# HG changeset patch
# User huffman
# Date 1117833835 -7200
# Node ID 36ee7f6af79fef7c6b3ea0f2532f77bf6bbdd473
# Parent  cfe047ad6384e1ee3333b72edc40ad234812a92d
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext

diff -r cfe047ad6384 -r 36ee7f6af79f src/HOLCF/Cfun.ML
--- a/src/HOLCF/Cfun.ML	Fri Jun 03 23:16:35 2005 +0200
+++ b/src/HOLCF/Cfun.ML	Fri Jun 03 23:23:55 2005 +0200
@@ -2,21 +2,12 @@
 (* legacy ML bindings *)
 
 val less_cfun_def = thm "less_cfun_def";
-val Rep_Cfun = thm "Rep_Cfun";
-val Rep_Cfun_inverse = thm "Rep_Cfun_inverse";
-val Abs_Cfun_inverse = thm "Abs_Cfun_inverse";
 val cfun_cong = thm "cfun_cong";
 val cfun_fun_cong = thm "cfun_fun_cong";
 val cfun_arg_cong = thm "cfun_arg_cong";
-val Abs_Cfun_inverse2 = thm "Abs_Cfun_inverse2";
-val Cfunapp2 = thm "Cfunapp2";
+val Abs_CFun_inverse2 = thm "Abs_CFun_inverse2";
 val beta_cfun = thm "beta_cfun";
 val eta_cfun = thm "eta_cfun";
-val inst_cfun_po = thm "inst_cfun_po";
-val less_cfun = thm "less_cfun";
-val minimal_cfun = thm "minimal_cfun";
-val UU_cfun_def = thm "UU_cfun_def";
-val least_cfun = thm "least_cfun";
 val cont_Rep_CFun2 = thm "cont_Rep_CFun2";
 val monofun_Rep_CFun2 = thm "monofun_Rep_CFun2";
 val contlub_Rep_CFun2 = thm "contlub_Rep_CFun2";
@@ -31,13 +22,13 @@
 val ch2ch_Rep_CFunR = thm "ch2ch_Rep_CFunR";
 val ch2ch_Rep_CFunL = thm "ch2ch_Rep_CFunL";
 val lub_cfun_mono = thm "lub_cfun_mono";
-val ex_lubcfun = thm "ex_lubcfun";
-val cont_lubcfun = thm "cont_lubcfun";
+val ex_lub_cfun = thm "ex_lub_cfun";
+val cont_lub_cfun = thm "cont_lub_cfun";
 val lub_cfun = thm "lub_cfun";
 val thelub_cfun = thm "thelub_cfun";
 val ext_cfun = thm "ext_cfun";
 val semi_monofun_Abs_CFun = thm "semi_monofun_Abs_CFun";
-val less_cfun2 = thm "less_cfun2";
+val less_cfun_ext = thm "less_cfun_ext";
 val Istrictify_def = thm "Istrictify_def";
 val strictify_def = thm "strictify_def";
 val ID_def = thm "ID_def";
@@ -52,20 +43,17 @@
 val cont2cont_Rep_CFun = thm "cont2cont_Rep_CFun";
 val cont2mono_LAM = thm "cont2mono_LAM";
 val cont2cont_LAM = thm "cont2cont_LAM";
-val cont2cont_eta = thm "cont2cont_eta";
 val cont_lemmas1 = [cont_const, cont_id, cont_Rep_CFun2,
                     cont2cont_Rep_CFun, cont2cont_LAM];
-val strict_Rep_CFun1 = thm "strict_Rep_CFun1";
 val Istrictify1 = thm "Istrictify1";
 val Istrictify2 = thm "Istrictify2";
-val monofun_Istrictify1 = thm "monofun_Istrictify1";
 val monofun_Istrictify2 = thm "monofun_Istrictify2";
-val contlub_Istrictify1 = thm "contlub_Istrictify1";
 val contlub_Istrictify2 = thm "contlub_Istrictify2";
 val cont_Istrictify1 = thm "cont_Istrictify1";
 val cont_Istrictify2 = thm "cont_Istrictify2";
 val strictify1 = thm "strictify1";
 val strictify2 = thm "strictify2";
+val Rep_CFun_strict1 = thm "Rep_CFun_strict1";
 val chfin_Rep_CFunR = thm "chfin_Rep_CFunR";
 val retraction_strict = thm "retraction_strict";
 val injection_eq = thm "injection_eq";
diff -r cfe047ad6384 -r 36ee7f6af79f src/HOLCF/Cfun.thy
--- a/src/HOLCF/Cfun.thy	Fri Jun 03 23:16:35 2005 +0200
+++ b/src/HOLCF/Cfun.thy	Fri Jun 03 23:23:55 2005 +0200
@@ -16,69 +16,29 @@
 subsection {* Definition of continuous function type *}
 
