# HG changeset patch
# User huffman
# Date 1118187639 -7200
# Node ID 1ff2965cc2e7f82dc4f2f1cb6785c983551e0c00
# Parent  45b12a01382fef1e4c9babdfae986922b5938aea
major cleanup: rewrote cpo proofs, removed obsolete lemmas, renamed some lemmas

diff -r 45b12a01382f -r 1ff2965cc2e7 src/HOLCF/Up.ML
--- a/src/HOLCF/Up.ML	Wed Jun 08 00:59:46 2005 +0200
+++ b/src/HOLCF/Up.ML	Wed Jun 08 01:40:39 2005 +0200
@@ -4,59 +4,28 @@
 val Iup_def = thm "Iup_def";
 val Ifup_def = thm "Ifup_def";
 val less_up_def = thm "less_up_def";
-val Abs_Up_inverse2 = thm "Abs_Up_inverse2";
-val Exh_Up = thm "Exh_Up";
-val inj_Abs_Up = thm "inj_Abs_Up";
-val inj_Rep_Up = thm "inj_Rep_Up";
-val inject_Iup = thm "inject_Iup";
-val defined_Iup = thm "defined_Iup";
-val upE = thm "upE";
 val Ifup1 = thm "Ifup1";
 val Ifup2 = thm "Ifup2";
-val less_up1a = thm "less_up1a";
-val less_up1b = thm "less_up1b";
-val less_up1c = thm "less_up1c";
 val refl_less_up = thm "refl_less_up";
 val antisym_less_up = thm "antisym_less_up";
 val trans_less_up = thm "trans_less_up";
-val inst_up_po = thm "inst_up_po";
 val minimal_up = thm "minimal_up";
-val UU_up_def = thm "UU_up_def";
 val least_up = thm "least_up";
-val less_up2b = thm "less_up2b";
-val less_up2c = thm "less_up2c";
-val monofun_Iup = thm "monofun_Iup";
-val monofun_Ifup1 = thm "monofun_Ifup1";
 val monofun_Ifup2 = thm "monofun_Ifup2";
 val up_lemma1 = thm "up_lemma1";
-val lub_up1a = thm "lub_up1a";
-val lub_up1b = thm "lub_up1b";
-val thelub_up1a = thm "thelub_up1a";
-val thelub_up1b = thm "thelub_up1b";
 val cpo_up = thm "cpo_up";
 val up_def = thm "up_def";
 val fup_def = thm "fup_def";
 val inst_up_pcpo = thm "inst_up_pcpo";
-val less_up3b = thm "less_up3b";
-val defined_Iup2 = thm "defined_Iup2";
-val contlub_Iup = thm "contlub_Iup";
 val cont_Iup = thm "cont_Iup";
-val contlub_Ifup1 = thm "contlub_Ifup1";
-val contlub_Ifup2 = thm "contlub_Ifup2";
 val cont_Ifup1 = thm "cont_Ifup1";
 val cont_Ifup2 = thm "cont_Ifup2";
 val Exh_Up1 = thm "Exh_Up1";
-val inject_up = thm "inject_up";
-val defined_up = thm "defined_up";
+val up_inject = thm "up_inject";
+val up_eq = thm "up_eq";
+val up_defined = thm "up_defined";
+val up_less = thm "up_less";
 val upE1 = thm "upE1";
 val fup1 = thm "fup1";
 val fup2 = thm "fup2";
-val less_up4b = thm "less_up4b";
-val less_up4c = thm "less_up4c";
-val thelub_up2a = thm "thelub_up2a";
-val thelub_up2b = thm "thelub_up2b";
-val up_lemma2 = thm "up_lemma2";
-val thelub_up2a_rev = thm "thelub_up2a_rev";
-val thelub_up2b_rev = thm "thelub_up2b_rev";
-val thelub_up3 = thm "thelub_up3";
 val fup3 = thm "fup3";
diff -r 45b12a01382f -r 1ff2965cc2e7 src/HOLCF/Up.thy
--- a/src/HOLCF/Up.thy	Wed Jun 08 00:59:46 2005 +0200
+++ b/src/HOLCF/Up.thy	Wed Jun 08 01:40:39 2005 +0200
@@ -15,114 +15,79 @@
 
 subsection {* Definition of new type for lifting *}
 
-typedef (Up) ('a) "u" = "UNIV :: (unit + 'a) set" ..
+typedef (Up) 'a u = "UNIV :: 'a option set" ..
 
