# HG changeset patch
# User huffman
# Date 1109804282 -3600
# Node ID eb3b1a5c304e65250b58c993801f5ff6d7f37636
# Parent  2454493bd77bc5a0be33f2dab3275c5bf96e3ed5
converted to new-style theory

diff -r 2454493bd77b -r eb3b1a5c304e src/HOLCF/Cfun1.ML
--- a/src/HOLCF/Cfun1.ML	Wed Mar 02 23:28:17 2005 +0100
+++ b/src/HOLCF/Cfun1.ML	Wed Mar 02 23:58:02 2005 +0100
@@ -1,90 +1,16 @@
-(*  Title:      HOLCF/Cfun1.ML
-    ID:         $Id$
-    Author:     Franz Regensburger
 
-The type ->  of continuous functions.
-*)
-
-(* ------------------------------------------------------------------------ *)
-(* derive old type definition rules for Abs_CFun & Rep_CFun                         *)
-(* Rep_CFun and Abs_CFun should be replaced by Rep_Cfun anf Abs_Cfun in future      *)
-(* ------------------------------------------------------------------------ *)
-Goal "Rep_CFun fo : CFun";
-by (rtac Rep_CFun 1);
-qed "Rep_Cfun";
-
-Goal "Abs_CFun (Rep_CFun fo) = fo";
-by (rtac Rep_CFun_inverse 1);
-qed "Rep_Cfun_inverse";
-
-Goal "f:CFun==>Rep_CFun(Abs_CFun f)=f";
-by (etac Abs_CFun_inverse 1);
-qed "Abs_Cfun_inverse";
-
-(* ------------------------------------------------------------------------ *)
-(* less_cfun is a partial order on type 'a -> 'b                            *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [less_cfun_def] "(f::'a->'b) << f";
-by (rtac refl_less 1);
-qed "refl_less_cfun";
-
-Goalw [less_cfun_def] 
-        "[|(f1::'a->'b) << f2; f2 << f1|] ==> f1 = f2";
-by (rtac injD 1);
-by (rtac antisym_less 2);
-by (atac 3);
-by (atac 2);
-by (rtac inj_inverseI 1);
-by (rtac Rep_Cfun_inverse 1);
-qed "antisym_less_cfun";
+(* legacy ML bindings *)
 
-Goalw [less_cfun_def] 
-        "[|(f1::'a->'b) << f2; f2 << f3|] ==> f1 << f3";
-by (etac trans_less 1);
-by (atac 1);
-qed "trans_less_cfun";
-
-(* ------------------------------------------------------------------------ *)
-(* lemmas about application of continuous functions                         *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| f=g; x=y |] ==> f$x = g$y";
-by (Asm_simp_tac 1);
-qed "cfun_cong";
-
-Goal "f=g ==> f$x = g$x";
-by (Asm_simp_tac 1);
-qed "cfun_fun_cong";
-
-Goal "x=y ==> f$x = f$y";
-by (Asm_simp_tac 1);
-qed "cfun_arg_cong";
-
-
-(* ------------------------------------------------------------------------ *)
-(* additional lemma about the isomorphism between -> and Cfun               *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "cont f ==> Rep_CFun (Abs_CFun f) = f";
-by (rtac Abs_Cfun_inverse 1);
-by (rewtac CFun_def);
-by (etac (mem_Collect_eq RS ssubst) 1);
-qed "Abs_Cfun_inverse2";
-
-(* ------------------------------------------------------------------------ *)
-(* simplification of application                                            *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "cont f ==> (Abs_CFun f)$x = f x";
-by (etac (Abs_Cfun_inverse2 RS fun_cong) 1);
-qed "Cfunapp2";
-
-(* ------------------------------------------------------------------------ *)
-(* beta - equality for continuous functions                                 *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "cont(c1) ==> (LAM x .c1 x)$u = c1 u";
-by (rtac Cfunapp2 1);
-by (atac 1);
-qed "beta_cfun";
+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 refl_less_cfun = thm "refl_less_cfun";
+val antisym_less_cfun = thm "antisym_less_cfun";
+val trans_less_cfun = thm "trans_less_cfun";
+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 beta_cfun = thm "beta_cfun";
diff -r 2454493bd77b -r eb3b1a5c304e src/HOLCF/Cfun1.thy
--- a/src/HOLCF/Cfun1.thy	Wed Mar 02 23:28:17 2005 +0100
+++ b/src/HOLCF/Cfun1.thy	Wed Mar 02 23:58:02 2005 +0100
@@ -1,19 +1,21 @@
 (*  Title:      HOLCF/Cfun1.thy
     ID:         $Id$
     Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
 
 Definition of the type ->  of continuous functions.
 
 *)
 
-Cfun1 = Cont +
+theory Cfun1 = Cont:
 
-default cpo
+defaultsort cpo
 
-typedef (CFun)  ('a, 'b) "->" (infixr 0) = "{f::'a => 'b. cont f}" (CfunI)
+typedef (CFun)  ('a, 'b) "->" (infixr 0) = "{f::'a => 'b. cont f}"
+by (rule exI, rule CfunI)
 
 (* to make << defineable *)
-instance "->"  :: (cpo,cpo)sq_ord
+instance "->"  :: (cpo,cpo)sq_ord ..
 
 syntax
 	Rep_CFun  :: "('a -> 'b) => ('a => 'b)" ("_$_" [999,1000] 999)
@@ -23,15 +25,114 @@
         less_cfun :: "[('a -> 'b),('a -> 'b)]=>bool"
 
 syntax (xsymbols)
-  "->"		:: [type, type] => type	("(_ \\<rightarrow>/ _)" [1,0]0)
+  "->"		:: "[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)
+					("(3\<Lambda>_./ _)" [0, 10] 10)
+  Rep_CFun      :: "('a -> 'b) => ('a => 'b)"  ("(_\<cdot>_)" [999,1000] 999)
 
 syntax (HTML output)
-  Rep_CFun      :: "('a -> 'b) => ('a => 'b)"  ("(_\\<cdot>_)" [999,1000] 999)
+  Rep_CFun      :: "('a -> 'b) => ('a => 'b)"  ("(_\<cdot>_)" [999,1000] 999)
+
+defs (overloaded)
+  less_cfun_def: "(op <<) == (% fo1 fo2. Rep_CFun fo1 << Rep_CFun fo2 )"
+
+(*  Title:      HOLCF/Cfun1.ML
+    ID:         $Id$
+    Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
+
+The type ->  of continuous functions.
+*)
+
+(* ------------------------------------------------------------------------ *)
+(* derive old type definition rules for Abs_CFun & Rep_CFun
+    *)
+(* Rep_CFun and Abs_CFun should be replaced by Rep_Cfun anf Abs_Cfun in future
+    *)
+(* ------------------------------------------------------------------------ *)
+
+lemma Rep_Cfun: "Rep_CFun fo : CFun"
+apply (rule Rep_CFun)
+done
+
+lemma Rep_Cfun_inverse: "Abs_CFun (Rep_CFun fo) = fo"
+apply (rule Rep_CFun_inverse)
+done
+
+lemma Abs_Cfun_inverse: "f:CFun==>Rep_CFun(Abs_CFun f)=f"
+apply (erule Abs_CFun_inverse)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* less_cfun is a partial order on type 'a -> 'b                            *)
+(* ------------------------------------------------------------------------ *)
+
+lemma refl_less_cfun: "(f::'a->'b) << f"
+
+apply (unfold less_cfun_def)
+apply (rule refl_less)
+done
 
-defs 
-  less_cfun_def "(op <<) == (% fo1 fo2. Rep_CFun fo1 << Rep_CFun fo2 )"
+lemma antisym_less_cfun: 
+        "[|(f1::'a->'b) << f2; f2 << f1|] ==> f1 = f2"
+apply (unfold less_cfun_def)
+apply (rule injD)
+apply (rule_tac [2] antisym_less)
+prefer 3 apply (assumption)
+prefer 2 apply (assumption)
+apply (rule inj_on_inverseI)
+apply (rule Rep_Cfun_inverse)
+done
+
+lemma trans_less_cfun: 
+        "[|(f1::'a->'b) << f2; f2 << f3|] ==> f1 << f3"
+apply (unfold less_cfun_def)
+apply (erule trans_less)
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* lemmas about application of continuous functions                         *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cfun_cong: "[| f=g; x=y |] ==> f$x = g$y"
+apply (simp (no_asm_simp))
+done
+
+lemma cfun_fun_cong: "f=g ==> f$x = g$x"
+apply (simp (no_asm_simp))
+done
+
+lemma cfun_arg_cong: "x=y ==> f$x = f$y"
+apply (simp (no_asm_simp))
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* additional lemma about the isomorphism between -> and 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
+
+(* ------------------------------------------------------------------------ *)
+(* simplification of application                                            *)
+(* ------------------------------------------------------------------------ *)
+
+lemma Cfunapp2: "cont f ==> (Abs_CFun f)$x = f x"
+apply (erule Abs_Cfun_inverse2 [THEN fun_cong])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* beta - equality for continuous functions                                 *)
+(* ------------------------------------------------------------------------ *)
+
+lemma beta_cfun: "cont(c1) ==> (LAM x .c1 x)$u = c1 u"
+apply (rule Cfunapp2)
+apply assumption
+done
 
 end
diff -r 2454493bd77b -r eb3b1a5c304e src/HOLCF/Cfun2.ML
--- a/src/HOLCF/Cfun2.ML	Wed Mar 02 23:28:17 2005 +0100
+++ b/src/HOLCF/Cfun2.ML	Wed Mar 02 23:58:02 2005 +0100
@@ -1,247 +1,30 @@
-(*  Title:      HOLCF/Cfun2
-    ID:         $Id$
-    Author:     Franz Regensburger
 