 typedef (CFun)  ('a, 'b) "->" (infixr 0) = "{f::'a => 'b. cont f}"
-by (rule exI, rule CfunI)
+by (rule exI, fast intro: cont_const)
 
 syntax
-	Rep_CFun  :: "('a -> 'b) => ('a => 'b)" ("_$_" [999,1000] 999)
-                                                (* application      *)
-        Abs_CFun  :: "('a => 'b) => ('a -> 'b)" (binder "LAM " 10)
-                                                (* abstraction      *)
-        less_cfun :: "[('a -> 'b),('a -> 'b)]=>bool"
+  Rep_CFun :: "('a -> 'b) => ('a => 'b)" ("_$_" [999,1000] 999)
+                                                (* application *)
+  Abs_CFun :: "('a => 'b) => ('a -> 'b)" (binder "LAM " 10)
+                                                (* abstraction *)
 
 syntax (xsymbols)
-  "->"		:: "[type, type] => type"      ("(_ \<rightarrow>/ _)" [1,0]0)
-  "LAM "	:: "[idts, 'a => 'b] => ('a -> 'b)"
+  "->"     :: "[type, type] => type"      ("(_ \<rightarrow>/ _)" [1,0]0)
+  "LAM "   :: "[idts, 'a => 'b] => ('a -> 'b)"
 					("(3\<Lambda>_./ _)" [0, 10] 10)
-  Rep_CFun      :: "('a -> 'b) => ('a => 'b)"  ("(_\<cdot>_)" [999,1000] 999)
+  Rep_CFun :: "('a -> 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999)
 
 syntax (HTML output)
-  Rep_CFun      :: "('a -> 'b) => ('a => 'b)"  ("(_\<cdot>_)" [999,1000] 999)
-
-text {*
-  Derive old type definition rules for @{term Abs_CFun} \& @{term Rep_CFun}.
-  @{term Rep_CFun} and @{term Abs_CFun} should be replaced by
-  @{term Rep_Cfun} and @{term Abs_Cfun} in future.
-*}
-
-lemma Rep_Cfun: "Rep_CFun fo : CFun"
-by (rule Rep_CFun)
-
-lemma Rep_Cfun_inverse: "Abs_CFun (Rep_CFun fo) = fo"
-by (rule Rep_CFun_inverse)
-
-lemma Abs_Cfun_inverse: "f:CFun==>Rep_CFun(Abs_CFun f)=f"
-by (erule Abs_CFun_inverse)
-
-text {* Additional lemma about the isomorphism between
-        @{typ "'a -> 'b"} and @{term Cfun} *}
-
-lemma Abs_Cfun_inverse2: "cont f ==> Rep_CFun (Abs_CFun f) = f"
-apply (rule Abs_Cfun_inverse)
-apply (unfold CFun_def)
-apply (erule mem_Collect_eq [THEN ssubst])
-done
-
-text {* Simplification of application *}
-
-lemma Cfunapp2: "cont f ==> (Abs_CFun f)$x = f x"
-by (erule Abs_Cfun_inverse2 [THEN fun_cong])
-
-text {* Beta - equality for continuous functions *}
-
-lemma beta_cfun: "cont(c1) ==> (LAM x .c1 x)$u = c1 u"
-by (rule Cfunapp2)
-
-text {* Eta - equality for continuous functions *}
-
-lemma eta_cfun: "(LAM x. f$x) = f"
-by (rule Rep_CFun_inverse)
+  Rep_CFun :: "('a -> 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999)
 
 subsection {* Class instances *}
 
 instance "->"  :: (cpo, cpo) sq_ord ..
 