 consts
-  Iup         :: "'a => ('a)u"
-  Ifup        :: "('a->'b)=>('a)u => 'b::pcpo"
+  Iup         :: "'a \<Rightarrow> 'a u"
+  Ifup        :: "('a \<rightarrow> 'b::pcpo) \<Rightarrow> 'a u \<Rightarrow> 'b"
 
 defs
-  Iup_def:     "Iup x == Abs_Up(Inr(x))"
-  Ifup_def:    "Ifup(f)(x)== case Rep_Up(x) of Inl(y) => UU | Inr(z) => f$z"
+  Iup_def:     "Iup x \<equiv> Abs_Up (Some x)"
+  Ifup_def:    "Ifup f x \<equiv> case Rep_Up x of None \<Rightarrow> \<bottom> | Some z \<Rightarrow> f\<cdot>z"
 
 lemma Abs_Up_inverse2: "Rep_Up (Abs_Up y) = y"
 by (simp add: Up_def Abs_Up_inverse)
 
-lemma Exh_Up: "z = Abs_Up(Inl ()) | (? x. z = Iup x)"
+lemma Exh_Up: "z = Abs_Up None \<or> (\<exists>x. z = Iup x)"
 apply (unfold Iup_def)
 apply (rule Rep_Up_inverse [THEN subst])
-apply (rule_tac s = "Rep_Up z" in sumE)
-apply (rule disjI1)
-apply (rule_tac f = "Abs_Up" in arg_cong)
-apply (rule unit_eq [THEN subst])
-apply assumption
-apply (rule disjI2)
-apply (rule exI)
-apply (rule_tac f = "Abs_Up" in arg_cong)
-apply assumption
+apply (case_tac "Rep_Up z")
+apply auto
 done
 
-lemma inj_Abs_Up: "inj(Abs_Up)"
+lemma inj_Abs_Up: "inj Abs_Up" (* worthless *)
 apply (rule inj_on_inverseI)
 apply (rule Abs_Up_inverse2)
 done
 
-lemma inj_Rep_Up: "inj(Rep_Up)"
+lemma inj_Rep_Up: "inj Rep_Up" (* worthless *)
 apply (rule inj_on_inverseI)
 apply (rule Rep_Up_inverse)
 done
 
-lemma inject_Iup [dest!]: "Iup x=Iup y ==> x=y"
-apply (unfold Iup_def)
-apply (rule inj_Inr [THEN injD])
-apply (rule inj_Abs_Up [THEN injD])
-apply assumption
-done
+lemma Iup_eq [simp]: "(Iup x = Iup y) = (x = y)"
+by (simp add: Iup_def Abs_Up_inject Up_def)
 
-lemma defined_Iup: "Iup x~=Abs_Up(Inl ())"
-apply (unfold Iup_def)
-apply (rule notI)
-apply (rule notE)
-apply (rule Inl_not_Inr)
-apply (rule sym)
-apply (erule inj_Abs_Up [THEN injD])
-done
+lemma Iup_defined [simp]: "Iup x \<noteq> Abs_Up None"
+by (simp add: Iup_def Abs_Up_inject Up_def)
 
-lemma upE: "[| p=Abs_Up(Inl ()) ==> Q; !!x. p=Iup(x)==>Q|] ==>Q"
-apply (rule Exh_Up [THEN disjE])
-apply fast
-apply (erule exE)
-apply fast
-done
+lemma upE: "\<lbrakk>p = Abs_Up None \<Longrightarrow> Q; \<And>x. p = Iup x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+by (rule Exh_Up [THEN disjE], auto)
 
-lemma Ifup1 [simp]: "Ifup(f)(Abs_Up(Inl ()))=UU"
-apply (unfold Ifup_def)
-apply (subst Abs_Up_inverse2)
-apply (subst sum_case_Inl)
-apply (rule refl)
-done
+lemma Ifup1 [simp]: "Ifup f (Abs_Up None) = \<bottom>"
+by (simp add: Ifup_def Abs_Up_inverse2)
 
-lemma Ifup2 [simp]: "Ifup(f)(Iup(x))=f$x"
-apply (unfold Ifup_def Iup_def)
-apply (subst Abs_Up_inverse2)
-apply (subst sum_case_Inr)
-apply (rule refl)
-done
+lemma Ifup2 [simp]: "Ifup f (Iup x) = f\<cdot>x"
+by (simp add: Ifup_def Iup_def Abs_Up_inverse2)
 
 subsection {* Ordering on type @{typ "'a u"} *}
 
 instance u :: (sq_ord) sq_ord ..
 
