integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
authoroheimb
Tue, 07 Sep 2004 16:02:42 +0200
changeset 15188 9d57263faf9e
parent 15187 8b74a39dba89
child 15189 9cfbc392918c
integrated Streams with ex/Stream.*; added FOCUS/Fstreams.thy
src/HOLCF/FOCUS/Buffer.ML
src/HOLCF/FOCUS/Buffer_adm.ML
src/HOLCF/FOCUS/FOCUS.ML
src/HOLCF/FOCUS/Fstream.ML
src/HOLCF/FOCUS/Fstream.thy
src/HOLCF/FOCUS/Fstreams.thy
src/HOLCF/FOCUS/ROOT.ML
src/HOLCF/FOCUS/Stream_adm.ML
src/HOLCF/IsaMakefile
src/HOLCF/ROOT.ML
src/HOLCF/Streams.thy
src/HOLCF/ex/Dagstuhl.ML
src/HOLCF/ex/Stream.ML
src/HOLCF/ex/Stream.thy
--- a/src/HOLCF/FOCUS/Buffer.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/Buffer.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -175,7 +175,7 @@
 by (Fast_tac 1);
 qed_spec_mp "BufAC_Asm_cong_lemma";
 Goal "\\<lbrakk>f \\<in> BufEq; ff \\<in> BufEq; s \\<in> BufAC_Asm\\<rbrakk> \\<Longrightarrow> f\\<cdot>s = ff\\<cdot>s";
-by (resolve_tac stream.take_lemmas 1);
+by (resolve_tac (thms "stream.take_lemmas") 1);
 by (eatac BufAC_Asm_cong_lemma 2 1);
 qed "BufAC_Asm_cong";
 
--- a/src/HOLCF/FOCUS/Buffer_adm.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/Buffer_adm.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -30,7 +30,7 @@
 	subgoal_tac "!d x. (s = Md d\\<leadsto>\\<bullet>\\<leadsto>x --> (? y. t = d\\<leadsto>y & (x,y):C)) = \
 		   \ (s = Md d\\<leadsto>\\<bullet>\\<leadsto>x --> ft\\<cdot>t = Def d & (x,rt\\<cdot>t):C)"  1,
 	Asm_simp_tac 1,
-	auto_tac (claset() addIs [surjectiv_scons RS sym], simpset())]);
+	auto_tac (claset() addIs [thm "surjectiv_scons" RS sym], simpset())]);
 
 val cont_BufAC_Cmt_F = prove_goal thy "down_cont BufAC_Cmt_F" (K [
 	auto_tac (claset(),simpset() addsimps [down_cont_def,BufAC_Cmt_F_def3])]);
@@ -56,7 +56,7 @@
 b y res_inst_tac [("x","Suc (Suc 0)")] exI 1;
 b y rtac conjI 1;
 b y  strip_tac 2;
-b y  dtac slen_mono 2;
+b y  dtac (thm "slen_mono") 2;
 b y  datac (thm "ile_trans") 1 2;
 b y ALLGOALS Force_tac;
 qed "BufAC_Asm_F_stream_antiP";
--- a/src/HOLCF/FOCUS/FOCUS.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/FOCUS.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -10,10 +10,10 @@
 AddSIs [ex_eqI, ex2_eqI];
 Addsimps[eq_UU_symf];
 Goal "(#x ~= 0) = (? a y. x = a~> y)";
-by (simp_tac (simpset() addsimps [slen_empty_eq, fstream_exhaust_eq]) 1);
+by (simp_tac (simpset() addsimps [thm "slen_empty_eq", fstream_exhaust_eq]) 1);
 qed "fstream_exhaust_slen_eq";
 
-Addsimps[slen_less_1_eq, fstream_exhaust_slen_eq,
-		   slen_fscons_eq, slen_fscons_less_eq];
+Addsimps[thm "slen_less_1_eq", fstream_exhaust_slen_eq,
+		   thm "slen_fscons_eq", thm "slen_fscons_less_eq"];
 Addsimps[thm "Suc_ile_eq"];
 AddEs  [strictI];
--- a/src/HOLCF/FOCUS/Fstream.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/Fstream.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -3,6 +3,8 @@
     Author: 	David von Oheimb, TU Muenchen
 *)
 
+Addsimps[eq_UU_iff RS sym];
+
 Goal "a = Def d \\<Longrightarrow> a\\<sqsubseteq>b \\<Longrightarrow> b = Def d";
 by (rtac (flat_eq RS iffD1 RS sym) 1);
 by (rtac Def_not_UU 1);
@@ -19,7 +21,7 @@
 
 qed_goal "fstream_exhaust" thy "x = UU |  (? a y. x = a~> y)" (fn _ => [
 	simp_tac (simpset() addsimps [fscons_def2]) 1,
-	cut_facts_tac [stream.exhaust] 1,
+	cut_facts_tac [thm "stream.exhaust"] 1,
 	fast_tac (claset() addDs [not_Undef_is_Def RS iffD1]) 1]);
 
 qed_goal "fstream_cases" thy "[| x = UU ==> P; !!a y. x = a~> y ==> P |] ==> P" 
@@ -34,7 +36,7 @@
 			  THEN hyp_subst_tac i THEN hyp_subst_tac (i+1);
 
 qed_goal "fstream_exhaust_eq" thy "(x ~= UU) = (? a y. x = a~> y)" (fn _ => [
-	simp_tac(simpset() addsimps [fscons_def2,stream_exhaust_eq]) 1,
+	simp_tac(simpset() addsimps [fscons_def2,thm "stream_exhaust_eq"]) 1,
 	fast_tac (claset() addDs [not_Undef_is_Def RS iffD1] addEs [DefE]) 1]);
 
 
@@ -52,17 +54,17 @@
 
 qed_goal "fstream_prefix" thy "a~> s << t ==> ? tt. t = a~> tt &  s << tt" (fn prems =>[
 	cut_facts_tac prems 1,
-	stream_case_tac "t" 1,
+	res_inst_tac [("x","t")] (thm "stream.casedist") 1,
 	 cut_facts_tac [fscons_not_empty] 1,
 	 fast_tac (HOL_cs addDs [eq_UU_iff RS iffD2]) 1,
 	asm_full_simp_tac (HOL_ss addsimps [fscons_def2]) 1,
-	dtac stream_flat_prefix 1,
+	dtac (thm "stream_flat_prefix") 1,
 	rtac Def_not_UU 1,
 	fast_tac HOL_cs 1]);
 
 qed_goal "fstream_prefix'" thy
 	"x << a~> z = (x = <> |  (? y. x = a~> y &  y << z))" (fn _ => [
-	simp_tac(HOL_ss addsimps [fscons_def2, Def_not_UU RS stream_prefix'])1,
+	simp_tac(HOL_ss addsimps [fscons_def2, Def_not_UU RS thm "stream_prefix'"])1,
 	Step_tac 1,
 	 ALLGOALS(etac swap),
 	 fast_tac (HOL_cs addIs [refl_less] addEs [DefE]) 2,
@@ -77,12 +79,12 @@
 
 section "ft & rt";
 
-bind_thm("ft_empty",hd stream.sel_rews);
+bind_thm("ft_empty",hd (thms "stream.sel_rews"));
 qed_goalw "ft_fscons" thy [fscons_def] "ft\\<cdot>(m~> s) = Def m" (fn _ => [
 	(Simp_tac 1)]);
 Addsimps[ft_fscons];
 
-bind_thm("rt_empty",hd (tl stream.sel_rews));
+bind_thm("rt_empty",hd (tl (thms "stream.sel_rews")));
 qed_goalw "rt_fscons" thy [fscons_def] "rt\\<cdot>(m~> s) = s" (fn _ => [
 	(Simp_tac 1)]);
 Addsimps[rt_fscons];
@@ -92,7 +94,7 @@
 	Safe_tac,
 	 etac subst 1,
 	 rtac exI 1,
-	 rtac (surjectiv_scons RS sym) 1,
+	 rtac (thm "surjectiv_scons" RS sym) 1,
 	Simp_tac 1]);
 Addsimps[ft_eq];
 
@@ -125,12 +127,12 @@
 
 qed_goal "slen_fscons_eq" thy 
 	"(Fin (Suc n) < #x) = (? a y. x = a~> y & Fin n < #y)" (fn _ => [
-	simp_tac (HOL_ss addsimps [fscons_def2, slen_scons_eq]) 1,
+	simp_tac (HOL_ss addsimps [fscons_def2, thm "slen_scons_eq"]) 1,
 	fast_tac (HOL_cs addDs [not_Undef_is_Def RS iffD1] addEs [DefE]) 1]);
 
 qed_goal "slen_fscons_eq_rev" thy 
 	"(#x < Fin (Suc (Suc n))) = (!a y. x ~= a~> y | #y < Fin (Suc n))" (K [
-	simp_tac (HOL_ss addsimps [fscons_def2, slen_scons_eq_rev]) 1,
+	simp_tac (HOL_ss addsimps [fscons_def2, thm "slen_scons_eq_rev"]) 1,
 	step_tac (HOL_cs addSEs [DefE]) 1,
 	step_tac (HOL_cs addSEs [DefE]) 1,
 	step_tac (HOL_cs addSEs [DefE]) 1,
@@ -153,7 +155,7 @@
 qed_goal "fstream_ind" thy 
 	"[| adm P; P <>; !!a s. P s ==> P (a~> s) |] ==> P x" (fn prems => [
 	cut_facts_tac prems 1,
-	etac stream.ind 1,
+	etac (thm "stream.ind") 1,
 	 atac 1,
 	fast_tac (HOL_cs addDs [not_Undef_is_Def RS iffD1] 
 			 addEs (tl (tl prems) RL [fscons_def2 RS subst])) 1]);
@@ -161,7 +163,7 @@
 qed_goal "fstream_ind2" thy 
   "[| adm P; P UU; !!a. P (a~> UU); !!a b s. P s ==> P (a~> b~> s) |] ==> P x" (fn prems => [
 	cut_facts_tac prems 1,
-	etac stream_ind2 1,
+	etac (thm "stream_ind2") 1,
 	  atac 1,
 	fast_tac (HOL_cs addDs [not_Undef_is_Def RS iffD1] 
 			 addIs ([hd(tl(tl prems))]RL[fscons_def2 RS subst]))1,
@@ -175,11 +177,11 @@
 section "fsfilter";
 
 qed_goalw "fsfilter_empty" thy [fsfilter_def] "A(C)UU = UU" (fn _ => [
-	rtac sfilter_empty 1]);
+	rtac (thm "sfilter_empty") 1]);
 
 qed_goalw "fsfilter_fscons" thy [fsfilter_def] 
 	"A(C)x~> xs = (if x:A then x~> (A(C)xs) else A(C)xs)" (fn _ => [
-	simp_tac (simpset() addsimps [fscons_def2,sfilter_scons,If_and_if]) 1]);
+	simp_tac (simpset() addsimps [fscons_def2,thm "sfilter_scons",If_and_if]) 1]);
 
 qed_goal "fsfilter_emptys" thy "{}(C)x = UU" (fn _ => [
 	res_inst_tac [("x","x")] fstream_ind 1,
@@ -189,7 +191,7 @@
 	asm_simp_tac (simpset()addsimps [fsfilter_fscons]) 1]);
 
 qed_goal "fsfilter_insert" thy "(insert a A)(C)a~> x = a~> ((insert a A)(C)x)" (fn _=> [
-	simp_tac (simpset() addsimps [fsfilter_fscons]) 1]);
+	simp_tac (simpset() addsimps [thm "fsfilter_fscons"]) 1]);
 
 qed_goal "fsfilter_single_in" thy "{a}(C)a~> x = a~> ({a}(C)x)" (fn _=> [
 	rtac fsfilter_insert 1]);
--- a/src/HOLCF/FOCUS/Fstream.thy	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/Fstream.thy	Tue Sep 07 16:02:42 2004 +0200
@@ -3,11 +3,12 @@
     Author: 	David von Oheimb, TU Muenchen
 
 FOCUS streams (with lifted elements)
+###TODO: integrate Fstreams.thy
 *)
 
 (* FOCUS flat streams *)
 