-Class Instance ->::(cpo,cpo)po
-*)
-
-(* for compatibility with old HOLCF-Version *)
-Goal "(op <<)=(%f1 f2. Rep_CFun f1 << Rep_CFun f2)";
-by (fold_goals_tac [less_cfun_def]);
-by (rtac refl 1);
-qed "inst_cfun_po";
-
-(* ------------------------------------------------------------------------ *)
-(* access to less_cfun in class po                                          *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "( f1 << f2 ) = (Rep_CFun(f1) << Rep_CFun(f2))";
-by (simp_tac (simpset() addsimps [inst_cfun_po]) 1);
-qed "less_cfun";
-
-(* ------------------------------------------------------------------------ *)
-(* Type 'a ->'b  is pointed                                                 *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "Abs_CFun(% x. UU) << f";
-by (stac less_cfun 1);
-by (stac Abs_Cfun_inverse2 1);
-by (rtac cont_const 1);
-by (rtac minimal_fun 1);
-qed "minimal_cfun";
-
-bind_thm ("UU_cfun_def",minimal_cfun RS minimal2UU RS sym);
-
-Goal "? x::'a->'b::pcpo.!y. x<<y";
-by (res_inst_tac [("x","Abs_CFun(% x. UU)")] exI 1);
-by (rtac (minimal_cfun RS allI) 1);
-qed "least_cfun";
-
-(* ------------------------------------------------------------------------ *)
-(* Rep_CFun yields continuous functions in 'a => 'b                             *)
-(* this is continuity of Rep_CFun in its 'second' argument                      *)
-(* cont_Rep_CFun2 ==> monofun_Rep_CFun2 & contlub_Rep_CFun2                            *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "cont(Rep_CFun(fo))";
-by (res_inst_tac [("P","cont")] CollectD 1);
-by (fold_goals_tac [CFun_def]);
-by (rtac Rep_Cfun 1);
-qed "cont_Rep_CFun2";
-
-bind_thm ("monofun_Rep_CFun2", cont_Rep_CFun2 RS cont2mono);
-(* monofun(Rep_CFun(?fo1)) *)
-
-
-bind_thm ("contlub_Rep_CFun2", cont_Rep_CFun2 RS cont2contlub);
-(* contlub(Rep_CFun(?fo1)) *)
-
-(* ------------------------------------------------------------------------ *)
-(* expanded thms cont_Rep_CFun2, contlub_Rep_CFun2                                 *)
-(* looks nice with mixfix syntac                                            *)
-(* ------------------------------------------------------------------------ *)
-
-bind_thm ("cont_cfun_arg", (cont_Rep_CFun2 RS contE RS spec RS mp));
-(* chain(?x1) ==> range (%i. ?fo3$(?x1 i)) <<| ?fo3$(lub (range ?x1))    *)
- 
-bind_thm ("contlub_cfun_arg", (contlub_Rep_CFun2 RS contlubE RS spec RS mp));
-(* chain(?x1) ==> ?fo4$(lub (range ?x1)) = lub (range (%i. ?fo4$(?x1 i))) *)
-
-
-(* ------------------------------------------------------------------------ *)
-(* Rep_CFun is monotone in its 'first' argument                                 *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [monofun] "monofun(Rep_CFun)";
-by (strip_tac 1);
-by (etac (less_cfun RS subst) 1);
-qed "monofun_Rep_CFun1";
-
-
-(* ------------------------------------------------------------------------ *)
-(* monotonicity of application Rep_CFun in mixfix syntax [_]_                   *)
-(* ------------------------------------------------------------------------ *)
-
-Goal  "f1 << f2 ==> f1$x << f2$x";
-by (res_inst_tac [("x","x")] spec 1);
-by (rtac (less_fun RS subst) 1);
-by (etac (monofun_Rep_CFun1 RS monofunE RS spec RS spec RS mp) 1);
-qed "monofun_cfun_fun";
-
-
-bind_thm ("monofun_cfun_arg", monofun_Rep_CFun2 RS monofunE RS spec RS spec RS mp);
-(* ?x2 << ?x1 ==> ?fo5$?x2 << ?fo5$?x1                                      *)
-
-Goal "chain Y ==> chain (%i. f\\<cdot>(Y i))";
-by (rtac chainI 1);
-by (rtac monofun_cfun_arg 1);
-by (etac chainE 1);
-qed "chain_monofun";
-
-
-(* ------------------------------------------------------------------------ *)
-(* monotonicity of Rep_CFun in both arguments in mixfix syntax [_]_             *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|f1<<f2;x1<<x2|] ==> f1$x1 << f2$x2";
-by (rtac trans_less 1);
-by (etac monofun_cfun_arg 1);
-by (etac monofun_cfun_fun 1);
-qed "monofun_cfun";
-
-
-Goal "f$x = UU ==> f$UU = UU";
-by (rtac (eq_UU_iff RS iffD2) 1);
-by (etac subst 1);
-by (rtac (minimal RS monofun_cfun_arg) 1);
-qed "strictI";
-
-
-(* ------------------------------------------------------------------------ *)
-(* ch2ch - rules for the type 'a -> 'b                                      *)
-(* use MF2 lemmas from Cont.ML                                              *)
-(* ------------------------------------------------------------------------ *)
+(* legacy ML bindings *)
 