 defs (overloaded)
-  less_up_def: "(op <<) == (%x1 x2. case Rep_Up(x1) of                 
-               Inl(y1) => True          
-             | Inr(y2) => (case Rep_Up(x2) of Inl(z1) => False       
-                                            | Inr(z2) => y2<<z2))"
+  less_up_def: "(op \<sqsubseteq>) \<equiv> (\<lambda>x1 x2. case Rep_Up x1 of
+               None \<Rightarrow> True
+             | Some y1 \<Rightarrow> (case Rep_Up x2 of None \<Rightarrow> False
+                                           | Some y2 \<Rightarrow> y1 \<sqsubseteq> y2))"
 
-lemma less_up1a [iff]: 
-        "Abs_Up(Inl ())<< z"
+lemma minimal_up [iff]: "Abs_Up None \<sqsubseteq> z"
 by (simp add: less_up_def Abs_Up_inverse2)
 
-lemma less_up1b [iff]: 
-        "~(Iup x) << (Abs_Up(Inl ()))"
+lemma not_Iup_less [iff]: "\<not> Iup x \<sqsubseteq> Abs_Up None"
 by (simp add: Iup_def less_up_def Abs_Up_inverse2)
 
-lemma less_up1c [iff]: 
-        "(Iup x) << (Iup y)=(x<<y)"
+lemma Iup_less [iff]: "(Iup x \<sqsubseteq> Iup y) = (x \<sqsubseteq> y)"
 by (simp add: Iup_def less_up_def Abs_Up_inverse2)
 
 subsection {* Type @{typ "'a u"} is a partial order *}
 
-lemma refl_less_up: "(p::'a u) << p"
-apply (rule_tac p = "p" in upE)
-apply auto
-done
+lemma refl_less_up: "(p::'a u) \<sqsubseteq> p"
+by (rule_tac p = "p" in upE, auto)
 
-lemma antisym_less_up: "[|(p1::'a u) << p2;p2 << p1|] ==> p1=p2"
+lemma antisym_less_up: "\<lbrakk>(p1::'a u) \<sqsubseteq> p2; p2 \<sqsubseteq> p1\<rbrakk> \<Longrightarrow> p1 = p2"
 apply (rule_tac p = "p1" in upE)
+apply (rule_tac p = "p2" in upE)
 apply simp
-apply (rule_tac p = "p2" in upE)
-apply (erule sym)
 apply simp
 apply (rule_tac p = "p2" in upE)
 apply simp
@@ -131,210 +96,104 @@
 apply simp
 done
 
-lemma trans_less_up: "[|(p1::'a u) << p2;p2 << p3|] ==> p1 << p3"
+lemma trans_less_up: "\<lbrakk>(p1::'a u) \<sqsubseteq> p2; p2 \<sqsubseteq> p3\<rbrakk> \<Longrightarrow> p1 \<sqsubseteq> p3"
 apply (rule_tac p = "p1" in upE)
 apply simp
 apply (rule_tac p = "p2" in upE)
 apply simp
 apply (rule_tac p = "p3" in upE)
-apply auto
-apply (blast intro: trans_less)
+apply simp
+apply (auto elim: trans_less)
 done
 
 instance u :: (cpo) po
 by intro_classes
   (assumption | rule refl_less_up antisym_less_up trans_less_up)+
 
-text {* for compatibility with old HOLCF-Version *}
-lemma inst_up_po: "(op <<)=(%x1 x2. case Rep_Up(x1) of                 
-                Inl(y1) => True  
-              | Inr(y2) => (case Rep_Up(x2) of Inl(z1) => False  
-                                             | Inr(z2) => y2<<z2))"
-apply (fold less_up_def)
-apply (rule refl)
-done
-
-subsection {* Monotonicity of @{term Iup} and @{term Ifup} *}
-
-lemma monofun_Iup: "monofun(Iup)"
-by (simp add: monofun_def)
-
-lemma monofun_Ifup1: "monofun(Ifup)"
-apply (rule monofunI)
-apply (rule less_fun [THEN iffD2, rule_format])
-apply (rule_tac p = "xa" in upE)
-apply simp
-apply simp
-apply (erule monofun_cfun_fun)
-done
-
-lemma monofun_Ifup2: "monofun(Ifup(f))"
-apply (rule monofunI)
-apply (rule_tac p = "x" in upE)
-apply simp
-apply simp
-apply (rule_tac p = "y" in upE)
-apply simp
-apply simp
-apply (erule monofun_cfun_arg)
-done
-
 subsection {* Type @{typ "'a u"} is a cpo *}
 