-Fstream = Streams + 
+Fstream = Stream + 
 
 default type
 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/FOCUS/Fstreams.thy	Tue Sep 07 16:02:42 2004 +0200
@@ -0,0 +1,357 @@
+(*  Title: 	HOLCF/FOCUS/Fstreams.thy
+    ID:         $Id$
+    Author: 	Borislav Gajanovic
+
+FOCUS flat streams (with lifted elements)
+###TODO: integrate this with Fstream.*
+*)
+
+theory Fstreams = Stream:
+
+defaultsort type
+
+types 'a fstream = "('a lift) stream"
+
+consts
+
+  fsingleton    :: "'a => 'a fstream"  ("<_>" [1000] 999)
+  fsfilter      :: "'a set \<Rightarrow> 'a fstream \<rightarrow> 'a fstream"
+  fsmap		:: "('a => 'b) => 'a fstream -> 'b fstream"
+
+  jth           :: "nat => 'a fstream => 'a"
+
+  first         :: "'a fstream => 'a"
+  last          :: "'a fstream => 'a"
+
+
+syntax
+
+  "@emptystream":: "'a fstream" 			("<>")
+  "@fsfilter"	:: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream"	("(_'(C')_)" [64,63] 63)
+
+syntax (xsymbols)
+
+  "@fsfilter"	:: "'a set \<Rightarrow> 'a fstream \<Rightarrow> 'a fstream" ("(_\<copyright>_)"
+								     [64,63] 63)
+translations
+
+  "<>"    => "\<bottom>"
+  "A(C)s" == "fsfilter A\<cdot>s"
+
+defs
+
+  fsingleton_def2:    "fsingleton  == %a. Def a && UU"
+
+  jth_def:            "jth == %n s. if Fin n < #s then THE a. i_th n s = Def a else arbitrary" 
+
+  first_def:          "first == %s. jth 0 s"
+  last_def:           "last == %s. case #s of Fin n => (if n~=0 then jth (THE k. Suc k = n) s else arbitrary)
+                              | Infty => arbitrary"
+
+  fsfilter_def:       "fsfilter A \<equiv> sfilter\<cdot>(flift2 (\<lambda>x. x\<in>A))"
+  fsmap_def:   	      "fsmap f  == smap$(flift2 f)"
+
+
+lemma ft_fsingleton[simp]: "ft$(<a>) = Def a"
+by (simp add: fsingleton_def2)
+
+lemma slen_fsingleton[simp]: "#(<a>) = Fin 1"
+by (simp add: fsingleton_def2 inat_defs)
+
+lemma slen_fstreams[simp]: "#(<a> ooo s) = iSuc (#s)"
+by (simp add: fsingleton_def2)
+
+lemma slen_fstreams2[simp]: "#(s ooo <a>) = iSuc (#s)"
+apply (case_tac "#s", auto)
+apply (insert slen_sconc [of _ s "Suc 0" "<a>"], auto)
+by (simp add: sconc_def)
+
+lemma j_th_0_fsingleton[simp]:"jth 0 (<a>) = a"
+apply (simp add: fsingleton_def2 jth_def)
+by (simp add: i_th_def Fin_0)
+
+lemma jth_0[simp]: "jth 0 (<a> ooo s) = a"  
+apply (simp add: fsingleton_def2 jth_def)
+by (simp add: i_th_def Fin_0)
+
+lemma first_sconc[simp]: "first (<a> ooo s) = a"
+by (simp add: first_def)
+
+lemma first_fsingleton[simp]: "first (<a>) = a"
+by (simp add: first_def)
+
+lemma jth_n[simp]: "Fin n = #s ==> jth n (s ooo <a>) = a"
+apply (simp add: jth_def, auto)
+apply (simp add: i_th_def rt_sconc1)
+by (simp add: inat_defs split: inat_splits)
+
+lemma last_sconc[simp]: "Fin n = #s ==> last (s ooo <a>) = a"
+apply (simp add: last_def)
+apply (simp add: inat_defs split:inat_splits)
+by (drule sym, auto)
+
+lemma last_fsingleton[simp]: "last (<a>) = a"
+by (simp add: last_def)
+
+lemma first_UU[simp]: "first UU = arbitrary"
+by (simp add: first_def jth_def)
+
+lemma last_UU[simp]:"last UU = arbitrary"
+by (simp add: last_def jth_def inat_defs)
+
+lemma last_infinite[simp]:"#s = Infty ==> last s = arbitrary"
+by (simp add: last_def)
+
+lemma jth_slen_lemma1:"n <= k & Fin n = #s ==> jth k s = arbitrary"
+by (simp add: jth_def inat_defs split:inat_splits, auto)
+
+lemma jth_UU[simp]:"jth n UU = arbitrary" 
+by (simp add: jth_def)
+
+lemma ext_last:"[|s ~= UU; Fin (Suc n) = #s|] ==> (stream_take n$s) ooo <(last s)> = s" 
+apply (simp add: last_def)
+apply (case_tac "#s", auto)
+apply (simp add: fsingleton_def2)
+apply (subgoal_tac "Def (jth n s) = i_th n s")
+apply (auto simp add: i_th_last)
+apply (drule slen_take_lemma1, auto)
+apply (simp add: jth_def)
+apply (case_tac "i_th n s = UU")
+apply auto
+apply (simp add: i_th_def)
+apply (case_tac "i_rt n s = UU", auto)
+apply (drule i_rt_slen [THEN iffD1])
+apply (drule slen_take_eq_rev [rule_format, THEN iffD2],auto)
+by (drule not_Undef_is_Def [THEN iffD1], auto)
+
+
+lemma fsingleton_lemma1[simp]: "(<a> = <b>) = (a=b)"
+by (simp add: fsingleton_def2)
+
+lemma fsingleton_lemma2[simp]: "<a> ~= <>"
+by (simp add: fsingleton_def2)
+
+lemma fsingleton_sconc:"<a> ooo s = Def a && s"
+by (simp add: fsingleton_def2)
+
+lemma fstreams_ind: 
+  "[| adm P; P <>; !!a s. P s ==> P (<a> ooo s) |] ==> P x"
+apply (simp add: fsingleton_def2)
+apply (rule stream.ind, auto)
+by (drule not_Undef_is_Def [THEN iffD1], auto)
+
+lemma fstreams_ind2:
+  "[| adm P; P <>; !!a. P (<a>); !!a b s. P s ==> P (<a> ooo <b> ooo s) |] ==> P x"
+apply (simp add: fsingleton_def2)
+apply (rule stream_ind2, auto)
+by (drule not_Undef_is_Def [THEN iffD1], auto)+
+
+lemma fstreams_take_Suc[simp]: "stream_take (Suc n)$(<a> ooo s) = <a> ooo stream_take n$s"
+by (simp add: fsingleton_def2)
+
+lemma fstreams_not_empty[simp]: "<a> ooo s ~= <>"
+by (simp add: fsingleton_def2)
+
+lemma fstreams_not_empty2[simp]: "s ooo <a> ~= <>"
+by (case_tac "s=UU", auto)
+
+lemma fstreams_exhaust: "x = UU | (EX a s. x = <a> ooo s)"
+apply (simp add: fsingleton_def2, auto)
+apply (erule contrapos_pp, auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+by (drule not_Undef_is_Def [THEN iffD1], auto)
+
+lemma fstreams_cases: "[| x = UU ==> P; !!a y. x = <a> ooo y ==> P |] ==> P"
+by (insert fstreams_exhaust [of x], auto)
+
+lemma fstreams_exhaust_eq: "(x ~= UU) = (? a y. x = <a> ooo y)"
+apply (simp add: fsingleton_def2, auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+by (drule not_Undef_is_Def [THEN iffD1], auto)
+
+lemma fstreams_inject: "(<a> ooo s = <b> ooo t) = (a=b & s=t)"
+by (simp add: fsingleton_def2)
+
+lemma fstreams_prefix: "<a> ooo s << t ==> EX tt. t = <a> ooo tt &  s << tt"
+apply (simp add: fsingleton_def2)
+apply (insert stream_prefix [of "Def a" s t], auto)
+by (drule stream.inverts, auto)
+
+lemma fstreams_prefix': "x << <a> ooo z = (x = <> |  (EX y. x = <a> ooo y &  y << z))"
+apply (auto, case_tac "x=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (simp add: fsingleton_def2, auto)
+apply (drule stream.inverts, auto)
+apply (drule ax_flat [rule_format], simp)
+apply (drule stream.inverts, auto)
+by (erule sconc_mono)
+
+lemma ft_fstreams[simp]: "ft$(<a> ooo s) = Def a"
+by (simp add: fsingleton_def2)
+
+lemma rt_fstreams[simp]: "rt$(<a> ooo s) = s"
+by (simp add: fsingleton_def2)
+
+lemma ft_eq[simp]: "(ft$s = Def a) = (EX t. s = <a> ooo t)"
+apply (rule stream.casedist [of s],auto)
+by (drule sym, auto simp add: fsingleton_def2)
+
+lemma surjective_fstreams: "(<d> ooo y = x) = (ft$x = Def d & rt$x = y)"
+by auto
+
+lemma fstreams_mono: "<a> ooo b << <a> ooo c ==> b << c"
+apply (simp add: fsingleton_def2)
+by (drule stream.inverts, auto)
+
+lemma fsmap_UU[simp]: "fsmap f$UU = UU"
+by (simp add: fsmap_def)
+
+lemma fsmap_fsingleton_sconc: "fsmap f$(<x> ooo xs) = <(f x)> ooo (fsmap f$xs)"
+by (simp add: fsmap_def fsingleton_def2 flift2_def)
+
+lemma fsmap_fsingleton[simp]: "fsmap f$(<x>) = <(f x)>"
+by (simp add: fsmap_def fsingleton_def2 flift2_def)
+
+
+declare range_composition[simp del]
+
+
+lemma fstreams_chain_lemma[rule_format]:
+  "ALL s x y. stream_take n$(s::'a fstream) << x & x << y & y << s & x ~= y --> stream_take (Suc n)$s << y"
+apply (induct_tac n, auto)
+apply (case_tac "s=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (case_tac "y=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (drule stream.inverts, auto)
+apply (simp add: less_lift_def, auto)
+apply (rule monofun_cfun, auto)
+apply (case_tac "s=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (erule_tac x="ya" in allE)
+apply (drule stream_prefix, auto)
+apply (case_tac "y=UU",auto)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (drule stream.inverts, auto)
+apply (simp add: less_lift_def)
+apply (drule stream.inverts, auto)+
+apply (erule_tac x="tt" in allE)
+apply (erule_tac x="yb" in allE, auto)
+apply (simp add: less_lift_def)
+by (rule monofun_cfun, auto)
+
+lemma fstreams_lub_lemma1: "[| chain Y; lub (range Y) = <a> ooo s |] ==> EX j t. Y j = <a> ooo t"
+apply (subgoal_tac "lub(range Y) ~= UU")
+apply (drule chain_UU_I_inverse2, auto)
+apply (drule_tac x="i" in is_ub_thelub, auto)
+by (drule fstreams_prefix' [THEN iffD1], auto)
+
+lemma fstreams_lub1: 
+ "[| chain Y; lub (range Y) = <a> ooo s |]
+     ==> (EX j t. Y j = <a> ooo t) & (EX X. chain X & (ALL i. EX j. <a> ooo X i << Y j) & lub (range X) = s)"
+apply (auto simp add: fstreams_lub_lemma1)
+apply (rule_tac x="%n. stream_take n$s" in exI, auto)
+apply (simp add: chain_stream_take)
+apply (induct_tac i, auto)
+apply (drule fstreams_lub_lemma1, auto)
+apply (rule_tac x="j" in exI, auto)
+apply (case_tac "max_in_chain j Y")
+apply (frule lub_finch1 [THEN thelubI], auto)
+apply (rule_tac x="j" in exI)
+apply (erule subst) back back
+apply (simp add: less_cprod_def sconc_mono)
+apply (simp add: max_in_chain_def, auto)
+apply (rule_tac x="ja" in exI)
+apply (subgoal_tac "Y j << Y ja")
+apply (drule fstreams_prefix, auto)+
+apply (rule sconc_mono)
+apply (rule fstreams_chain_lemma, auto)
+apply (subgoal_tac "Y ja << (LUB i. (Y i))", clarsimp)
+apply (drule fstreams_mono, simp)
+apply (rule is_ub_thelub, simp)
+apply (blast intro: chain_mono3)
+by (rule stream_reach2)
+
+
+lemma lub_Pair_not_UU_lemma: 
+  "[| chain Y; lub (range Y) = ((a::'a::flat), b); a ~= UU; b ~= UU |] 
+      ==> EX j c d. Y j = (c, d) & c ~= UU & d ~= UU"
+apply (frule thelub_cprod, clarsimp)
+apply (drule chain_UU_I_inverse2, clarsimp)
+apply (case_tac "Y i", clarsimp)
+apply (case_tac "max_in_chain i Y")
+apply (drule maxinch_is_thelub, auto)
+apply (rule_tac x="i" in exI, auto)
+apply (simp add: max_in_chain_def, auto)
+apply (subgoal_tac "Y i << Y j",auto)
+apply (simp add: less_cprod_def, clarsimp)
+apply (drule ax_flat [rule_format], auto)
+apply (case_tac "snd (Y j) = UU",auto)
+apply (case_tac "Y j", auto)
+apply (rule_tac x="j" in exI)
+apply (case_tac "Y j",auto)
+by (drule chain_mono3, auto)
+
+lemma fstreams_lub_lemma2: 
+  "[| chain Y; lub (range Y) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] ==> EX j t. Y j = (a, <m> ooo t)"
+apply (frule lub_Pair_not_UU_lemma, auto)
+apply (drule_tac x="j" in is_ub_thelub, auto)
+apply (simp add: less_cprod_def, clarsimp)
+apply (drule ax_flat [rule_format], clarsimp)
+by (drule fstreams_prefix' [THEN iffD1], auto)
+
+lemma fstreams_lub2:
+  "[| chain Y; lub (range Y) = (a, <m> ooo ms); (a::'a::flat) ~= UU |] 
+      ==> (EX j t. Y j = (a, <m> ooo t)) & (EX X. chain X & (ALL i. EX j. (a, <m> ooo X i) << Y j) & lub (range X) = ms)"
+apply (auto simp add: fstreams_lub_lemma2)
+apply (rule_tac x="%n. stream_take n$ms" in exI, auto)
+apply (simp add: chain_stream_take)
+apply (induct_tac i, auto)
+apply (drule fstreams_lub_lemma2, auto)
+apply (rule_tac x="j" in exI, auto)
+apply (simp add: less_cprod_def)
+apply (case_tac "max_in_chain j Y")
+apply (frule lub_finch1 [THEN thelubI], auto)
+apply (rule_tac x="j" in exI)
+apply (erule subst) back back
+apply (simp add: less_cprod_def sconc_mono)
+apply (simp add: max_in_chain_def, auto)
+apply (rule_tac x="ja" in exI)
+apply (subgoal_tac "Y j << Y ja")
+apply (simp add: less_cprod_def, auto)
+apply (drule trans_less)
+apply (simp add: ax_flat, auto)
+apply (drule fstreams_prefix, auto)+
+apply (rule sconc_mono)
+apply (subgoal_tac "tt ~= tta" "tta << ms")
+apply (blast intro: fstreams_chain_lemma)
+apply (frule lub_cprod [THEN thelubI], auto)
+apply (subgoal_tac "snd (Y ja) << (LUB i. snd (Y i))", clarsimp)
+apply (drule fstreams_mono, simp)
+apply (rule is_ub_thelub chainI)
+apply (simp add: chain_def less_cprod_def)
+apply (subgoal_tac "fst (Y j) ~= fst (Y ja) | snd (Y j) ~= snd (Y ja)", simp)
+apply (drule ax_flat[rule_format], simp)+
+apply (drule prod_eqI, auto)
+apply (simp add: chain_mono3)
+by (rule stream_reach2)
+
+
+lemma cpo_cont_lemma:
+  "[| monofun (f::'a::cpo => 'b::cpo); (!Y. chain Y --> f (lub(range Y)) << (LUB i. f (Y i))) |] ==> cont f"
+apply (rule monocontlub2cont, auto)
+apply (simp add: contlub, auto)
+apply (erule_tac x="Y" in allE, auto)
+apply (simp add: po_eq_conv)
+apply (frule cpo,auto)
+apply (frule is_lubD1)
+apply (frule ub2ub_monofun, auto)
+apply (drule thelubI, auto)
+apply (rule is_lub_thelub, auto)
+by (erule ch2ch_monofun, simp)
+
+end
+
+
+
+
--- a/src/HOLCF/FOCUS/ROOT.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/ROOT.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -9,5 +9,6 @@
 val banner = "HOLCF/FOCUS";
 writeln banner;
 