 defs (overloaded)
-  less_cfun_def: "(op <<) == (% fo1 fo2. Rep_CFun fo1 << Rep_CFun fo2 )"
+  less_cfun_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>f g. Rep_CFun f \<sqsubseteq> Rep_CFun g)"
 
 lemma adm_CFun: "adm (\<lambda>f. f \<in> CFun)"
 by (simp add: CFun_def, rule admI, rule cont_lub_fun)
@@ -101,275 +61,190 @@
 lemmas cont_Abs_CFun = 
   typedef_cont_Abs [OF type_definition_CFun less_cfun_def adm_CFun]
 
-lemmas strict_Rep_CFun =
-  typedef_strict_Rep [OF type_definition_CFun less_cfun_def UU_CFun]
+lemmas Rep_CFun_strict =
+  typedef_Rep_strict [OF type_definition_CFun less_cfun_def UU_CFun]
+
+lemmas Abs_CFun_strict =
+  typedef_Abs_strict [OF type_definition_CFun less_cfun_def UU_CFun]
 
-lemmas strict_Abs_CFun =
-  typedef_strict_Abs [OF type_definition_CFun less_cfun_def UU_CFun]
+text {* Additional lemma about the isomorphism between
+        @{typ "'a -> 'b"} and @{term CFun} *}
+
+lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f"
+by (simp add: Abs_CFun_inverse CFun_def)
 
-text {* for compatibility with old HOLCF-Version *}
-lemma inst_cfun_po: "(op <<)=(%f1 f2. Rep_CFun f1 << Rep_CFun f2)"
-apply (fold less_cfun_def)
-apply (rule refl)
-done
+text {* Beta-equality for continuous functions *}
+
+lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
+by (simp add: Abs_CFun_inverse2)
+
+text {* Eta-equality for continuous functions *}
+
+lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
+by (rule Rep_CFun_inverse)
+
+text {* Extensionality for continuous functions *}
+
+lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
+by (simp add: Rep_CFun_inject [symmetric] ext)
 
 text {* lemmas about application of continuous functions *}
 
-lemma cfun_cong: "[| f=g; x=y |] ==> f$x = g$y"
+lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y"
 by simp
 
-lemma cfun_fun_cong: "f=g ==> f$x = g$x"
+lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x"
 by simp
 
-lemma cfun_arg_cong: "x=y ==> f$x = f$y"
+lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
 by simp
 
-text {* access to @{term less_cfun} in class po *}
+subsection {* Continuity of application *}
 
-lemma less_cfun: "( f1 << f2 ) = (Rep_CFun(f1) << Rep_CFun(f2))"
-by (simp add: inst_cfun_po)
-
-subsection {* Type @{typ "'a -> 'b"} is pointed *}
+lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
+by (rule cont_Rep_CFun [THEN cont2cont_CF1L])
 
-lemma minimal_cfun: "Abs_CFun(% x. UU) << f"
-apply (subst less_cfun)
-apply (subst Abs_Cfun_inverse2)
-apply (rule cont_const)
-apply (rule minimal_fun)
-done
-
-lemmas UU_cfun_def = minimal_cfun [THEN minimal2UU, symmetric, standard]
-
-lemma least_cfun: "? x::'a->'b::pcpo.!y. x<<y"
-apply (rule_tac x = "Abs_CFun (% x. UU) " in exI)
-apply (rule minimal_cfun [THEN allI])
+lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
+apply (rule_tac P = "cont" in CollectD)
+apply (fold CFun_def)
+apply (rule Rep_CFun)
 done
 
-subsection {* Monotonicity of application *}
+lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono]
+lemmas contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub]
 
-text {*
-  @{term Rep_CFun} yields continuous functions in @{typ "'a => 'b"}.
-  This is continuity of @{term Rep_CFun} in its 'second' argument:
-  @{prop "cont_Rep_CFun2 ==> monofun_Rep_CFun2 & contlub_Rep_CFun2"}
-*}
+lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard]
+lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard]
+lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
+lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard]
+
+text {* contlub, cont properties of @{term Rep_CFun} in each argument *}
+
+lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(lub (range Y)) = (\<Squnion>i. f\<cdot>(Y i))"
+by (rule contlub_Rep_CFun2 [THEN contlubE])
 