-Goal "chain(Y) ==> chain(%i. f$(Y i))";
-by (etac (monofun_Rep_CFun2 RS ch2ch_MF2R) 1);
-qed "ch2ch_Rep_CFunR";
-
-
-bind_thm ("ch2ch_Rep_CFunL", monofun_Rep_CFun1 RS ch2ch_MF2L);
-(* chain(?F) ==> chain (%i. ?F i$?x)                                  *)
-
-
-(* ------------------------------------------------------------------------ *)
-(*  the lub of a chain of continous functions is monotone                   *)
-(* use MF2 lemmas from Cont.ML                                              *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "chain(F) ==> monofun(% x. lub(range(% j.(F j)$x)))";
-by (rtac lub_MF2_mono 1);
-by (rtac monofun_Rep_CFun1 1);
-by (rtac (monofun_Rep_CFun2 RS allI) 1);
-by (atac 1);
-qed "lub_cfun_mono";
-
-(* ------------------------------------------------------------------------ *)
-(* a lemma about the exchange of lubs for type 'a -> 'b                     *)
-(* use MF2 lemmas from Cont.ML                                              *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| chain(F); chain(Y) |] ==>\
-\               lub(range(%j. lub(range(%i. F(j)$(Y i))))) =\
-\               lub(range(%i. lub(range(%j. F(j)$(Y i)))))";
-by (rtac ex_lubMF2 1);
-by (rtac monofun_Rep_CFun1 1);
-by (rtac (monofun_Rep_CFun2 RS allI) 1);
-by (atac 1);
-by (atac 1);
-qed "ex_lubcfun";
-
-(* ------------------------------------------------------------------------ *)
-(* the lub of a chain of cont. functions is continuous                      *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "chain(F) ==> cont(% x. lub(range(% j. F(j)$x)))";
-by (rtac monocontlub2cont 1);
-by (etac lub_cfun_mono 1);
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (stac (contlub_cfun_arg RS ext) 1);
-by (atac 1);
-by (etac ex_lubcfun 1);
-by (atac 1);
-qed "cont_lubcfun";
-
-(* ------------------------------------------------------------------------ *)
-(* type 'a -> 'b is chain complete                                          *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "chain(CCF) ==> range(CCF) <<| (LAM x. lub(range(% i. CCF(i)$x)))";
-by (rtac is_lubI 1);
-by (rtac ub_rangeI 1);
-by (stac less_cfun 1);
-by (stac Abs_Cfun_inverse2 1);
-by (etac cont_lubcfun 1);
-by (rtac (lub_fun RS is_lubD1 RS ub_rangeD) 1);
-by (etac (monofun_Rep_CFun1 RS ch2ch_monofun) 1);
-by (stac less_cfun 1);
-by (stac Abs_Cfun_inverse2 1);
-by (etac cont_lubcfun 1);
-by (rtac (lub_fun RS is_lub_lub) 1);
-by (etac (monofun_Rep_CFun1 RS ch2ch_monofun) 1);
-by (etac (monofun_Rep_CFun1 RS ub2ub_monofun) 1);
-qed "lub_cfun";
-
-bind_thm ("thelub_cfun", lub_cfun RS thelubI);
-(* 
-chain(?CCF1) ==>  lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i$x)))
-*)
-
-Goal "chain(CCF::nat=>('a->'b)) ==> ? x. range(CCF) <<| x";
-by (rtac exI 1);
-by (etac lub_cfun 1);
-qed "cpo_cfun";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Extensionality in 'a -> 'b                                               *)
-(* ------------------------------------------------------------------------ *)
-
-val prems = Goal "(!!x. f$x = g$x) ==> f = g";
-by (res_inst_tac [("t","f")] (Rep_Cfun_inverse RS subst) 1);
-by (res_inst_tac [("t","g")] (Rep_Cfun_inverse RS subst) 1);
-by (res_inst_tac [("f","Abs_CFun")] arg_cong 1);
-by (rtac ext 1);
-by (resolve_tac prems 1);
-qed "ext_cfun";
-
-(* ------------------------------------------------------------------------ *)
-(* Monotonicity of Abs_CFun                                                     *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[| cont(f); cont(g); f<<g|] ==> Abs_CFun(f)<<Abs_CFun(g)";
-by (rtac (less_cfun RS iffD2) 1);
-by (stac Abs_Cfun_inverse2 1);
-by (assume_tac 1);
-by (stac Abs_Cfun_inverse2 1);
-by (assume_tac 1);
-by (assume_tac 1);
-qed "semi_monofun_Abs_CFun";
-
-(* ------------------------------------------------------------------------ *)
-(* Extenionality wrt. << in 'a -> 'b                                        *)
-(* ------------------------------------------------------------------------ *)
-
-val prems = Goal "(!!x. f$x << g$x) ==> f << g";
-by (res_inst_tac [("t","f")] (Rep_Cfun_inverse RS subst) 1);
-by (res_inst_tac [("t","g")] (Rep_Cfun_inverse RS subst) 1);
-by (rtac semi_monofun_Abs_CFun 1);
-by (rtac cont_Rep_CFun2 1);
-by (rtac cont_Rep_CFun2 1);
-by (rtac (less_fun RS iffD2) 1);
-by (rtac allI 1);
-by (resolve_tac prems 1);
-qed "less_cfun2";
-
-
+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";
+val cont_cfun_arg = thm "cont_cfun_arg";
+val contlub_cfun_arg = thm "contlub_cfun_arg";
+val monofun_Rep_CFun1 = thm "monofun_Rep_CFun1";
+val monofun_cfun_fun = thm "monofun_cfun_fun";
+val monofun_cfun_arg = thm "monofun_cfun_arg";
+val chain_monofun = thm "chain_monofun";
+val monofun_cfun = thm "monofun_cfun";
+val strictI = thm "strictI";
+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 lub_cfun = thm "lub_cfun";
+val thelub_cfun = thm "thelub_cfun";
+val cpo_cfun = thm "cpo_cfun";
+val ext_cfun = thm "ext_cfun";
+val semi_monofun_Abs_CFun = thm "semi_monofun_Abs_CFun";
+val less_cfun2 = thm "less_cfun2";
diff -r 2454493bd77b -r eb3b1a5c304e src/HOLCF/Cfun2.thy
--- a/src/HOLCF/Cfun2.thy	Wed Mar 02 23:28:17 2005 +0100
+++ b/src/HOLCF/Cfun2.thy	Wed Mar 02 23:58:02 2005 +0100
@@ -1,13 +1,267 @@
 (*  Title:      HOLCF/Cfun2.thy
     ID:         $Id$
     Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
 
 Class Instance ->::(cpo,cpo)po
 
 *)
 
-Cfun2 = Cfun1 + 
+theory Cfun2 = Cfun1:
+
+instance "->"::(cpo,cpo)po
+apply (intro_classes)
+apply (rule refl_less_cfun)
+apply (rule antisym_less_cfun, assumption+)
+apply (rule trans_less_cfun, assumption+)
+done
+
+(*  Title:      HOLCF/Cfun2
+    ID:         $Id$
+    Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
+
+Class Instance ->::(cpo,cpo)po
+*)
+
+(* 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
+
+(* ------------------------------------------------------------------------ *)
+(* access to less_cfun in class po                                          *)
+(* ------------------------------------------------------------------------ *)
+
+lemma less_cfun: "( f1 << f2 ) = (Rep_CFun(f1) << Rep_CFun(f2))"
+apply (simp (no_asm) add: inst_cfun_po)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Type 'a ->'b  is pointed                                                 *)
+(* ------------------------------------------------------------------------ *)
+
+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])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Rep_CFun yields continuous functions in 'a => 'b                             *)
+(* this is continuity of Rep_CFun in its 'second' argument                      *)
+(* cont_Rep_CFun2 ==> monofun_Rep_CFun2 & contlub_Rep_CFun2                            *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_Rep_CFun2: "cont(Rep_CFun(fo))"
+apply (rule_tac P = "cont" in CollectD)
+apply (fold CFun_def)
+apply (rule Rep_Cfun)
+done
+
+lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
+(* monofun(Rep_CFun(?fo1)) *)
+
+
+lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard]
+(* contlub(Rep_CFun(?fo1)) *)
+
+(* ------------------------------------------------------------------------ *)
+(* expanded thms cont_Rep_CFun2, contlub_Rep_CFun2                                 *)
+(* looks nice with mixfix syntac                                            *)
+(* ------------------------------------------------------------------------ *)
+
+lemmas cont_cfun_arg = cont_Rep_CFun2 [THEN contE, THEN spec, THEN mp]
+(* chain(?x1) ==> range (%i. ?fo3$(?x1 i)) <<| ?fo3$(lub (range ?x1))    *)
+ 
+lemmas contlub_cfun_arg = contlub_Rep_CFun2 [THEN contlubE, THEN spec, THEN mp]
+(* chain(?x1) ==> ?fo4$(lub (range ?x1)) = lub (range (%i. ?fo4$(?x1 i))) *)
+
+
+(* ------------------------------------------------------------------------ *)
+(* Rep_CFun is monotone in its 'first' argument                                 *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monofun_Rep_CFun1: "monofun(Rep_CFun)"
+apply (unfold monofun)
+apply (intro strip)
+apply (erule less_cfun [THEN subst])
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* monotonicity of application Rep_CFun in mixfix syntax [_]_                   *)
+(* ------------------------------------------------------------------------ *)
+
+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, THEN spec, THEN spec, THEN mp])
+done
+
+
+lemmas monofun_cfun_arg = monofun_Rep_CFun2 [THEN monofunE, THEN spec, THEN spec, THEN mp, standard]
+(* ?x2 << ?x1 ==> ?fo5$?x2 << ?fo5$?x1                                      *)
+
+lemma chain_monofun: "chain Y ==> chain (%i. f\<cdot>(Y i))"
+apply (rule chainI)
+apply (rule monofun_cfun_arg)
+apply (erule chainE)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* monotonicity of Rep_CFun in both arguments in mixfix syntax [_]_             *)
+(* ------------------------------------------------------------------------ *)
+
+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)
+done
+
 