-text {* Some kind of surjectivity lemma *}
-
-lemma up_lemma1: "z=Iup(x) ==> Iup(Ifup(LAM x. x)(z)) = z"
-by simp
-
-lemma lub_up1a: "[|chain(Y);EX i x. Y(i)=Iup(x)|]  
-      ==> range(Y) <<| Iup(lub(range(%i.(Ifup (LAM x. x) (Y(i))))))"
+lemma is_lub_Iup:
+  "range S <<| x \<Longrightarrow> range (\<lambda>i. Iup (S i)) <<| Iup x"
 apply (rule is_lubI)
 apply (rule ub_rangeI)
-apply (rule_tac p = "Y (i) " in upE)
-apply (rule_tac s = "Abs_Up (Inl ())" and t = "Y (i) " in subst)
-apply (erule sym)
-apply (rule less_up1a)
-apply (rule_tac t = "Y (i) " in up_lemma1 [THEN subst])
-apply assumption
-apply (rule less_up1c [THEN iffD2])
-apply (rule is_ub_thelub)
-apply (erule monofun_Ifup2 [THEN ch2ch_monofun])
-apply (rule_tac p = "u" in upE)
-apply (erule exE)
-apply (erule exE)
-apply (rule_tac P = "Y (i) <<Abs_Up (Inl ())" in notE)
-apply (rule_tac s = "Iup (x) " and t = "Y (i) " in ssubst)
-apply assumption
-apply (rule less_up1b)
-apply (erule subst)
-apply (erule ub_rangeD)
-apply (rule_tac t = "u" in up_lemma1 [THEN subst])
-apply assumption
-apply (rule less_up1c [THEN iffD2])
-apply (rule is_lub_thelub)
-apply (erule monofun_Ifup2 [THEN ch2ch_monofun])
-apply (erule monofun_Ifup2 [THEN ub2ub_monofun])
-done
-
-lemma lub_up1b: "[|chain(Y); ALL i x. Y(i)~=Iup(x)|] ==> range(Y) <<| Abs_Up (Inl ())"
-apply (rule is_lubI)
+apply (subst Iup_less)
+apply (erule is_ub_lub)
+apply (rule_tac p="u" in upE)
+apply (drule ub_rangeD)
+apply simp
+apply simp
+apply (erule is_lub_lub)
 apply (rule ub_rangeI)
-apply (rule_tac p = "Y (i) " in upE)
-apply (rule_tac s = "Abs_Up (Inl ())" and t = "Y (i) " in ssubst)
-apply assumption
-apply (rule refl_less)
+apply (drule_tac i=i in ub_rangeD)
 apply simp
-apply (rule less_up1a)
-done
-
-lemmas thelub_up1a = lub_up1a [THEN thelubI, standard]
-(*
-[| chain ?Y1; EX i x. ?Y1 i = Iup x |] ==>
- lub (range ?Y1) = Iup (lub (range (%i. Iup (LAM x. x) (?Y1 i))))
-*)
-
-lemmas thelub_up1b = lub_up1b [THEN thelubI, standard]
-(*
-[| chain ?Y1; ! i x. ?Y1 i ~= Iup x |] ==>
- lub (range ?Y1) = UU_up
-*)
-
-text {* New versions where @{typ "'a"} does not have to be a pcpo *}
-
-lemma up_lemma1a: "EX x. z=Iup(x) ==> Iup(THE a. Iup a = z) = z"
-apply (erule exE)
-apply (rule theI)
-apply (erule sym)
-apply simp
-apply (erule inject_Iup)
 done
 
 text {* Now some lemmas about chains of @{typ "'a u"} elements *}
 
-lemma up_chain_lemma1:
-  "[| chain Y; EX x. Y j = Iup x |] ==> EX x. Y (i + j) = Iup x"
+lemma up_lemma1: "z \<noteq> Abs_Up None \<Longrightarrow> Iup (THE a. Iup a = z) = z"
+by (rule_tac p="z" in upE, simp_all)
+
+lemma up_lemma2:
+  "\<lbrakk>chain Y; Y j \<noteq> Abs_Up None\<rbrakk> \<Longrightarrow> Y (i + j) \<noteq> Abs_Up None"
+apply (erule contrapos_nn)
 apply (drule_tac x="j" and y="i + j" in chain_mono3)
 apply (rule le_add2)
-apply (rule_tac p="Y (i + j)" in upE)
-apply auto
+apply (rule_tac p="Y j" in upE)
+apply assumption
+apply simp
 done
 
