--- a/src/HOLCF/Cfun.thy Mon Oct 11 16:14:15 2010 -0700
+++ b/src/HOLCF/Cfun.thy Mon Oct 11 16:24:44 2010 -0700
@@ -6,7 +6,7 @@
header {* The type of continuous functions *}
theory Cfun
-imports Pcpodef Ffun Product_Cpo
+imports Pcpodef Fun_Cpo Product_Cpo
begin
default_sort cpo
--- a/src/HOLCF/Ffun.thy Mon Oct 11 16:14:15 2010 -0700
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,273 +0,0 @@
-(* Title: HOLCF/FunCpo.thy
- Author: Franz Regensburger
-*)
-
-header {* Class instances for the full function space *}
-
-theory Ffun
-imports Cont
-begin
-
-subsection {* Full function space is a partial order *}
-
-instantiation "fun" :: (type, below) below
-begin
-
-definition
- below_fun_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x)"
-
-instance ..
-end
-
-instance "fun" :: (type, po) po
-proof
- fix f :: "'a \<Rightarrow> 'b"
- show "f \<sqsubseteq> f"
- by (simp add: below_fun_def)
-next
- fix f g :: "'a \<Rightarrow> 'b"
- assume "f \<sqsubseteq> g" and "g \<sqsubseteq> f" thus "f = g"
- by (simp add: below_fun_def fun_eq_iff below_antisym)
-next
- fix f g h :: "'a \<Rightarrow> 'b"
- assume "f \<sqsubseteq> g" and "g \<sqsubseteq> h" thus "f \<sqsubseteq> h"
- unfolding below_fun_def by (fast elim: below_trans)
-qed
-
-text {* make the symbol @{text "<<"} accessible for type fun *}
-
-lemma expand_fun_below: "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
-by (simp add: below_fun_def)
-
-lemma below_fun_ext: "(\<And>x. f x \<sqsubseteq> g x) \<Longrightarrow> f \<sqsubseteq> g"
-by (simp add: below_fun_def)
-
-subsection {* Full function space is chain complete *}
-
-text {* function application is monotone *}
-
-lemma monofun_app: "monofun (\<lambda>f. f x)"
-by (rule monofunI, simp add: below_fun_def)
-
-text {* chains of functions yield chains in the po range *}
-
-lemma ch2ch_fun: "chain S \<Longrightarrow> chain (\<lambda>i. S i x)"
-by (simp add: chain_def below_fun_def)
-
-lemma ch2ch_lambda: "(\<And>x. chain (\<lambda>i. S i x)) \<Longrightarrow> chain S"
-by (simp add: chain_def below_fun_def)
-
-text {* upper bounds of function chains yield upper bound in the po range *}
-
-lemma ub2ub_fun:
- "range S <| u \<Longrightarrow> range (\<lambda>i. S i x) <| u x"
-by (auto simp add: is_ub_def below_fun_def)
-
-text {* Type @{typ "'a::type => 'b::cpo"} is chain complete *}
-
-lemma is_lub_lambda:
- assumes f: "\<And>x. range (\<lambda>i. Y i x) <<| f x"
- shows "range Y <<| f"
-apply (rule is_lubI)
-apply (rule ub_rangeI)
-apply (rule below_fun_ext)
-apply (rule is_ub_lub [OF f])
-apply (rule below_fun_ext)
-apply (rule is_lub_lub [OF f])
-apply (erule ub2ub_fun)
-done
-
-lemma lub_fun:
- "chain (S::nat \<Rightarrow> 'a::type \<Rightarrow> 'b::cpo)
- \<Longrightarrow> range S <<| (\<lambda>x. \<Squnion>i. S i x)"
-apply (rule is_lub_lambda)
-apply (rule cpo_lubI)
-apply (erule ch2ch_fun)
-done
-
-lemma thelub_fun:
- "chain (S::nat \<Rightarrow> 'a::type \<Rightarrow> 'b::cpo)
- \<Longrightarrow> (\<Squnion>i. S i) = (\<lambda>x. \<Squnion>i. S i x)"
-by (rule lub_fun [THEN thelubI])
-
-lemma cpo_fun:
- "chain (S::nat \<Rightarrow> 'a::type \<Rightarrow> 'b::cpo) \<Longrightarrow> \<exists>x. range S <<| x"
-by (rule exI, erule lub_fun)
-
-instance "fun" :: (type, cpo) cpo
-by intro_classes (rule cpo_fun)
-
-instance "fun" :: (finite, finite_po) finite_po ..