-instance "->"::(cpo,cpo)po (refl_less_cfun,antisym_less_cfun,trans_less_cfun)
+lemma strictI: "f$x = UU ==> f$UU = UU"
+apply (rule eq_UU_iff [THEN iffD2])
+apply (erule subst)
+apply (rule minimal [THEN monofun_cfun_arg])
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* ch2ch - rules for the type 'a -> 'b                                      *)
+(* use MF2 lemmas from Cont.ML                                              *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ch2ch_Rep_CFunR: "chain(Y) ==> chain(%i. f$(Y i))"
+apply (erule monofun_Rep_CFun2 [THEN ch2ch_MF2R])
+done
+
+
+lemmas ch2ch_Rep_CFunL = monofun_Rep_CFun1 [THEN ch2ch_MF2L, standard]
+(* chain(?F) ==> chain (%i. ?F i$?x)                                  *)
+
+
+(* ------------------------------------------------------------------------ *)
+(*  the lub of a chain of continous functions is monotone                   *)
+(* use MF2 lemmas from Cont.ML                                              *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_cfun_mono: "chain(F) ==> monofun(% x. lub(range(% j.(F j)$x)))"
+apply (rule lub_MF2_mono)
+apply (rule monofun_Rep_CFun1)
+apply (rule monofun_Rep_CFun2 [THEN allI])
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* a lemma about the exchange of lubs for type 'a -> 'b                     *)
+(* use MF2 lemmas from Cont.ML                                              *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ex_lubcfun: "[| chain(F); chain(Y) |] ==> 
+                lub(range(%j. lub(range(%i. F(j)$(Y i))))) = 
+                lub(range(%i. lub(range(%j. F(j)$(Y i)))))"
+apply (rule ex_lubMF2)
+apply (rule monofun_Rep_CFun1)
+apply (rule monofun_Rep_CFun2 [THEN allI])
+apply assumption
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the lub of a chain of cont. functions is continuous                      *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_lubcfun: "chain(F) ==> cont(% x. lub(range(% j. F(j)$x)))"
+apply (rule monocontlub2cont)
+apply (erule lub_cfun_mono)
+apply (rule contlubI)
+apply (intro strip)
+apply (subst contlub_cfun_arg [THEN ext])
+apply assumption
+apply (erule ex_lubcfun)
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* type 'a -> 'b is chain complete                                          *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_cfun: "chain(CCF) ==> range(CCF) <<| (LAM x. lub(range(% i. CCF(i)$x)))"
+apply (rule is_lubI)
+apply (rule ub_rangeI)
+apply (subst less_cfun)
+apply (subst Abs_Cfun_inverse2)
+apply (erule cont_lubcfun)
+apply (rule lub_fun [THEN is_lubD1, THEN ub_rangeD])
+apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
+apply (subst less_cfun)
+apply (subst Abs_Cfun_inverse2)
+apply (erule cont_lubcfun)
+apply (rule lub_fun [THEN is_lub_lub])
+apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
+apply (erule monofun_Rep_CFun1 [THEN ub2ub_monofun])
+done
+
+lemmas thelub_cfun = lub_cfun [THEN thelubI, standard]
+(* 
+chain(?CCF1) ==>  lub (range ?CCF1) = (LAM x. lub (range (%i. ?CCF1 i$x)))
+*)
+
+lemma cpo_cfun: "chain(CCF::nat=>('a->'b)) ==> ? x. range(CCF) <<| x"
+apply (rule exI)
+apply (erule lub_cfun)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* Extensionality in 'a -> 'b                                               *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ext_cfun: "(!!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_tac f = "Abs_CFun" in arg_cong)
+apply (rule ext)
+apply simp
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Monotonicity of Abs_CFun                                                     *)
+(* ------------------------------------------------------------------------ *)
+
+lemma semi_monofun_Abs_CFun: "[| cont(f); cont(g); f<<g|] ==> Abs_CFun(f)<<Abs_CFun(g)"
+apply (rule less_cfun [THEN iffD2])
+apply (subst Abs_Cfun_inverse2)
+apply assumption
+apply (subst Abs_Cfun_inverse2)
+apply assumption
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Extenionality wrt. << in '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 (rule allI)
+apply simp
+done
 
 end
diff -r 2454493bd77b -r eb3b1a5c304e src/HOLCF/Cfun3.ML
--- a/src/HOLCF/Cfun3.ML	Wed Mar 02 23:28:17 2005 +0100
+++ b/src/HOLCF/Cfun3.ML	Wed Mar 02 23:58:02 2005 +0100
@@ -1,498 +1,52 @@
-(*  Title:      HOLCF/Cfun3
-    ID:         $Id$
-    Author:     Franz Regensburger
 
-Class instance of  -> for class pcpo
-*)
-
-(* for compatibility with old HOLCF-Version *)
-Goal "UU = Abs_CFun(%x. UU)";
-by (simp_tac (HOL_ss addsimps [UU_def,UU_cfun_def]) 1);
-qed "inst_cfun_pcpo";
-
-(* ------------------------------------------------------------------------ *)
-(* the contlub property for Rep_CFun its 'first' argument                       *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "contlub(Rep_CFun)";
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (rtac (expand_fun_eq RS iffD2) 1);
-by (strip_tac 1);
-by (stac thelub_cfun 1);
-by (atac 1);
-by (stac Cfunapp2 1);
-by (etac cont_lubcfun 1);
-by (stac thelub_fun 1);
-by (etac (monofun_Rep_CFun1 RS ch2ch_monofun) 1);
-by (rtac refl 1);
-qed "contlub_Rep_CFun1";
-
-
-(* ------------------------------------------------------------------------ *)
-(* the cont property for Rep_CFun in its first argument                        *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "cont(Rep_CFun)";
-by (rtac monocontlub2cont 1);
-by (rtac monofun_Rep_CFun1 1);
-by (rtac contlub_Rep_CFun1 1);
-qed "cont_Rep_CFun1";
-
-
-(* ------------------------------------------------------------------------ *)
-(* contlub, cont properties of Rep_CFun in its first argument in mixfix _[_]   *)
-(* ------------------------------------------------------------------------ *)
-
-Goal 
-"chain(FY) ==>\
-\ lub(range FY)$x = lub(range (%i. FY(i)$x))";
-by (rtac trans 1);
-by (etac (contlub_Rep_CFun1 RS contlubE RS spec RS mp RS fun_cong) 1);
-by (stac thelub_fun 1);
-by (etac (monofun_Rep_CFun1 RS ch2ch_monofun) 1);
-by (rtac refl 1);
-qed "contlub_cfun_fun";
-
-
-Goal 
-"chain(FY) ==>\
-\ range(%i. FY(i)$x) <<| lub(range FY)$x";
-by (rtac thelubE 1);
-by (etac ch2ch_Rep_CFunL 1);
-by (etac (contlub_cfun_fun RS sym) 1);
-qed "cont_cfun_fun";
-
-
-(* ------------------------------------------------------------------------ *)
-(* contlub, cont  properties of Rep_CFun in both argument in mixfix _[_]       *)
-(* ------------------------------------------------------------------------ *)
-
-Goal 
-"[|chain(FY);chain(TY)|] ==>\
-\ (lub(range FY))$(lub(range TY)) = lub(range(%i. FY(i)$(TY i)))";
-by (rtac contlub_CF2 1);
-by (rtac cont_Rep_CFun1 1);
-by (rtac allI 1);
-by (rtac cont_Rep_CFun2 1);
-by (atac 1);
-by (atac 1);
-qed "contlub_cfun";
-
-Goal 
-"[|chain(FY);chain(TY)|] ==>\
-\ range(%i.(FY i)$(TY i)) <<| (lub (range FY))$(lub(range TY))";
-by (rtac thelubE 1);
-by (rtac (monofun_Rep_CFun1 RS ch2ch_MF2LR) 1);
-by (rtac allI 1);
-by (rtac monofun_Rep_CFun2 1);
-by (atac 1);
-by (atac 1);
-by (etac (contlub_cfun RS sym) 1);
-by (atac 1);
-qed "cont_cfun";
-
-
-(* ------------------------------------------------------------------------ *)
-(* cont2cont lemma for Rep_CFun                                               *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "[|cont(%x. ft x);cont(%x. tt x)|] ==> cont(%x. (ft x)$(tt x))";
-by (best_tac (claset() addIs [cont2cont_app2, cont_const, cont_Rep_CFun1,
-	                      cont_Rep_CFun2]) 1);
-qed "cont2cont_Rep_CFun";
-
-
-
-(* ------------------------------------------------------------------------ *)
-(* cont2mono Lemma for %x. LAM y. c1(x)(y)                                  *)
-(* ------------------------------------------------------------------------ *)
-
-val [p1,p2] = Goal  
- "[| !!x. cont(c1 x); !!y. monofun(%x. c1 x y)|] ==> monofun(%x. LAM y. c1 x y)";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (stac less_cfun 1);
-by (stac less_fun 1);
-by (rtac allI 1);
-by (stac beta_cfun 1);
-by (rtac p1 1);
-by (stac beta_cfun 1);
-by (rtac p1 1);
-by (etac (p2 RS monofunE RS spec RS spec RS mp) 1);
-qed "cont2mono_LAM";
-
-(* ------------------------------------------------------------------------ *)
-(* cont2cont Lemma for %x. LAM y. c1 x y)                                 *)
-(* ------------------------------------------------------------------------ *)
-
-val [p1,p2] = Goal  
- "[| !!x. cont(c1 x); !!y. cont(%x. c1 x y) |] ==> cont(%x. LAM y. c1 x y)";
-by (rtac monocontlub2cont 1);
-by (rtac (p1 RS cont2mono_LAM) 1);
-by (rtac (p2 RS cont2mono) 1);
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (stac thelub_cfun 1);
-by (rtac (p1 RS cont2mono_LAM RS ch2ch_monofun) 1);
-by (rtac (p2 RS cont2mono) 1);
-by (atac 1);
-by (res_inst_tac [("f","Abs_CFun")] arg_cong 1);
-by (rtac ext 1);
-by (stac (p1 RS beta_cfun RS ext) 1);
-by (etac (p2 RS cont2contlub RS contlubE RS spec RS mp) 1);
-qed "cont2cont_LAM";
-
-(* ------------------------------------------------------------------------ *)
-(* cont2cont tactic                                                       *)
-(* ------------------------------------------------------------------------ *)
-
-val cont_lemmas1 = [cont_const, cont_id, cont_Rep_CFun2,
-                    cont2cont_Rep_CFun,cont2cont_LAM];
-
-Addsimps cont_lemmas1;
-
-(* HINT: 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)));*)
-
-(* ------------------------------------------------------------------------ *)
-(* function application _[_]  is strict in its first arguments              *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "(UU::'a::cpo->'b)$x = (UU::'b)";
-by (stac inst_cfun_pcpo 1);
-by (stac beta_cfun 1);
-by (Simp_tac 1);
-by (rtac refl 1);
-qed "strict_Rep_CFun1";
-
-
-(* ------------------------------------------------------------------------ *)
-(* results about strictify                                                  *)
-(* ------------------------------------------------------------------------ *)
-
-Goalw [Istrictify_def]
-        "Istrictify(f)(UU)= (UU)";
-by (Simp_tac 1);
-qed "Istrictify1";
-
-Goalw [Istrictify_def]
-        "~x=UU ==> Istrictify(f)(x)=f$x";
-by (Asm_simp_tac 1);
-qed "Istrictify2";
-
-Goal "monofun(Istrictify)";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (rtac (less_fun RS iffD2) 1);
-by (strip_tac 1);
-by (res_inst_tac [("Q","xa=UU")] (excluded_middle RS disjE) 1);
-by (stac Istrictify2 1);
-by (atac 1);
-by (stac Istrictify2 1);
-by (atac 1);
-by (rtac monofun_cfun_fun 1);
-by (atac 1);
-by (hyp_subst_tac 1);
-by (stac Istrictify1 1);
-by (stac Istrictify1 1);
-by (rtac refl_less 1);
-qed "monofun_Istrictify1";
-
-Goal "monofun(Istrictify(f))";
-by (rtac monofunI 1);
-by (strip_tac 1);
-by (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1);
-by (stac Istrictify2 1);
-by (etac notUU_I 1);
-by (atac 1);
-by (stac Istrictify2 1);
-by (atac 1);
-by (rtac monofun_cfun_arg 1);
-by (atac 1);
-by (hyp_subst_tac 1);
-by (stac Istrictify1 1);
-by (rtac minimal 1);
-qed "monofun_Istrictify2";
-
-
-Goal "contlub(Istrictify)";
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (rtac (expand_fun_eq RS iffD2) 1);
-by (strip_tac 1);
-by (stac thelub_fun 1);
-by (etac (monofun_Istrictify1 RS ch2ch_monofun) 1);
-by (res_inst_tac [("Q","x=UU")] (excluded_middle RS disjE) 1);
-by (stac Istrictify2 1);
-by (atac 1);
-by (stac (Istrictify2 RS ext) 1);
-by (atac 1);
-by (stac thelub_cfun 1);
-by (atac 1);
-by (stac beta_cfun 1);
-by (rtac cont_lubcfun 1);
-by (atac 1);
-by (rtac refl 1);
-by (hyp_subst_tac 1);
-by (stac Istrictify1 1);
-by (stac (Istrictify1 RS ext) 1);
-by (rtac (chain_UU_I_inverse RS sym) 1);
-by (rtac (refl RS allI) 1);
-qed "contlub_Istrictify1";
+(* legacy ML bindings *)
 
