src/HOLCF/Bifinite.thy

tuned;

(*  Title:      HOLCF/Bifinite.thy
    ID:         $Id$
    Author:     Brian Huffman
*)

header {* Bifinite domains and approximation *}

theory Bifinite
imports Deflation
begin

subsection {* Omega-profinite and bifinite domains *}

class profinite = cpo +
  fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a"
  assumes chain_approx [simp]: "chain approx"
  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 approx_less: "approx i\<cdot>x \<sqsubseteq> x"
proof -
  have "chain (\<lambda>i. approx i\<cdot>x)" by simp
  hence "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by (rule is_ub_thelub)
  thus "approx i\<cdot>x \<sqsubseteq> x" by simp
qed

lemma finite_deflation_approx: "finite_deflation (approx i)"
proof
  fix x :: 'a
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
    by (rule approx_idem)
  show "approx i\<cdot>x \<sqsubseteq> x"
    by (rule approx_less)
  show "finite {x. approx i\<cdot>x = x}"
    by (rule finite_fixes_approx)
qed

interpretation approx: finite_deflation ["approx i"]
by (rule finite_deflation_approx)

lemma deflation_approx: "deflation (approx i)"
by (rule approx.deflation_axioms)

lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda> x. x)"
by (rule ext_cfun, simp add: contlub_cfun_fun)

lemma approx_strict [simp]: "approx i\<cdot>\<bottom> = \<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 deflation_less_comp1 [OF deflation_approx deflation_approx])
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 deflation_less_comp2 [OF deflation_approx deflation_approx])
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 finite_image_approx: "finite ((\<lambda>x. approx n\<cdot>x) ` A)"
by (rule approx.finite_image)

lemma finite_range_approx: "finite (range (\<lambda>x. approx i\<cdot>x))"
by (rule approx.finite_range)

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

lemma profinite_compact_eq_approx: "compact x \<Longrightarrow> \<exists>i. approx i\<cdot>x = x"
by (rule admD2, simp_all)

lemma profinite_compact_iff: "compact x \<longleftrightarrow> (\<exists>n. approx n\<cdot>x = x)"
 apply (rule iffI)
  apply (erule profinite_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 profinite_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_cfun_lemma:
  assumes a: "finite (range (\<lambda>x. a\<cdot>x))"
  assumes b: "finite (range (\<lambda>y. b\<cdot>y))"
  shows "finite (range (\<lambda>f. \<Lambda> x. b\<cdot>(f\<cdot>(a\<cdot>x))))"  (is "finite (range ?h)")
proof (rule finite_imageD)
  let ?f = "\<lambda>g. range (\<lambda>x. (a\<cdot>x, g\<cdot>x))"
  show "finite (?f ` range ?h)"
  proof (rule finite_subset)
    let ?B = "Pow (range (\<lambda>x. a\<cdot>x) \<times> range (\<lambda>y. b\<cdot>y))"
    show "?f ` range ?h \<subseteq> ?B"
      by clarsimp
    show "finite ?B"
      by (simp add: a b)
  qed
  show "inj_on ?f (range ?h)"
  proof (rule inj_onI, rule ext_cfun, clarsimp)
    fix x f g
    assume "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) = range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
    hence "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) \<subseteq> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
      by (rule equalityD1)
    hence "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) \<in> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
      by (simp add: subset_eq)
    then obtain y where "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) = (a\<cdot>y, b\<cdot>(g\<cdot>(a\<cdot>y)))"
      by (rule rangeE)
    thus "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
      by clarsimp
  qed
qed

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 proof
  show "chain (approx :: nat \<Rightarrow> ('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow> 'b))"
    unfolding approx_cfun_def by simp
next
  fix x :: "'a \<rightarrow> 'b"
  show "(\<Squnion>i. approx i\<cdot>x) = x"
    unfolding approx_cfun_def
    by (simp add: lub_distribs eta_cfun)
next
  fix i :: nat and x :: "'a \<rightarrow> 'b"
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
    unfolding approx_cfun_def by simp
next
  fix i :: nat
  show "finite {x::'a \<rightarrow> 'b. approx i\<cdot>x = x}"
    apply (rule finite_range_imp_finite_fixes)
    apply (simp add: approx_cfun_def)
    apply (intro finite_range_cfun_lemma finite_range_approx)
    done
qed

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