replace lub (range Y) with (LUB i. Y i)
(* Title: HOLCF/Cont.thy
ID: $Id$
Author: Franz Regensburger
Results about continuity and monotonicity.
*)
header {* Continuity and monotonicity *}
theory Cont
imports Pcpo
begin
text {*
Now we change the default class! Form now on all untyped type variables are
of default class po
*}
defaultsort po
subsection {* Definitions *}
definition
monofun :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" -- "monotonicity" where
"monofun f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"
definition
contlub :: "('a::cpo \<Rightarrow> 'b::cpo) \<Rightarrow> bool" -- "first cont. def" where
"contlub f = (\<forall>Y. chain Y \<longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i)))"
definition
cont :: "('a::cpo \<Rightarrow> 'b::cpo) \<Rightarrow> bool" -- "secnd cont. def" where
"cont f = (\<forall>Y. chain Y \<longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i))"
lemma contlubI:
"\<lbrakk>\<And>Y. chain Y \<Longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))\<rbrakk> \<Longrightarrow> contlub f"
by (simp add: contlub_def)
lemma contlubE:
"\<lbrakk>contlub f; chain Y\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))"
by (simp add: contlub_def)
lemma contI:
"\<lbrakk>\<And>Y. chain Y \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)\<rbrakk> \<Longrightarrow> cont f"
by (simp add: cont_def)
lemma contE:
"\<lbrakk>cont f; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)"
by (simp add: cont_def)
lemma monofunI:
"\<lbrakk>\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y\<rbrakk> \<Longrightarrow> monofun f"
by (simp add: monofun_def)
lemma monofunE:
"\<lbrakk>monofun f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq> f y"
by (simp add: monofun_def)
subsection {* @{prop "monofun f \<and> contlub f \<equiv> cont f"} *}
text {* monotone functions map chains to chains *}
lemma ch2ch_monofun: "\<lbrakk>monofun f; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. f (Y i))"
apply (rule chainI)
apply (erule monofunE)
apply (erule chainE)
done
text {* monotone functions map upper bound to upper bounds *}
lemma ub2ub_monofun:
"\<lbrakk>monofun f; range Y <| u\<rbrakk> \<Longrightarrow> range (\<lambda>i. f (Y i)) <| f u"
apply (rule ub_rangeI)
apply (erule monofunE)
apply (erule ub_rangeD)
done
lemma ub2ub_monofun':
"\<lbrakk>monofun f; S <| u\<rbrakk> \<Longrightarrow> f ` S <| f u"
apply (rule ub_imageI)
apply (erule monofunE)
apply (erule (1) is_ubD)
done
text {* monotone functions map directed sets to directed sets *}
lemma dir2dir_monofun:
assumes f: "monofun f"
assumes S: "directed S"
shows "directed (f ` S)"
proof (rule directedI)
from directedD1 [OF S]
obtain x where "x \<in> S" ..
hence "f x \<in> f ` S" by simp
thus "\<exists>x. x \<in> f ` S" ..
next
fix x assume "x \<in> f ` S"
then obtain a where x: "x = f a" and a: "a \<in> S" ..
fix y assume "y \<in> f ` S"
then obtain b where y: "y = f b" and b: "b \<in> S" ..
from directedD2 [OF S a b]
obtain c where "c \<in> S" and "a \<sqsubseteq> c \<and> b \<sqsubseteq> c" ..
hence "f c \<in> f ` S" and "x \<sqsubseteq> f c \<and> y \<sqsubseteq> f c"
using monofunE [OF f] x y by simp_all
thus "\<exists>z\<in>f ` S. x \<sqsubseteq> z \<and> y \<sqsubseteq> z" ..