-
-instance "fun" :: (type, discrete_cpo) discrete_cpo
-proof
- fix f g :: "'a \<Rightarrow> 'b"
- show "f \<sqsubseteq> g \<longleftrightarrow> f = g"
- unfolding expand_fun_below fun_eq_iff
- by simp
-qed
-
-text {* chain-finite function spaces *}
-
-lemma maxinch2maxinch_lambda:
- "(\<And>x. max_in_chain n (\<lambda>i. S i x)) \<Longrightarrow> max_in_chain n S"
-unfolding max_in_chain_def fun_eq_iff by simp
-
-lemma maxinch_mono:
- "\<lbrakk>max_in_chain i Y; i \<le> j\<rbrakk> \<Longrightarrow> max_in_chain j Y"
-unfolding max_in_chain_def
-proof (intro allI impI)
- fix k
- assume Y: "\<forall>n\<ge>i. Y i = Y n"
- assume ij: "i \<le> j"
- assume jk: "j \<le> k"
- from ij jk have ik: "i \<le> k" by simp
- from Y ij have Yij: "Y i = Y j" by simp
- from Y ik have Yik: "Y i = Y k" by simp
- from Yij Yik show "Y j = Y k" by auto
-qed
-
-instance "fun" :: (finite, chfin) chfin
-proof
- fix Y :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"
- let ?n = "\<lambda>x. LEAST n. max_in_chain n (\<lambda>i. Y i x)"
- assume "chain Y"
- hence "\<And>x. chain (\<lambda>i. Y i x)"
- by (rule ch2ch_fun)
- hence "\<And>x. \<exists>n. max_in_chain n (\<lambda>i. Y i x)"
- by (rule chfin)
- hence "\<And>x. max_in_chain (?n x) (\<lambda>i. Y i x)"
- by (rule LeastI_ex)
- hence "\<And>x. max_in_chain (Max (range ?n)) (\<lambda>i. Y i x)"
- by (rule maxinch_mono [OF _ Max_ge], simp_all)
- hence "max_in_chain (Max (range ?n)) Y"
- by (rule maxinch2maxinch_lambda)
- thus "\<exists>n. max_in_chain n Y" ..
-qed
-
-subsection {* Full function space is pointed *}
-
-lemma minimal_fun: "(\<lambda>x. \<bottom>) \<sqsubseteq> f"
-by (simp add: below_fun_def)
-
-lemma least_fun: "\<exists>x::'a::type \<Rightarrow> 'b::pcpo. \<forall>y. x \<sqsubseteq> y"
-apply (rule_tac x = "\<lambda>x. \<bottom>" in exI)
-apply (rule minimal_fun [THEN allI])
-done
-
-instance "fun" :: (type, pcpo) pcpo
-by intro_classes (rule least_fun)
-
-text {* for compatibility with old HOLCF-Version *}
-lemma inst_fun_pcpo: "\<bottom> = (\<lambda>x. \<bottom>)"
-by (rule minimal_fun [THEN UU_I, symmetric])
-
-text {* function application is strict in the left argument *}
-lemma app_strict [simp]: "\<bottom> x = \<bottom>"
-by (simp add: inst_fun_pcpo)
-
-text {*
- The following results are about application for functions in @{typ "'a=>'b"}
-*}
-
-lemma monofun_fun_fun: "f \<sqsubseteq> g \<Longrightarrow> f x \<sqsubseteq> g x"
-by (simp add: below_fun_def)
-
-lemma monofun_fun_arg: "\<lbrakk>monofun f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq> f y"
-by (rule monofunE)
-
-lemma monofun_fun: "\<lbrakk>monofun f; monofun g; f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq> g y"
-by (rule below_trans [OF monofun_fun_arg monofun_fun_fun])
-
-subsection {* Propagation of monotonicity and continuity *}
-
-text {* the lub of a chain of monotone functions is monotone *}
-
-lemma monofun_lub_fun:
- "\<lbrakk>chain (F::nat \<Rightarrow> 'a \<Rightarrow> 'b::cpo); \<forall>i. monofun (F i)\<rbrakk>
- \<Longrightarrow> monofun (\<Squnion>i. F i)"
-apply (rule monofunI)
-apply (simp add: thelub_fun)
-apply (rule lub_mono)
-apply (erule ch2ch_fun)
-apply (erule ch2ch_fun)
-apply (simp add: monofunE)
-done
-