-lemma up_chain_lemma2:
-  "[| chain Y; EX x. Y j = Iup x |] ==>
-    Iup (THE a. Iup a = Y (i + j)) = Y (i + j)"
-apply (drule_tac i=i in up_chain_lemma1)
-apply assumption
-apply (erule up_lemma1a)
-done
+lemma up_lemma3:
+  "\<lbrakk>chain Y; Y j \<noteq> Abs_Up None\<rbrakk> \<Longrightarrow> Iup (THE a. Iup a = Y (i + j)) = Y (i + j)"
+by (rule up_lemma1 [OF up_lemma2])
 
-lemma up_chain_lemma3:
-  "[| chain Y; EX x. Y j = Iup x |] ==> chain (%i. THE a. Iup a = Y (i + j))"
+lemma up_lemma4:
+  "\<lbrakk>chain Y; Y j \<noteq> Abs_Up None\<rbrakk> \<Longrightarrow> chain (\<lambda>i. THE a. Iup a = Y (i + j))"
 apply (rule chainI)
-apply (rule less_up1c [THEN iffD1])
-apply (simp only: up_chain_lemma2)
+apply (rule Iup_less [THEN iffD1])
+apply (subst up_lemma3, assumption+)+
 apply (simp add: chainE)
 done
 
-lemma up_chain_lemma4:
-  "[| chain Y; EX x. Y j = Iup x |] ==>
-    (%i. Y (i + j)) = (%i. Iup (THE a. Iup a = Y (i + j)))"
-apply (rule ext)
-apply (rule up_chain_lemma2 [symmetric])
-apply assumption+
-done
+lemma up_lemma5:
+  "\<lbrakk>chain Y; Y j \<noteq> Abs_Up None\<rbrakk> \<Longrightarrow>
+    (\<lambda>i. Y (i + j)) = (\<lambda>i. Iup (THE a. Iup a = Y (i + j)))"
+by (rule ext, rule up_lemma3 [symmetric])
 
-lemma is_lub_range_shift:
-  "[| chain S; range (%i. S (i + j)) <<| x |] ==> range S <<| x"
-apply (rule is_lubI)
-apply (rule ub_rangeI)
-apply (rule trans_less)
-apply (erule chain_mono3)
-apply (rule le_add1)
-apply (erule is_ub_lub)
-apply (erule is_lub_lub)
-apply (rule ub_rangeI)
-apply (erule ub_rangeD)
+lemma up_lemma6:
+  "\<lbrakk>chain Y; Y j \<noteq> Abs_Up None\<rbrakk>  
+      \<Longrightarrow> range Y <<| Iup (\<Squnion>i. THE a. Iup a = Y(i + j))"
+apply (rule_tac j1="j" in is_lub_range_shift [THEN iffD1])
+apply assumption
+apply (subst up_lemma5, assumption+)
+apply (rule is_lub_Iup)
+apply (rule thelubE [OF _ refl])
+apply (rule up_lemma4, assumption+)
 done
 
-lemma is_lub_Iup:
-  "range S <<| x \<Longrightarrow> range (%i. Iup (S i)) <<| Iup x"
-apply (rule is_lubI)
-apply (rule ub_rangeI)
-apply (subst less_up1c)
-apply (erule is_ub_lub)
-apply (rule_tac p=u in upE)
-apply (drule ub_rangeD)
-apply (simp only: less_up1b)
-apply (simp only: less_up1c)
-apply (erule is_lub_lub)
-apply (rule ub_rangeI)
-apply (drule_tac i=i in ub_rangeD)
-apply (simp only: less_up1c)
+lemma up_chain_cases:
+  "chain Y \<Longrightarrow>
+   (\<exists>A. chain A \<and> lub (range Y) = Iup (lub (range A)) \<and>
+   (\<exists>j. \<forall>i. Y (i + j) = Iup (A i))) \<or> (Y = (\<lambda>i. Abs_Up None))"
+apply (rule disjCI)
+apply (simp add: expand_fun_eq)
+apply (erule exE, rename_tac j)
+apply (rule_tac x="\<lambda>i. THE a. Iup a = Y (i + j)" in exI)
+apply (rule conjI)
+apply (simp add: up_lemma4)
+apply (rule conjI)
+apply (simp add: up_lemma6 [THEN thelubI])
+apply (rule_tac x=j in exI)
+apply (simp add: up_lemma3)
 done
 