qed
text {* left to right: @{prop "monofun f \<and> contlub f \<Longrightarrow> cont f"} *}
lemma monocontlub2cont: "\<lbrakk>monofun f; contlub f\<rbrakk> \<Longrightarrow> cont f"
apply (rule contI)
apply (rule thelubE)
apply (erule (1) ch2ch_monofun)
apply (erule (1) contlubE [symmetric])
done
text {* first a lemma about binary chains *}
lemma binchain_cont:
"\<lbrakk>cont f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> range (\<lambda>i::nat. f (if i = 0 then x else y)) <<| f y"
apply (subgoal_tac "f (\<Squnion>i::nat. if i = 0 then x else y) = f y")
apply (erule subst)
apply (erule contE)
apply (erule bin_chain)
apply (rule_tac f=f in arg_cong)
apply (erule lub_bin_chain [THEN thelubI])
done
text {* right to left: @{prop "cont f \<Longrightarrow> monofun f \<and> contlub f"} *}
text {* part1: @{prop "cont f \<Longrightarrow> monofun f"} *}
lemma cont2mono: "cont f \<Longrightarrow> monofun f"
apply (rule monofunI)
apply (drule (1) binchain_cont)
apply (drule_tac i=0 in is_ub_lub)
apply simp
done
lemmas ch2ch_cont = cont2mono [THEN ch2ch_monofun]
text {* right to left: @{prop "cont f \<Longrightarrow> monofun f \<and> contlub f"} *}
text {* part2: @{prop "cont f \<Longrightarrow> contlub f"} *}
lemma cont2contlub: "cont f \<Longrightarrow> contlub f"
apply (rule contlubI)
apply (rule thelubI [symmetric])
apply (erule (1) contE)
done
lemmas cont2contlubE = cont2contlub [THEN contlubE]
lemma contI2:
assumes mono: "monofun f"
assumes less: "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk>
\<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))"
shows "cont f"
apply (rule monocontlub2cont)
apply (rule mono)
apply (rule contlubI)
apply (rule antisym_less)
apply (rule less, assumption)
apply (erule ch2ch_monofun [OF mono])
apply (rule is_lub_thelub)
apply (erule ch2ch_monofun [OF mono])
apply (rule ub2ub_monofun [OF mono])
apply (rule is_lubD1)
apply (erule cpo_lubI)
done
subsection {* Continuity of basic functions *}
text {* The identity function is continuous *}
lemma cont_id: "cont (\<lambda>x. x)"
apply (rule contI)
apply (erule cpo_lubI)
done
text {* constant functions are continuous *}
lemma cont_const: "cont (\<lambda>x. c)"
apply (rule contI)
apply (rule lub_const)
done
text {* if-then-else is continuous *}
lemma cont_if [simp]:
"\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. if b then f x else g x)"
by (induct b) simp_all
subsection {* Finite chains and flat pcpos *}
text {* monotone functions map finite chains to finite chains *}
lemma monofun_finch2finch:
"\<lbrakk>monofun f; finite_chain Y\<rbrakk> \<Longrightarrow> finite_chain (\<lambda>n. f (Y n))"
apply (unfold finite_chain_def)
apply (simp add: ch2ch_monofun)
apply (force simp add: max_in_chain_def)
done
text {* The same holds for continuous functions *}
lemma cont_finch2finch:
"\<lbrakk>cont f; finite_chain Y\<rbrakk> \<Longrightarrow> finite_chain (\<lambda>n. f (Y n))"
by (rule cont2mono [THEN monofun_finch2finch])
lemma chfindom_monofun2cont: "monofun f \<Longrightarrow> cont (f::'a::chfin \<Rightarrow> 'b::cpo)"
apply (rule monocontlub2cont)
apply assumption
apply (rule contlubI)
apply (frule chfin2finch)
apply (clarsimp simp add: finite_chain_def)
apply (subgoal_tac "max_in_chain i (\<lambda>i. f (Y i))")
apply (simp add: maxinch_is_thelub ch2ch_monofun)
apply (force simp add: max_in_chain_def)
done
text {* some properties of flat *}
lemma flatdom_strict2mono: "f \<bottom> = \<bottom> \<Longrightarrow> monofun (f::'a::flat \<Rightarrow> 'b::pcpo)"
apply (rule monofunI)
apply (drule ax_flat)
apply auto
done
lemma flatdom_strict2cont: "f \<bottom> = \<bottom> \<Longrightarrow> cont (f::'a::flat \<Rightarrow> 'b::pcpo)"
by (rule flatdom_strict2mono [THEN chfindom_monofun2cont])
text {* functions with discrete domain *}
lemma cont_discrete_cpo [simp]: "cont (f::'a::discrete_cpo \<Rightarrow> 'b::cpo)"
apply (rule contI)
apply (drule discrete_chain_const, clarify)
apply (simp add: lub_const)
done
end