfolding lower_le_def Rep_PDUnit

by (simp, rule Rep_pd_basis_nonempty [folded ex_in_conv])

lemma PDUnit_lower_mono: "x \<sqsubseteq> y \<Longrightarrow> PDUnit x \<le>\<flat> PDUnit y"

unfolding lower_le_def Rep_PDUnit by fast

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

unfolding lower_le_def Rep_PDPlus by fast

lemma PDPlus_lower_less: "t \<le>\<flat> PDPlus t u"

unfolding lower_le_def Rep_PDPlus by fast

lemma lower_le_PDUnit_PDUnit_iff [simp]:

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

unfolding lower_le_def Rep_PDUnit by fast

lemma lower_le_PDUnit_PDPlus_iff:

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

unfolding lower_le_def Rep_PDPlus Rep_PDUnit by fast

lemma lower_le_PDPlus_iff: "(PDPlus t u \<le>\<flat> v) = (t \<le>\<flat> v \<and> u \<le>\<flat> v)"

unfolding lower_le_def Rep_PDPlus by fast

lemma lower_le_induct [induct set: lower_le]:

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

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

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

  assumes 3: "\<And>t u v. \<lbrakk>P t v; P u v\<rbrakk> \<Longrightarrow> P (PDPlus t u) v"

  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_induct)

apply (simp add: 1)

apply (simp add: lower_le_PDUnit_PDPlus_iff)

apply (simp add: 2)

apply (subst PDPlus_commute)

apply (simp add: 2)

apply (simp add: lower_le_PDPlus_iff 3)

done

lemma approx_pd_lower_chain:

  "approx_pd n t \<le>\<flat> approx_pd (Suc n) t"

apply (induct t rule: pd_basis_induct)

apply (simp add: compact_basis.take_chain)

apply (simp add: PDPlus_lower_mono)

done

lemma approx_pd_lower_le: "approx_pd i t \<le>\<flat> t"

apply (induct t rule: pd_basis_induct)

apply (simp add: compact_basis.take_less)

apply (simp add: PDPlus_lower_mono)

done

lemma approx_pd_lower_mono:

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

apply (erule lower_le_induct)

apply (simp add: compact_basis.take_mono)

apply (simp add: lower_le_PDUnit_PDPlus_iff)

apply (simp add: lower_le_PDPlus_iff)

done

subsection {* Type definition *}

cpodef (open) 'a lower_pd =

  "{S::'a pd_basis cset. lower_le.ideal (Rep_cset S)}"

by (rule lower_le.cpodef_ideal_lemma)

lemma ideal_Rep_lower_pd: "lower_le.ideal (Rep_cset (Rep_lower_pd xs))"

by (rule Rep_lower_pd [unfolded mem_Collect_eq])

definition

  lower_principal :: "'a pd_basis \<Rightarrow> 'a lower_pd" where

  "lower_principal t = Abs_lower_pd (Abs_cset {u. u \<le>\<flat> t})"

lemma Rep_lower_principal:

  "Rep_cset (Rep_lower_pd (lower_principal t)) = {u. u \<le>\<flat> t}"

unfolding lower_principal_def

by (simp add: Abs_lower_pd_inverse lower_le.ideal_principal)

interpretation lower_pd:

  ideal_completion

    [lower_le approx_pd lower_principal "\<lambda>x. Rep_cset (Rep_lower_pd x)"]

apply unfold_locales

apply (rule approx_pd_lower_le)

apply (rule approx_pd_idem)

apply (erule approx_pd_lower_mono)

apply (rule approx_pd_lower_chain)

apply (rule finite_range_approx_pd)

apply (rule approx_pd_covers)

apply (rule ideal_Rep_lower_pd)

apply (simp add: cont2contlubE [OF cont_Rep_lower_pd] Rep_cset_lub)

apply (rule Rep_lower_principal)

apply (simp only: less_lower_pd_def sq_le_cset_def)

done

text {* Lower powerdomain is pointed *}

lemma lower_pd_minimal: "lower_principal (PDUnit compact_bot) \<sqsubseteq> ys"

by (induct ys rule: lower_pd.principal_induct, simp, simp)

instance lower_pd :: (bifinite) pcpo

by intro_classes (fast intro: lower_pd_minimal)

lemma inst_lower_pd_pcpo: "\<bottom> = lower_principal (PDUnit compact_bot)"

by (rule lower_pd_minimal [THEN UU_I, symmetric])

text {* Lower powerdomain is profinite *}

instantiation lower_pd :: (profinite) profinite

begin

definition

  approx_lower_pd_def: "approx = lower_pd.completion_approx"

instance

apply (intro_classes, unfold approx_lower_pd_def)

apply (rule lower_pd.chain_completion_approx)

apply (rule lower_pd.lub_completion_approx)

apply (rule lower_pd.completion_approx_idem)

apply (rule lower_pd.finite_fixes_completion_approx)

done

end

instance lower_pd :: (bifinite) bifinite ..

lemma approx_lower_principal [simp]:

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

unfolding approx_lower_pd_def

by (rule lower_pd.completion_approx_principal)

lemma approx_eq_lower_principal:

  "\<exists>t\<in>Rep_cset (Rep_lower_pd xs).