-text {* the lub of a chain of continuous functions is continuous *}
-
-lemma cont_lub_fun:
- "\<lbrakk>chain F; \<forall>i. cont (F i)\<rbrakk> \<Longrightarrow> cont (\<Squnion>i. F i)"
-apply (rule contI2)
-apply (erule monofun_lub_fun)
-apply (simp add: cont2mono)
-apply (simp add: thelub_fun cont2contlubE)
-apply (simp add: diag_lub ch2ch_fun ch2ch_cont)
-done
-
-lemma cont2cont_lub:
- "\<lbrakk>chain F; \<And>i. cont (F i)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i x)"
-by (simp add: thelub_fun [symmetric] cont_lub_fun)
-
-lemma mono2mono_fun: "monofun f \<Longrightarrow> monofun (\<lambda>x. f x y)"
-apply (rule monofunI)
-apply (erule (1) monofun_fun_arg [THEN monofun_fun_fun])
-done
-
-lemma cont2cont_fun: "cont f \<Longrightarrow> cont (\<lambda>x. f x y)"
-apply (rule contI2)
-apply (erule cont2mono [THEN mono2mono_fun])
-apply (simp add: cont2contlubE)
-apply (simp add: thelub_fun ch2ch_cont)
-done
-
-text {* Note @{text "(\<lambda>x. \<lambda>y. f x y) = f"} *}
-
-lemma mono2mono_lambda:
- assumes f: "\<And>y. monofun (\<lambda>x. f x y)" shows "monofun f"
-apply (rule monofunI)
-apply (rule below_fun_ext)
-apply (erule monofunE [OF f])
-done
-
-lemma cont2cont_lambda [simp]:
- assumes f: "\<And>y. cont (\<lambda>x. f x y)" shows "cont f"
-apply (rule contI2)
-apply (simp add: mono2mono_lambda cont2mono f)
-apply (rule below_fun_ext)
-apply (simp add: thelub_fun cont2contlubE [OF f])
-done
-
-text {* What D.A.Schmidt calls continuity of abstraction; never used here *}
-
-lemma contlub_lambda:
- "(\<And>x::'a::type. chain (\<lambda>i. S i x::'b::cpo))
- \<Longrightarrow> (\<lambda>x. \<Squnion>i. S i x) = (\<Squnion>i. (\<lambda>x. S i x))"
-by (simp add: thelub_fun ch2ch_lambda)
-
-lemma contlub_abstraction:
- "\<lbrakk>chain Y; \<forall>y. cont (\<lambda>x.(c::'a::cpo\<Rightarrow>'b::type\<Rightarrow>'c::cpo) x y)\<rbrakk> \<Longrightarrow>
- (\<lambda>y. \<Squnion>i. c (Y i) y) = (\<Squnion>i. (\<lambda>y. c (Y i) y))"
-apply (rule thelub_fun [symmetric])
-apply (simp add: ch2ch_cont)
-done
-
-lemma mono2mono_app:
- "\<lbrakk>monofun f; \<forall>x. monofun (f x); monofun t\<rbrakk> \<Longrightarrow> monofun (\<lambda>x. (f x) (t x))"
-apply (rule monofunI)
-apply (simp add: monofun_fun monofunE)
-done
-
-lemma cont2cont_app:
- "\<lbrakk>cont f; \<forall>x. cont (f x); cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x) (t x))"
-apply (erule cont_apply [where t=t])
-apply (erule spec)
-apply (erule cont2cont_fun)
-done
-
-lemmas cont2cont_app2 = cont2cont_app [rule_format]
-
-lemma cont2cont_app3: "\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. f (t x))"
-by (rule cont2cont_app2 [OF cont_const])
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Fun_Cpo.thy Mon Oct 11 16:24:44 2010 -0700
@@ -0,0 +1,274 @@
+(* Title: HOLCF/Fun_Cpo.thy
+ Author: Franz Regensburger
+ Author: Brian Huffman
+*)
+
+header {* Class instances for the full function space *}
+
+theory Fun_Cpo
+imports Cont
+begin
+
+subsection {* Full function space is a partial order *}
+
+instantiation "fun" :: (type, below) below
+begin
+
+definition
+ below_fun_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>f g. \<forall>x. f x \<sqsubseteq> g x)"
+
+instance ..
