(*  Title:      HOLCF/Deflation.thy

    Author:     Brian Huffman

*)

header {* Continuous Deflations and Embedding-Projection Pairs *}

theory Deflation

imports Cfun

begin

defaultsort cpo

subsection {* Continuous deflations *}

locale deflation =

  fixes d :: "'a \<rightarrow> 'a"

  assumes idem: "\<And>x. d\<cdot>(d\<cdot>x) = d\<cdot>x"

  assumes less: "\<And>x. d\<cdot>x \<sqsubseteq> x"

begin

lemma less_ID: "d \<sqsubseteq> ID"

by (rule less_cfun_ext, simp add: less)

text {* The set of fixed points is the same as the range. *}

lemma fixes_eq_range: "{x. d\<cdot>x = x} = range (\<lambda>x. d\<cdot>x)"

by (auto simp add: eq_sym_conv idem)

lemma range_eq_fixes: "range (\<lambda>x. d\<cdot>x) = {x. d\<cdot>x = x}"

by (auto simp add: eq_sym_conv idem)

text {*

  The pointwise ordering on deflation functions coincides with

  the subset ordering of their sets of fixed-points.

*}

lemma lessI:

  assumes f: "\<And>x. d\<cdot>x = x \<Longrightarrow> f\<cdot>x = x" shows "d \<sqsubseteq> f"

proof (rule less_cfun_ext)

  fix x

  from less have "f\<cdot>(d\<cdot>x) \<sqsubseteq> f\<cdot>x" by (rule monofun_cfun_arg)

  also from idem have "f\<cdot>(d\<cdot>x) = d\<cdot>x" by (rule f)

  finally show "d\<cdot>x \<sqsubseteq> f\<cdot>x" .

qed

lemma lessD: "\<lbrakk>f \<sqsubseteq> d; f\<cdot>x = x\<rbrakk> \<Longrightarrow> d\<cdot>x = x"

proof (rule antisym_less)

  from less show "d\<cdot>x \<sqsubseteq> x" .

next

  assume "f \<sqsubseteq> d"

  hence "f\<cdot>x \<sqsubseteq> d\<cdot>x" by (rule monofun_cfun_fun)

  also assume "f\<cdot>x = x"

  finally show "x \<sqsubseteq> d\<cdot>x" .

qed

end

lemma adm_deflation: "adm (\<lambda>d. deflation d)"

by (simp add: deflation_def)

lemma deflation_ID: "deflation ID"

by (simp add: deflation.intro)

lemma deflation_UU: "deflation \<bottom>"

by (simp add: deflation.intro)

lemma deflation_less_iff:

  "\<lbrakk>deflation p; deflation q\<rbrakk> \<Longrightarrow> p \<sqsubseteq> q \<longleftrightarrow> (\<forall>x. p\<cdot>x = x \<longrightarrow> q\<cdot>x = x)"

 apply safe

  apply (simp add: deflation.lessD)

 apply (simp add: deflation.lessI)

done

text {*

  The composition of two deflations is equal to

  the lesser of the two (if they are comparable).

*}

lemma deflation_less_comp1:

  assumes "deflation f"

  assumes "deflation g"

  shows "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>(g\<cdot>x) = f\<cdot>x"

proof (rule antisym_less)

  interpret g: deflation g by fact

  from g.less show "f\<cdot>(g\<cdot>x) \<sqsubseteq> f\<cdot>x" by (rule monofun_cfun_arg)

next

  interpret f: deflation f by fact

  assume "f \<sqsubseteq> g" hence "f\<cdot>x \<sqsubseteq> g\<cdot>x" by (rule monofun_cfun_fun)

  hence "f\<cdot>(f\<cdot>x) \<sqsubseteq> f\<cdot>(g\<cdot>x)" by (rule monofun_cfun_arg)

  also have "f\<cdot>(f\<cdot>x) = f\<cdot>x" by (rule f.idem)

  finally show "f\<cdot>x \<sqsubseteq> f\<cdot>(g\<cdot>x)" .

qed

lemma deflation_less_comp2:

  "\<lbrakk>deflation f; deflation g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> g\<cdot>(f\<cdot>x) = f\<cdot>x"

by (simp only: deflation.lessD deflation.idem)

subsection {* Deflations with finite range *}

lemma finite_range_imp_finite_fixes:

  "finite (range f) \<Longrightarrow> finite {x. f x = x}"

proof -

  have "{x. f x = x} \<subseteq> range f"

    by (clarify, erule subst, rule rangeI)

  moreover assume "finite (range f)"

  ultimately show "finite {x. f x = x}"

    by (rule finite_subset)

qed

locale finite_deflation = deflation +

  assumes finite_fixes: "finite {x. d\<cdot>x = x}"

begin

lemma finite_range: "finite (range (\<lambda>x. d\<cdot>x))"

by (simp add: range_eq_fixes finite_fixes)

lemma finite_image: "finite ((\<lambda>x. d\<cdot>x) ` A)"

by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range])

lemma compact: "compact (d\<cdot>x)"

proof (rule compactI2)

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

  assume Y: "chain Y"

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

  proof (rule finite_range_imp_finch)

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

      using Y by simp

    have "range (\<lambda>i. d\<cdot>(Y i)) \<subseteq> range (\<lambda>x. d\<cdot>x)"

      by clarsimp

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

      using finite_range by (rule finite_subset)

  qed

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

    by (simp add: finite_chain_def maxinch_is_thelub Y)

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

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

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

    by (rule monofun_cfun_arg)

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

    by (simp add: contlub_cfun_arg Y idem)

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

    using j by simp

  hence "d\<cdot>x \<sqsubseteq> Y j"

    using less by (rule trans_less)

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

qed

end

subsection {* Continuous embedding-projection pairs *}

locale ep_pair =

  fixes e :: "'a \<rightarrow> 'b" and p :: "'b \<rightarrow> 'a"

  assumes e_inverse [simp]: "\<And>x. p\<cdot>(e\<cdot>x) = x"

  and e_p_less: "\<And>y. e\<cdot>(p\<cdot>y) \<sqsubseteq> y"

begin

lemma e_less_iff [simp]: "e\<cdot>x \<sqsubseteq> e\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y"

proof

  assume "e\<cdot>x \<sqsubseteq> e\<cdot>y"

  hence "p\<cdot>(e\<cdot>x) \<sqsubseteq> p\<cdot>(e\<cdot>y)" by (rule monofun_cfun_arg)

  thus "x \<sqsubseteq> y" by simp

next

  assume "x \<sqsubseteq> y"

  thus "e\<cdot>x \<sqsubseteq> e\<cdot>y" by (rule monofun_cfun_arg)

qed

lemma e_eq_iff [simp]: "e\<cdot>x = e\<cdot>y \<longleftrightarrow> x = y"

unfolding po_eq_conv e_less_iff ..

lemma p_eq_iff:

  "\<lbrakk>e\<cdot>(p\<cdot>x) = x; e\<cdot>(p\<cdot>y) = y\<rbrakk> \<Longrightarrow> p\<cdot>x = p\<cdot>y \<longleftrightarrow> x = y"

by (safe, erule subst, erule subst, simp)

lemma p_inverse: "(\<exists>x. y = e\<cdot>x) = (e\<cdot>(p\<cdot>y) = y)"

by (auto, rule exI, erule sym)

lemma e_less_iff_less_p: "e\<cdot>x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> p\<cdot>y"

proof

  assume "e\<cdot>x \<sqsubseteq> y"

  then have "p\<cdot>(e\<cdot>x) \<sqsubseteq> p\<cdot>y" by (rule monofun_cfun_arg)

  then show "x \<sqsubseteq> p\<cdot>y" by simp

next

  assume "x \<sqsubseteq> p\<cdot>y"

  then have "e\<cdot>x \<sqsubseteq> e\<cdot>(p\<cdot>y)" by (rule monofun_cfun_arg)

  then show "e\<cdot>x \<sqsubseteq> y" using e_p_less by (rule trans_less)

qed

lemma compact_e_rev: "compact (e\<cdot>x) \<Longrightarrow> compact x"

proof -

  assume "compact (e\<cdot>x)"

  hence "adm (\<lambda>y. \<not> e\<cdot>x \<sqsubseteq> y)" by (rule compactD)

  hence "adm (\<lambda>y. \<not> e\<cdot>x \<sqsubseteq> e\<cdot>y)" by (rule adm_subst [OF cont_Rep_CFun2])

  hence "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" by simp

  thus "compact x" by (rule compactI)

qed

lemma compact_e: "compact x \<Longrightarrow> compact (e\<cdot>x)"

proof -

  assume "compact x"

  hence "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" by (rule compactD)

  hence "adm (\<lambda>y. \<not> x \<sqsubseteq> p\<cdot>y)" by (rule adm_subst [OF cont_Rep_CFun2])

  hence "adm (\<lambda>y. \<not> e\<cdot>x \<sqsubseteq> y)" by (simp add: e_less_iff_less_p)

  thus "compact (e\<cdot>x)" by (rule compactI)

qed

lemma compact_e_iff: "compact (e\<cdot>x) \<longleftrightarrow> compact x"

by (rule iffI [OF compact_e_rev compact_e])

