(* Title: HOL/HOLCF/Deflation.thy
Author: Brian Huffman
*)
section {* Continuous deflations and ep-pairs *}
theory Deflation
imports Plain_HOLCF
begin
default_sort cpo
subsection {* Continuous deflations *}
locale deflation =
fixes d :: "'a \<rightarrow> 'a"
assumes idem: "\<And>x. d\<cdot>(d\<cdot>x) = d\<cdot>x"
assumes below: "\<And>x. d\<cdot>x \<sqsubseteq> x"
begin
lemma below_ID: "d \<sqsubseteq> ID"
by (rule cfun_belowI, simp add: below)
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 belowI:
assumes f: "\<And>x. d\<cdot>x = x \<Longrightarrow> f\<cdot>x = x" shows "d \<sqsubseteq> f"
proof (rule cfun_belowI)
fix x
from below 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 belowD: "\<lbrakk>f \<sqsubseteq> d; f\<cdot>x = x\<rbrakk> \<Longrightarrow> d\<cdot>x = x"
proof (rule below_antisym)
from below 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 deflation_strict: "deflation d \<Longrightarrow> d\<cdot>\<bottom> = \<bottom>"
by (rule deflation.below [THEN bottomI])
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_bottom: "deflation \<bottom>"
by (simp add: deflation.intro)
lemma deflation_below_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.belowD)
apply (simp add: deflation.belowI)
done
text {*
The composition of two deflations is equal to
the lesser of the two (if they are comparable).
*}
lemma deflation_below_comp1:
assumes "deflation f"
assumes "deflation g"
shows "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>(g\<cdot>x) = f\<cdot>x"
proof (rule below_antisym)
interpret g: deflation g by fact
from g.below 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_below_comp2:
"\<lbrakk>deflation f; deflation g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> g\<cdot>(f\<cdot>x) = f\<cdot>x"
by (simp only: deflation.belowD 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 below by (rule below_trans)
thus "\<exists>j. d\<cdot>x \<sqsubseteq> Y j" ..
qed
end
lemma finite_deflation_intro:
"deflation d \<Longrightarrow> finite {x. d\<cdot>x = x} \<Longrightarrow> finite_deflation d"
by (intro finite_deflation.intro finite_deflation_axioms.intro)
lemma finite_deflation_imp_deflation:
"finite_deflation d \<Longrightarrow> deflation d"
unfolding finite_deflation_def by simp
lemma finite_deflation_bottom: "finite_deflation \<bottom>"
by default simp_all
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_below: "\<And>y. e\<cdot>(p\<cdot>y) \<sqsubseteq> y"
begin
lemma e_below_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_below_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_below_iff_below_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_below by (rule below_trans)
qed
lemma compact_e_rev: "compact (e\<cdot>x) \<Longrightarrow> compact x"
proof -
assume "compact (e\<cdot>x)"
hence "adm (\<lambda>y. e\<cdot>x \<notsqsubseteq> y)" by (rule compactD)
hence "adm (\<lambda>y. e\<cdot>x \<notsqsubseteq> e\<cdot>y)" by (rule adm_subst [OF cont_Rep_cfun2])
hence "adm (\<lambda>y. x \<notsqsubseteq> 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. x \<notsqsubseteq> y)" by (rule compactD)
hence "adm (\<lambda>y. x \<notsqsubseteq> p\<cdot>y)" by (rule adm_subst [OF cont_Rep_cfun2])
hence "adm (\<lambda>y. e\<cdot>x \<notsqsubseteq> y)" by (simp add: e_below_iff_below_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_below)
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_below_iff_below_p below)
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_below_iff_below_p below)
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.below)
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_below = this
show ?thesis
proof
fix x
show "(p oo d oo e)\<cdot>x \<sqsubseteq> x"
by (rule p_d_e_below)
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 below_antisym)
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_below)
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 (rule finite_deflation_intro)
have "deflation d" ..
thus "deflation (p oo d oo e)"
using d by (rule deflation_p_d_e)
next
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
end
subsection {* Uniqueness of ep-pairs *}
lemma ep_pair_unique_e_lemma:
assumes 1: "ep_pair e1 p" and 2: "ep_pair e2 p"
shows "e1 \<sqsubseteq> e2"
proof (rule cfun_belowI)
fix x
have "e1\<cdot>(p\<cdot>(e2\<cdot>x)) \<sqsubseteq> e2\<cdot>x"
by (rule ep_pair.e_p_below [OF 1])
thus "e1\<cdot>x \<sqsubseteq> e2\<cdot>x"
by (simp only: ep_pair.e_inverse [OF 2])
qed
lemma ep_pair_unique_e:
"\<lbrakk>ep_pair e1 p; ep_pair e2 p\<rbrakk> \<Longrightarrow> e1 = e2"
by (fast intro: below_antisym elim: ep_pair_unique_e_lemma)
lemma ep_pair_unique_p_lemma:
assumes 1: "ep_pair e p1" and 2: "ep_pair e p2"
shows "p1 \<sqsubseteq> p2"
proof (rule cfun_belowI)
fix x
have "e\<cdot>(p1\<cdot>x) \<sqsubseteq> x"
by (rule ep_pair.e_p_below [OF 1])
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: ep_pair.e_inverse [OF 2])
qed
lemma ep_pair_unique_p:
"\<lbrakk>ep_pair e p1; ep_pair e p2\<rbrakk> \<Longrightarrow> p1 = p2"
by (fast intro: below_antisym 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_below)
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_below)
finally show "(e2 oo e1)\<cdot>((p1 oo p2)\<cdot>y) \<sqsubseteq> y"
by simp
qed
locale pcpo_ep_pair = ep_pair e p
for e :: "'a::pcpo \<rightarrow> 'b::pcpo"
and 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_below)
finally show "e\<cdot>\<bottom> = \<bottom>" by simp
qed
lemma e_bottom_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