+end
+
+instance "fun" :: (type, po) po
+proof
+ fix f :: "'a \<Rightarrow> 'b"
+ show "f \<sqsubseteq> f"
+ by (simp add: below_fun_def)
+next
+ fix f g :: "'a \<Rightarrow> 'b"
+ assume "f \<sqsubseteq> g" and "g \<sqsubseteq> f" thus "f = g"
+ by (simp add: below_fun_def fun_eq_iff below_antisym)
+next
+ fix f g h :: "'a \<Rightarrow> 'b"
+ assume "f \<sqsubseteq> g" and "g \<sqsubseteq> h" thus "f \<sqsubseteq> h"
+ unfolding below_fun_def by (fast elim: below_trans)
+qed
+
+text {* make the symbol @{text "<<"} accessible for type fun *}
+
+lemma expand_fun_below: "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
+by (simp add: below_fun_def)
+
+lemma below_fun_ext: "(\<And>x. f x \<sqsubseteq> g x) \<Longrightarrow> f \<sqsubseteq> g"
+by (simp add: below_fun_def)
+
+subsection {* Full function space is chain complete *}
+
+text {* function application is monotone *}
+
+lemma monofun_app: "monofun (\<lambda>f. f x)"
+by (rule monofunI, simp add: below_fun_def)
+
+text {* chains of functions yield chains in the po range *}
+
+lemma ch2ch_fun: "chain S \<Longrightarrow> chain (\<lambda>i. S i x)"
+by (simp add: chain_def below_fun_def)
+
+lemma ch2ch_lambda: "(\<And>x. chain (\<lambda>i. S i x)) \<Longrightarrow> chain S"
+by (simp add: chain_def below_fun_def)
+
+text {* upper bounds of function chains yield upper bound in the po range *}
+
+lemma ub2ub_fun:
+ "range S <| u \<Longrightarrow> range (\<lambda>i. S i x) <| u x"
+by (auto simp add: is_ub_def below_fun_def)
+
+text {* Type @{typ "'a::type => 'b::cpo"} is chain complete *}
+
+lemma is_lub_lambda:
+ assumes f: "\<And>x. range (\<lambda>i. Y i x) <<| f x"
+ shows "range Y <<| f"
+apply (rule is_lubI)
+apply (rule ub_rangeI)
+apply (rule below_fun_ext)
+apply (rule is_ub_lub [OF f])
+apply (rule below_fun_ext)
+apply (rule is_lub_lub [OF f])
+apply (erule ub2ub_fun)
+done
+
+lemma lub_fun:
+ "chain (S::nat \<Rightarrow> 'a::type \<Rightarrow> 'b::cpo)
+ \<Longrightarrow> range S <<| (\<lambda>x. \<Squnion>i. S i x)"
+apply (rule is_lub_lambda)
+apply (rule cpo_lubI)
+apply (erule ch2ch_fun)
+done
+
+lemma thelub_fun:
+ "chain (S::nat \<Rightarrow> 'a::type \<Rightarrow> 'b::cpo)
+ \<Longrightarrow> (\<Squnion>i. S i) = (\<lambda>x. \<Squnion>i. S i x)"
+by (rule lub_fun [THEN thelubI])
+
+lemma cpo_fun:
+ "chain (S::nat \<Rightarrow> 'a::type \<Rightarrow> 'b::cpo) \<Longrightarrow> \<exists>x. range S <<| x"
+by (rule exI, erule lub_fun)
+
+instance "fun" :: (type, cpo) cpo
+by intro_classes (rule cpo_fun)
+
+instance "fun" :: (finite, finite_po) finite_po ..