-Goal "contlub(Istrictify(f::'a -> 'b))";
-by (rtac contlubI 1);
-by (strip_tac 1);
-by (case_tac "lub(range(Y))=(UU::'a)" 1);
-by (asm_simp_tac (simpset() addsimps [Istrictify1, chain_UU_I_inverse, chain_UU_I, Istrictify1]) 1);
-by (stac Istrictify2 1);
-by (atac 1);
-by (res_inst_tac [("s","lub(range(%i. f$(Y i)))")] trans 1);
-by (rtac contlub_cfun_arg 1);
-by (atac 1);
-by (rtac lub_equal2 1);
-by (best_tac (claset() addIs [ch2ch_monofun, monofun_Istrictify2]) 3);
-by (best_tac (claset() addIs [ch2ch_monofun, monofun_Rep_CFun2]) 2);
-by (rtac (chain_mono2 RS exE) 1);
-by (atac 2);
-by (etac chain_UU_I_inverse2 1);
-by (blast_tac (claset() addIs [Istrictify2 RS sym]) 1);
-qed "contlub_Istrictify2";
-
-
-bind_thm ("cont_Istrictify1", contlub_Istrictify1 RS 
-        (monofun_Istrictify1 RS monocontlub2cont)); 
-
-bind_thm ("cont_Istrictify2", contlub_Istrictify2 RS 
-        (monofun_Istrictify2 RS monocontlub2cont)); 
-
-
-Goalw [strictify_def] "strictify$f$UU=UU";
-by (stac beta_cfun 1);
-by (simp_tac (simpset() addsimps [cont_Istrictify2,cont_Istrictify1, cont2cont_CF1L]) 1);
-by (stac beta_cfun 1);
-by (rtac cont_Istrictify2 1);
-by (rtac Istrictify1 1);
-qed "strictify1";
-
-Goalw [strictify_def] "~x=UU ==> strictify$f$x=f$x";
-by (stac beta_cfun 1);
-by (simp_tac (simpset() addsimps [cont_Istrictify2,cont_Istrictify1, cont2cont_CF1L]) 1);
-by (stac beta_cfun 1);
-by (rtac cont_Istrictify2 1);
-by (etac Istrictify2 1);
-qed "strictify2";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Instantiate the simplifier                                               *)
-(* ------------------------------------------------------------------------ *)
-
-Addsimps [minimal,refl_less,beta_cfun,strict_Rep_CFun1,strictify1, strictify2];
-
-
-(* ------------------------------------------------------------------------ *)
-(* use cont_tac as autotac.                                                *)
-(* ------------------------------------------------------------------------ *)
-
-(* HINT: cont_tac is now installed in simplifier in Lift.ML ! *)
-(*simpset_ref() := simpset() addsolver (K (DEPTH_SOLVE_1 o cont_tac));*)
-
-(* ------------------------------------------------------------------------ *)
-(* some lemmata for functions with flat/chfin domain/range types	    *)
-(* ------------------------------------------------------------------------ *)
-
-Goal "chain (Y::nat => 'a::cpo->'b::chfin) \
-\     ==> !s. ? n. lub(range(Y))$s = Y n$s";
-by (rtac allI 1);
-by (stac contlub_cfun_fun 1);
-by (atac 1);
-by (fast_tac (HOL_cs addSIs [thelubI,chfin,lub_finch2,chfin2finch,ch2ch_Rep_CFunL])1);
-qed "chfin_Rep_CFunR";
-
-(* ------------------------------------------------------------------------ *)
-(* continuous isomorphisms are strict                                       *)
-(* a prove for embedding projection pairs is similar                        *)
-(* ------------------------------------------------------------------------ *)
-
-Goal  
-"!!f g.[|!y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a) |] \
-\ ==> f$UU=UU & g$UU=UU";
-by (rtac conjI 1);
-by (rtac UU_I 1);
-by (res_inst_tac [("s","f$(g$(UU::'b))"),("t","UU::'b")] subst 1);
-by (etac spec 1);
-by (rtac (minimal RS monofun_cfun_arg) 1);
-by (rtac UU_I 1);
-by (res_inst_tac [("s","g$(f$(UU::'a))"),("t","UU::'a")] subst 1);
-by (etac spec 1);
-by (rtac (minimal RS monofun_cfun_arg) 1);
-qed "iso_strict";
-
-
-Goal "[|!x. rep$(ab$x)=x;!y. ab$(rep$y)=y; z~=UU|] ==> rep$z ~= UU";
-by (etac contrapos_nn 1);
-by (dres_inst_tac [("f","ab")] cfun_arg_cong 1);
-by (etac box_equals 1);
-by (fast_tac HOL_cs 1);
-by (etac (iso_strict RS conjunct1) 1);
-by (atac 1);
-qed "isorep_defined";
-
-Goal "[|!x. rep$(ab$x) = x;!y. ab$(rep$y)=y ; z~=UU|] ==> ab$z ~= UU";
-by (etac contrapos_nn 1);
-by (dres_inst_tac [("f","rep")] cfun_arg_cong 1);
-by (etac box_equals 1);
-by (fast_tac HOL_cs 1);
-by (etac (iso_strict RS conjunct2) 1);
-by (atac 1);
-qed "isoabs_defined";
-
-(* ------------------------------------------------------------------------ *)
-(* propagation of flatness and chainfiniteness by continuous isomorphisms   *)
-(* ------------------------------------------------------------------------ *)
+val Istrictify_def = thm "Istrictify_def";
+val strictify_def = thm "strictify_def";
+val ID_def = thm "ID_def";
+val oo_def = thm "oo_def";
+val inst_cfun_pcpo = thm "inst_cfun_pcpo";
+val contlub_Rep_CFun1 = thm "contlub_Rep_CFun1";
+val cont_Rep_CFun1 = thm "cont_Rep_CFun1";
+val contlub_cfun_fun = thm "contlub_cfun_fun";
+val cont_cfun_fun = thm "cont_cfun_fun";
+val contlub_cfun = thm "contlub_cfun";
+val cont_cfun = thm "cont_cfun";
+val cont2cont_Rep_CFun = thm "cont2cont_Rep_CFun";
+val cont2mono_LAM = thm "cont2mono_LAM";
+val cont2cont_LAM = thm "cont2cont_LAM";
+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 chfin_Rep_CFunR = thm "chfin_Rep_CFunR";
+val iso_strict = thm "iso_strict";
+val isorep_defined = thm "isorep_defined";
+val isoabs_defined = thm "isoabs_defined";
+val chfin2chfin = thm "chfin2chfin";
+val flat2flat = thm "flat2flat";
+val flat_codom = thm "flat_codom";
+val ID1 = thm "ID1";
+val cfcomp1 = thm "cfcomp1";
+val cfcomp2 = thm "cfcomp2";
+val ID2 = thm "ID2";
+val ID3 = thm "ID3";
+val assoc_oo = thm "assoc_oo";
 
