(*  Title:      HOLCF/Bifinite.thy

    ID:         $Id$

    Author:     Brian Huffman

*)

header {* Bifinite domains and approximation *}

theory Bifinite

imports Cfun

begin

subsection {* Omega-profinite and bifinite domains *}

class profinite = cpo +

  fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a"

  assumes chain_approx_app: "chain (\<lambda>i. approx i\<cdot>x)"

  assumes lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x"

  assumes approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"

  assumes finite_fixes_approx: "finite {x. approx i\<cdot>x = x}"

class bifinite = profinite + pcpo

lemma finite_range_imp_finite_fixes:

  "finite {x. \<exists>y. x = f y} \<Longrightarrow> finite {x. f x = x}"

apply (subgoal_tac "{x. f x = x} \<subseteq> {x. \<exists>y. x = f y}")

apply (erule (1) finite_subset)

apply (clarify, erule subst, rule exI, rule refl)

done

lemma chain_approx [simp]: "chain approx"

apply (rule chainI)

apply (rule less_cfun_ext)

apply (rule chainE)

apply (rule chain_approx_app)

done

lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda> x. x)"

by (rule ext_cfun, simp add: contlub_cfun_fun)

lemma approx_less: "approx i\<cdot>x \<sqsubseteq> x"

apply (subgoal_tac "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)", simp)

apply (rule is_ub_thelub, simp)

done

lemma approx_strict [simp]: "approx i\<cdot>(\<bottom>::'a::bifinite) = \<bottom>"

by (rule UU_I, rule approx_less)

lemma approx_approx1:

  "i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>x"

apply (rule antisym_less)

apply (rule monofun_cfun_arg [OF approx_less])

apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])

apply (rule monofun_cfun_arg)

apply (rule monofun_cfun_fun)

apply (erule chain_mono [OF chain_approx])

done

lemma approx_approx2:

  "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>x"

apply (rule antisym_less)

apply (rule approx_less)

apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])

apply (rule monofun_cfun_fun)

apply (erule chain_mono [OF chain_approx])

done

lemma approx_approx [simp]:

  "approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>x"

apply (rule_tac x=i and y=j in linorder_le_cases)

apply (simp add: approx_approx1 min_def)

apply (simp add: approx_approx2 min_def)

done

lemma idem_fixes_eq_range:

  "\<forall>x. f (f x) = f x \<Longrightarrow> {x. f x = x} = {y. \<exists>x. y = f x}"

by (auto simp add: eq_sym_conv)

lemma finite_approx: "finite {y. \<exists>x. y = approx n\<cdot>x}"

using finite_fixes_approx by (simp add: idem_fixes_eq_range)

lemma finite_image_approx: "finite ((\<lambda>x. approx n\<cdot>x) ` A)"

by (rule finite_subset [OF _ finite_fixes_approx [where i=n]]) auto

lemma finite_range_approx: "finite (range (\<lambda>x. approx n\<cdot>x))"

by (rule finite_image_approx)

lemma compact_approx [simp]: "compact (approx n\<cdot>x)"

proof (rule compactI2)

  fix Y::"nat \<Rightarrow> 'a"

  assume Y: "chain Y"

  have "finite_chain (\<lambda>i. approx n\<cdot>(Y i))"

  proof (rule finite_range_imp_finch)

    show "chain (\<lambda>i. approx n\<cdot>(Y i))"

      using Y by simp

    have "range (\<lambda>i. approx n\<cdot>(Y i)) \<subseteq> {x. approx n\<cdot>x = x}"

      by clarsimp

    thus "finite (range (\<lambda>i. approx n\<cdot>(Y i)))"

      using finite_fixes_approx by (rule finite_subset)

  qed

  hence "\<exists>j. (\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)"

    by (simp add: finite_chain_def maxinch_is_thelub Y)

  then obtain j where j: "(\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" ..