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

unfolding approx_lower_pd_def

by (rule lower_pd.completion_approx_eq_principal)

subsection {* Monadic unit and plus *}

definition

  lower_unit :: "'a \<rightarrow> 'a lower_pd" where

  "lower_unit = compact_basis.basis_fun (\<lambda>a. lower_principal (PDUnit a))"

definition

  lower_plus :: "'a lower_pd \<rightarrow> 'a lower_pd \<rightarrow> 'a lower_pd" where

  "lower_plus = lower_pd.basis_fun (\<lambda>t. lower_pd.basis_fun (\<lambda>u.

      lower_principal (PDPlus t u)))"

abbreviation

  lower_add :: "'a lower_pd \<Rightarrow> 'a lower_pd \<Rightarrow> 'a lower_pd"

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

  "xs +\<flat> ys == lower_plus\<cdot>xs\<cdot>ys"

syntax

  "_lower_pd" :: "args \<Rightarrow> 'a lower_pd" ("{_}\<flat>")

translations

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

  "{x}\<flat>" == "CONST lower_unit\<cdot>x"

lemma lower_unit_Rep_compact_basis [simp]:

  "{Rep_compact_basis a}\<flat> = lower_principal (PDUnit a)"

unfolding lower_unit_def

by (simp add: compact_basis.basis_fun_principal PDUnit_lower_mono)

lemma lower_plus_principal [simp]:

  "lower_principal t +\<flat> lower_principal u = lower_principal (PDPlus t u)"

unfolding lower_plus_def

by (simp add: lower_pd.basis_fun_principal

    lower_pd.basis_fun_mono PDPlus_lower_mono)

lemma approx_lower_unit [simp]:

  "approx n\<cdot>{x}\<flat> = {approx n\<cdot>x}\<flat>"

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

apply (simp add: approx_Rep_compact_basis)

done

lemma approx_lower_plus [simp]:

  "approx n\<cdot>(xs +\<flat> ys) = (approx n\<cdot>xs) +\<flat> (approx n\<cdot>ys)"

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

lemma lower_plus_assoc: "(xs +\<flat> ys) +\<flat> zs = xs +\<flat> (ys +\<flat> zs)"

apply (induct xs ys arbitrary: zs rule: lower_pd.principal_induct2, simp, simp)

apply (rule_tac x=zs in lower_pd.principal_induct, simp)

apply (simp add: PDPlus_assoc)

done

lemma lower_plus_commute: "xs +\<flat> ys = ys +\<flat> xs"

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

apply (simp add: PDPlus_commute)

done

lemma lower_plus_absorb: "xs +\<flat> xs = xs"

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

apply (simp add: PDPlus_absorb)

done

interpretation aci_lower_plus: ab_semigroup_idem_mult ["op +\<flat>"]

  by unfold_locales

    (rule lower_plus_assoc lower_plus_commute lower_plus_absorb)+

lemma lower_plus_left_commute: "xs +\<flat> (ys +\<flat> zs) = ys +\<flat> (xs +\<flat> zs)"

by (rule aci_lower_plus.mult_left_commute)

lemma lower_plus_left_absorb: "xs +\<flat> (xs +\<flat> ys) = xs +\<flat> ys"

by (rule aci_lower_plus.mult_left_idem)

lemmas lower_plus_aci = aci_lower_plus.mult_ac_idem

lemma lower_plus_less1: "xs \<sqsubseteq> xs +\<flat> ys"

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

apply (simp add: PDPlus_lower_less)

done

lemma lower_plus_less2: "ys \<sqsubseteq> xs +\<flat> ys"

by (subst lower_plus_commute, rule lower_plus_less1)

lemma lower_plus_least: "\<lbrakk>xs \<sqsubseteq> zs; ys \<sqsubseteq> zs\<rbrakk> \<Longrightarrow> xs +\<flat> ys \<sqsubseteq> zs"

apply (subst lower_plus_absorb [of zs, symmetric])

apply (erule (1) monofun_cfun [OF monofun_cfun_arg])

done

lemma lower_plus_less_iff:

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

apply safe

apply (erule trans_less [OF lower_plus_less1])

apply (erule trans_less [OF lower_plus_less2])

apply (erule (1) lower_plus_least)

done

lemma lower_unit_less_plus_iff:

  "{x}\<flat> \<sqsubseteq> ys +\<flat> zs \<longleftrightarrow> {x}\<flat> \<sqsubseteq> ys \<or> {x}\<flat> \<sqsubseteq> zs"

 apply (rule iffI)

  apply (subgoal_tac

    "adm (\<lambda>f. f\<cdot>{x}\<flat> \<sqsubseteq> f\<cdot>ys \<or> f\<cdot>{x}\<flat> \<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 lower_pd.compact_imp_principal, simp)

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

    apply (clarify, simp add: lower_le_PDUnit_PDPlus_iff)

   apply simp

  apply simp

 apply (erule disjE)

  apply (erule trans_less [OF _ lower_plus_less1])

 apply (erule trans_less [OF _ lower_plus_less2])

done

lemma lower_unit_less_iff [simp]: "{x}\<flat> \<sqsubseteq> {y}\<flat> \<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 lower_pd_less_simps =

  lower_unit_less_iff

  lower_plus_less_iff

  lower_unit_less_plus_iff

lemma fooble:

  fixes f :: "'a::po \<Rightarrow> 'b::po"

  assumes f: "\<And>x y. f x \<sqsubseteq> f y \<longleftrightarrow> x \<sqsubseteq> y"

  shows "f x = f y \<longleftrightarrow> x = y"

unfolding po_eq_conv by (simp add: f)

lemma lower_unit_eq_iff [simp]: "{x}\<flat> = {y}\<flat> \<longleftrightarrow> x = y"

by (rule lower_unit_less_iff [THEN fooble])

lemma lower_unit_strict [simp]: "{\<bottom>}\<flat> = \<bottom>"

unfolding inst_lower_pd_pcpo Rep_compact_bot [symmetric] by simp

lemma lower_unit_strict_iff [simp]: "{x}\<flat> = \<bottom> \<longleftrightarrow> x = \<bottom>"

unfolding lower_unit_strict [symmetric] by (rule lower_unit_eq_iff)

lemma lower_plus_strict_iff [simp]:

  "xs +\<flat> ys = \<bottom> \<longleftrightarrow> xs = \<bottom> \<and> ys = \<bottom>"

apply safe

apply (rule UU_I, erule subst, rule lower_plus_less1)

apply (rule UU_I, erule subst, rule lower_plus_less2)

apply (rule lower_plus_absorb)

done

lemma lower_plus_strict1 [simp]: "\<bottom> +\<flat> ys = ys"

apply (rule antisym_less [OF _ lower_plus_less2])

apply (simp add: lower_plus_least)

done

lemma lower_plus_strict2 [simp]: "xs +\<flat> \<bottom> = xs"

apply (rule antisym_less [OF _ lower_plus_less1])

apply (simp add: lower_plus_least)

done

lemma compact_lower_unit_iff [simp]: "compact {x}\<flat> \<longleftrightarrow> compact x"

unfolding profinite_compact_iff by simp

lemma compact_lower_plus [simp]:

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

by (auto dest!: lower_pd.compact_imp_principal)

subsection {* Induction rules *}

lemma lower_pd_induct1:

  assumes P: "adm P"

  assumes unit: "\<And>x. P {x}\<flat>"

  assumes insert:

    "\<And>x ys. \<lbrakk>P {x}\<flat>; P ys\<rbrakk> \<Longrightarrow> P ({x}\<flat> +\<flat> ys)"

  shows "P (xs::'a lower_pd)"

apply (induct xs rule: lower_pd.principal_induct, rule P)

apply (induct_tac a rule: pd_basis_induct1)

apply (simp only: lower_unit_Rep_compact_basis [symmetric])

apply (rule unit)

apply (simp only: lower_unit_Rep_compact_basis [symmetric]

                  lower_plus_principal [symmetric])

apply (erule insert [OF unit])

done

lemma lower_pd_induct:

  assumes P: "adm P"

  assumes unit: "\<And>x. P {x}\<flat>"

  assumes plus: "\<And>xs ys. \<lbrakk>P xs; P ys\<rbrakk> \<Longrightarrow> P (xs +\<flat> ys)"

  shows "P (xs::'a lower_pd)"

apply (induct xs rule: lower_pd.principal_induct, rule P)

apply (induct_tac a rule: pd_basis_induct)

apply (simp only: lower_unit_Rep_compact_basis [symmetric] unit)

apply (simp only: lower_plus_principal [symmetric] plus)

done

subsection {* Monadic bind *}

definition

  lower_bind_basis ::

  "'a pd_basis \<Rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where

  "lower_bind_basis = fold_pd

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

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

lemma ACI_lower_bind:

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

apply unfold_locales

apply (simp add: lower_plus_assoc)

apply (simp add: lower_plus_commute)

apply (simp add: lower_plus_absorb eta_cfun)

done

lemma lower_bind_basis_simps [simp]:

  "lower_bind_basis (PDUnit a) =

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

  "lower_bind_basis (PDPlus t u) =

    (\<Lambda> f. lower_bind_basis t\<cdot>f +\<flat> lower_bind_basis u\<cdot>f)"

unfolding lower_bind_basis_def

apply -

apply (rule fold_pd_PDUnit [OF ACI_lower_bind])

apply (rule fold_pd_PDPlus [OF ACI_lower_bind])

done

lemma lower_bind_basis_mono:

  "t \<le>\<flat> u \<Longrightarrow> lower_bind_basis t \<sqsubseteq> lower_bind_basis u"

unfolding expand_cfun_less

apply (erule lower_le_induct, safe)

apply (simp add: monofun_cfun)

apply (simp add: rev_trans_less [OF lower_plus_less1])

apply (simp add: lower_plus_less_iff)

done

definition

  lower_bind :: "'a lower_pd \<rightarrow> ('a \<rightarrow> 'b lower_pd) \<rightarrow> 'b lower_pd" where

  "lower_bind = lower_pd.basis_fun lower_bind_basis"

lemma lower_bind_principal [simp]:

  "lower_bind\<cdot>(lower_principal t) = lower_bind_basis t"

unfolding lower_bind_def

apply (rule lower_pd.basis_fun_principal)

apply (erule lower_bind_basis_mono)

done

lemma lower_bind_unit [simp]:

  "lower_bind\<cdot>{x}\<flat>\<cdot>f = f\<cdot>x"

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

lemma lower_bind_plus [simp]:

  "lower_bind\<cdot>(xs +\<flat> ys)\<cdot>f = lower_bind\<cdot>xs\<cdot>f +\<flat> lower_bind\<cdot>ys\<cdot>f"

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

lemma lower_bind_strict [simp]: "lower_bind\<cdot>\<bottom>\<cdot>f = f\<cdot>\<bottom>"

unfolding lower_unit_strict [symmetric] by (rule lower_bind_unit)

subsection {* Map and join *}

definition

  lower_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a lower_pd \<rightarrow> 'b lower_pd" where

  "lower_map = (\<Lambda> f xs. lower_bind\<cdot>xs\<cdot>(\<Lambda> x. {f\<cdot>x}\<flat>))"

definition

  lower_join :: "'a lower_pd lower_pd \<rightarrow> 'a lower_pd" where

  "lower_join = (\<Lambda> xss. lower_bind\<cdot>xss\<cdot>(\<Lambda> xs. xs))"

lemma lower_map_unit [simp]:

  "lower_map\<cdot>f\<cdot>{x}\<flat> = {f\<cdot>x}\<flat>"

unfolding lower_map_def by simp

lemma lower_map_plus [simp]:

  "lower_map\<cdot>f\<cdot>(xs +\<flat> ys) = lower_map\<cdot>f\<cdot>xs +\<flat> lower_map\<cdot>f\<cdot>ys"

unfolding lower_map_def by simp

lemma lower_join_unit [simp]:

  "lower_join\<cdot>{xs}\<flat> = xs"

unfolding lower_join_def by simp

lemma lower_join_plus [simp]:

  "lower_join\<cdot>(xss +\<flat> yss) = lower_join\<cdot>xss +\<flat> lower_join\<cdot>yss"

unfolding lower_join_def by simp

lemma lower_map_ident: "lower_map\<cdot>(\<Lambda> x. x)\<cdot>xs = xs"

by (induct xs rule: lower_pd_induct, simp_all)

lemma lower_map_map:

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

by (induct xs rule: lower_pd_induct, simp_all)

lemma lower_join_map_unit:

  "lower_join\<cdot>(lower_map\<cdot>lower_unit\<cdot>xs) = xs"

by (induct xs rule: lower_pd_induct, simp_all)

lemma lower_join_map_join:

  "lower_join\<cdot>(lower_map\<cdot>lower_join\<cdot>xsss) = lower_join\<cdot>(lower_join\<cdot>xsss)"

by (induct xsss rule: lower_pd_induct, simp_all)

lemma lower_join_map_map:

  "lower_join\<cdot>(lower_map\<cdot>(lower_map\<cdot>f)\<cdot>xss) =

   lower_map\<cdot>f\<cdot>(lower_join\<cdot>xss)"

by (induct xss rule: lower_pd_induct, simp_all)

lemma lower_map_approx: "lower_map\<cdot>(approx n)\<cdot>xs = approx n\<cdot>xs"

by (induct xs rule: lower_pd_induct, simp_all)

end