-Goal "!!f g.[|! Y::nat=>'a. chain Y --> (? n. max_in_chain n Y); \
-\ !y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a::chfin) |] \
-\ ==> ! Y::nat=>'b. chain Y --> (? n. max_in_chain n Y)";
-by (rewtac max_in_chain_def);
-by (strip_tac 1);
-by (rtac exE 1);
-by (res_inst_tac [("P","chain(%i. g$(Y i))")] mp 1);
-by (etac spec 1);
-by (etac ch2ch_Rep_CFunR 1);
-by (rtac exI 1);
-by (strip_tac 1);
-by (res_inst_tac [("s","f$(g$(Y x))"),("t","Y(x)")] subst 1);
-by (etac spec 1);
-by (res_inst_tac [("s","f$(g$(Y j))"),("t","Y(j)")] subst 1);
-by (etac spec 1);
-by (rtac cfun_arg_cong 1);
-by (rtac mp 1);
-by (etac spec 1);
-by (atac 1);
-qed "chfin2chfin";
-
-
-Goal "!!f g.[|!x y::'a. x<<y --> x=UU | x=y; \
-\ !y. f$(g$y)=(y::'b); !x. g$(f$x)=(x::'a)|] ==> !x y::'b. x<<y --> x=UU | x=y";
-by (strip_tac 1);
-by (rtac disjE 1);
-by (res_inst_tac [("P","g$x<<g$y")] mp 1);
-by (etac monofun_cfun_arg 2);
-by (dtac spec 1);
-by (etac spec 1);
-by (rtac disjI1 1);
-by (rtac trans 1);
-by (res_inst_tac [("s","f$(g$x)"),("t","x")] subst 1);
-by (etac spec 1);
-by (etac cfun_arg_cong 1);
-by (rtac (iso_strict RS conjunct1) 1);
-by (atac 1);
-by (atac 1);
-by (rtac disjI2 1);
-by (res_inst_tac [("s","f$(g$x)"),("t","x")] subst 1);
-by (etac spec 1);
-by (res_inst_tac [("s","f$(g$y)"),("t","y")] subst 1);
-by (etac spec 1);
-by (etac cfun_arg_cong 1);
-qed "flat2flat";
-
-(* ------------------------------------------------------------------------- *)
-(* a result about functions with flat codomain                               *)
-(* ------------------------------------------------------------------------- *)
-
-Goal "f$(x::'a)=(c::'b::flat) ==> f$(UU::'a)=(UU::'b) | (!z. f$(z::'a)=c)";
-by (case_tac "f$(x::'a)=(UU::'b)" 1);
-by (rtac disjI1 1);
-by (rtac UU_I 1);
-by (res_inst_tac [("s","f$(x)"),("t","UU::'b")] subst 1);
-by (atac 1);
-by (rtac (minimal RS monofun_cfun_arg) 1);
-by (case_tac "f$(UU::'a)=(UU::'b)" 1);
-by (etac disjI1 1);
-by (rtac disjI2 1);
-by (rtac allI 1);
-by (hyp_subst_tac 1);
-by (res_inst_tac [("a","f$(UU::'a)")] (refl RS box_equals) 1);
-by (res_inst_tac [("fo5","f")] ((minimal RS monofun_cfun_arg) RS (ax_flat RS spec RS spec RS mp) RS disjE) 1);
-by (contr_tac 1);
-by (atac 1);
-by (res_inst_tac [("fo5","f")] ((minimal RS monofun_cfun_arg) RS (ax_flat RS spec RS spec RS mp) RS disjE) 1);
-by (contr_tac 1);
-by (atac 1);
-qed "flat_codom";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Access to definitions                                                    *)
-(* ------------------------------------------------------------------------ *)
-
-
-Goalw [ID_def] "ID$x=x";
-by (stac beta_cfun 1);
-by (rtac cont_id 1);
-by (rtac refl 1);
-qed "ID1";
-
-Goalw [oo_def] "(f oo g)=(LAM x. f$(g$x))";
-by (stac beta_cfun 1);
-by (Simp_tac 1);
-by (stac beta_cfun 1);
-by (Simp_tac 1);
-by (rtac refl 1);
-qed "cfcomp1";
-
-Goal  "(f oo g)$x=f$(g$x)";
-by (stac cfcomp1 1);
-by (stac beta_cfun 1);
-by (Simp_tac 1);
-by (rtac refl 1);
-qed "cfcomp2";
-
-
-(* ------------------------------------------------------------------------ *)
-(* Show that interpretation of (pcpo,_->_) is a category                    *)
-(* The class of objects is interpretation of syntactical class pcpo         *)
-(* The class of arrows  between objects 'a and 'b is interpret. of 'a -> 'b *)
-(* The identity arrow is interpretation of ID                               *)
-(* The composition of f and g is interpretation of oo                       *)
-(* ------------------------------------------------------------------------ *)
-
-
-Goal "f oo ID = f ";
-by (rtac ext_cfun 1);
-by (stac cfcomp2 1);
-by (stac ID1 1);
-by (rtac refl 1);
-qed "ID2";
-
-Goal "ID oo f = f ";
-by (rtac ext_cfun 1);
-by (stac cfcomp2 1);
-by (stac ID1 1);
-by (rtac refl 1);
-qed "ID3";
-
-
-Goal "f oo (g oo h) = (f oo g) oo h";
-by (rtac ext_cfun 1);
-by (res_inst_tac [("s","f$(g$(h$x))")] trans  1);
-by (stac cfcomp2 1);
-by (stac cfcomp2 1);
-by (rtac refl 1);
-by (stac cfcomp2 1);
-by (stac cfcomp2 1);
-by (rtac refl 1);
-qed "assoc_oo";
-
-(* ------------------------------------------------------------------------ *)
-(* Merge the different rewrite rules for the simplifier                     *)
-(* ------------------------------------------------------------------------ *)
-
-Addsimps ([ID1,ID2,ID3,cfcomp2]);
-
-
+structure Cfun3 =
+struct
+  val thy = the_context ();
+  val Istrictify_def = Istrictify_def;
+  val strictify_def = strictify_def;
+  val ID_def = ID_def;
+  val oo_def = oo_def;
+end;
diff -r 2454493bd77b -r eb3b1a5c304e src/HOLCF/Cfun3.thy
--- a/src/HOLCF/Cfun3.thy	Wed Mar 02 23:28:17 2005 +0100
+++ b/src/HOLCF/Cfun3.thy	Wed Mar 02 23:58:02 2005 +0100
@@ -1,25 +1,29 @@
 (*  Title:      HOLCF/Cfun3.thy
     ID:         $Id$
     Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
 
 Class instance of  -> for class pcpo
 
 *)
 
