(* Title: HOL/Complete_Partial_Order.thy
Author: Brian Huffman, Portland State University
Author: Alexander Krauss, TU Muenchen
*)
section \<open>Chain-complete partial orders and their fixpoints\<close>
theory Complete_Partial_Order
imports Product_Type
begin
subsection \<open>Monotone functions\<close>
text \<open>Dictionary-passing version of @{const Orderings.mono}.\<close>
definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)) \<Longrightarrow> monotone orda ordb f"
unfolding monotone_def by iprover
lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
unfolding monotone_def by iprover
subsection \<open>Chains\<close>
text \<open>
A chain is a totally-ordered set. Chains are parameterized over
the order for maximal flexibility, since type classes are not enough.
\<close>
definition chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
where "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
lemma chainI:
assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x"
shows "chain ord S"
using assms unfolding chain_def by fast
lemma chainD:
assumes "chain ord S" and "x \<in> S" and "y \<in> S"
shows "ord x y \<or> ord y x"
using assms unfolding chain_def by fast
lemma chainE:
assumes "chain ord S" and "x \<in> S" and "y \<in> S"
obtains "ord x y" | "ord y x"
using assms unfolding chain_def by fast
lemma chain_empty: "chain ord {}"
by (simp add: chain_def)
lemma chain_equality: "chain op = A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
by (auto simp add: chain_def)
lemma chain_subset: "chain ord A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> chain ord B"
by (rule chainI) (blast dest: chainD)
lemma chain_imageI:
assumes chain: "chain le_a Y"
and mono: "\<And>x y. x \<in> Y \<Longrightarrow> y \<in> Y \<Longrightarrow> le_a x y \<Longrightarrow> le_b (f x) (f y)"
shows "chain le_b (f ` Y)"
by (blast intro: chainI dest: chainD[OF chain] mono)
subsection \<open>Chain-complete partial orders\<close>
text \<open>
A \<open>ccpo\<close> has a least upper bound for any chain. In particular, the
empty set is a chain, so every \<open>ccpo\<close> must have a bottom element.
\<close>
class ccpo = order + Sup +
assumes ccpo_Sup_upper: "chain (op \<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> Sup A"
assumes ccpo_Sup_least: "chain (op \<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z"
begin
lemma chain_singleton: "Complete_Partial_Order.chain op \<le> {x}"
by (rule chainI) simp
lemma ccpo_Sup_singleton [simp]: "\<Squnion>{x} = x"
by (rule antisym) (auto intro: ccpo_Sup_least ccpo_Sup_upper simp add: chain_singleton)
subsection \<open>Transfinite iteration of a function\<close>
context notes [[inductive_internals]]
begin
inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"
for f :: "'a \<Rightarrow> 'a"
where
step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
| Sup: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
end
lemma iterates_le_f: "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
by (induct x rule: iterates.induct)
(force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+
lemma chain_iterates:
assumes f: "monotone (op \<le>) (op \<le>) f"
shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")
proof (rule chainI)
fix x y
assume "x \<in> ?C" "y \<in> ?C"
then show "x \<le> y \<or> y \<le> x"
proof (induct x arbitrary: y rule: iterates.induct)
fix x y
assume y: "y \<in> ?C"
and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
from y show "f x \<le> y \<or> y \<le> f x"
proof (induct y rule: iterates.induct)
case (step y)
with IH f show ?case by (auto dest: monotoneD)
next
case (Sup M)
then have chM: "chain (op \<le>) M"
and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
show "f x \<le> Sup M \<or> Sup M \<le> f x"
proof (cases "\<exists>z\<in>M. f x \<le> z")
case True
then have "f x \<le> Sup M"
apply rule
apply (erule order_trans)
apply (rule ccpo_Sup_upper[OF chM])
apply assumption
done
then show ?thesis ..
next
case False
with IH' show ?thesis
by (auto intro: ccpo_Sup_least[OF chM])
qed
qed
next
case (Sup M y)
show ?case
proof (cases "\<exists>x\<in>M. y \<le> x")
case True
then have "y \<le> Sup M"
apply rule
apply (erule order_trans)
apply (rule ccpo_Sup_upper[OF Sup(1)])
apply assumption
done
then show ?thesis ..