+use_thy "Fstreams";
 use_thy "FOCUS";
 use_thy "Buffer_adm";
--- a/src/HOLCF/FOCUS/Stream_adm.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/FOCUS/Stream_adm.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -24,12 +24,12 @@
 by ( dtac spec 1);
 by ( Safe_tac);
 by ( rtac exI 1);
-by ( rtac slen_strict_mono 1);
+by ( rtac (thm "slen_strict_mono") 1);
 by (   etac spec 1);
 by (  atac 1);
 by ( atac 1);
 by (dtac (not_ex RS iffD1) 1);
-by (stac slen_infinite 1);
+by (stac (thm "slen_infinite") 1);
 by (etac thin_rl 1);
 by (dtac spec 1);
 by (fold_goals_tac [thm "ile_def"]);
@@ -67,16 +67,16 @@
 by (dres_inst_tac [("P","%x. x")] subst 1 THEN atac 1);
 by (etac allE 1 THEN dtac mp 1 THEN rtac ile_lemma 1);
 by ( etac (thm "ile_trans") 1);
-by ( etac slen_mono 1);
+by ( etac (thm "slen_mono") 1);
 by (etac ssubst 1);
 by (safe_tac HOL_cs);
-by ( eatac (ile_lemma RS slen_take_lemma3 RS subst) 2 1);
+by ( eatac (ile_lemma RS thm "slen_take_lemma3" RS subst) 2 1);
 by (etac allE 1);
 by (etac allE 1);
 by (dtac mp 1);
-by ( etac slen_rt_mult 1);
+by ( etac (thm "slen_rt_mult") 1);
 by (dtac mp 1);
-by (etac monofun_rt_mult 1);
+by (etac (thm "monofun_rt_mult") 1);
 by (mp_tac 1); 
 by (atac 1);
 qed "stream_monoP2I";
@@ -123,9 +123,9 @@
         etac allE 1 THEN mp_tac 1,
 	dres_inst_tac [("P","%x. x")] subst 1 THEN atac 1,
 	etac conjE 1 THEN rtac conjI 1,
-	 etac (slen_take_lemma3 RS ssubst) 1 THEN atac 1,
+	 etac (thm "slen_take_lemma3" RS ssubst) 1 THEN atac 1,
 	 atac 1,
-	etac allE 1 THEN etac allE 1 THEN dtac mp 1 THEN etac monofun_rt_mult 1,
+	etac allE 1 THEN etac allE 1 THEN dtac mp 1 THEN etac (thm "monofun_rt_mult") 1,
 	mp_tac 1,
 	atac 1]);
 
--- a/src/HOLCF/IsaMakefile	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/IsaMakefile	Tue Sep 07 16:02:42 2004 +0200
@@ -39,7 +39,7 @@
   Up1.thy Up2.ML Up2.thy Up3.ML Up3.thy adm.ML cont_consts.ML \
   domain/axioms.ML domain/extender.ML domain/interface.ML \
   domain/library.ML domain/syntax.ML domain/theorems.ML holcf_logic.ML \
-  ex/Stream.thy ex/Stream.ML Streams.thy
+  ex/Stream.thy
 	@$(ISATOOL) usedir -b -r $(OUT)/HOL HOLCF
 
 
@@ -68,7 +68,8 @@
 
 HOLCF-FOCUS: HOLCF $(LOG)/HOLCF-FOCUS.gz
 
-$(LOG)/HOLCF-FOCUS.gz: $(OUT)/HOLCF FOCUS/Fstream.thy FOCUS/Fstream.ML \
+$(LOG)/HOLCF-FOCUS.gz: $(OUT)/HOLCF FOCUS/Fstreams.thy \
+  FOCUS/Fstream.thy FOCUS/Fstream.ML \
   FOCUS/FOCUS.thy FOCUS/FOCUS.ML \
   FOCUS/Stream_adm.thy FOCUS/Stream_adm.ML  ../HOL/Library/Continuity.thy \
   FOCUS/Buffer.thy FOCUS/Buffer.ML FOCUS/Buffer_adm.thy FOCUS/Buffer_adm.ML
--- a/src/HOLCF/ROOT.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/ROOT.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -24,6 +24,5 @@
 use "domain/interface.ML";
 
 path_add "~~/src/HOLCF/ex";
-use_thy "Streams";
 