-lemma lub_up1c: "[|chain(Y); EX x. Y(j)=Iup(x)|]  
-      ==> range(Y) <<| Iup(lub(range(%i. THE a. Iup a = Y(i + j))))"
-apply (rule_tac j=j in is_lub_range_shift)
-apply assumption
-apply (subst up_chain_lemma4)
-apply assumption+
-apply (rule is_lub_Iup)
-apply (rule thelubE [OF _ refl])
-apply (rule up_chain_lemma3)
-apply assumption+
-done
-
-lemmas thelub_up1c = lub_up1c [THEN thelubI, standard]
-
-lemma cpo_up: "chain(Y::nat=>('a)u) ==> EX x. range(Y) <<|x"
-apply (case_tac "EX i x. Y i = Iup x")
-apply (erule exE)
-apply (rule exI)
-apply (erule lub_up1c)
-apply assumption
-apply (rule exI)
-apply (erule lub_up1b)
-apply simp
+lemma cpo_up: "chain (Y::nat \<Rightarrow> 'a u) \<Longrightarrow> \<exists>x. range Y <<| x"
+apply (frule up_chain_cases, safe)
+apply (rule_tac x="Iup (lub (range A))" in exI)
+apply (erule_tac j1="j" in is_lub_range_shift [THEN iffD1])
+apply (simp add: is_lub_Iup thelubE)
+apply (rule_tac x="Abs_Up None" in exI)
+apply (rule lub_const)
 done
 
 instance u :: (cpo) cpo
@@ -342,13 +201,8 @@
 
 subsection {* Type @{typ "'a u"} is pointed *}
 
-lemma minimal_up: "Abs_Up(Inl ()) << z"
-by (rule less_up1a)
-
-lemmas UU_up_def = minimal_up [THEN minimal2UU, symmetric, standard]
-
-lemma least_up: "EX x::'a u. ALL y. x<<y"
-apply (rule_tac x = "Abs_Up (Inl ())" in exI)
+lemma least_up: "EX x::'a u. ALL y. x\<sqsubseteq>y"
+apply (rule_tac x = "Abs_Up None" in exI)
 apply (rule minimal_up [THEN allI])
 done
 
@@ -356,244 +210,106 @@
 by intro_classes (rule least_up)
 
 text {* for compatibility with old HOLCF-Version *}
-lemma inst_up_pcpo: "UU = Abs_Up(Inl ())"
-by (simp add: UU_def UU_up_def)
+lemma inst_up_pcpo: "\<bottom> = Abs_Up None"
+by (rule minimal_up [THEN UU_I, symmetric])
 
 text {* some lemmas restated for class pcpo *}
 
-lemma less_up3b: "~ Iup(x) << UU"
+lemma less_up3b: "~ Iup(x) \<sqsubseteq> \<bottom>"
 apply (subst inst_up_pcpo)
-apply (rule less_up1b)
+apply simp
 done
 
-lemma defined_Iup2 [iff]: "Iup(x) ~= UU"
+lemma defined_Iup2 [iff]: "Iup(x) ~= \<bottom>"
 apply (subst inst_up_pcpo)
-apply (rule defined_Iup)
+apply (rule Iup_defined)
 done
 
 subsection {* Continuity of @{term Iup} and @{term Ifup} *}
 
 text {* continuity for @{term Iup} *}
 
-lemma cont_Iup [iff]: "cont(Iup)"
+lemma cont_Iup: "cont Iup"
 apply (rule contI)
 apply (rule is_lub_Iup)
 apply (erule thelubE [OF _ refl])
 done
 
-lemmas contlub_Iup = cont_Iup [THEN cont2contlub]
-
 text {* continuity for @{term Ifup} *}
 
-lemma contlub_Ifup1: "contlub(Ifup)"
-apply (rule contlubI)
-apply (rule trans)
-apply (rule_tac [2] thelub_fun [symmetric])
-apply (erule_tac [2] monofun_Ifup1 [THEN ch2ch_monofun])
-apply (rule ext)
-apply (rule_tac p = "x" in upE)
-apply simp
-apply (rule lub_const [THEN thelubI, symmetric])
-apply simp
-apply (erule contlub_cfun_fun)
+lemma cont_Ifup1: "cont (\<lambda>f. Ifup f x)"
+apply (rule contI)
+apply (rule_tac p="x" in upE)
+apply (simp add: lub_const)
+apply (simp add: cont_cfun_fun)
 done
 