next
case False with Sup
show ?thesis by (auto intro: ccpo_Sup_least)
qed
qed
qed
lemma bot_in_iterates: "Sup {} \<in> iterates f"
by (auto intro: iterates.Sup simp add: chain_empty)
subsection \<open>Fixpoint combinator\<close>
definition fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
where "fixp f = Sup (iterates f)"
lemma iterates_fixp:
assumes f: "monotone (op \<le>) (op \<le>) f"
shows "fixp f \<in> iterates f"
unfolding fixp_def
by (simp add: iterates.Sup chain_iterates f)
lemma fixp_unfold:
assumes f: "monotone (op \<le>) (op \<le>) f"
shows "fixp f = f (fixp f)"
proof (rule antisym)
show "fixp f \<le> f (fixp f)"
by (intro iterates_le_f iterates_fixp f)
have "f (fixp f) \<le> Sup (iterates f)"
by (intro ccpo_Sup_upper chain_iterates f iterates.step iterates_fixp)
then show "f (fixp f) \<le> fixp f"
by (simp only: fixp_def)
qed
lemma fixp_lowerbound:
assumes f: "monotone (op \<le>) (op \<le>) f"
and z: "f z \<le> z"
shows "fixp f \<le> z"
unfolding fixp_def
proof (rule ccpo_Sup_least[OF chain_iterates[OF f]])
fix x
assume "x \<in> iterates f"
then show "x \<le> z"
proof (induct x rule: iterates.induct)
case (step x)
from f \<open>x \<le> z\<close> have "f x \<le> f z" by (rule monotoneD)
also note z
finally show "f x \<le> z" .
next
case (Sup M)
then show ?case
by (auto intro: ccpo_Sup_least)
qed
qed
end
subsection \<open>Fixpoint induction\<close>
setup \<open>Sign.map_naming (Name_Space.mandatory_path "ccpo")\<close>
definition admissible :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
where "admissible lub ord P \<longleftrightarrow> (\<forall>A. chain ord A \<longrightarrow> A \<noteq> {} \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
lemma admissibleI:
assumes "\<And>A. chain ord A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)"
shows "ccpo.admissible lub ord P"
using assms unfolding ccpo.admissible_def by fast
lemma admissibleD:
assumes "ccpo.admissible lub ord P"
assumes "chain ord A"
assumes "A \<noteq> {}"
assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
shows "P (lub A)"
using assms by (auto simp: ccpo.admissible_def)
setup \<open>Sign.map_naming Name_Space.parent_path\<close>
lemma (in ccpo) fixp_induct:
assumes adm: "ccpo.admissible Sup (op \<le>) P"
assumes mono: "monotone (op \<le>) (op \<le>) f"
assumes bot: "P (Sup {})"
assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
shows "P (fixp f)"
unfolding fixp_def
using adm chain_iterates[OF mono]
proof (rule ccpo.admissibleD)
show "iterates f \<noteq> {}"
using bot_in_iterates by auto
next
fix x
assume "x \<in> iterates f"
then show "P x"
proof (induct rule: iterates.induct)
case prems: (step x)
from this(2) show ?case by (rule step)
next
case (Sup M)
then show ?case by (cases "M = {}") (auto intro: step bot ccpo.admissibleD adm)
qed
qed
lemma admissible_True: "ccpo.admissible lub ord (\<lambda>x. True)"
unfolding ccpo.admissible_def by simp
(*lemma admissible_False: "\<not> ccpo.admissible lub ord (\<lambda>x. False)"
unfolding ccpo.admissible_def chain_def by simp
*)
lemma admissible_const: "ccpo.admissible lub ord (\<lambda>x. t)"
by (auto intro: ccpo.admissibleI)
lemma admissible_conj:
assumes "ccpo.admissible lub ord (\<lambda>x. P x)"
assumes "ccpo.admissible lub ord (\<lambda>x. Q x)"
shows "ccpo.admissible lub ord (\<lambda>x. P x \<and> Q x)"
using assms unfolding ccpo.admissible_def by simp
lemma admissible_all:
assumes "\<And>y. ccpo.admissible lub ord (\<lambda>x. P x y)"
shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y. P x y)"
using assms unfolding ccpo.admissible_def by fast
lemma admissible_ball:
assumes "\<And>y. y \<in> A \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x y)"
shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y\<in>A. P x y)"
using assms unfolding ccpo.admissible_def by fast
lemma chain_compr: "chain ord A \<Longrightarrow> chain ord {x \<in> A. P x}"
unfolding chain_def by fast
context ccpo
begin
lemma admissible_disj:
fixes P Q :: "'a \<Rightarrow> bool"
assumes P: "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x)"
assumes Q: "ccpo.admissible Sup (op \<le>) (\<lambda>x. Q x)"
shows "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x \<or> Q x)"
proof (rule ccpo.admissibleI)
fix A :: "'a set"
assume chain: "chain (op \<le>) A"
assume A: "A \<noteq> {}" and P_Q: "\<forall>x\<in>A. P x \<or> Q x"
have "(\<exists>x\<in>A. P x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<exists>x\<in>A. Q x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"
(is "?P \<or> ?Q" is "?P1 \<and> ?P2 \<or> _")
proof (rule disjCI)
assume "\<not> ?Q"
then consider "\<forall>x\<in>A. \<not> Q x" | a where "a \<in> A" "\<forall>y\<in>A. a \<le> y \<longrightarrow> \<not> Q y"
by blast
then show ?P
proof cases
case 1
with P_Q have "\<forall>x\<in>A. P x" by blast
with A show ?P by blast
next
case 2
note a = \<open>a \<in> A\<close>
show ?P
proof
from P_Q 2 have *: "\<forall>y\<in>A. a \<le> y \<longrightarrow> P y" by blast
with a have "P a" by blast
with a show ?P1 by blast
show ?P2
proof
fix x
assume x: "x \<in> A"
with chain a show "\<exists>y\<in>A. x \<le> y \<and> P y"
proof (rule chainE)
assume le: "a \<le> x"
with * a x have "P x" by blast
with x le show ?thesis by blast
next
assume "a \<ge> x"
with a \<open>P a\<close> show ?thesis by blast
qed
qed
qed
qed
qed
moreover
have "Sup A = Sup {x \<in> A. P x}" if "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y" for P
proof (rule antisym)
have chain_P: "chain (op \<le>) {x \<in> A. P x}"
by (rule chain_compr [OF chain])
show "Sup A \<le> Sup {x \<in> A. P x}"
apply (rule ccpo_Sup_least [OF chain])
apply (drule that [rule_format])
apply clarify
apply (erule order_trans)
apply (simp add: ccpo_Sup_upper [OF chain_P])
done
show "Sup {x \<in> A. P x} \<le> Sup A"
apply (rule ccpo_Sup_least [OF chain_P])
apply clarify
apply (simp add: ccpo_Sup_upper [OF chain])
done
qed
ultimately
consider "\<exists>x. x \<in> A \<and> P x" "Sup A = Sup {x \<in> A. P x}"
| "\<exists>x. x \<in> A \<and> Q x" "Sup A = Sup {x \<in> A. Q x}"
by blast
then show "P (Sup A) \<or> Q (Sup A)"
apply cases
apply simp_all
apply (rule disjI1)
apply (rule ccpo.admissibleD [OF P chain_compr [OF chain]]; simp)
apply (rule disjI2)
apply (rule ccpo.admissibleD [OF Q chain_compr [OF chain]]; simp)
done
qed
end
instance complete_lattice \<subseteq> ccpo
by standard (fast intro: Sup_upper Sup_least)+
lemma lfp_eq_fixp:
assumes mono: "mono f"
shows "lfp f = fixp f"
proof (rule antisym)
from mono have f': "monotone (op \<le>) (op \<le>) f"
unfolding mono_def monotone_def .
show "lfp f \<le> fixp f"
by (rule lfp_lowerbound, subst fixp_unfold [OF f'], rule order_refl)
show "fixp f \<le> lfp f"
by (rule fixp_lowerbound [OF f']) (simp add: lfp_fixpoint [OF mono])
qed
hide_const (open) iterates fixp
end