 print_depth 10;  
--- a/src/HOLCF/Streams.thy	Tue Sep 07 15:59:16 2004 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,564 +0,0 @@
-(*  Title: 	HOLCF/Streams.thy
-    ID:         $Id$
-    Author: 	Borislav Gajanovic and David von Oheimb
-
-Stream domains with concatenation.
-TODO: HOLCF/ex/Stream.* should be integrated into this file.
-*)
-
-theory Streams = Stream :
-
-(* ----------------------------------------------------------------------- *)
-
-lemma stream_neq_UU: "x~=UU ==> EX a as. x=a&&as & a~=UU"
-by (simp add: stream_exhaust_eq,auto)
-
-lemma stream_prefix1: "[| x<<y; xs<<ys |] ==> x&&xs << y&&ys"
-by (insert stream_prefix' [of y "x&&xs" ys],force)
-
-lemma stream_take_le_mono : "k<=n ==> stream_take k$s1 << stream_take n$s1"
-apply (insert chain_stream_take [of s1])
-by (drule chain_mono3,auto)
-
-lemma mono_stream_take: "s1 << s2 ==> stream_take n$s1 << stream_take n$s2"
-by (simp add: monofun_cfun_arg)
-
-lemma stream_take_prefix [simp]: "stream_take n$s << s"
-apply (subgoal_tac "s=(LUB n. stream_take n$s)")
- apply (erule ssubst, rule is_ub_thelub)
- apply (simp only: chain_stream_take)
-by (simp only: stream_reach2)
-
-lemma stream_take_take_less:"stream_take k$(stream_take n$s) << stream_take k$s"
-by (rule monofun_cfun_arg,auto)
-
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)"
-apply (rule stream.casedist [of s1])
- apply (rule stream.casedist [of s2],simp+)
-by (rule stream.casedist [of s2],auto)
-
-lemma slen_take_lemma4 [rule_format]: 
-  "!s. stream_take n$s ~= s --> #(stream_take n$s) = Fin n"
-apply (induct_tac n,auto simp add: Fin_0)
-apply (case_tac "s=UU",simp)
-by (drule stream_neq_UU,auto)
-
-lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n"; 
-apply (case_tac "stream_take n$s = s")
- apply (simp add: slen_take_eq_rev)
-by (simp add: slen_take_lemma4)
-
-lemma stream_take_idempotent [simp]: 
- "stream_take n$(stream_take n$s) = stream_take n$s"
-apply (case_tac "stream_take n$s = s")
-apply (auto,insert slen_take_lemma4 [of n s]);
-by (auto,insert slen_take_lemma1 [of "stream_take n$s" n],simp)
-
-lemma stream_take_take_Suc [simp]: "stream_take n$(stream_take (Suc n)$s) = 
-                                    stream_take n$s"
-apply (simp add: po_eq_conv,auto)
- apply (simp add: stream_take_take_less)
-apply (subgoal_tac "stream_take n$s = stream_take n$(stream_take n$s)")
- apply (erule ssubst)
- apply (rule_tac monofun_cfun_arg)
- apply (insert chain_stream_take [of s])
-by (simp add: chain_def,simp)
-
-lemma mono_stream_take_pred: 
-  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
-                       stream_take n$s1 << stream_take n$s2"
-by (drule mono_stream_take [of _ _ n],simp)
-
-lemma stream_take_lemma10 [rule_format]:
-  "ALL k<=n. stream_take n$s1 << stream_take n$s2 
-                             --> stream_take k$s1 << stream_take k$s2"
-apply (induct_tac n,simp,clarsimp)
-apply (case_tac "k=Suc n",blast)
-apply (erule_tac x="k" in allE)
-by (drule mono_stream_take_pred,simp)
-
-lemma stream_take_finite [simp]: "stream_finite (stream_take n$s)"
-apply (simp add: stream.finite_def)
-by (rule_tac x="n" in exI,simp)
-
-lemma slen_stream_take_finite [simp]: "#(stream_take n$s) ~= \<infinity>"
-by (simp add: slen_def)
-
-lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==> 
-                     stream_take n$s ~= stream_take (Suc n)$s"
-apply auto
-apply (subgoal_tac "stream_take n$s ~=s")
- apply (insert slen_take_lemma4 [of n s],auto)
-apply (rule stream.casedist [of s],simp)
-apply (simp add: inat_defs split:inat_splits)
-by (simp add: slen_take_lemma4)
-
-
-(* ----------------------------------------------------------------------- *)
-
-consts
- 
-  i_rt :: "nat => 'a stream => 'a stream" (* chops the first i elements *)
-  i_th :: "nat => 'a stream => 'a"        (* the i-th element *)
-
-  sconc         :: "'a stream => 'a stream => 'a stream" (infixr "ooo" 65) 
-  constr_sconc  :: "'a stream => 'a stream => 'a stream" (* constructive *)
-  constr_sconc' :: "nat => 'a stream => 'a stream => 'a stream" 
-
-defs
-  i_rt_def: "i_rt == %i s. iterate i rt s"  
-  i_th_def: "i_th == %i s. ft$(i_rt i s)" 
-
-  sconc_def: "s1 ooo s2 == case #s1 of 
-                       Fin n => (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))
-                     | \<infinity>     => s1" 
-
-  constr_sconc_def: "constr_sconc s1 s2 == case #s1 of 
-                                             Fin n => constr_sconc' n s1 s2 
-                                           | \<infinity>    => s1"
-primrec 
-  constr_sconc'_0:   "constr_sconc' 0 s1 s2 = s2"
-  constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 && 
-                                                    constr_sconc' n (rt$s1) s2"
-
-
-(* ----------------------------------------------------------------------- *)
-   section "i_rt"
-(* ----------------------------------------------------------------------- *)
-
-lemma i_rt_UU [simp]: "i_rt n UU = UU"
-apply (simp add: i_rt_def)
-by (rule iterate.induct,auto)
-
-lemma i_rt_0 [simp]: "i_rt 0 s = s"
-by (simp add: i_rt_def)
-
-lemma i_rt_Suc [simp]: "a ~= UU ==> i_rt (Suc n) (a&&s) = i_rt n s"
-by (simp add: i_rt_def iterate_Suc2 del: iterate_Suc)
-
-lemma i_rt_Suc_forw: "i_rt (Suc n) s = i_rt n (rt$s)"
-by (simp only: i_rt_def iterate_Suc2)
-
-lemma i_rt_Suc_back:"i_rt (Suc n) s = rt$(i_rt n s)"
-by (simp only: i_rt_def,auto)
-
-lemma i_rt_mono: "x << s ==> i_rt n x  << i_rt n s"
-by (simp add: i_rt_def monofun_rt_mult)
-
-lemma i_rt_ij_lemma: "Fin (i + j) <= #x ==> Fin j <= #(i_rt i x)"
-by (simp add: i_rt_def slen_rt_mult)
-
-lemma slen_i_rt_mono: "#s2 <= #s1 ==> #(i_rt n s2) <= #(i_rt n s1)"
-apply (induct_tac n,auto)
-apply (simp add: i_rt_Suc_back)
-by (drule slen_rt_mono,simp)
-
-lemma i_rt_take_lemma1 [rule_format]: "ALL s. i_rt n (stream_take n$s) = UU"
-apply (induct_tac n); 
- apply (simp add: i_rt_Suc_back,auto)
-apply (case_tac "s=UU",auto)
-by (drule stream_neq_UU,simp add: i_rt_Suc_forw,auto)
-
-lemma i_rt_slen: "(i_rt n s = UU) = (stream_take n$s = s)"
-apply auto
- apply (insert i_rt_ij_lemma [of n "Suc 0" s]);
- apply (subgoal_tac "#(i_rt n s)=0")
-  apply (case_tac "stream_take n$s = s",simp+)
-  apply (insert slen_take_eq [of n s],simp)
-  apply (simp add: inat_defs split:inat_splits)
- apply (simp add: slen_take_eq )
-by (simp, insert i_rt_take_lemma1 [of n s],simp)
-
-lemma i_rt_lemma_slen: "#s=Fin n ==> i_rt n s = UU"
-by (simp add: i_rt_slen slen_take_lemma1)
-
-lemma stream_finite_i_rt [simp]: "stream_finite (i_rt n s) = stream_finite s"
-apply (induct_tac n, auto)
- apply (rule stream.casedist [of "s"], auto simp del: i_rt_Suc)
-by (simp add: i_rt_Suc_back stream_finite_rt_eq)+
-
-lemma take_i_rt_len_lemma: "ALL sl x j t. Fin sl = #x & n <= sl &
-                            #(stream_take n$x) = Fin t & #(i_rt n x)= Fin j 
-                                              --> Fin (j + t) = #x"
-apply (induct_tac n,auto)
- apply (simp add: inat_defs)
-apply (case_tac "x=UU",auto)
- apply (simp add: inat_defs)
-apply (drule stream_neq_UU,auto)
-apply (subgoal_tac "EX k. Fin k = #as",clarify)
- apply (erule_tac x="k" in allE)
- apply (erule_tac x="as" in allE,auto)
- apply (erule_tac x="THE p. Suc p = t" in allE,auto)
-   apply (simp add: inat_defs split:inat_splits)
-  apply (simp add: inat_defs split:inat_splits)
-  apply (simp only: the_equality)
- apply (simp add: inat_defs split:inat_splits)
- apply force
-by (simp add: inat_defs split:inat_splits)
-
-lemma take_i_rt_len: 
-"[| Fin sl = #x; n <= sl; #(stream_take n$x) = Fin t; #(i_rt n x) = Fin j |] ==>
-    Fin (j + t) = #x"
-by (blast intro: take_i_rt_len_lemma [rule_format])
-
-
-(* ----------------------------------------------------------------------- *)
-   section "i_th"
-(* ----------------------------------------------------------------------- *)
-
-lemma i_th_i_rt_step:
-"[| i_th n s1 << i_th n s2; i_rt (Suc n) s1 << i_rt (Suc n) s2 |] ==> 
-   i_rt n s1 << i_rt n s2"
-apply (simp add: i_th_def i_rt_Suc_back)
-apply (rule stream.casedist [of "i_rt n s1"],simp)
-apply (rule stream.casedist [of "i_rt n s2"],auto)
-by (drule stream_prefix1,auto)
-
-lemma i_th_stream_take_Suc [rule_format]: 
- "ALL s. i_th n (stream_take (Suc n)$s) = i_th n s"
-apply (induct_tac n,auto)
- apply (simp add: i_th_def)
- apply (case_tac "s=UU",auto)
- apply (drule stream_neq_UU,auto)
-apply (case_tac "s=UU",simp add: i_th_def)
-apply (drule stream_neq_UU,auto)
-by (simp add: i_th_def i_rt_Suc_forw)
-
-lemma last_lemma10: "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==> 
-                     i_th n s1 << i_th n s2"
-apply (rule i_th_stream_take_Suc [THEN subst])
-apply (rule i_th_stream_take_Suc [THEN subst]) back
-apply (simp add: i_th_def)
-apply (rule monofun_cfun_arg)
-by (erule i_rt_mono)
-
-lemma i_th_last: "i_th n s && UU = i_rt n (stream_take (Suc n)$s)"
-apply (insert surjectiv_scons [of "i_rt n (stream_take (Suc n)$s)"])
-apply (rule i_th_stream_take_Suc [THEN subst])
-apply (simp add: i_th_def  i_rt_Suc_back [symmetric])
-by (simp add: i_rt_take_lemma1)
-
-lemma i_th_last_eq: 
-"i_th n s1 = i_th n s2 ==> i_rt n (stream_take (Suc n)$s1) = i_rt n (stream_take (Suc n)$s2)"
-apply (insert i_th_last [of n s1])
-apply (insert i_th_last [of n s2])
-by auto
-
-lemma i_th_prefix_lemma:
-"[| k <= n; stream_take (Suc n)$s1 << stream_take (Suc n)$s2 |] ==> 
-    i_th k s1 << i_th k s2"
-apply (subgoal_tac "stream_take (Suc k)$s1 << stream_take (Suc k)$s2")
- apply (simp add: last_lemma10)
-by (blast intro: stream_take_lemma10)
-
-lemma take_i_rt_prefix_lemma1: 
-  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
-   i_rt (Suc n) s1 << i_rt (Suc n) s2 ==> 
-   i_rt n s1 << i_rt n s2 & stream_take n$s1 << stream_take n$s2"
-apply auto
- apply (insert i_th_prefix_lemma [of n n s1 s2])
- apply (rule i_th_i_rt_step,auto)
-by (drule mono_stream_take_pred,simp)
-
-lemma take_i_rt_prefix_lemma: 
-"[| stream_take n$s1 << stream_take n$s2; i_rt n s1 << i_rt n s2 |] ==> s1 << s2"
-apply (case_tac "n=0",simp)
-apply (insert neq0_conv [of n])
-apply (insert not0_implies_Suc [of n],auto)
-apply (subgoal_tac "stream_take 0$s1 << stream_take 0$s2 & 
-                    i_rt 0 s1 << i_rt 0 s2")
- defer 1
- apply (rule zero_induct,blast)
- apply (blast dest: take_i_rt_prefix_lemma1)
-by simp
-
-lemma streams_prefix_lemma: "(s1 << s2) = 
-  (stream_take n$s1 << stream_take n$s2 & i_rt n s1 << i_rt n s2)"; 
-apply auto
-  apply (simp add: monofun_cfun_arg)
- apply (simp add: i_rt_mono)
