(* Title: HOL/HOLCF/Pcpo.thy
Author: Franz Regensburger
*)
section \<open>Classes cpo and pcpo\<close>
theory Pcpo
imports Porder
begin
subsection \<open>Complete partial orders\<close>
text \<open>The class cpo of chain complete partial orders\<close>
class cpo = po +
assumes cpo: "chain S \<Longrightarrow> \<exists>x. range S <<| x"
begin
text \<open>in cpo's everthing equal to THE lub has lub properties for every chain\<close>
lemma cpo_lubI: "chain S \<Longrightarrow> range S <<| (\<Squnion>i. S i)"
by (fast dest: cpo elim: is_lub_lub)
lemma thelubE: "\<lbrakk>chain S; (\<Squnion>i. S i) = l\<rbrakk> \<Longrightarrow> range S <<| l"
by (blast dest: cpo intro: is_lub_lub)
text \<open>Properties of the lub\<close>
lemma is_ub_thelub: "chain S \<Longrightarrow> S x \<sqsubseteq> (\<Squnion>i. S i)"
by (blast dest: cpo intro: is_lub_lub [THEN is_lub_rangeD1])
lemma is_lub_thelub: "\<lbrakk>chain S; range S <| x\<rbrakk> \<Longrightarrow> (\<Squnion>i. S i) \<sqsubseteq> x"
by (blast dest: cpo intro: is_lub_lub [THEN is_lubD2])
lemma lub_below_iff: "chain S \<Longrightarrow> (\<Squnion>i. S i) \<sqsubseteq> x \<longleftrightarrow> (\<forall>i. S i \<sqsubseteq> x)"
by (simp add: is_lub_below_iff [OF cpo_lubI] is_ub_def)
lemma lub_below: "\<lbrakk>chain S; \<And>i. S i \<sqsubseteq> x\<rbrakk> \<Longrightarrow> (\<Squnion>i. S i) \<sqsubseteq> x"
by (simp add: lub_below_iff)
lemma below_lub: "\<lbrakk>chain S; x \<sqsubseteq> S i\<rbrakk> \<Longrightarrow> x \<sqsubseteq> (\<Squnion>i. S i)"
by (erule below_trans, erule is_ub_thelub)
lemma lub_range_mono: "\<lbrakk>range X \<subseteq> range Y; chain Y; chain X\<rbrakk> \<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)"
apply (erule lub_below)
apply (subgoal_tac "\<exists>j. X i = Y j")
apply clarsimp
apply (erule is_ub_thelub)
apply auto
done
lemma lub_range_shift: "chain Y \<Longrightarrow> (\<Squnion>i. Y (i + j)) = (\<Squnion>i. Y i)"
apply (rule below_antisym)
apply (rule lub_range_mono)
apply fast
apply assumption
apply (erule chain_shift)
apply (rule lub_below)
apply assumption
apply (rule_tac i="i" in below_lub)
apply (erule chain_shift)
apply (erule chain_mono)
apply (rule le_add1)
done
lemma maxinch_is_thelub: "chain Y \<Longrightarrow> max_in_chain i Y = ((\<Squnion>i. Y i) = Y i)"
apply (rule iffI)
apply (fast intro!: lub_eqI lub_finch1)
apply (unfold max_in_chain_def)
apply (safe intro!: below_antisym)
apply (fast elim!: chain_mono)
apply (drule sym)
apply (force elim!: is_ub_thelub)
done
text \<open>the \<open>\<sqsubseteq>\<close> relation between two chains is preserved by their lubs\<close>
lemma lub_mono: "\<lbrakk>chain X; chain Y; \<And>i. X i \<sqsubseteq> Y i\<rbrakk> \<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)"
by (fast elim: lub_below below_lub)
text \<open>the = relation between two chains is preserved by their lubs\<close>
lemma lub_eq: "(\<And>i. X i = Y i) \<Longrightarrow> (\<Squnion>i. X i) = (\<Squnion>i. Y i)"
by simp
lemma ch2ch_lub:
assumes 1: "\<And>j. chain (\<lambda>i. Y i j)"
assumes 2: "\<And>i. chain (\<lambda>j. Y i j)"
shows "chain (\<lambda>i. \<Squnion>j. Y i j)"
apply (rule chainI)
apply (rule lub_mono [OF 2 2])
apply (rule chainE [OF 1])
done
lemma diag_lub:
assumes 1: "\<And>j. chain (\<lambda>i. Y i j)"
assumes 2: "\<And>i. chain (\<lambda>j. Y i j)"
shows "(\<Squnion>i. \<Squnion>j. Y i j) = (\<Squnion>i. Y i i)"
proof (rule below_antisym)
have 3: "chain (\<lambda>i. Y i i)"
apply (rule chainI)
apply (rule below_trans)
apply (rule chainE [OF 1])
apply (rule chainE [OF 2])
done
have 4: "chain (\<lambda>i. \<Squnion>j. Y i j)"
by (rule ch2ch_lub [OF 1 2])
show "(\<Squnion>i. \<Squnion>j. Y i j) \<sqsubseteq> (\<Squnion>i. Y i i)"
apply (rule lub_below [OF 4])
