(*  Title:      HOLCF/Bifinite.thy

     ID:         $Id$

     Author:     Brian Huffman

 *)

 header {* Bifinite domains and approximation *}

 theory Bifinite

 imports Cfun

 begin

 subsection {* Bifinite domains *}

 axclass approx < pcpo

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

 axclass bifinite < approx

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

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

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

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

 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 :: nat \<Rightarrow> 'a::bifinite \<rightarrow> 'a)"

 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::'a::bifinite). x)"

 by (rule ext_cfun, simp add: contlub_cfun_fun)

 lemma approx_less: "approx i\<cdot>x \<sqsubseteq> (x::'a::bifinite)"

 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::'a::bifinite)"

 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_mono3 [OF chain_approx])

 done

 lemma approx_approx2:

   "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>(x::'a::bifinite)"

 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_mono3 [OF chain_approx])

 done

 lemma approx_approx [simp]:

   "approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>(x::'a::bifinite)"

 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::'a::bifinite. \<exists>x. y = approx n\<cdot>x}"

 using finite_fixes_approx by (simp add: idem_fixes_eq_range)

 lemma finite_range_approx:

   "finite (range (\<lambda>x::'a::bifinite. approx n\<cdot>x))"

 by (simp add: image_def finite_approx)

 lemma compact_approx [simp]:

   fixes x :: "'a::bifinite"

   shows "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:

   fixes x :: "'a::bifinite"

   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::'a::bifinite) = (\<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::'a::bifinite)"

 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:

   fixes x y :: "'a::bifinite"

   shows "(\<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 [rule_format], 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

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

 defs (overloaded)

   approx_cfun_def:

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

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

  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

 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