text {* Deflations from ep-pairs *}

lemma deflation_e_p: "deflation (e oo p)"

by (simp add: deflation.intro e_p_less)

lemma deflation_e_d_p:

  assumes "deflation d"

  shows "deflation (e oo d oo p)"

proof

  interpret deflation d by fact

  fix x :: 'b

  show "(e oo d oo p)\<cdot>((e oo d oo p)\<cdot>x) = (e oo d oo p)\<cdot>x"

    by (simp add: idem)

  show "(e oo d oo p)\<cdot>x \<sqsubseteq> x"

    by (simp add: e_less_iff_less_p less)

qed

lemma finite_deflation_e_d_p:

  assumes "finite_deflation d"

  shows "finite_deflation (e oo d oo p)"

proof

  interpret finite_deflation d by fact

  fix x :: 'b

  show "(e oo d oo p)\<cdot>((e oo d oo p)\<cdot>x) = (e oo d oo p)\<cdot>x"

    by (simp add: idem)

  show "(e oo d oo p)\<cdot>x \<sqsubseteq> x"

    by (simp add: e_less_iff_less_p less)

  have "finite ((\<lambda>x. e\<cdot>x) ` (\<lambda>x. d\<cdot>x) ` range (\<lambda>x. p\<cdot>x))"

    by (simp add: finite_image)

  hence "finite (range (\<lambda>x. (e oo d oo p)\<cdot>x))"

    by (simp add: image_image)

  thus "finite {x. (e oo d oo p)\<cdot>x = x}"

    by (rule finite_range_imp_finite_fixes)

qed

lemma deflation_p_d_e:

  assumes "deflation d"

  assumes d: "\<And>x. d\<cdot>x \<sqsubseteq> e\<cdot>(p\<cdot>x)"

  shows "deflation (p oo d oo e)"

proof -

  interpret d: deflation d by fact

  {

    fix x

    have "d\<cdot>(e\<cdot>x) \<sqsubseteq> e\<cdot>x"

      by (rule d.less)

    hence "p\<cdot>(d\<cdot>(e\<cdot>x)) \<sqsubseteq> p\<cdot>(e\<cdot>x)"

      by (rule monofun_cfun_arg)

    hence "(p oo d oo e)\<cdot>x \<sqsubseteq> x"

      by simp

  }

  note p_d_e_less = this

  show ?thesis

  proof

    fix x

    show "(p oo d oo e)\<cdot>x \<sqsubseteq> x"

      by (rule p_d_e_less)

  next

    fix x

    show "(p oo d oo e)\<cdot>((p oo d oo e)\<cdot>x) = (p oo d oo e)\<cdot>x"

    proof (rule antisym_less)

      show "(p oo d oo e)\<cdot>((p oo d oo e)\<cdot>x) \<sqsubseteq> (p oo d oo e)\<cdot>x"

        by (rule p_d_e_less)

      have "p\<cdot>(d\<cdot>(d\<cdot>(d\<cdot>(e\<cdot>x)))) \<sqsubseteq> p\<cdot>(d\<cdot>(e\<cdot>(p\<cdot>(d\<cdot>(e\<cdot>x)))))"

        by (intro monofun_cfun_arg d)

      hence "p\<cdot>(d\<cdot>(e\<cdot>x)) \<sqsubseteq> p\<cdot>(d\<cdot>(e\<cdot>(p\<cdot>(d\<cdot>(e\<cdot>x)))))"

        by (simp only: d.idem)

      thus "(p oo d oo e)\<cdot>x \<sqsubseteq> (p oo d oo e)\<cdot>((p oo d oo e)\<cdot>x)"

        by simp

    qed

  qed

qed

lemma finite_deflation_p_d_e:

  assumes "finite_deflation d"

  assumes d: "\<And>x. d\<cdot>x \<sqsubseteq> e\<cdot>(p\<cdot>x)"

  shows "finite_deflation (p oo d oo e)"

proof -

  interpret d: finite_deflation d by fact

  show ?thesis

  proof (intro_locales)

    have "deflation d" ..