apply (rule lub_below [OF 2])
apply (rule below_lub [OF 3])
apply (rule below_trans)
apply (rule chain_mono [OF 1 max.cobounded1])
apply (rule chain_mono [OF 2 max.cobounded2])
done
show "(\<Squnion>i. Y i i) \<sqsubseteq> (\<Squnion>i. \<Squnion>j. Y i j)"
apply (rule lub_mono [OF 3 4])
apply (rule is_ub_thelub [OF 2])
done
qed
lemma ex_lub:
assumes 1: "\<And>j. chain (\<lambda>i. Y i j)"
assumes 2: "\<And>i. chain (\<lambda>j. Y i j)"
shows "(\<Squnion>i. \<Squnion>j. Y i j) = (\<Squnion>j. \<Squnion>i. Y i j)"
by (simp add: diag_lub 1 2)
end
subsection \<open>Pointed cpos\<close>
text \<open>The class pcpo of pointed cpos\<close>
class pcpo = cpo +
assumes least: "\<exists>x. \<forall>y. x \<sqsubseteq> y"
begin
definition bottom :: "'a" ("\<bottom>")
where "bottom = (THE x. \<forall>y. x \<sqsubseteq> y)"
lemma minimal [iff]: "\<bottom> \<sqsubseteq> x"
unfolding bottom_def
apply (rule the1I2)
apply (rule ex_ex1I)
apply (rule least)
apply (blast intro: below_antisym)
apply simp
done
end
text \<open>Old "UU" syntax:\<close>
syntax UU :: logic
translations "UU" \<rightharpoonup> "CONST bottom"
text \<open>Simproc to rewrite @{term "\<bottom> = x"} to @{term "x = \<bottom>"}.\<close>
setup \<open>Reorient_Proc.add (fn Const(\<^const_name>\<open>bottom\<close>, _) => true | _ => false)\<close>
simproc_setup reorient_bottom ("\<bottom> = x") = Reorient_Proc.proc
text \<open>useful lemmas about @{term \<bottom>}\<close>
lemma below_bottom_iff [simp]: "x \<sqsubseteq> \<bottom> \<longleftrightarrow> x = \<bottom>"
by (simp add: po_eq_conv)
lemma eq_bottom_iff: "x = \<bottom> \<longleftrightarrow> x \<sqsubseteq> \<bottom>"
by simp
lemma bottomI: "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
by (subst eq_bottom_iff)
lemma lub_eq_bottom_iff: "chain Y \<Longrightarrow> (\<Squnion>i. Y i) = \<bottom> \<longleftrightarrow> (\<forall>i. Y i = \<bottom>)"
by (simp only: eq_bottom_iff lub_below_iff)
subsection \<open>Chain-finite and flat cpos\<close>
text \<open>further useful classes for HOLCF domains\<close>
class chfin = po +
assumes chfin: "chain Y \<Longrightarrow> \<exists>n. max_in_chain n Y"
begin
subclass cpo
apply standard
apply (frule chfin)
apply (blast intro: lub_finch1)
done
lemma chfin2finch: "chain Y \<Longrightarrow> finite_chain Y"
by (simp add: chfin finite_chain_def)
end
class flat = pcpo +
assumes ax_flat: "x \<sqsubseteq> y \<Longrightarrow> x = \<bottom> \<or> x = y"
begin
subclass chfin
proof
fix Y
assume *: "chain Y"
show "\<exists>n. max_in_chain n Y"
apply (unfold max_in_chain_def)
apply (cases "\<forall>i. Y i = \<bottom>")
apply simp
apply simp
apply (erule exE)
apply (rule_tac x="i" in exI)
apply clarify
using * apply (blast dest: chain_mono ax_flat)
done
qed
lemma flat_below_iff: "x \<sqsubseteq> y \<longleftrightarrow> x = \<bottom> \<or> x = y"
by (safe dest!: ax_flat)
lemma flat_eq: "a \<noteq> \<bottom> \<Longrightarrow> a \<sqsubseteq> b = (a = b)"
by (safe dest!: ax_flat)
end
subsection \<open>Discrete cpos\<close>
class discrete_cpo = below +
assumes discrete_cpo [simp]: "x \<sqsubseteq> y \<longleftrightarrow> x = y"
begin
subclass po
by standard simp_all
text \<open>In a discrete cpo, every chain is constant\<close>
lemma discrete_chain_const:
assumes S: "chain S"
shows "\<exists>x. S = (\<lambda>i. x)"
proof (intro exI ext)
fix i :: nat
from S le0 have "S 0 \<sqsubseteq> S i" by (rule chain_mono)
then have "S 0 = S i" by simp
then show "S i = S 0" by (rule sym)
qed
subclass chfin
proof
fix S :: "nat \<Rightarrow> 'a"
assume S: "chain S"
then have "\<exists>x. S = (\<lambda>i. x)"
by (rule discrete_chain_const)
then have "max_in_chain 0 S"
by (auto simp: max_in_chain_def)
then show "\<exists>i. max_in_chain i S" ..
qed
end
end