-lemma cont_Rep_CFun2: "cont (Rep_CFun fo)"
-apply (rule_tac P = "cont" in CollectD)
-apply (fold CFun_def)
-apply (rule Rep_Cfun)
-done
+lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(lub (range Y))"
+by (rule cont_Rep_CFun2 [THEN contE])
+
+lemma contlub_cfun_fun: "chain F \<Longrightarrow> lub (range F)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
+by (rule contlub_Rep_CFun1 [THEN contlubE])
 
-lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
- -- {* @{thm monofun_Rep_CFun2} *} (* monofun(Rep_CFun(?fo)) *)
+lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| lub (range F)\<cdot>x"
+by (rule cont_Rep_CFun1 [THEN contE])
 
-lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard]
- -- {* @{thm contlub_Rep_CFun2} *} (* contlub(Rep_CFun(?fo)) *)
+text {* Extensionality wrt. @{term "op <<"} in @{typ "'a -> 'b"} *}
 
-text {* expanded thms @{thm [source] cont_Rep_CFun2}, @{thm [source] contlub_Rep_CFun2} look nice with mixfix syntax *}
+lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
+by (simp add: less_cfun_def less_fun_def)
 
-lemmas cont_cfun_arg = cont_Rep_CFun2 [THEN contE, THEN spec, THEN mp]
-  -- {* @{thm cont_cfun_arg} *} (* chain(x1) ==> range (%i. fo3$(x1 i)) <<| fo3$(lub (range ?x1))    *)
- 
-lemmas contlub_cfun_arg = contlub_Rep_CFun2 [THEN contlubE, THEN spec, THEN mp]
- -- {* @{thm contlub_cfun_arg} *} (* chain(?x1) ==> ?fo4$(lub (range ?x1)) = lub (range (%i. ?fo4$(?x1 i))) *)
+text {* monotonicity of application *}
+
+lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
+by (simp add: less_cfun_def less_fun_def)
 
-text {* @{term Rep_CFun} is monotone in its 'first' argument *}
+lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
+by (rule monofun_Rep_CFun2 [THEN monofunE])
 
-lemma monofun_Rep_CFun1: "monofun(Rep_CFun)"
-by (rule cont_Rep_CFun [THEN cont2mono])
+lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
+by (rule trans_less [OF monofun_cfun_fun monofun_cfun_arg])
 
-text {* monotonicity of application @{term Rep_CFun} in mixfix syntax @{text "[_]_"} *}
+text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
 
-lemma monofun_cfun_fun: "f1 << f2 ==> f1$x << f2$x"
-apply (rule_tac x = "x" in spec)
-apply (rule less_fun [THEN subst])
-apply (erule monofun_Rep_CFun1 [THEN monofunE [rule_format]])
-done
+lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
+by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
+
+lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
+by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
 
-lemmas monofun_cfun_arg = monofun_Rep_CFun2 [THEN monofunE [rule_format], standard]
- -- {* @{thm monofun_cfun_arg} *} (* ?x2 << ?x1 ==> ?fo5$?x2 << ?fo5$?x1 *)
+lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
+by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])
 
-lemma chain_monofun: "chain Y ==> chain (%i. f\<cdot>(Y i))"
+lemma ch2ch_Rep_CFun: "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
 apply (rule chainI)
-apply (rule monofun_cfun_arg)
+apply (rule monofun_cfun)
+apply (erule chainE)
 apply (erule chainE)
 done
 
-text {* monotonicity of @{term Rep_CFun} in both arguments in mixfix syntax @{text "[_]_"} *}
+text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
 
-lemma monofun_cfun: "[|f1<<f2;x1<<x2|] ==> f1$x1 << f2$x2"
-apply (rule trans_less)
-apply (erule monofun_cfun_arg)
-apply (erule monofun_cfun_fun)
+lemma contlub_cfun: 
+  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
+apply (simp only: contlub_cfun_fun)
+apply (simp only: contlub_cfun_arg)
+apply (rule diag_lub)
+apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
+apply (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
 done
 
-lemma strictI: "f$x = UU ==> f$UU = UU"
-apply (rule eq_UU_iff [THEN iffD2])
+lemma cont_cfun: 
+  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
+apply (rule thelubE)
+apply (simp only: ch2ch_Rep_CFun)
+apply (simp only: contlub_cfun)
+done
+
+text {* strictness *}
+
+lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
+apply (rule UU_I)
 apply (erule subst)
 apply (rule minimal [THEN monofun_cfun_arg])
 done
 