    thus "deflation (p oo d oo e)"

      using d by (rule deflation_p_d_e)

  next

    show "finite_deflation_axioms (p oo d oo e)"

    proof

      have "finite ((\<lambda>x. d\<cdot>x) ` range (\<lambda>x. e\<cdot>x))"

        by (rule d.finite_image)

      hence "finite ((\<lambda>x. p\<cdot>x) ` (\<lambda>x. d\<cdot>x) ` range (\<lambda>x. e\<cdot>x))"

        by (rule finite_imageI)

      hence "finite (range (\<lambda>x. (p oo d oo e)\<cdot>x))"

        by (simp add: image_image)

      thus "finite {x. (p oo d oo e)\<cdot>x = x}"

        by (rule finite_range_imp_finite_fixes)

    qed

  qed

qed

end

subsection {* Uniqueness of ep-pairs *}

lemma ep_pair_unique_e_lemma:

  assumes "ep_pair e1 p" and "ep_pair e2 p"

  shows "e1 \<sqsubseteq> e2"

proof (rule less_cfun_ext)

  interpret e1: ep_pair e1 p by fact

  interpret e2: ep_pair e2 p by fact

  fix x

  have "e1\<cdot>(p\<cdot>(e2\<cdot>x)) \<sqsubseteq> e2\<cdot>x"

    by (rule e1.e_p_less)

  thus "e1\<cdot>x \<sqsubseteq> e2\<cdot>x"

    by (simp only: e2.e_inverse)

qed

lemma ep_pair_unique_e:

  "\<lbrakk>ep_pair e1 p; ep_pair e2 p\<rbrakk> \<Longrightarrow> e1 = e2"

by (fast intro: antisym_less elim: ep_pair_unique_e_lemma)

lemma ep_pair_unique_p_lemma:

  assumes "ep_pair e p1" and "ep_pair e p2"

  shows "p1 \<sqsubseteq> p2"

proof (rule less_cfun_ext)

  interpret p1: ep_pair e p1 by fact

  interpret p2: ep_pair e p2 by fact

  fix x

  have "e\<cdot>(p1\<cdot>x) \<sqsubseteq> x"

    by (rule p1.e_p_less)

  hence "p2\<cdot>(e\<cdot>(p1\<cdot>x)) \<sqsubseteq> p2\<cdot>x"

    by (rule monofun_cfun_arg)

  thus "p1\<cdot>x \<sqsubseteq> p2\<cdot>x"

    by (simp only: p2.e_inverse)

qed

lemma ep_pair_unique_p:

  "\<lbrakk>ep_pair e p1; ep_pair e p2\<rbrakk> \<Longrightarrow> p1 = p2"

by (fast intro: antisym_less elim: ep_pair_unique_p_lemma)

subsection {* Composing ep-pairs *}

lemma ep_pair_ID_ID: "ep_pair ID ID"

by default simp_all

lemma ep_pair_comp:

  assumes "ep_pair e1 p1" and "ep_pair e2 p2"

  shows "ep_pair (e2 oo e1) (p1 oo p2)"

proof

  interpret ep1: ep_pair e1 p1 by fact

  interpret ep2: ep_pair e2 p2 by fact

  fix x y

  show "(p1 oo p2)\<cdot>((e2 oo e1)\<cdot>x) = x"

    by simp

  have "e1\<cdot>(p1\<cdot>(p2\<cdot>y)) \<sqsubseteq> p2\<cdot>y"

    by (rule ep1.e_p_less)

  hence "e2\<cdot>(e1\<cdot>(p1\<cdot>(p2\<cdot>y))) \<sqsubseteq> e2\<cdot>(p2\<cdot>y)"

    by (rule monofun_cfun_arg)

  also have "e2\<cdot>(p2\<cdot>y) \<sqsubseteq> y"

    by (rule ep2.e_p_less)

  finally show "(e2 oo e1)\<cdot>((p1 oo p2)\<cdot>y) \<sqsubseteq> y"

    by simp

qed

locale pcpo_ep_pair = ep_pair +

  constrains e :: "'a::pcpo \<rightarrow> 'b::pcpo"

  constrains p :: "'b::pcpo \<rightarrow> 'a::pcpo"

begin

lemma e_strict [simp]: "e\<cdot>\<bottom> = \<bottom>"

proof -

  have "\<bottom> \<sqsubseteq> p\<cdot>\<bottom>" by (rule minimal)

  hence "e\<cdot>\<bottom> \<sqsubseteq> e\<cdot>(p\<cdot>\<bottom>)" by (rule monofun_cfun_arg)

  also have "e\<cdot>(p\<cdot>\<bottom>) \<sqsubseteq> \<bottom>" by (rule e_p_less)

  finally show "e\<cdot>\<bottom> = \<bottom>" by simp

qed

lemma e_defined_iff [simp]: "e\<cdot>x = \<bottom> \<longleftrightarrow> x = \<bottom>"

by (rule e_eq_iff [where y="\<bottom>", unfolded e_strict])

lemma e_defined: "x \<noteq> \<bottom> \<Longrightarrow> e\<cdot>x \<noteq> \<bottom>"

by simp

lemma p_strict [simp]: "p\<cdot>\<bottom> = \<bottom>"

by (rule e_inverse [where x="\<bottom>", unfolded e_strict])

lemmas stricts = e_strict p_strict

end

end