-by (erule take_i_rt_prefix_lemma,simp)
-
-lemma streams_prefix_lemma1:
- "[| stream_take n$s1 = stream_take n$s2; i_rt n s1 = i_rt n s2 |] ==> s1 = s2"
-apply (simp add: po_eq_conv,auto)
- apply (insert streams_prefix_lemma)
- by blast+
-
-
-(* ----------------------------------------------------------------------- *)
-   section "sconc"
-(* ----------------------------------------------------------------------- *)
-
-lemma UU_sconc [simp]: " UU ooo s = s "
-by (simp add: sconc_def inat_defs)
-
-lemma scons_neq_UU: "a~=UU ==> a && s ~=UU"
-by auto
-
-lemma singleton_sconc [rule_format, simp]: "x~=UU --> (x && UU) ooo y = x && y"
-apply (simp add: sconc_def inat_defs split:inat_splits,auto)
-apply (rule someI2_ex,auto)
- apply (rule_tac x="x && y" in exI,auto)
-apply (simp add: i_rt_Suc_forw)
-apply (case_tac "xa=UU",simp)
-by (drule stream_neq_UU,auto)
-
-lemma ex_sconc [rule_format]: 
-  "ALL k y. #x = Fin k --> (EX w. stream_take k$w = x & i_rt k w = y)"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of x],auto)
-  apply (simp add: stream.finite_def)
-  apply (drule slen_take_lemma1,blast)
- apply (simp add: inat_defs split:inat_splits)+
-apply (erule_tac x="y" in allE,auto)
-by (rule_tac x="a && w" in exI,auto)
-
-lemma rt_sconc1: "Fin n = #x ==> i_rt n (x ooo y) = y"; 
-apply (simp add: sconc_def inat_defs split:inat_splits , arith?,auto)
-apply (rule someI2_ex,auto)
-by (drule ex_sconc,simp)
-
-lemma sconc_inj2: "\<lbrakk>Fin n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z"
-apply (frule_tac y=y in rt_sconc1)
-by (auto elim: rt_sconc1)
-
-lemma sconc_UU [simp]:"s ooo UU = s"
-apply (case_tac "#s")
- apply (simp add: sconc_def inat_defs)
- apply (rule someI2_ex)
-  apply (rule_tac x="s" in exI)
-  apply auto
-   apply (drule slen_take_lemma1,auto)
-  apply (simp add: i_rt_lemma_slen)
- apply (drule slen_take_lemma1,auto)
- apply (simp add: i_rt_slen)
-by (simp add: sconc_def inat_defs)
-
-lemma stream_take_sconc [simp]: "Fin n = #x ==> stream_take n$(x ooo y) = x"
-apply (simp add: sconc_def)
-apply (simp add: inat_defs split:inat_splits,auto)
-apply (rule someI2_ex,auto)
-by (drule ex_sconc,simp)
-
-lemma scons_sconc [rule_format,simp]: "a~=UU --> (a && x) ooo y = a && x ooo y"
-apply (case_tac "#x",auto)
- apply (simp add: sconc_def) 
- apply (rule someI2_ex)
-  apply (drule ex_sconc,simp)
- apply (rule someI2_ex,auto)
-  apply (simp add: i_rt_Suc_forw)
-  apply (rule_tac x="a && x" in exI,auto)
- apply (case_tac "xa=UU",auto)
-  apply (drule_tac s="stream_take nat$x" in scons_neq_UU)
-  apply (simp add: i_rt_Suc_forw)
- apply (drule stream_neq_UU,clarsimp)
- apply (drule streams_prefix_lemma1,simp+)
-by (simp add: sconc_def)
-
-lemma ft_sconc: "x ~= UU ==> ft$(x ooo y) = ft$x"
-by (rule stream.casedist [of x],auto)
-
-lemma sconc_assoc: "(x ooo y) ooo z = x ooo y ooo z"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)
-  apply (simp add: stream.finite_def del: scons_sconc)
-  apply (drule slen_take_lemma1,auto simp del: scons_sconc)
- apply (case_tac "a = UU", auto)
-by (simp add: sconc_def)
-
-
-(* ----------------------------------------------------------------------- *)
-
-lemma sconc_mono: "y << y' ==> x ooo y << x ooo y'"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of "x"])
-   apply (auto simp add: stream.finite_def)
-  apply (drule slen_take_lemma1,blast)
- by (simp add: stream_prefix',auto simp add: sconc_def)
-
-lemma sconc_mono1 [simp]: "x << x ooo y"
-by (rule sconc_mono [of UU, simplified])
-
-(* ----------------------------------------------------------------------- *)
-
-lemma empty_sconc [simp]: "(x ooo y = UU) = (x = UU & y = UU)"
-apply (case_tac "#x",auto)
-   by (insert sconc_mono1 [of x y],auto);
-
-(* ----------------------------------------------------------------------- *)
-
-lemma rt_sconc [rule_format, simp]: "s~=UU --> rt$(s ooo x) = rt$s ooo x"
-by (rule stream.casedist,auto)
-
-(* ----------------------------------------------------------------------- *)
-
-lemma sconc_lemma [rule_format, simp]: "ALL s. stream_take n$s ooo i_rt n s = s"
-apply (induct_tac n,auto)
-apply (case_tac "s=UU",auto)
-by (drule stream_neq_UU,auto)
-
-(* ----------------------------------------------------------------------- *)
-   subsection "pointwise equality"
-(* ----------------------------------------------------------------------- *)
-
-lemma ex_last_stream_take_scons: "stream_take (Suc n)$s = 
-                     stream_take n$s ooo i_rt n (stream_take (Suc n)$s)"
-by (insert sconc_lemma [of n "stream_take (Suc n)$s"],simp)
-
-lemma i_th_stream_take_eq: 
-"!!n. ALL n. i_th n s1 = i_th n s2 ==> stream_take n$s1 = stream_take n$s2"
-apply (induct_tac n,auto)
-apply (subgoal_tac "stream_take (Suc na)$s1 =
-                    stream_take na$s1 ooo i_rt na (stream_take (Suc na)$s1)")
- apply (subgoal_tac "i_rt na (stream_take (Suc na)$s1) = 
-                    i_rt na (stream_take (Suc na)$s2)")
-  apply (subgoal_tac "stream_take (Suc na)$s2 = 
-                    stream_take na$s2 ooo i_rt na (stream_take (Suc na)$s2)")
-   apply (insert ex_last_stream_take_scons,simp)
-  apply blast
- apply (erule_tac x="na" in allE)
- apply (insert i_th_last_eq [of _ s1 s2])
-by blast+
-
-lemma pointwise_eq_lemma[rule_format]: "ALL n. i_th n s1 = i_th n s2 ==> s1 = s2"
-by (insert i_th_stream_take_eq [THEN stream.take_lemmas],blast)
-
-(* ----------------------------------------------------------------------- *)
-   subsection "finiteness"
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_sconc_finite1:
-  "[| #(x ooo y) = Infty; Fin n = #x |] ==> #y = Infty"
-apply (case_tac "#y ~= Infty",auto)
-apply (simp only: slen_infinite [symmetric])
-apply (drule_tac y=y in rt_sconc1)
-apply (insert stream_finite_i_rt [of n "x ooo y"])
-by (simp add: slen_infinite)
-
-lemma slen_sconc_infinite1: "#x=Infty ==> #(x ooo y) = Infty"
-by (simp add: sconc_def)
-
-lemma slen_sconc_infinite2: "#y=Infty ==> #(x ooo y) = Infty"
-apply (case_tac "#x")
- apply (simp add: sconc_def)
- apply (rule someI2_ex)
-  apply (drule ex_sconc,auto)
- apply (erule contrapos_pp)
- apply (insert stream_finite_i_rt)
- apply (simp add: slen_infinite ,auto)
-by (simp add: sconc_def)
-
-lemma sconc_finite: "(#x~=Infty & #y~=Infty) = (#(x ooo y)~=Infty)"
-apply auto
-  apply (case_tac "#x",auto)
-  apply (erule contrapos_pp,simp)
-  apply (erule slen_sconc_finite1,simp)
- apply (drule slen_sconc_infinite1 [of _ y],simp)
-by (drule slen_sconc_infinite2 [of _ x],simp)
-
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_sconc_mono3: "[| Fin n = #x; Fin k = #(x ooo y) |] ==> n <= k"
-apply (insert slen_mono [of "x" "x ooo y"])
-by (simp add: inat_defs split: inat_splits)
-
-(* ----------------------------------------------------------------------- *)
-   subsection "finite slen"
-(* ----------------------------------------------------------------------- *)
-
-lemma slen_sconc: "[| Fin n = #x; Fin m = #y |] ==> #(x ooo y) = Fin (n + m)"
-apply (case_tac "#(x ooo y)")
- apply (frule_tac y=y in rt_sconc1)
- apply (insert take_i_rt_len [of "THE j. Fin j = #(x ooo y)" "x ooo y" n n m],simp)
- apply (insert slen_sconc_mono3 [of n x _ y],simp)
-by (insert sconc_finite [of x y],auto)
-
-(* ----------------------------------------------------------------------- *)
-   subsection "flat prefix"
-(* ----------------------------------------------------------------------- *)
-
-lemma sconc_prefix: "(s1::'a::flat stream) << s2 ==> EX t. s1 ooo t = s2"
-apply (case_tac "#s1")
- apply (subgoal_tac "stream_take nat$s1 = stream_take nat$s2");
-  apply (rule_tac x="i_rt nat s2" in exI)
-  apply (simp add: sconc_def)
-  apply (rule someI2_ex)
-   apply (drule ex_sconc)
-   apply (simp,clarsimp,drule streams_prefix_lemma1)
-   apply (simp+,rule slen_take_lemma3 [rule_format, of _ s1 s2]);
-  apply (simp+,rule_tac x="UU" in exI)
-apply (insert slen_take_lemma3 [rule_format, of _ s1 s2]);
-by (rule stream.take_lemmas,simp)
-
-(* ----------------------------------------------------------------------- *)
-   subsection "continuity"
-(* ----------------------------------------------------------------------- *)
-
-lemma chain_sconc: "chain S ==> chain (%i. (x ooo S i))"
-by (simp add: chain_def,auto simp add: sconc_mono)
-
-lemma chain_scons: "chain S ==> chain (%i. a && S i)"
-apply (simp add: chain_def,auto)
-by (rule monofun_cfun_arg,simp)
-
-lemma contlub_scons: "contlub (%x. a && x)"
-by (simp add: contlub_Rep_CFun2)
-
-lemma contlub_scons_lemma: "chain S ==> (LUB i. a && S i) = a && (LUB i. S i)"
-apply (insert contlub_scons [of a])
-by (simp only: contlub)
-
-lemma finite_lub_sconc: "chain Y ==> (stream_finite x) ==> 
-                        (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
-apply (rule stream_finite_ind [of x])
- apply (auto)
-apply (subgoal_tac "(LUB i. a && (s ooo Y i)) = a && (LUB i. s ooo Y i)")
- by (force,blast dest: contlub_scons_lemma chain_sconc)
-
-lemma contlub_sconc_lemma: 
-  "chain Y ==> (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
-apply (case_tac "#x=Infty")
- apply (simp add: sconc_def)
- prefer 2
- apply (drule finite_lub_sconc,auto simp add: slen_infinite)
-apply (simp add: slen_def)
-apply (insert lub_const [of x] unique_lub [of _ x _])
-by (auto simp add: lub)
-
-lemma contlub_sconc: "contlub (%y. x ooo y)"; 
-by (rule contlubI, insert contlub_sconc_lemma [of _ x], simp);
-
-lemma monofun_sconc: "monofun (%y. x ooo y)"
-by (simp add: monofun sconc_mono)
-
-lemma cont_sconc: "cont (%y. x ooo y)"
-apply (rule  monocontlub2cont)
- apply (rule monofunI, simp add: sconc_mono)
-by (rule contlub_sconc);
-
-
-(* ----------------------------------------------------------------------- *)
-   section "constr_sconc"
-(* ----------------------------------------------------------------------- *)
-
-lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s"
-by (simp add: constr_sconc_def inat_defs)
-
-lemma "x ooo y = constr_sconc x y"
-apply (case_tac "#x")
- apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)
-  defer 1
-  apply (simp add: constr_sconc_def del: scons_sconc)
-  apply (case_tac "#s")
-   apply (simp add: inat_defs)
-   apply (case_tac "a=UU",auto simp del: scons_sconc)
-   apply (simp)
-  apply (simp add: sconc_def)
- apply (simp add: constr_sconc_def)
-apply (simp add: stream.finite_def)
-by (drule slen_take_lemma1,auto)
-
-end
--- a/src/HOLCF/ex/Dagstuhl.ML	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/ex/Dagstuhl.ML	Tue Sep 07 16:02:42 2004 +0200
@@ -30,12 +30,12 @@
 val lemma5=result();
 