-subsection {* Type @{typ "'a -> 'b"} is a cpo *}
-
-text {* ch2ch - rules for the type @{typ "'a -> 'b"} use MF2 lemmas from Cont.thy *}
-
-lemma ch2ch_Rep_CFunR: "chain(Y) ==> chain(%i. f$(Y i))"
-by (erule monofun_Rep_CFun2 [THEN ch2ch_MF2R])
+text {* the lub of a chain of continous functions is monotone *}
 
-lemmas ch2ch_Rep_CFunL = monofun_Rep_CFun1 [THEN ch2ch_MF2L, standard]
- -- {* @{thm ch2ch_Rep_CFunL} *} (* chain(?F) ==> chain (%i. ?F i$?x) *)
-
-text {* the lub of a chain of continous functions is monotone: uses MF2 lemmas from Cont.thy *}
-
-lemma lub_cfun_mono: "chain(F) ==> monofun(% x. lub(range(% j.(F j)$x)))"
-apply (rule lub_MF2_mono)
-apply (rule monofun_Rep_CFun1)
-apply (rule monofun_Rep_CFun2 [THEN allI])
-apply assumption
+lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
+apply (drule ch2ch_monofun [OF monofun_Rep_CFun])
+apply (simp add: thelub_fun [symmetric])
+apply (erule monofun_lub_fun)
+apply (simp add: monofun_Rep_CFun2)
 done
 
 text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"}: uses MF2 lemmas from Cont.thy *}
 
-lemma ex_lubcfun: "[| chain(F); chain(Y) |] ==> 
+lemma ex_lub_cfun: "[| chain(F); chain(Y) |] ==> 
                 lub(range(%j. lub(range(%i. F(j)$(Y i))))) = 
                 lub(range(%i. lub(range(%j. F(j)$(Y i)))))"
-apply (rule ex_lubMF2)
-apply (rule monofun_Rep_CFun1)
-apply (rule monofun_Rep_CFun2 [THEN allI])
-apply assumption
-apply assumption
-done
+by (simp add: diag_lub ch2ch_Rep_CFunL ch2ch_Rep_CFunR)
 
 text {* the lub of a chain of cont. functions is continuous *}
 
-lemma cont_lubcfun: "chain(F) ==> cont(% x. lub(range(% j. F(j)$x)))"
-apply (rule monocontlub2cont)
-apply (erule lub_cfun_mono)
-apply (rule contlubI [rule_format])
-apply (subst contlub_cfun_arg [THEN ext])
-apply assumption
-apply (erule ex_lubcfun)
-apply assumption
+lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
+apply (rule cont2cont_lub)
+apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
+apply (rule cont_Rep_CFun2)
 done
 
 text {* type @{typ "'a -> 'b"} is chain complete *}
 
-lemma lub_cfun: "chain(CCF) ==> range(CCF) <<| (LAM x. lub(range(% i. CCF(i)$x)))"
-apply (rule is_lubI)
-apply (rule ub_rangeI)
-apply (subst less_cfun)
-apply (subst Abs_Cfun_inverse2)
-apply (erule cont_lubcfun)
-apply (rule lub_fun [THEN is_lubD1, THEN ub_rangeD])
-apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
-apply (subst less_cfun)
-apply (subst Abs_Cfun_inverse2)
-apply (erule cont_lubcfun)
-apply (rule lub_fun [THEN is_lub_lub])
-apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
-apply (erule monofun_Rep_CFun1 [THEN ub2ub_monofun])
+lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (LAM x. LUB i. F i$x)"
+apply (subst thelub_fun [symmetric])
+apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
+apply (erule typedef_is_lub [OF type_definition_CFun less_cfun_def adm_CFun])
 done
 
 lemmas thelub_cfun = lub_cfun [THEN thelubI, standard]
- -- {* @{thm thelub_cfun} *} (* 
-chain(?CCF1) ==>  lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i$x)))
-*)
+ -- {* @{thm thelub_cfun} *} (* chain F \<Longrightarrow> lub (range F) = (\<Lambda>x. \<Squnion>i. F i\<cdot>x) *)
 
