--- a/src/HOL/Complete_Partial_Order.thy Fri Aug 05 16:36:03 2016 +0200
+++ b/src/HOL/Complete_Partial_Order.thy Fri Aug 05 18:14:28 2016 +0200
@@ -1,12 +1,12 @@
-(* Title: HOL/Complete_Partial_Order.thy
- Author: Brian Huffman, Portland State University
- Author: Alexander Krauss, TU Muenchen
+(* 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
+ imports Product_Type
begin
subsection \<open>Monotone functions\<close>
@@ -14,131 +14,139 @@
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))"
+ 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 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
+ unfolding monotone_def by iprover
subsection \<open>Chains\<close>
-text \<open>A chain is a totally-ordered set. Chains are parameterized over
+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)"
+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
+ 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
+ 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
+ using assms unfolding chain_def by fast
lemma chain_empty: "chain ord {}"
-by(simp add: chain_def)
+ 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)
+ 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_subset:
- "\<lbrakk> chain ord A; B \<subseteq> A \<rbrakk>
- \<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)
-lemma chain_imageI:
- assumes chain: "chain le_a Y"
- and mono: "\<And>x y. \<lbrakk> x \<in> Y; y \<in> Y; le_a x y \<rbrakk> \<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 ccpo has a least upper bound for any chain. In particular, the
- empty set is a chain, so every ccpo must have a bottom element.
+ 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: "\<lbrakk>chain (op \<le>) A; x \<in> A\<rbrakk> \<Longrightarrow> x \<le> Sup A"
- assumes ccpo_Sup_least: "\<lbrakk>chain (op \<le>) A; \<And>x. x \<in> A \<Longrightarrow> x \<le> z\<rbrakk> \<Longrightarrow> Sup A \<le> z"
+ 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
+ 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)
+ 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
+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"
+ 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 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"
+ 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"
+ 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)
+ 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"
+ case True
+ then have "f x \<le> Sup M"
apply rule
apply (erule order_trans)
- by (rule ccpo_Sup_upper[OF chM])
- thus ?thesis ..
+ 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])
+ 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"
+ case True
+ then have "y \<le> Sup M"
apply rule
apply (erule order_trans)
- by (rule ccpo_Sup_upper[OF Sup(1)])
- thus ?thesis ..
+ 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)
@@ -147,19 +155,19 @@
qed
lemma bot_in_iterates: "Sup {} \<in> iterates f"
-by(auto intro: iterates.Sup simp add: chain_empty)
+ 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)"
+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)
+ 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"
@@ -169,35 +177,45 @@
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)
- thus "f (fixp f) \<le> fixp f"
- unfolding fixp_def .
+ 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
+ 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"
- thus "x \<le> z"
+ fix x
+ assume "x \<in> iterates f"
+ then show "x \<le> z"
proof (induct x rule: iterates.induct)
- fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD)
- also note z finally show "f x \<le> z" .
- qed (auto intro: ccpo_Sup_least)
+ 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 = (\<forall>A. chain ord A \<longrightarrow> (A \<noteq> {}) \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
+ 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
+ using assms unfolding ccpo.admissible_def by fast
lemma admissibleD:
assumes "ccpo.admissible lub ord P"
@@ -205,7 +223,7 @@
assumes "A \<noteq> {}"
assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
shows "P (lub A)"
-using assms by (auto simp: ccpo.admissible_def)
+ using assms by (auto simp: ccpo.admissible_def)
setup \<open>Sign.map_naming Name_Space.parent_path\<close>
@@ -215,44 +233,54 @@
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]
+ unfolding fixp_def
+ using adm chain_iterates[OF mono]
proof (rule ccpo.admissibleD)
- show "iterates f \<noteq> {}" using bot_in_iterates by auto
- fix x assume "x \<in> iterates f"
- thus "P x"
- by (induct rule: iterates.induct)
- (case_tac "M = {}", auto intro: step bot ccpo.admissibleD adm)
+ 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
+ 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)
+ 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
+ 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
+ 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
+ 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
+ unfolding chain_def by fast
-context ccpo begin
+context ccpo
+begin
lemma admissible_disj_lemma:
assumes A: "chain (op \<le>)A"
@@ -280,17 +308,24 @@
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 A: "chain (op \<le>) A"
- assume "A \<noteq> {}"
- and "\<forall>x\<in>A. P x \<or> Q x"
- hence "(\<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)"
- using chainD[OF A] by blast
- hence "(\<exists>x. x \<in> A \<and> P x) \<and> Sup A = Sup {x \<in> A. P x} \<or> (\<exists>x. x \<in> A \<and> Q x) \<and> Sup A = Sup {x \<in> A. Q x}"
+ fix A :: "'a set"
+ assume A: "chain (op \<le>) A"
+ assume "A \<noteq> {}" and "\<forall>x\<in>A. P x \<or> Q x"
+ then 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)"
+ using chainD[OF A] by blast (* slow *)
+ then have "(\<exists>x. x \<in> A \<and> P x) \<and> Sup A = Sup {x \<in> A. P x} \<or> (\<exists>x. x \<in> A \<and> Q x) \<and> Sup A = Sup {x \<in> A. Q x}"
using admissible_disj_lemma [OF A] by blast
- thus "P (Sup A) \<or> Q (Sup A)"
- apply (rule disjE, simp_all)
- apply (rule disjI1, rule ccpo.admissibleD [OF P chain_compr [OF A]], simp, simp)
- apply (rule disjI2, rule ccpo.admissibleD [OF Q chain_compr [OF A]], simp, simp)
+ then show "P (Sup A) \<or> Q (Sup A)"
+ apply (rule disjE)
+ apply simp_all
+ apply (rule disjI1)
+ apply (rule ccpo.admissibleD [OF P chain_compr [OF A]])
+ apply simp
+ apply simp
+ apply (rule disjI2)
+ apply (rule ccpo.admissibleD [OF Q chain_compr [OF A]])
+ apply simp
+ apply simp
done
qed
@@ -300,7 +335,8 @@
by standard (fast intro: Sup_upper Sup_least)+
lemma lfp_eq_fixp:
- assumes f: "mono f" shows "lfp f = fixp f"
+ assumes f: "mono f"
+ shows "lfp f = fixp f"
proof (rule antisym)
from f have f': "monotone (op \<le>) (op \<le>) f"
unfolding mono_def monotone_def .