 val prems = goal Dagstuhl.thy "YS = YYS";
-by (resolve_tac stream.take_lemmas 1);
+by (resolve_tac (thms "stream.take_lemmas") 1);
 by (induct_tac "n" 1);
-by (simp_tac (simpset() addsimps stream.rews) 1);
+by (simp_tac (simpset() addsimps (thms "stream.rews")) 1);
 by (stac YS_def2 1);
 by (stac YYS_def2 1);
-by (asm_simp_tac (simpset() addsimps stream.rews) 1);
+by (asm_simp_tac (simpset() addsimps (thms "stream.rews")) 1);
 by (rtac (lemma5 RS sym RS subst) 1);
 by (rtac refl 1);
 val wir_moel=result();
--- a/src/HOLCF/ex/Stream.ML	Tue Sep 07 15:59:16 2004 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,594 +0,0 @@
-(*  Title: 	HOLCF/ex//Stream.ML
-    ID:         $Id$
-    Author: 	Franz Regensburger and David von Oheimb, TU Muenchen
-
-General Stream domain.
-*)
-
-fun stream_case_tac s i = res_inst_tac [("x",s)] stream.casedist i
-			  THEN hyp_subst_tac i THEN hyp_subst_tac (i+1);
-
-
-val [stream_con_rew1,stream_con_rew2] = stream.con_rews;
-
-Addsimps stream.rews;
-Addsimps[eq_UU_iff RS sym];
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about scons                                                    *)
-(* ----------------------------------------------------------------------- *)
-
-section "scons";
-
-Goal "(a && s = UU) = (a = UU)";
-by Safe_tac;
-by (etac contrapos_pp 1);
-by (safe_tac (claset() addSIs stream.con_rews));
-qed "scons_eq_UU";
-
-Goal "[| a && x = UU; a ~= UU |] ==> R";
-by (dtac stream_con_rew2 1);
-by (contr_tac 1);
-qed "scons_not_empty";
-
-Goal "(x ~= UU) = (EX a y. a ~= UU &  x = a && y)";
-by (cut_facts_tac [stream.exhaust] 1);
-by (best_tac (claset() addDs [stream_con_rew2]) 1);
-qed "stream_exhaust_eq";
-
-Goal 
-"[| a && s << t; a ~= UU  |] ==> EX b tt. t = b && tt &  b ~= UU &  s << tt";
-by (stream_case_tac "t" 1);
-by (fast_tac (claset() addDs [eq_UU_iff RS iffD2, stream_con_rew2]) 1);
-by (fast_tac (claset() addDs stream.inverts) 1);
-qed "stream_prefix";
-
-Goal "[| x && xs << y && ys; (x::'a::flat) ~= UU|] ==> x = y & xs << ys";
-by (cut_inst_tac [("x1","x::'a::flat"), ("x","y::'a::flat")] 
-                 (ax_flat RS spec RS spec) 1);
-by (forward_tac stream.inverts 1);
-by Safe_tac;
-by (dtac (hd stream.con_rews RS subst) 1);
-by (fast_tac (claset() addDs [eq_UU_iff RS iffD2, stream_con_rew2]) 1);
-qed "stream_flat_prefix";
-
-Goal "b ~= UU ==> x << b && z = \
-\           (x = UU |  (EX a y. x = a && y &  a ~= UU &  a << b &  y << z))";
-by Safe_tac;
-by (stream_case_tac "x" 1);
-by (safe_tac (claset() addSDs stream.inverts 
-                addSIs [monofun_cfun, monofun_cfun_arg]));
-by (Fast_tac 1);
-qed "stream_prefix'";
-
-Goal "[| a ~= UU; b ~= UU |] ==> (a && s = b && t) = (a = b &  s = t)";
-by (best_tac (claset() addSEs stream.injects) 1);
-qed "stream_inject_eq";	
-Addsimps [stream_inject_eq];
-
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream_when                                              *)
-(* ----------------------------------------------------------------------- *)
-
-section "stream_when";
-
-Goal "stream_when$UU$s=UU";
-by (stream_case_tac "s" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "stream_when_strictf";
-	
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about ft and rt                                                *)
-(* ----------------------------------------------------------------------- *)
-
-section "ft & rt";
-
-(*
-Goal "ft$s=UU --> s=UU";
-by (stream_case_tac "s" 1);
-by (REPEAT (asm_simp_tac (simpset() addsimps stream.rews) 1));
-qed "stream_ft_lemma1";
-*)
-
-Goal "s~=UU ==> ft$s~=UU";
-by (stream_case_tac "s" 1);
-by  (Blast_tac 1);
-by (Asm_simp_tac 1);
-qed "ft_defin";
-
-Goal "rt$s~=UU ==> s~=UU";
-by Auto_tac;
-qed "rt_strict_rev";
-
-Goal "(ft$s)&&(rt$s)=s";
-by (stream_case_tac "s" 1);
-by (ALLGOALS Asm_simp_tac);
-qed "surjectiv_scons";
-
-Goal "ALL x s. x << s --> iterate i rt x << iterate i rt s";
-by (induct_tac "i" 1);
-by (Simp_tac 1);
-by (strip_tac 1);
-by (stream_case_tac "x" 1);
-by (rtac (minimal RS (monofun_iterate2 RS monofunE RS spec RS spec RS mp))1);
-by (dtac stream_prefix 1);
-by Safe_tac;
-by (asm_simp_tac (HOL_ss addsimps iterate_Suc2::stream.rews) 1);
-qed_spec_mp "monofun_rt_mult";
-
-			
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream_take                                              *)
-(* ----------------------------------------------------------------------- *)
-
-section "stream_take";
-
-Goal  "(LUB i. stream_take i$s) = s";
-by (subgoal_tac "(LUB i. stream_take i$s) = fix$stream_copy$s" 1);
-by (simp_tac (simpset() addsimps [fix_def2, stream.take_def,
-                                  contlub_cfun_fun, chain_iterate]) 2);
-by (asm_full_simp_tac (simpset() addsimps [stream.reach]) 1);
-qed "stream_reach2";
-
-Goal "chain (%i. stream_take i$s)";
-by (rtac chainI 1);
-by (subgoal_tac "ALL i s::'a stream. stream_take i$s << stream_take (Suc i)$s" 1);
-by (Fast_tac 1);
-by (rtac allI 1);
-by (induct_tac "ia" 1);
-by (Simp_tac 1);
-by (rtac allI 1);
-by (stream_case_tac "s" 1);
-by (Simp_tac 1);
-by (asm_simp_tac (simpset() addsimps [monofun_cfun_arg]) 1);
-qed "chain_stream_take";
-
-
-Goalw [stream.take_def]
-	"stream_take n$x = x ==> stream_take (Suc n)$x = x";
-by (rtac antisym_less 1);
-by (subgoal_tac "iterate (Suc n) stream_copy UU$x << fix$stream_copy$x" 1);
-by (asm_full_simp_tac (simpset() addsimps [stream.reach]) 1);
-by (rtac monofun_cfun_fun 1);
-by (stac fix_def2 1);
-by (rtac is_ub_thelub 1);
-by (rtac chain_iterate 1);
-by (etac subst 1 THEN rtac monofun_cfun_fun 1);
-by (rtac (rewrite_rule [chain_def] chain_iterate RS spec) 1);
-qed "stream_take_more";
-
-(*
-Goal 
- "ALL  s2. stream_take n$s2 = s2 --> stream_take (Suc n)$s2=s2";
-by (induct_tac "n" 1);
-by (simp_tac (simpset() addsimps stream_rews) 1);
-by (strip_tac 1);
-by (hyp_subst_tac  1);
-by (simp_tac (simpset() addsimps stream_rews) 1);
-by (rtac allI 1);
-by (res_inst_tac [("s","s2")] streamE 1);
-by (asm_simp_tac (simpset() addsimps stream_rews) 1);
-by (asm_simp_tac (simpset() addsimps stream_rews) 1);
-by (strip_tac 1 );
-by (subgoal_tac "stream_take n1$xs = xs" 1);
-by (rtac ((hd stream_inject) RS conjunct2) 2);
-by (atac 4);
-by (atac 2);
-by (atac 2);
-by (rtac cfun_arg_cong 1);
-by (Blast_tac 1);
-qed "stream_take_lemma2";
-*)
-
-Goal 
-"ALL x xs. x~=UU --> stream_take n$(x && xs) = x && xs --> stream_take n$xs=xs";
-by (induct_tac "n" 1);
-by (Asm_simp_tac 1);
-by (strip_tac 1);
-by (res_inst_tac [("P","x && xs=UU")] notE 1);
-by (eresolve_tac stream.con_rews 1);
-by (etac sym 1);
-by (strip_tac 1);
-by (rtac stream_take_more 1);
-by (res_inst_tac [("a1","x")] ((hd stream.injects) RS conjunct2) 1);
-by (assume_tac 3);
-by (etac (hd(tl(tl(stream.take_rews))) RS subst) 1);
-by (atac 1);
-qed_spec_mp "stream_take_lemma3";
-
-Goal 
- "ALL x xs. stream_take n$xs=xs --> stream_take (Suc n)$(x && xs) = x && xs";
-by (induct_tac "n" 1);
-by Auto_tac;  
-qed_spec_mp "stream_take_lemma4";
-
-
-(* ------------------------------------------------------------------------- *)
-(* special induction rules                                                   *)
-(* ------------------------------------------------------------------------- *)
-
-section "induction";
-
-
-val prems = Goalw [stream.finite_def]
- "[| stream_finite x;  \
-\    P UU; \
-\    !!a s. [| a ~= UU; P s |] \
-\ ==> P (a && s) |] ==> P x";
-by (cut_facts_tac prems 1);
-by (etac exE 1);
-by (etac subst 1);
-by (rtac stream.finite_ind 1);
-by (atac 1);
-by (eresolve_tac prems 1);
-by (atac 1);
-qed "stream_finite_ind";
-
-val major::prems = Goal
-"[| P UU;\
-\   !! x    .    x ~= UU                  ==> P (x      && UU); \
-\   !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )  \
-\   |] ==> !s. P (stream_take n$s)";
-by (res_inst_tac [("n","n")] nat_induct2 1);
-by (asm_simp_tac (simpset() addsimps [major]) 1);
-by (rtac allI 1);
-by (stream_case_tac "s" 1);
-by (asm_simp_tac (simpset() addsimps [major]) 1);
-by (asm_simp_tac (simpset() addsimps prems) 1);
-by (rtac allI 1);
-by (stream_case_tac "s" 1);
-by (asm_simp_tac (simpset() addsimps [major]) 1);
-by (res_inst_tac [("x","sa")] stream.casedist 1);
-by (asm_simp_tac (simpset() addsimps prems) 1);
-by (asm_simp_tac (simpset() addsimps prems) 1);
-qed "stream_finite_ind2";
-
-
-val prems = Goal
-"[| adm P; P UU; !!a. a ~= UU ==> P (a && UU); \
-\   !!a b s. [| a ~= UU; b ~= UU; P s |] ==> P (a && b && s)\
-\ |] ==> P x";
-by (rtac (stream.reach RS subst) 1);
-by (rtac (adm_impl_admw RS wfix_ind) 1);
-by (rtac adm_subst 1);
-by (cont_tacR 1);
-by (resolve_tac prems 1);
-by (rtac allI 1);
-by (rtac (rewrite_rule [stream.take_def] stream_finite_ind2 RS spec) 1);
-by (resolve_tac prems 1);
-by (eresolve_tac prems 1);
-by (eresolve_tac prems 1);
-by (atac 1);
-by (atac 1);
-qed "stream_ind2";
-
-
-(* ----------------------------------------------------------------------- *)
-(* simplify use of coinduction                                             *)
-(* ----------------------------------------------------------------------- *)
-
-section "coinduction";
-
-
-Goalw [stream.bisim_def]
-"!s1 s2. R s1 s2 --> ft$s1 = ft$s2 &  R (rt$s1) (rt$s2) ==> stream_bisim R";
-by (strip_tac 1);
-by (etac allE 1);
-by (etac allE 1);
-by (dtac mp 1);
-by (atac 1);
-by (etac conjE 1);
-by (case_tac "x = UU & x' = UU" 1);
-by (rtac disjI1 1);
-by (Blast_tac 1);
-by (rtac disjI2 1);
-by (rtac disjE 1);
-by (etac (de_Morgan_conj RS subst) 1);
-by (res_inst_tac [("x","ft$x")] exI 1);
-by (res_inst_tac [("x","rt$x")] exI 1);
-by (res_inst_tac [("x","rt$x'")] exI 1);
-by (rtac conjI 1);
-by (etac ft_defin 1);
-by (asm_simp_tac (simpset() addsimps [surjectiv_scons]) 1);
-by (eres_inst_tac [("s","ft$x"),("t","ft$x'")] subst 1);
-by (simp_tac (simpset() addsimps [surjectiv_scons]) 1);
-by (res_inst_tac [("x","ft$x'")] exI 1);
-by (res_inst_tac [("x","rt$x")] exI 1);
-by (res_inst_tac [("x","rt$x'")] exI 1);
-by (rtac conjI 1);
-by (etac ft_defin 1);