 subsection {* Miscellaneous *}
 
-text {* Extensionality in @{typ "'a -> 'b"} *}
-
-lemma ext_cfun: "(!!x. f$x = g$x) ==> f = g"
-apply (rule Rep_CFun_inject [THEN iffD1])
-apply (rule ext)
-apply simp
-done
-
 text {* Monotonicity of @{term Abs_CFun} *}
 
 lemma semi_monofun_Abs_CFun: "[| cont(f); cont(g); f<<g|] ==> Abs_CFun(f)<<Abs_CFun(g)"
-by (simp add: less_cfun Abs_Cfun_inverse2)
-
-text {* Extensionality wrt. @{term "op <<"} in @{typ "'a -> 'b"} *}
-
-lemma less_cfun2: "(!!x. f$x << g$x) ==> f << g"
-apply (rule_tac t = "f" in Rep_Cfun_inverse [THEN subst])
-apply (rule_tac t = "g" in Rep_Cfun_inverse [THEN subst])
-apply (rule semi_monofun_Abs_CFun)
-apply (rule cont_Rep_CFun2)
-apply (rule cont_Rep_CFun2)
-apply (rule less_fun [THEN iffD2])
-apply simp
-done
+by (simp add: less_cfun_def Abs_CFun_inverse2)
 
 text {* for compatibility with old HOLCF-Version *}
-lemma inst_cfun_pcpo: "UU = Abs_CFun(%x. UU)"
-apply (simp add: UU_def UU_cfun_def)
-done
+lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
+by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)
 
 subsection {* Continuity of application *}
 
-text {* the contlub property for @{term Rep_CFun} its 'first' argument *}
-
-lemma contlub_Rep_CFun1: "contlub(Rep_CFun)"
-by (rule cont_Rep_CFun [THEN cont2contlub])
-
-text {* the cont property for @{term Rep_CFun} in its first argument *}
-
-lemma cont_Rep_CFun1: "cont(Rep_CFun)"
-by (rule cont_Rep_CFun)
-
-text {* contlub, cont properties of @{term Rep_CFun} in its first argument in mixfix @{text "_[_]"} *}
-
-lemma contlub_cfun_fun: 
-"chain(FY) ==> 
-  lub(range FY)$x = lub(range (%i. FY(i)$x))"
-apply (rule trans)
-apply (erule contlub_Rep_CFun1 [THEN contlubE, THEN spec, THEN mp, THEN fun_cong])
-apply (subst thelub_fun)
-apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
-apply (rule refl)
-done
-
-lemma cont_cfun_fun: 
-"chain(FY) ==> 
-  range(%i. FY(i)$x) <<| lub(range FY)$x"
-apply (rule thelubE)
-apply (erule ch2ch_Rep_CFunL)
-apply (erule contlub_cfun_fun [symmetric])
-done
-
-text {* contlub, cont  properties of @{term Rep_CFun} in both argument in mixfix @{text "_[_]"} *}
-
-lemma contlub_cfun: 
-"[|chain(FY);chain(TY)|] ==> 
-  (lub(range FY))$(lub(range TY)) = lub(range(%i. FY(i)$(TY i)))"
-apply (rule contlub_CF2)
-apply (rule cont_Rep_CFun1)
-apply (rule allI)
-apply (rule cont_Rep_CFun2)
-apply assumption
-apply assumption
-done
-
-lemma cont_cfun: 
-"[|chain(FY);chain(TY)|] ==> 
-  range(%i.(FY i)$(TY i)) <<| (lub (range FY))$(lub(range TY))"
-apply (rule thelubE)
-apply (rule monofun_Rep_CFun1 [THEN ch2ch_MF2LR])
-apply (rule allI)
-apply (rule monofun_Rep_CFun2)
-apply assumption
-apply assumption
-apply (erule contlub_cfun [symmetric])
-apply assumption
-done
-
 text {* cont2cont lemma for @{term Rep_CFun} *}
 