-lemma contlub_Ifup2: "contlub(Ifup(f))"
-apply (rule contlubI)
-apply (case_tac "EX i x. Y i = Iup x")
-apply (erule exE)
-apply (subst thelub_up1c)
-apply assumption
-apply assumption
-apply simp
-prefer 2
-apply (subst thelub_up1b)
-apply assumption
-apply simp
-apply simp
-apply (rule chain_UU_I_inverse [symmetric])
-apply (rule allI)
-apply (rule_tac p = "Y(i)" in upE)
-apply simp
+lemma monofun_Ifup2: "monofun (\<lambda>x. Ifup f x)"
+apply (rule monofunI)
+apply (rule_tac p="x" in upE)
 apply simp
-apply (subst contlub_cfun_arg)
-apply  (erule up_chain_lemma3)
-apply  assumption
-apply (rule trans)
-prefer 2
-apply (rule_tac j=i in lub_range_shift)
-apply (erule monofun_Ifup2 [THEN ch2ch_monofun])
-apply (rule lub_equal [rule_format])
-apply (rule chain_monofun)
-apply (erule up_chain_lemma3)
-apply assumption
-apply (rule monofun_Ifup2 [THEN ch2ch_monofun])
-apply (erule chain_shift)
-apply (drule_tac i=k in up_chain_lemma1)
-apply assumption
-apply clarify
+apply (rule_tac p="y" in upE)
 apply simp
-apply (subst the_equality)
-apply (rule refl)
-apply (erule inject_Iup)
-apply (rule refl)
+apply (simp add: monofun_cfun_arg)
 done
 
-lemma cont_Ifup1: "cont(Ifup)"
-apply (rule monocontlub2cont)
-apply (rule monofun_Ifup1)
-apply (rule contlub_Ifup1)
-done
-
-lemma cont_Ifup2: "cont(Ifup(f))"
-apply (rule monocontlub2cont)
-apply (rule monofun_Ifup2)
-apply (rule contlub_Ifup2)
+lemma cont_Ifup2: "cont (\<lambda>x. Ifup f x)"
+apply (rule contI)
+apply (frule up_chain_cases, safe)
+apply (rule_tac j1="j" in is_lub_range_shift [THEN iffD1])
+apply (erule monofun_Ifup2 [THEN ch2ch_monofun])
+apply (simp add: cont_cfun_arg)
+apply (simp add: thelub_const lub_const)
 done
 
 subsection {* Continuous versions of constants *}
 
 constdefs  
-        up  :: "'a -> ('a)u"
-       "up  == (LAM x. Iup(x))"
-        fup :: "('a->'c)-> ('a)u -> 'c::pcpo"
-       "fup == (LAM f p. Ifup(f)(p))"
+  up  :: "'a \<rightarrow> 'a u"
+  "up \<equiv> \<Lambda> x. Iup x"
+
+  fup :: "('a \<rightarrow> 'b::pcpo) \<rightarrow> 'a u \<rightarrow> 'b"
+  "fup \<equiv> \<Lambda> f p. Ifup f p"
 
 translations
-"case l of up$x => t1" == "fup$(LAM x. t1)$l"
+"case l of up\<cdot>x => t1" == "fup\<cdot>(LAM x. t1)\<cdot>l"
 
 text {* continuous versions of lemmas for @{typ "('a)u"} *}
 
-lemma Exh_Up1: "z = UU | (EX x. z = up$x)"
-apply (unfold up_def)
-apply simp
-apply (subst inst_up_pcpo)
-apply (rule Exh_Up)
-done
-
-lemma inject_up: "up$x=up$y ==> x=y"
-apply (unfold up_def)
-apply (rule inject_Iup)
-apply auto
-done
-
-lemma defined_up [simp]: " up$x ~= UU"
-by (simp add: up_def)
-
-lemma upE1: 
-        "[| p=UU ==> Q; !!x. p=up$x==>Q|] ==>Q"
-apply (unfold up_def)
-apply (rule upE)
-apply (simp only: inst_up_pcpo)
-apply fast
-apply simp
-done
-
-lemmas up_conts = cont_lemmas1 cont_Iup cont_Ifup1 cont_Ifup2 cont2cont_CF1L
-
-lemma fup1 [simp]: "fup$f$UU=UU"
-apply (unfold up_def fup_def)
-apply (subst inst_up_pcpo)
-apply (subst beta_cfun)
-apply (intro up_conts)
-apply (subst beta_cfun)
-apply (rule cont_Ifup2)
-apply simp
-done
-
-lemma fup2 [simp]: "fup$f$(up$x)=f$x"
-apply (unfold up_def fup_def)
-apply (simplesubst beta_cfun)
-apply (rule cont_Iup)
-apply (subst beta_cfun)
-apply (intro up_conts)
-apply (subst beta_cfun)
-apply (rule cont_Ifup2)
-apply simp
+lemma Exh_Up1: "z = \<bottom> \<or> (\<exists>x. z = up\<cdot>x)"
+apply (rule_tac p="z" in upE)
+apply (simp add: inst_up_pcpo)
+apply (simp add: up_def cont_Iup)
 done
 