+
+instance "fun" :: (type, discrete_cpo) discrete_cpo
+proof
+ fix f g :: "'a \<Rightarrow> 'b"
+ show "f \<sqsubseteq> g \<longleftrightarrow> f = g"
+ unfolding expand_fun_below fun_eq_iff
+ by simp
+qed
+
+text {* chain-finite function spaces *}
+
+lemma maxinch2maxinch_lambda:
+ "(\<And>x. max_in_chain n (\<lambda>i. S i x)) \<Longrightarrow> max_in_chain n S"
+unfolding max_in_chain_def fun_eq_iff by simp
+
+lemma maxinch_mono:
+ "\<lbrakk>max_in_chain i Y; i \<le> j\<rbrakk> \<Longrightarrow> max_in_chain j Y"
+unfolding max_in_chain_def
+proof (intro allI impI)
+ fix k
+ assume Y: "\<forall>n\<ge>i. Y i = Y n"
+ assume ij: "i \<le> j"
+ assume jk: "j \<le> k"
+ from ij jk have ik: "i \<le> k" by simp
+ from Y ij have Yij: "Y i = Y j" by simp
+ from Y ik have Yik: "Y i = Y k" by simp
+ from Yij Yik show "Y j = Y k" by auto
+qed
+
+instance "fun" :: (finite, chfin) chfin
+proof
+ fix Y :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"
+ let ?n = "\<lambda>x. LEAST n. max_in_chain n (\<lambda>i. Y i x)"
+ assume "chain Y"
+ hence "\<And>x. chain (\<lambda>i. Y i x)"
+ by (rule ch2ch_fun)
+ hence "\<And>x. \<exists>n. max_in_chain n (\<lambda>i. Y i x)"
+ by (rule chfin)
+ hence "\<And>x. max_in_chain (?n x) (\<lambda>i. Y i x)"
+ by (rule LeastI_ex)
+ hence "\<And>x. max_in_chain (Max (range ?n)) (\<lambda>i. Y i x)"
+ by (rule maxinch_mono [OF _ Max_ge], simp_all)
+ hence "max_in_chain (Max (range ?n)) Y"
+ by (rule maxinch2maxinch_lambda)
+ thus "\<exists>n. max_in_chain n Y" ..
+qed
+
+subsection {* Full function space is pointed *}
+
+lemma minimal_fun: "(\<lambda>x. \<bottom>) \<sqsubseteq> f"
+by (simp add: below_fun_def)
+
+lemma least_fun: "\<exists>x::'a::type \<Rightarrow> 'b::pcpo. \<forall>y. x \<sqsubseteq> y"
+apply (rule_tac x = "\<lambda>x. \<bottom>" in exI)
+apply (rule minimal_fun [THEN allI])
+done
+
+instance "fun" :: (type, pcpo) pcpo
+by intro_classes (rule least_fun)
+
+text {* for compatibility with old HOLCF-Version *}
+lemma inst_fun_pcpo: "\<bottom> = (\<lambda>x. \<bottom>)"
+by (rule minimal_fun [THEN UU_I, symmetric])
+
+text {* function application is strict in the left argument *}
+lemma app_strict [simp]: "\<bottom> x = \<bottom>"
+by (simp add: inst_fun_pcpo)
+
+text {*
+ The following results are about application for functions in @{typ "'a=>'b"}
+*}
+
+lemma monofun_fun_fun: "f \<sqsubseteq> g \<Longrightarrow> f x \<sqsubseteq> g x"
+by (simp add: below_fun_def)
+
+lemma monofun_fun_arg: "\<lbrakk>monofun f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq> f y"
+by (rule monofunE)
+
+lemma monofun_fun: "\<lbrakk>monofun f; monofun g; f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq> g y"
+by (rule below_trans [OF monofun_fun_arg monofun_fun_fun])
+
+subsection {* Propagation of monotonicity and continuity *}
+
+text {* the lub of a chain of monotone functions is monotone *}
+
+lemma monofun_lub_fun:
+ "\<lbrakk>chain (F::nat \<Rightarrow> 'a \<Rightarrow> 'b::cpo); \<forall>i. monofun (F i)\<rbrakk>
+ \<Longrightarrow> monofun (\<Squnion>i. F i)"
+apply (rule monofunI)