-Cfun3 = Cfun2 +
+theory Cfun3 = Cfun2:
+
+instance "->" :: (cpo,cpo)cpo
+by (intro_classes, rule cpo_cfun)
 
-instance "->" :: (cpo,cpo)cpo              (cpo_cfun)
-instance "->" :: (cpo,pcpo)pcpo            (least_cfun)
+instance "->" :: (cpo,pcpo)pcpo
+by (intro_classes, rule least_cfun)
 
-default pcpo
+defaultsort pcpo
 
 consts  
         Istrictify   :: "('a->'b)=>'a=>'b"
         strictify    :: "('a->'b)->'a->'b"
 defs
 
-Istrictify_def  "Istrictify f x == if x=UU then UU else f$x"    
-strictify_def   "strictify == (LAM f x. Istrictify f x)"
+Istrictify_def:  "Istrictify f x == if x=UU then UU else f$x"    
+strictify_def:   "strictify == (LAM f x. Istrictify f x)"
 
 consts
         ID      :: "('a::cpo) -> 'a"
@@ -31,7 +35,512 @@
 
 defs
 
-  ID_def        "ID ==(LAM x. x)"
-  oo_def        "cfcomp == (LAM f g x. f$(g$x))" 
+  ID_def:        "ID ==(LAM x. x)"
+  oo_def:        "cfcomp == (LAM f g x. f$(g$x))" 
+
+(*  Title:      HOLCF/Cfun3
+    ID:         $Id$
+    Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
+
+Class instance of  -> for class pcpo
+*)
+
+(* for compatibility with old HOLCF-Version *)
+lemma inst_cfun_pcpo: "UU = Abs_CFun(%x. UU)"
+apply (simp add: UU_def UU_cfun_def)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the contlub property for Rep_CFun its 'first' argument                       *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlub_Rep_CFun1: "contlub(Rep_CFun)"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule expand_fun_eq [THEN iffD2])
+apply (intro strip)
+apply (subst thelub_cfun)
+apply assumption
+apply (subst Cfunapp2)
+apply (erule cont_lubcfun)
+apply (subst thelub_fun)
+apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
+apply (rule refl)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* the cont property for Rep_CFun in its first argument                        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_Rep_CFun1: "cont(Rep_CFun)"
+apply (rule monocontlub2cont)
+apply (rule monofun_Rep_CFun1)
+apply (rule contlub_Rep_CFun1)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* contlub, cont properties of Rep_CFun in its first argument in mixfix _[_]   *)
+(* ------------------------------------------------------------------------ *)
+
+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
+
+
+(* ------------------------------------------------------------------------ *)
+(* contlub, cont  properties of Rep_CFun in both argument in mixfix _[_]       *)
+(* ------------------------------------------------------------------------ *)
+
+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
+
+
+(* ------------------------------------------------------------------------ *)
+(* cont2cont lemma for 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
+
+
+
+(* ------------------------------------------------------------------------ *)
+(* cont2mono Lemma for %x. LAM y. c1(x)(y)                                  *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont2mono_LAM:
+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)
+apply (intro strip)
+apply (subst less_cfun)
+apply (subst less_fun)
+apply (rule allI)
+apply (subst beta_cfun)
+apply (rule p1)
+apply (subst beta_cfun)
+apply (rule p1)
+apply (erule p2 [THEN monofunE, THEN spec, THEN spec, THEN mp])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* cont2cont Lemma for %x. LAM y. c1 x y)                                 *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont2cont_LAM:
+assumes p1: "!!x. cont(c1 x)"
+assumes p2: "!!y. cont(%x. c1 x y)"
+shows "cont(%x. LAM y. c1 x y)"
+apply (rule monocontlub2cont)
+apply (rule p1 [THEN cont2mono_LAM])
+apply (rule p2 [THEN cont2mono])
+apply (rule contlubI)
+apply (intro strip)
+apply (subst thelub_cfun)
+apply (rule p1 [THEN cont2mono_LAM, THEN ch2ch_monofun])
+apply (rule p2 [THEN cont2mono])
+apply assumption
+apply (rule_tac f = "Abs_CFun" in arg_cong)
+apply (rule ext)
+apply (subst p1 [THEN beta_cfun, THEN ext])
+apply (erule p2 [THEN cont2contlub, THEN contlubE, THEN spec, THEN mp])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* cont2cont tactic                                                       *)
+(* ------------------------------------------------------------------------ *)
+
+lemmas cont_lemmas1 = cont_const cont_id cont_Rep_CFun2
+                    cont2cont_Rep_CFun cont2cont_LAM
+
+declare cont_lemmas1 [simp]
+
+(* HINT: 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)));*)
+
+(* ------------------------------------------------------------------------ *)
+(* function application _[_]  is strict in its first arguments              *)
+(* ------------------------------------------------------------------------ *)
+
+lemma strict_Rep_CFun1: "(UU::'a::cpo->'b)$x = (UU::'b)"
+apply (subst inst_cfun_pcpo)
+apply (subst beta_cfun)
+apply (simp (no_asm))
+apply (rule refl)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* results about strictify                                                  *)
+(* ------------------------------------------------------------------------ *)
+
+lemma Istrictify1: 
+        "Istrictify(f)(UU)= (UU)"
+apply (unfold Istrictify_def)
+apply (simp (no_asm))
+done
+
+lemma Istrictify2: 
+        "~x=UU ==> Istrictify(f)(x)=f$x"
+apply (unfold Istrictify_def)
+apply (simp (no_asm_simp))
+done
+
+lemma monofun_Istrictify1: "monofun(Istrictify)"
+apply (rule monofunI)
+apply (intro strip)
+apply (rule less_fun [THEN iffD2])
+apply (intro strip)
+apply (rule_tac Q = "xa=UU" in excluded_middle [THEN disjE])
+apply (subst Istrictify2)
+apply assumption
+apply (subst Istrictify2)
+apply assumption
+apply (rule monofun_cfun_fun)
+apply assumption
+apply (erule ssubst)
+apply (subst Istrictify1)
+apply (subst Istrictify1)
+apply (rule refl_less)
+done
+
+lemma monofun_Istrictify2: "monofun(Istrictify(f))"
+apply (rule monofunI)
+apply (intro strip)
+apply (rule_tac Q = "x=UU" in excluded_middle [THEN disjE])
+apply (simplesubst Istrictify2)
+apply (erule notUU_I)
+apply assumption
+apply (subst Istrictify2)
+apply assumption
+apply (rule monofun_cfun_arg)
+apply assumption
+apply (erule ssubst)
+apply (subst Istrictify1)
+apply (rule minimal)
+done
+
+
+lemma contlub_Istrictify1: "contlub(Istrictify)"
+apply (rule contlubI)
+apply (intro strip)
+apply (rule expand_fun_eq [THEN iffD2])
+apply (intro strip)
+apply (subst thelub_fun)
+apply (erule monofun_Istrictify1 [THEN ch2ch_monofun])
+apply (rule_tac Q = "x=UU" in excluded_middle [THEN disjE])
+apply (subst Istrictify2)
+apply assumption
+apply (subst Istrictify2 [THEN ext])
+apply assumption
+apply (subst thelub_cfun)
+apply assumption
+apply (subst beta_cfun)
+apply (rule cont_lubcfun)
+apply assumption
+apply (rule refl)
+apply (erule ssubst)
+apply (subst Istrictify1)
+apply (subst Istrictify1 [THEN ext])
+apply (rule chain_UU_I_inverse [symmetric])
+apply (rule refl [THEN allI])
+done
+
+lemma contlub_Istrictify2: "contlub(Istrictify(f::'a -> 'b))"
+apply (rule contlubI)
+apply (intro strip)
+apply (case_tac "lub (range (Y))= (UU::'a) ")
+apply (simp (no_asm_simp) add: Istrictify1 chain_UU_I_inverse chain_UU_I Istrictify1)
+apply (subst Istrictify2)
+apply assumption
+apply (rule_tac s = "lub (range (%i. f$ (Y i))) " in trans)
+apply (rule contlub_cfun_arg)
+apply assumption
+apply (rule lub_equal2)
+prefer 3 apply (best intro: ch2ch_monofun monofun_Istrictify2)
+prefer 2 apply (best intro: ch2ch_monofun monofun_Rep_CFun2)
+apply (rule chain_mono2 [THEN exE])
+prefer 2 apply (assumption)
+apply (erule chain_UU_I_inverse2)
+apply (blast intro: Istrictify2 [symmetric])
+done
+
+