  assume "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. Y i)"

  hence "approx n\<cdot>(approx n\<cdot>x) \<sqsubseteq> approx n\<cdot>(\<Squnion>i. Y i)"

    by (rule monofun_cfun_arg)

  hence "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. approx n\<cdot>(Y i))"

    by (simp add: contlub_cfun_arg Y)

  hence "approx n\<cdot>x \<sqsubseteq> approx n\<cdot>(Y j)"

    using j by simp

  hence "approx n\<cdot>x \<sqsubseteq> Y j"

    using approx_less by (rule trans_less)

  thus "\<exists>j. approx n\<cdot>x \<sqsubseteq> Y j" ..

qed

lemma bifinite_compact_eq_approx:

  assumes x: "compact x"

  shows "\<exists>i. approx i\<cdot>x = x"

proof -

  have chain: "chain (\<lambda>i. approx i\<cdot>x)" by simp

  have less: "x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by simp

  obtain i where i: "x \<sqsubseteq> approx i\<cdot>x"

    using compactD2 [OF x chain less] ..

  with approx_less have "approx i\<cdot>x = x"

    by (rule antisym_less)

  thus "\<exists>i. approx i\<cdot>x = x" ..

qed

lemma bifinite_compact_iff:

  "compact x \<longleftrightarrow> (\<exists>n. approx n\<cdot>x = x)"

 apply (rule iffI)

  apply (erule bifinite_compact_eq_approx)

 apply (erule exE)

 apply (erule subst)

 apply (rule compact_approx)

done

lemma approx_induct:

  assumes adm: "adm P" and P: "\<And>n x. P (approx n\<cdot>x)"

  shows "P x"

proof -

  have "P (\<Squnion>n. approx n\<cdot>x)"

    by (rule admD [OF adm], simp, simp add: P)

  thus "P x" by simp

qed

lemma bifinite_less_ext: "(\<And>i. approx i\<cdot>x \<sqsubseteq> approx i\<cdot>y) \<Longrightarrow> x \<sqsubseteq> y"

apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> (\<Squnion>i. approx i\<cdot>y)", simp)

apply (rule lub_mono, simp, simp, simp)

done

subsection {* Instance for continuous function space *}

lemma finite_range_lemma:

  fixes h :: "'a::cpo \<rightarrow> 'b::cpo"

  fixes k :: "'c::cpo \<rightarrow> 'd::cpo"

  shows "\<lbrakk>finite {y. \<exists>x. y = h\<cdot>x}; finite {y. \<exists>x. y = k\<cdot>x}\<rbrakk>

    \<Longrightarrow> finite {g. \<exists>f. g = (\<Lambda> x. k\<cdot>(f\<cdot>(h\<cdot>x)))}"

 apply (rule_tac f="\<lambda>g. {(h\<cdot>x, y) |x y. y = g\<cdot>x}" in finite_imageD)

  apply (rule_tac B="Pow ({y. \<exists>x. y = h\<cdot>x} \<times> {y. \<exists>x. y = k\<cdot>x})"

           in finite_subset)

   apply (rule image_subsetI)

   apply (clarsimp, fast)

  apply simp

 apply (rule inj_onI)

 apply (clarsimp simp add: expand_set_eq)

 apply (rule ext_cfun, simp)

 apply (drule_tac x="h\<cdot>x" in spec)

 apply (drule_tac x="k\<cdot>(f\<cdot>(h\<cdot>x))" in spec)

 apply (drule iffD1, fast)

 apply clarsimp

done

instantiation "->" :: (profinite, profinite) profinite

begin

definition

  approx_cfun_def:

    "approx = (\<lambda>n. \<Lambda> f x. approx n\<cdot>(f\<cdot>(approx n\<cdot>x)))"

instance

 apply (intro_classes, unfold approx_cfun_def)

    apply simp

   apply (simp add: lub_distribs eta_cfun)

  apply simp

 apply simp

 apply (rule finite_range_imp_finite_fixes)

 apply (intro finite_range_lemma finite_approx)

done

end

instance "->" :: (profinite, bifinite) bifinite ..

lemma approx_cfun: "approx n\<cdot>f\<cdot>x = approx n\<cdot>(f\<cdot>(approx n\<cdot>x))"

by (simp add: approx_cfun_def)

end