-lemma cont2cont_Rep_CFun: "[|cont(%x. ft x);cont(%x. tt x)|] ==> cont(%x. (ft x)$(tt x))"
-apply (best intro: cont2cont_app2 cont_const cont_Rep_CFun1 cont_Rep_CFun2)
-done
+lemma cont2cont_Rep_CFun:
+  "\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x)\<cdot>(t x))"
+by (best intro: cont2cont_app2 cont_const cont_Rep_CFun cont_Rep_CFun2)
 
 text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
 
@@ -377,10 +252,10 @@
 assumes p1: "!!x. cont(c1 x)"
 assumes p2: "!!y. monofun(%x. c1 x y)"
 shows "monofun(%x. LAM y. c1 x y)"
-apply (rule monofunI [rule_format])
-apply (rule less_cfun2)
-apply (simp add: beta_cfun p1)
-apply (erule p2 [THEN monofunE [rule_format]])
+apply (rule monofunI)
+apply (rule less_cfun_ext)
+apply (simp add: p1)
+apply (erule p2 [THEN monofunE])
 done
 
 text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *}
@@ -394,11 +269,6 @@
 apply (simp add: p2 cont2cont_CF1L_rev)
 done
 
-text {* cont2cont Lemma for @{term "%x. LAM y. c1 x$y"} *}
-
-lemma cont2cont_eta: "cont c1 ==> cont (%x. LAM y. c1 x$y)"
-by (simp only: eta_cfun)
-
 text {* cont2cont tactic *}
 
 lemmas cont_lemmas1 =
@@ -427,19 +297,13 @@
 Addsimprocs [cont_proc];
 *}
 
-text {* HINT: @{text cont_tac} is now installed in simplifier in Lift.ML ! *}
-
 (*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*)
 (*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*)
 
-text {* function application @{text "_[_]"} is strict in its first arguments *}
+text {* function application is strict in its first argument *}
 
-lemma strict_Rep_CFun1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
-by (simp add: inst_cfun_pcpo beta_cfun)
-
-text {* Instantiate the simplifier *}
-
-declare beta_cfun [simp]
+lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
+by (simp add: Rep_CFun_strict)
 
 text {* some lemmata for functions with flat/chfin domain/range types *}
 
@@ -498,7 +362,7 @@
   "\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y
     \<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y"
 apply clarify
-apply (drule_tac fo=g in monofun_cfun_arg)
+apply (drule_tac f=g in monofun_cfun_arg)
 apply (drule ax_flat [rule_format])
 apply (erule disjE)
 apply (simp add: injection_defined_rev)
@@ -585,32 +449,23 @@
 lemma Istrictify2: "x \<noteq> \<bottom> \<Longrightarrow> Istrictify f x = f\<cdot>x"
 by (simp add: Istrictify_def)
 
-lemma monofun_Istrictify1: "monofun (\<lambda>f. Istrictify f x)"
-apply (rule monofunI [rule_format])
-apply (simp add: Istrictify_def monofun_cfun_fun)
+lemma cont_Istrictify1: "cont (\<lambda>f. Istrictify f x)"
+apply (case_tac "x = \<bottom>")
+apply (simp add: Istrictify1)
+apply (simp add: Istrictify2)
 done
 
 lemma monofun_Istrictify2: "monofun (\<lambda>x. Istrictify f x)"
-apply (rule monofunI [rule_format])
+apply (rule monofunI)
 apply (simp add: Istrictify_def monofun_cfun_arg)
 apply clarify
 apply (simp add: eq_UU_iff)
 done
 
-lemma contlub_Istrictify1: "contlub (\<lambda>f. Istrictify f x)"
-apply (rule contlubI [rule_format])
-apply (case_tac "x = \<bottom>")
-apply (simp add: Istrictify1)
-apply (simp add: thelub_const)
-apply (simp add: Istrictify2)
-apply (erule contlub_cfun_fun)
-done
-
 lemma contlub_Istrictify2: "contlub (\<lambda>x. Istrictify f x)"
-apply (rule contlubI [rule_format])
+apply (rule contlubI)
 apply (case_tac "lub (range Y) = \<bottom>")
-apply (simp add: Istrictify1 chain_UU_I)
-apply (simp add: thelub_const)
+apply (simp add: Istrictify1 chain_UU_I thelub_const)
 apply (simp add: Istrictify2)
 apply (simp add: contlub_cfun_arg)
 apply (rule lub_equal2)
@@ -622,9 +477,6 @@
 apply (erule monofun_Istrictify2 [THEN ch2ch_monofun])
 done
 
-lemmas cont_Istrictify1 =
-  monocontlub2cont [OF monofun_Istrictify1 contlub_Istrictify1, standard]
-
 lemmas cont_Istrictify2 =
   monocontlub2cont [OF monofun_Istrictify2 contlub_Istrictify2, standard]