-by (asm_simp_tac (simpset() addsimps [surjectiv_scons]) 1);
-by (res_inst_tac [("s","ft$x"),("t","ft$x'")] ssubst 1);
-by (etac sym 1);
-by (simp_tac (simpset() addsimps [surjectiv_scons]) 1);
-qed "stream_coind_lemma2";
-
-(* ----------------------------------------------------------------------- *)
-(* theorems about stream_finite                                            *)
-(* ----------------------------------------------------------------------- *)
-
-section "stream_finite";
-
-
-Goalw [stream.finite_def] "stream_finite UU";
-by (rtac exI 1);
-by (Simp_tac 1);
-qed "stream_finite_UU";
-
-Goal  "~  stream_finite s ==> s ~= UU";
-by (blast_tac (claset() addIs [stream_finite_UU]) 1);
-qed "stream_finite_UU_rev";
-
-Goalw [stream.finite_def] "stream_finite xs ==> stream_finite (x && xs)";
-by (blast_tac (claset() addIs [stream_take_lemma4]) 1);
-qed "stream_finite_lemma1";
-
-Goalw [stream.finite_def]
- "[| x ~= UU; stream_finite (x && xs) |] ==> stream_finite xs";
-by (blast_tac (claset() addIs [stream_take_lemma3]) 1);
-qed "stream_finite_lemma2";
-
-Goal "stream_finite (rt$s) = stream_finite s";
-by (stream_case_tac "s" 1);
-by (simp_tac (simpset() addsimps [stream_finite_UU]) 1);
-by (Asm_simp_tac 1);
-by (fast_tac (claset() addIs [stream_finite_lemma1] 
-                       addDs [stream_finite_lemma2]) 1);
-qed "stream_finite_rt_eq";
-
-Goal "stream_finite s ==> !t. t<<s --> stream_finite t";
-by (eres_inst_tac [("x","s")] stream_finite_ind 1);
-by (strip_tac 1); 
-by (dtac UU_I 1);
-by (asm_simp_tac (simpset() addsimps [stream_finite_UU]) 1);
-by (strip_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [stream_prefix']) 1);
-by (fast_tac (claset() addSIs [stream_finite_UU, stream_finite_lemma1]) 1);
-qed_spec_mp "stream_finite_less";
-
-Goal "adm (%x. ~  stream_finite x)";
-by (rtac admI2 1);
-by (best_tac (claset() addIs [stream_finite_less, is_ub_thelub]) 1);
-qed "adm_not_stream_finite";
-
-section "smap";
-
-bind_thm ("smap_unfold", fix_prover2 thy smap_def 
-	"smap = (\\<Lambda> f s. case s of x && l => f\\<cdot>x && smap\\<cdot>f\\<cdot>l)");
-
-Goal "smap\\<cdot>f\\<cdot>\\<bottom> = \\<bottom>";
-by (stac smap_unfold 1);
-by (Simp_tac 1);
-qed "smap_empty";
-
-Goal "x~=\\<bottom> ==> smap\\<cdot>f\\<cdot>(x&&xs) = (f\\<cdot>x)&&(smap\\<cdot>f\\<cdot>xs)";
-by (rtac trans 1);
-by  (stac smap_unfold 1);
-by  (Asm_simp_tac 1);
-by (rtac refl 1);
-qed "smap_scons";
-
-Addsimps [smap_empty, smap_scons];
-
-(* ------------------------------------------------------------------------- *)
-
-section "slen";
-
-Goalw [slen_def, stream.finite_def] "#\\<bottom> = 0";
-by (Simp_tac 1);
-by (stac Least_equality 1);
-by    Auto_tac;
-by (simp_tac(simpset() addsimps [thm "Fin_0"]) 1);
-qed "slen_empty";
-
-Goalw [slen_def] "x ~= \\<bottom> ==> #(x&&xs) = iSuc (#xs)";
-by (res_inst_tac [("Q1","stream_finite (x && xs)")] (split_if RS iffD2) 1);
-by Safe_tac;
-by (rtac (split_if RS iffD2) 2);
-by  Safe_tac;
-by   (fast_tac (claset() addIs [stream_finite_lemma1]) 2);
-by  (rtac (thm "iSuc_Infty" RS sym) 2);
-by (rtac (split_if RS iffD2) 1);
-by (Simp_tac 1);
-by Safe_tac;
-by  (eatac stream_finite_lemma2 1 2);
-by (rewtac stream.finite_def);
-by (Clarify_tac 1);
-by (eatac Least_Suc2 1 1);
-by  (rtac not_sym 1);
-by  Auto_tac;
-qed "slen_scons"; 
-
-Addsimps [slen_empty, slen_scons];
-
-Goal "(#x < Fin (Suc 0)) = (x = UU)";
-by (stream_case_tac "x" 1);
-by (auto_tac (claset(), simpset() delsimps [thm "iSuc_Fin"] addsimps
- [thm "Fin_0", thm "iSuc_Fin" RS sym, thm "i0_iless_iSuc", thm "iSuc_mono"]));
-qed "slen_less_1_eq";
-
-Goal "(#x = 0) = (x = \\<bottom>)";
-by Auto_tac;
-by (stream_case_tac "x" 1);
-by (rotate_tac ~1 2);
-by Auto_tac;
-qed "slen_empty_eq";
-
-Goal "(Fin (Suc n) < #x) = (? a y. x = a && y &  a ~= \\<bottom> &  Fin n < #y)";
-by (stream_case_tac "x" 1);
-by (auto_tac (claset(), simpset() delsimps [thm "iSuc_Fin"] addsimps
-                [thm "iSuc_Fin" RS sym, thm "iSuc_mono"]));
-by  (dtac sym 1);
-by  Auto_tac;
-qed "slen_scons_eq";
-
-
-Goal "#x = iSuc n --> (? a y. x = a&&y &  a ~= \\<bottom> &  #y = n)";
-by (stream_case_tac "x" 1);
-by Auto_tac;
-qed_spec_mp "slen_iSuc";
-
-
-Goal 
-"(#x < Fin (Suc (Suc n))) = (!a y. x ~= a && y |  a = \\<bottom> |  #y < Fin (Suc n))";
-by (stac (thm "iSuc_Fin" RS sym) 1);
-by (stac (thm "iSuc_Fin" RS sym) 1);
-by (safe_tac HOL_cs);
-by  (rotate_tac ~1 1);
-by  (asm_full_simp_tac (HOL_ss addsimps [slen_scons, thm "iSuc_mono"]) 1);
-by  (Asm_full_simp_tac 1);
-by (stream_case_tac "x" 1);
-by ( simp_tac (HOL_ss addsimps [slen_empty, thm "i0_iless_iSuc"]) 1);
-by (asm_simp_tac (HOL_ss addsimps [slen_scons, thm "iSuc_mono"]) 1);
-by (etac allE 1);
-by (etac allE 1);
-by (safe_tac HOL_cs);
-by (  contr_tac 2);
-by ( fast_tac HOL_cs 1);
-by (Asm_full_simp_tac 1);
-qed "slen_scons_eq_rev";
-
-Goal "!x. (Fin n < #x) = (stream_take n\\<cdot>x ~= x)";
-by (induct_tac "n" 1);
-by  (simp_tac(simpset() addsimps [slen_empty_eq, thm "Fin_0"]) 1);
-by  (Fast_tac 1);
-by (rtac allI 1);
-by (asm_simp_tac (simpset() addsimps [slen_scons_eq]) 1);
-by (etac thin_rl 1);
-by (safe_tac HOL_cs);
-by  (Asm_full_simp_tac 1);
-by (stream_case_tac "x" 1);
-by (rotate_tac ~1 2);
-by Auto_tac;
-qed_spec_mp "slen_take_eq";
-
-Goalw [thm "ile_def"] "(#x <= Fin n) = (stream_take n\\<cdot>x = x)";
-by (simp_tac (simpset() addsimps [slen_take_eq]) 1);
-qed "slen_take_eq_rev";
-
-Goal "#x = Fin n ==> stream_take n\\<cdot>x = x";
-by (asm_simp_tac (simpset() addsimps [slen_take_eq_rev RS sym]) 1);
-qed "slen_take_lemma1";
-
-Goal "!x. ~stream_finite x --> #(stream_take i\\<cdot>x) = Fin i";
-by (induct_tac "i" 1);
-by  (simp_tac(simpset() addsimps [thm "Fin_0"]) 1);
-by (rtac allI 1);
-by (stream_case_tac "x" 1);
-by  (auto_tac (claset()addSIs [stream_finite_lemma1], simpset() delsimps [thm "iSuc_Fin"] addsimps [thm "iSuc_Fin" RS sym]));
-by (force_tac (claset(), simpset() addsimps [stream.finite_def]) 1);
-qed_spec_mp "slen_take_lemma2";
-
-Goal "stream_finite x = (#x ~= Infty)";
-by (rtac iffI 1);
-by  (etac stream_finite_ind 1);
-by   (Simp_tac 1);
-by  (etac (slen_scons RS ssubst) 1);
-by  (stac (thm "iSuc_Infty" RS sym) 1);
-by  (etac contrapos_nn 1);
-by  (etac (thm "iSuc_inject" RS iffD1) 1);
-by (case_tac "#x" 1);
-by (auto_tac (claset()addSDs [slen_take_lemma1],
-        simpset() addsimps [stream.finite_def]));
-qed "slen_infinite";
-
-Goal "s << t ==> #s <= #t";
-by (case_tac "stream_finite t" 1);
-by  (EVERY'[dtac (slen_infinite RS subst), dtac notnotD] 2);
-by  (Asm_simp_tac 2);
-by (etac rev_mp 1);
-by (res_inst_tac [("x","s")] allE 1);
-by  (atac 2);
-by (etac (stream_finite_ind) 1);
-by  (Simp_tac 1);
-by (rtac allI 1);
-by (stream_case_tac "x" 1);
-by  (asm_simp_tac (HOL_ss addsimps [slen_empty, thm "i0_lb"]) 1);
-by (asm_simp_tac (HOL_ss addsimps [slen_scons]) 1);
-by (fast_tac(claset() addSIs [thm "iSuc_ile_mono" RS iffD2] addSDs stream.inverts) 1);
-qed "slen_mono";
-
-Goal "!(x::'a::flat stream) y. Fin n <= #x --> x << y --> \
-\ stream_take n\\<cdot>x = stream_take n\\<cdot>y";
-by (induct_tac "n" 1);
-by ( simp_tac(simpset() addsimps [slen_empty_eq]) 1);
-by (strip_tac 1);
-by (stream_case_tac "x" 1);
-by  (asm_full_simp_tac (HOL_ss addsimps [slen_empty, 
-              thm "iSuc_Fin" RS sym, thm "not_iSuc_ilei0"]) 1);
-by (fatac stream_prefix 1 1);
-by (safe_tac (claset() addSDs [stream_flat_prefix]));
-by (Simp_tac 1);
-by (rtac cfun_arg_cong 1);
-by (rotate_tac 3 1);
-by (asm_full_simp_tac (simpset() delsimps [thm "iSuc_Fin"] addsimps 
-        [thm "iSuc_Fin" RS sym, thm "iSuc_ile_mono"]) 1);
-qed_spec_mp "slen_take_lemma3";
-
-Goal "stream_finite t ==> \
-\!s. #(s::'a::flat stream) = #t &  s << t --> s = t";
-by (etac stream_finite_ind 1);
-by  (fast_tac (claset() addDs [eq_UU_iff RS iffD2]) 1);
-by Safe_tac;
-by (stream_case_tac "sa" 1);
-by  (dtac sym 1);
-by  (Asm_full_simp_tac 1);
-by (safe_tac (claset() addSDs [stream_flat_prefix]));
-by (rtac cfun_arg_cong 1);
-by (etac allE 1);
-by (etac mp 1);
-by (Asm_full_simp_tac 1);
-qed "slen_strict_mono_lemma";
-
-Goal "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t";
-by (rtac (thm "ilessI1") 1);
-by  (etac slen_mono 1);
-by (dtac slen_strict_mono_lemma 1);
-by (Fast_tac 1);
-qed "slen_strict_mono";
-
-Goal "!x. Fin (i + j) <= #x --> Fin j <= #(iterate i rt x)";
-by (induct_tac "i" 1);
-by  (Simp_tac 1);
-by (strip_tac 1);
-by (stream_case_tac "x" 1);
-by  (asm_full_simp_tac (simpset() delsimps [thm "iSuc_Fin"] 
-             addsimps [thm "iSuc_Fin" RS sym]) 1);
-by (stac iterate_Suc2 1);
-by (rotate_tac 2 1);
-by (asm_full_simp_tac  (simpset() delsimps [thm "iSuc_Fin"] 
-    addsimps [thm "iSuc_Fin" RS sym, thm "iSuc_ile_mono"]) 1);
-qed_spec_mp "slen_rt_mult";
-
-
-(* ------------------------------------------------------------------------- *)
-
-section "sfilter";
-
-bind_thm ("sfilter_unfold", fix_prover2 thy sfilter_def 
-	"sfilter = (\\<Lambda> p s. case s of x && xs => \
-\  If p\\<cdot>x then x && sfilter\\<cdot>p\\<cdot>xs else sfilter\\<cdot>p\\<cdot>xs fi)");
-
-Goal "sfilter\\<cdot>\\<bottom> = \\<bottom>";
-by (rtac ext_cfun 1);
-by (stac sfilter_unfold 1);
-by (stream_case_tac "x" 1);
-by  Auto_tac;
-qed "strict_sfilter";
-
-Goal "sfilter\\<cdot>f\\<cdot>\\<bottom> = \\<bottom>";
-by (stac sfilter_unfold 1);
-by (Simp_tac 1);
-qed "sfilter_empty";
-
-Goal "x ~= \\<bottom> ==> sfilter\\<cdot>f\\<cdot>(x && xs) = \
-\ If f\\<cdot>x then x && sfilter\\<cdot>f\\<cdot>xs else sfilter\\<cdot>f\\<cdot>xs fi";
-by (rtac trans 1);
-by (stac sfilter_unfold 1);
-by (Asm_simp_tac 1);
-by (rtac refl 1);
-qed "sfilter_scons";
-
--- a/src/HOLCF/ex/Stream.thy	Tue Sep 07 15:59:16 2004 +0200
+++ b/src/HOLCF/ex/Stream.thy	Tue Sep 07 16:02:42 2004 +0200
@@ -1,29 +1,1012 @@
 (*  Title: 	HOLCF/ex/Stream.thy
     ID:         $Id$
-    Author: 	Franz Regensburger, David von Oheimb
+    Author: 	Franz Regensburger, David von Oheimb, Borislav Gajanovic
 