-lemma less_up4b: "~ up$x << UU"
-by (simp add: up_def fup_def less_up3b)
+lemma up_inject: "up\<cdot>x = up\<cdot>y \<Longrightarrow> x = y"
+by (simp add: up_def cont_Iup)
 
-lemma less_up4c: "(up$x << up$y) = (x<<y)"
-by (simp add: up_def fup_def)
+lemma up_eq: "(up\<cdot>x = up\<cdot>y) = (x = y)"
+by (rule iffI, erule up_inject, simp)
+
+lemma up_defined [simp]: " up\<cdot>x \<noteq> \<bottom>"
+by (simp add: up_def cont_Iup inst_up_pcpo)
 
-lemma thelub_up2a: 
-"[| chain(Y); EX i x. Y(i) = up$x |] ==> 
-       lub(range(Y)) = up$(lub(range(%i. fup$(LAM x. x)$(Y i))))"
-apply (unfold up_def fup_def)
-apply (subst beta_cfun)
-apply (rule cont_Iup)
-apply (subst beta_cfun)
-apply (intro up_conts)
-apply (subst beta_cfun [THEN ext])
-apply (rule cont_Ifup2)
-apply (rule thelub_up1a)
-apply assumption
-apply (erule exE)
-apply (erule exE)
-apply (rule exI)
-apply (rule exI)
-apply (erule box_equals)
-apply (rule refl)
-apply simp
-done
+lemma not_up_less_UU [simp]: "\<not> up\<cdot>x \<sqsubseteq> \<bottom>"
+by (simp add: eq_UU_iff [symmetric])
 
-lemma thelub_up2b: 
-"[| chain(Y); ! i x. Y(i) ~= up$x |] ==> lub(range(Y)) = UU"
-apply (unfold up_def fup_def)
-apply (subst inst_up_pcpo)
-apply (erule thelub_up1b)
-apply simp
+lemma up_less: "(up\<cdot>x \<sqsubseteq> up\<cdot>y) = (x \<sqsubseteq> y)"
+by (simp add: up_def cont_Iup)
+
+lemma upE1: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x. p = up\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
+apply (rule_tac p="p" in upE)
+apply (simp add: inst_up_pcpo)
+apply (simp add: up_def cont_Iup)
 done
 
-lemma up_lemma2: "(EX x. z = up$x) = (z~=UU)"
-apply (rule iffI)
-apply (erule exE)
-apply simp
-apply (rule_tac p = "z" in upE1)
-apply simp
-apply (erule exI)
-done
-
-lemma thelub_up2a_rev:
-  "[| chain(Y); lub(range(Y)) = up$x |] ==> EX i x. Y(i) = up$x"
-apply (rule exE)
-apply (rule chain_UU_I_inverse2)
-apply (rule up_lemma2 [THEN iffD1])
-apply (erule exI)
-apply (rule exI)
-apply (rule up_lemma2 [THEN iffD2])
-apply assumption
-done
-
-lemma thelub_up2b_rev:
-  "[| chain(Y); lub(range(Y)) = UU |] ==> ! i x.  Y(i) ~= up$x"
-by (blast dest!: chain_UU_I [THEN spec] exI [THEN up_lemma2 [THEN iffD1]])
+lemma fup1 [simp]: "fup\<cdot>f\<cdot>\<bottom> = \<bottom>"
+by (simp add: fup_def cont_Ifup1 cont_Ifup2 inst_up_pcpo)
 
-lemma thelub_up3: "chain(Y) ==> lub(range(Y)) = UU |  
-                   lub(range(Y)) = up$(lub(range(%i. fup$(LAM x. x)$(Y i))))"
-apply (rule disjE)
-apply (rule_tac [2] disjI1)
-apply (rule_tac [2] thelub_up2b)
-prefer 2 apply (assumption)
-prefer 2 apply (assumption)
-apply (rule_tac [2] disjI2)
-apply (rule_tac [2] thelub_up2a)
-prefer 2 apply (assumption)
-prefer 2 apply (assumption)
-apply fast
-done
+lemma fup2 [simp]: "fup\<cdot>f\<cdot>(up\<cdot>x) = f\<cdot>x"
+by (simp add: up_def fup_def cont_Iup cont_Ifup1 cont_Ifup2 )
 
-lemma fup3: "fup$up$x=x"
-apply (rule_tac p = "x" in upE1)
-apply simp
-apply simp
-done
-
-text {* for backward compatibility *}
-
-lemmas less_up2b = less_up1b
-lemmas less_up2c = less_up1c
+lemma fup3: "fup\<cdot>up\<cdot>x = x"
+by (rule_tac p=x in upE1, simp_all)
 
 end