+apply (simp add: thelub_fun)
+apply (rule lub_mono)
+apply (erule ch2ch_fun)
+apply (erule ch2ch_fun)
+apply (simp add: monofunE)
+done
+
+text {* the lub of a chain of continuous functions is continuous *}
+
+lemma cont_lub_fun:
+ "\<lbrakk>chain F; \<forall>i. cont (F i)\<rbrakk> \<Longrightarrow> cont (\<Squnion>i. F i)"
+apply (rule contI2)
+apply (erule monofun_lub_fun)
+apply (simp add: cont2mono)
+apply (simp add: thelub_fun cont2contlubE)
+apply (simp add: diag_lub ch2ch_fun ch2ch_cont)
+done
+
+lemma cont2cont_lub:
+ "\<lbrakk>chain F; \<And>i. cont (F i)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i x)"
+by (simp add: thelub_fun [symmetric] cont_lub_fun)
+
+lemma mono2mono_fun: "monofun f \<Longrightarrow> monofun (\<lambda>x. f x y)"
+apply (rule monofunI)
+apply (erule (1) monofun_fun_arg [THEN monofun_fun_fun])
+done
+
+lemma cont2cont_fun: "cont f \<Longrightarrow> cont (\<lambda>x. f x y)"
+apply (rule contI2)
+apply (erule cont2mono [THEN mono2mono_fun])
+apply (simp add: cont2contlubE)
+apply (simp add: thelub_fun ch2ch_cont)
+done
+
+text {* Note @{text "(\<lambda>x. \<lambda>y. f x y) = f"} *}
+
+lemma mono2mono_lambda:
+ assumes f: "\<And>y. monofun (\<lambda>x. f x y)" shows "monofun f"
+apply (rule monofunI)
+apply (rule below_fun_ext)
+apply (erule monofunE [OF f])
+done
+
+lemma cont2cont_lambda [simp]:
+ assumes f: "\<And>y. cont (\<lambda>x. f x y)" shows "cont f"
+apply (rule contI2)
+apply (simp add: mono2mono_lambda cont2mono f)
+apply (rule below_fun_ext)
+apply (simp add: thelub_fun cont2contlubE [OF f])
+done
+
+text {* What D.A.Schmidt calls continuity of abstraction; never used here *}
+
+lemma contlub_lambda:
+ "(\<And>x::'a::type. chain (\<lambda>i. S i x::'b::cpo))
+ \<Longrightarrow> (\<lambda>x. \<Squnion>i. S i x) = (\<Squnion>i. (\<lambda>x. S i x))"
+by (simp add: thelub_fun ch2ch_lambda)
+
+lemma contlub_abstraction:
+ "\<lbrakk>chain Y; \<forall>y. cont (\<lambda>x.(c::'a::cpo\<Rightarrow>'b::type\<Rightarrow>'c::cpo) x y)\<rbrakk> \<Longrightarrow>
+ (\<lambda>y. \<Squnion>i. c (Y i) y) = (\<Squnion>i. (\<lambda>y. c (Y i) y))"
+apply (rule thelub_fun [symmetric])
+apply (simp add: ch2ch_cont)
+done
+
+lemma mono2mono_app:
+ "\<lbrakk>monofun f; \<forall>x. monofun (f x); monofun t\<rbrakk> \<Longrightarrow> monofun (\<lambda>x. (f x) (t x))"
+apply (rule monofunI)
+apply (simp add: monofun_fun monofunE)
+done
+
+lemma cont2cont_app:
+ "\<lbrakk>cont f; \<forall>x. cont (f x); cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x) (t x))"
+apply (erule cont_apply [where t=t])
+apply (erule spec)
+apply (erule cont2cont_fun)
+done
+
+lemmas cont2cont_app2 = cont2cont_app [rule_format]
+
+lemma cont2cont_app3: "\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. f (t x))"
+by (rule cont2cont_app2 [OF cont_const])
+
+end
--- a/src/HOLCF/IsaMakefile Mon Oct 11 16:14:15 2010 -0700
+++ b/src/HOLCF/IsaMakefile Mon Oct 11 16:24:44 2010 -0700
@@ -48,9 +48,9 @@
Deflation.thy \
Domain.thy \
Domain_Aux.thy \
- Ffun.thy \
Fixrec.thy \
Fix.thy \
+ Fun_Cpo.thy \
HOLCF.thy \
Lift.thy \
LowerPD.thy \