+lemmas cont_Istrictify1 = contlub_Istrictify1 [THEN monofun_Istrictify1 [THEN monocontlub2cont], standard]
+
+lemmas cont_Istrictify2 = contlub_Istrictify2 [THEN monofun_Istrictify2 [THEN monocontlub2cont], standard]
+
+
+lemma strictify1: "strictify$f$UU=UU"
+apply (unfold strictify_def)
+apply (subst beta_cfun)
+apply (simp (no_asm) add: cont_Istrictify2 cont_Istrictify1 cont2cont_CF1L)
+apply (subst beta_cfun)
+apply (rule cont_Istrictify2)
+apply (rule Istrictify1)
+done
+
+lemma strictify2: "~x=UU ==> strictify$f$x=f$x"
+apply (unfold strictify_def)
+apply (subst beta_cfun)
+apply (simp (no_asm) add: cont_Istrictify2 cont_Istrictify1 cont2cont_CF1L)
+apply (subst beta_cfun)
+apply (rule cont_Istrictify2)
+apply (erule Istrictify2)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* Instantiate the simplifier                                               *)
+(* ------------------------------------------------------------------------ *)
+
+declare minimal [simp] refl_less [simp] beta_cfun [simp] strict_Rep_CFun1 [simp] strictify1 [simp] strictify2 [simp]
+
+
+(* ------------------------------------------------------------------------ *)
+(* use cont_tac as autotac.                                                *)
+(* ------------------------------------------------------------------------ *)
+
+(* HINT: cont_tac is now installed in simplifier in Lift.ML ! *)
+(*simpset_ref() := simpset() addsolver (K (DEPTH_SOLVE_1 o cont_tac));*)
+
+(* ------------------------------------------------------------------------ *)
+(* some lemmata for functions with flat/chfin domain/range types	    *)
+(* ------------------------------------------------------------------------ *)
+
+lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)  
+      ==> !s. ? n. lub(range(Y))$s = Y n$s"
+apply (rule allI)
+apply (subst contlub_cfun_fun)
+apply assumption
+apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* continuous isomorphisms are strict                                       *)
+(* a prove for embedding projection pairs is similar                        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma iso_strict: 
+"!!f g.[|!y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a) |]  
+  ==> f$UU=UU & g$UU=UU"
+apply (rule conjI)
+apply (rule UU_I)
+apply (rule_tac s = "f$ (g$ (UU::'b))" and t = "UU::'b" in subst)
+apply (erule spec)
+apply (rule minimal [THEN monofun_cfun_arg])
+apply (rule UU_I)
+apply (rule_tac s = "g$ (f$ (UU::'a))" and t = "UU::'a" in subst)
+apply (erule spec)
+apply (rule minimal [THEN monofun_cfun_arg])
+done
+
+
+lemma isorep_defined: "[|!x. rep$(ab$x)=x;!y. ab$(rep$y)=y; z~=UU|] ==> rep$z ~= UU"
+apply (erule contrapos_nn)
+apply (drule_tac f = "ab" in cfun_arg_cong)
+apply (erule box_equals)
+apply fast
+apply (erule iso_strict [THEN conjunct1])
+apply assumption
+done
+
+lemma isoabs_defined: "[|!x. rep$(ab$x) = x;!y. ab$(rep$y)=y ; z~=UU|] ==> ab$z ~= UU"
+apply (erule contrapos_nn)
+apply (drule_tac f = "rep" in cfun_arg_cong)
+apply (erule box_equals)
+apply fast
+apply (erule iso_strict [THEN conjunct2])
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* propagation of flatness and chainfiniteness by continuous isomorphisms   *)
+(* ------------------------------------------------------------------------ *)
+
+lemma chfin2chfin: "!!f g.[|! Y::nat=>'a. chain Y --> (? n. max_in_chain n Y);  
+  !y. f$(g$y)=(y::'b) ; !x. g$(f$x)=(x::'a::chfin) |]  
+  ==> ! Y::nat=>'b. chain Y --> (? n. max_in_chain n Y)"
+apply (unfold max_in_chain_def)
+apply (intro strip)
+apply (rule exE)
+apply (rule_tac P = "chain (%i. g$ (Y i))" in mp)
+apply (erule spec)
+apply (erule ch2ch_Rep_CFunR)
+apply (rule exI)
+apply (intro strip)
+apply (rule_tac s = "f$ (g$ (Y x))" and t = "Y (x) " in subst)
+apply (erule spec)
+apply (rule_tac s = "f$ (g$ (Y j))" and t = "Y (j) " in subst)
+apply (erule spec)
+apply (rule cfun_arg_cong)
+apply (rule mp)
+apply (erule spec)
+apply assumption
+done
+
+
+lemma flat2flat: "!!f g.[|!x y::'a. x<<y --> x=UU | x=y;  
+  !y. f$(g$y)=(y::'b); !x. g$(f$x)=(x::'a)|] ==> !x y::'b. x<<y --> x=UU | x=y"
+apply (intro strip)
+apply (rule disjE)
+apply (rule_tac P = "g$x<<g$y" in mp)
+apply (erule_tac [2] monofun_cfun_arg)
+apply (drule spec)
+apply (erule spec)
+apply (rule disjI1)
+apply (rule trans)
+apply (rule_tac s = "f$ (g$x) " and t = "x" in subst)
+apply (erule spec)
+apply (erule cfun_arg_cong)
+apply (rule iso_strict [THEN conjunct1])
+apply assumption
+apply assumption
+apply (rule disjI2)
+apply (rule_tac s = "f$ (g$x) " and t = "x" in subst)
+apply (erule spec)
+apply (rule_tac s = "f$ (g$y) " and t = "y" in subst)
+apply (erule spec)
+apply (erule cfun_arg_cong)
+done
+
+(* ------------------------------------------------------------------------- *)
+(* a result about functions with flat codomain                               *)
+(* ------------------------------------------------------------------------- *)
+
+lemma flat_codom: "f$(x::'a)=(c::'b::flat) ==> f$(UU::'a)=(UU::'b) | (!z. f$(z::'a)=c)"
+apply (case_tac "f$ (x::'a) = (UU::'b) ")
+apply (rule disjI1)
+apply (rule UU_I)
+apply (rule_tac s = "f$ (x) " and t = "UU::'b" in subst)
+apply assumption
+apply (rule minimal [THEN monofun_cfun_arg])
+apply (case_tac "f$ (UU::'a) = (UU::'b) ")
+apply (erule disjI1)
+apply (rule disjI2)
+apply (rule allI)
+apply (erule subst)
+apply (rule_tac a = "f$ (UU::'a) " in refl [THEN box_equals])
+apply (rule_tac fo5 = "f" in minimal [THEN monofun_cfun_arg, THEN ax_flat [THEN spec, THEN spec, THEN mp], THEN disjE])
+apply simp
+apply assumption
+apply (rule_tac fo5 = "f" in minimal [THEN monofun_cfun_arg, THEN ax_flat [THEN spec, THEN spec, THEN mp], THEN disjE])
+apply simp
+apply assumption
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* Access to definitions                                                    *)
+(* ------------------------------------------------------------------------ *)
+
+
+lemma ID1: "ID$x=x"
+apply (unfold ID_def)
+apply (subst beta_cfun)
+apply (rule cont_id)
+apply (rule refl)
+done
+
+lemma cfcomp1: "(f oo g)=(LAM x. f$(g$x))"
+apply (unfold oo_def)
+apply (subst beta_cfun)
+apply (simp (no_asm))
+apply (subst beta_cfun)
+apply (simp (no_asm))
+apply (rule refl)
+done
+
+lemma cfcomp2: "(f oo g)$x=f$(g$x)"
+apply (subst cfcomp1)
+apply (subst beta_cfun)
+apply (simp (no_asm))
+apply (rule refl)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* Show that interpretation of (pcpo,_->_) is a category                    *)
+(* The class of objects is interpretation of syntactical class pcpo         *)
+(* The class of arrows  between objects 'a and 'b is interpret. of 'a -> 'b *)
+(* The identity arrow is interpretation of ID                               *)
+(* The composition of f and g is interpretation of oo                       *)
+(* ------------------------------------------------------------------------ *)
+
+
+lemma ID2: "f oo ID = f "
+apply (rule ext_cfun)
+apply (subst cfcomp2)
+apply (subst ID1)
+apply (rule refl)
+done
+
+lemma ID3: "ID oo f = f "
+apply (rule ext_cfun)
+apply (subst cfcomp2)
+apply (subst ID1)
+apply (rule refl)
+done
+
+
+lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
+apply (rule ext_cfun)
+apply (rule_tac s = "f$ (g$ (h$x))" in trans)
+apply (subst cfcomp2)
+apply (subst cfcomp2)
+apply (rule refl)
+apply (subst cfcomp2)
+apply (subst cfcomp2)
+apply (rule refl)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Merge the different rewrite rules for the simplifier                     *)
+(* ------------------------------------------------------------------------ *)
+
+declare  ID1[simp] ID2[simp] ID3[simp] cfcomp2[simp]
 
 end