 General Stream domain.
-TODO: should be integrated with HOLCF/Streams
 *)
 
-Stream = HOLCF + Nat_Infinity +
+theory Stream = HOLCF + Nat_Infinity:
 
-domain 'a stream = "&&" (ft::'a) (lazy rt::'a stream) (infixr 65)
+domain 'a stream = "&&" (ft::'a) (lazy rt::"'a stream") (infixr 65)
 
 consts
 
-  smap		:: "('a -> 'b) -> 'a stream -> 'b stream"
-  sfilter	:: "('a -> tr) -> 'a stream -> 'a stream"
-  slen		:: "'a stream => inat"			("#_" [1000] 1000)
+  smap		:: "('a \<rightarrow> 'b) \<rightarrow> 'a stream \<rightarrow> 'b stream"
+  sfilter	:: "('a \<rightarrow> tr) \<rightarrow> 'a stream \<rightarrow> 'a stream"
+  slen		:: "'a stream \<Rightarrow> inat"			("#_" [1000] 1000)
 
 defs
 
-  smap_def	"smap	 == fix\\<cdot>(\\<Lambda> h f s. case s of x && xs => f\\<cdot>x && h\\<cdot>f\\<cdot>xs)"
-  sfilter_def	"sfilter == fix\\<cdot>(\\<Lambda> h p s. case s of x && xs => 
-				     If p\\<cdot>x then x && h\\<cdot>p\\<cdot>xs else h\\<cdot>p\\<cdot>xs fi)"
-  slen_def	"#s == if stream_finite s 
-		      then Fin (LEAST n. stream_take n\\<cdot>s = s) else \\<infinity>"
+  smap_def:	"smap	 \<equiv> fix\<cdot>(\<Lambda> h f s. case s of x && xs \<Rightarrow> f\<cdot>x && h\<cdot>f\<cdot>xs)"
+  sfilter_def:	"sfilter \<equiv> fix\<cdot>(\<Lambda> h p s. case s of x && xs \<Rightarrow> 
+				     If p\<cdot>x then x && h\<cdot>p\<cdot>xs else h\<cdot>p\<cdot>xs fi)"
+  slen_def:	"#s \<equiv> if stream_finite s 
+		      then Fin (LEAST n. stream_take n\<cdot>s = s) else \<infinity>"
+
+(* concatenation *)
+
+consts
+ 
+  i_rt :: "nat => 'a stream => 'a stream" (* chops the first i elements *)
+  i_th :: "nat => 'a stream => 'a"        (* the i-th element *)
+
+  sconc         :: "'a stream => 'a stream => 'a stream" (infixr "ooo" 65) 
+  constr_sconc  :: "'a stream => 'a stream => 'a stream" (* constructive *)
+  constr_sconc' :: "nat => 'a stream => 'a stream => 'a stream" 
+
+defs
+  i_rt_def: "i_rt == %i s. iterate i rt s"  
+  i_th_def: "i_th == %i s. ft$(i_rt i s)" 
+
+  sconc_def: "s1 ooo s2 == case #s1 of 
+                       Fin n \<Rightarrow> (SOME s. (stream_take n$s=s1) & (i_rt n s = s2))
+                     | \<infinity>     \<Rightarrow> s1" 
+
+  constr_sconc_def: "constr_sconc s1 s2 == case #s1 of 
+                                             Fin n \<Rightarrow> constr_sconc' n s1 s2 
+                                           | \<infinity>    \<Rightarrow> s1"
+primrec 
+  constr_sconc'_0:   "constr_sconc' 0 s1 s2 = s2"
+  constr_sconc'_Suc: "constr_sconc' (Suc n) s1 s2 = ft$s1 && 
+                                                    constr_sconc' n (rt$s1) s2"
+
+
+declare stream.rews [simp add]
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about scons                                                    *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "scons"
+
+lemma scons_eq_UU: "(a && s = UU) = (a = UU)"
+by (auto, erule contrapos_pp, simp)
+
+lemma scons_not_empty: "[| a && x = UU; a ~= UU |] ==> R"
+by auto
+
+lemma stream_exhaust_eq: "(x ~= UU) = (EX a y. a ~= UU &  x = a && y)"
+by (auto,insert stream.exhaust [of x],auto)
+
+lemma stream_neq_UU: "x~=UU ==> EX a as. x=a&&as & a~=UU"
+by (simp add: stream_exhaust_eq,auto)
+
+lemma stream_inject_eq [simp]:
+  "[| a ~= UU; b ~= UU |] ==> (a && s = b && t) = (a = b &  s = t)"
+by (insert stream.injects [of a s b t], auto)
+
+lemma stream_prefix: 
+  "[| a && s << t; a ~= UU  |] ==> EX b tt. t = b && tt &  b ~= UU &  s << tt"
+apply (insert stream.exhaust [of t], auto)
+apply (drule eq_UU_iff [THEN iffD2], simp)
+by (drule stream.inverts, auto)
+
+lemma stream_prefix': 
+  "b ~= UU ==> x << b && z = 
+   (x = UU |  (EX a y. x = a && y &  a ~= UU &  a << b &  y << z))"
+apply (case_tac "x=UU",auto)
+apply (drule stream_exhaust_eq [THEN iffD1],auto)
+apply (drule stream.inverts,auto)
+by (intro monofun_cfun,auto)
+
+(*
+lemma stream_prefix1: "[| x<<y; xs<<ys |] ==> x&&xs << y&&ys"
+by (insert stream_prefix' [of y "x&&xs" ys],force)
+*)
+
+lemma stream_flat_prefix: 
+  "[| x && xs << y && ys; (x::'a::flat) ~= UU|] ==> x = y & xs << ys"
+apply (case_tac "y=UU",auto)
+apply (drule eq_UU_iff [THEN iffD2],auto)
+apply (drule stream.inverts,auto)
+apply (drule ax_flat [rule_format],simp)
+by (drule stream.inverts,auto)
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about stream_when                                              *)
+(* ----------------------------------------------------------------------- *)
+
+section "stream_when"
+
+
+lemma stream_when_strictf: "stream_when$UU$s=UU"
+by (rule stream.casedist [of s], auto)
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about ft and rt                                                *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "ft & rt"
+
+
+lemma ft_defin: "s~=UU ==> ft$s~=UU"
+by (drule stream_exhaust_eq [THEN iffD1],auto)
+
+lemma rt_strict_rev: "rt$s~=UU ==> s~=UU"
+by auto
+
+lemma surjectiv_scons: "(ft$s)&&(rt$s)=s"
+by (rule stream.casedist [of s], auto)
+
+lemma monofun_rt_mult: "x << s ==> iterate i rt x << iterate i rt s"
+by (insert monofun_iterate2 [of i "rt"], simp add: monofun, auto)
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about stream_take                                              *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "stream_take";
+
+
+lemma stream_reach2: "(LUB i. stream_take i$s) = s"
+apply (insert stream.reach [of s], erule subst) back
+apply (simp add: fix_def2 stream.take_def)
+apply (insert contlub_cfun_fun [of "%i. iterate i stream_copy UU" s,THEN sym])
+by (simp add: chain_iterate)
+
+lemma chain_stream_take: "chain (%i. stream_take i$s)"
+apply (rule chainI) 
+apply (rule monofun_cfun_fun)
+apply (simp add: stream.take_def del: iterate_Suc)
+by (rule chainE, simp add: chain_iterate)
+
+lemma stream_take_prefix [simp]: "stream_take n$s << s"
+apply (insert stream_reach2 [of s])
+apply (erule subst) back
+apply (rule is_ub_thelub)
+by (simp only: chain_stream_take)
+
+lemma stream_take_more [rule_format]: 
+  "ALL x. stream_take n$x = x --> stream_take (Suc n)$x = x"
+apply (induct_tac n,auto)
+apply (case_tac "x=UU",auto)
+by (drule stream_exhaust_eq [THEN iffD1],auto)
+
+lemma stream_take_lemma3 [rule_format]: 
+  "ALL x xs. x~=UU --> stream_take n$(x && xs) = x && xs --> stream_take n$xs=xs"
+apply (induct_tac n,clarsimp)
+apply (drule sym, erule scons_not_empty, simp)
+apply (clarify, rule stream_take_more)
+apply (erule_tac x="x" in allE)
+by (erule_tac x="xs" in allE,simp)
+
+lemma stream_take_lemma4: 
+  "ALL x xs. stream_take n$xs=xs --> stream_take (Suc n)$(x && xs) = x && xs"
+by auto
+
+lemma stream_take_idempotent [rule_format, simp]: 
+ "ALL s. stream_take n$(stream_take n$s) = stream_take n$s"
+apply (induct_tac n, auto)
+apply (case_tac "s=UU", auto)
+by (drule stream_exhaust_eq [THEN iffD1], auto)
+
+lemma stream_take_take_Suc [rule_format, simp]: 
+  "ALL s. stream_take n$(stream_take (Suc n)$s) = 
+                                    stream_take n$s"
+apply (induct_tac n, auto)
+apply (case_tac "s=UU", auto)
+by (drule stream_exhaust_eq [THEN iffD1], auto)
+
+lemma mono_stream_take_pred: 
+  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
+                       stream_take n$s1 << stream_take n$s2"
+by (insert monofun_cfun_arg [of "stream_take (Suc n)$s1" 
+  "stream_take (Suc n)$s2" "stream_take n"], auto)
+(*
+lemma mono_stream_take_pred: 
+  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
+                       stream_take n$s1 << stream_take n$s2"
+by (drule mono_stream_take [of _ _ n],simp)
+*)
+
+lemma stream_take_lemma10 [rule_format]:
+  "ALL k<=n. stream_take n$s1 << stream_take n$s2 
+                             --> stream_take k$s1 << stream_take k$s2"
+apply (induct_tac n,simp,clarsimp)
+apply (case_tac "k=Suc n",blast)
+apply (erule_tac x="k" in allE)
+by (drule mono_stream_take_pred,simp)
+
+lemma stream_take_le_mono : "k<=n ==> stream_take k$s1 << stream_take n$s1"
+apply (insert chain_stream_take [of s1])
+by (drule chain_mono3,auto)
+
+lemma mono_stream_take: "s1 << s2 ==> stream_take n$s1 << stream_take n$s2"
+by (simp add: monofun_cfun_arg)
+
+(*
+lemma stream_take_prefix [simp]: "stream_take n$s << s"
+apply (subgoal_tac "s=(LUB n. stream_take n$s)")
+ apply (erule ssubst, rule is_ub_thelub)
+ apply (simp only: chain_stream_take)
+by (simp only: stream_reach2)
+*)
+
+lemma stream_take_take_less:"stream_take k$(stream_take n$s) << stream_take k$s"
+by (rule monofun_cfun_arg,auto)
+
+
+(* ------------------------------------------------------------------------- *)
+(* special induction rules                                                   *)
+(* ------------------------------------------------------------------------- *)
+
+
+section "induction"
+
+lemma stream_finite_ind: 
+ "[| stream_finite x; P UU; !!a s. [| a ~= UU; P s |] ==> P (a && s) |] ==> P x"
+apply (simp add: stream.finite_def,auto)
+apply (erule subst)
+by (drule stream.finite_ind [of P _ x], auto)
+
+lemma stream_finite_ind2: 
+"[| P UU; !! x. x ~= UU ==> P (x && UU); !! y z s. [| y ~= UU; z ~= UU; P s |] ==> P (y && z && s )|] ==> 
+                                 !s. P (stream_take n$s)"
+apply (rule nat_induct2 [of _ n],auto)
+apply (case_tac "s=UU",clarsimp)
+apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
+apply (case_tac "s=UU",clarsimp)
+apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
+apply (case_tac "y=UU",clarsimp)
+by (drule stream_exhaust_eq [THEN iffD1],clarsimp)
+
+lemma stream_ind2: 
+"[| adm P; P UU; !!a. a ~= UU ==> P (a && UU); !!a b s. [| a ~= UU; b ~= UU; P s |] ==> P (a && b && s) |] ==> P x"
+apply (insert stream.reach [of x],erule subst)
+apply (frule adm_impl_admw, rule wfix_ind, auto)
+apply (rule adm_subst [THEN adm_impl_admw],auto)
+apply (insert stream_finite_ind2 [of P])
+by (simp add: stream.take_def)
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* simplify use of coinduction                                             *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "coinduction"
+
+lemma stream_coind_lemma2: "!s1 s2. R s1 s2 --> ft$s1 = ft$s2 &  R (rt$s1) (rt$s2) ==> stream_bisim R"
+apply (simp add: stream.bisim_def,clarsimp)
+apply (case_tac "x=UU",clarsimp)
+apply (erule_tac x="UU" in allE,simp)
+apply (case_tac "x'=UU",simp)
+apply (drule stream_exhaust_eq [THEN iffD1],auto)+
+apply (case_tac "x'=UU",auto)
+apply (erule_tac x="a && y" in allE)
+apply (erule_tac x="UU" in allE)+
+apply (auto,drule stream_exhaust_eq [THEN iffD1],clarsimp)
+apply (erule_tac x="a && y" in allE)
+apply (erule_tac x="aa && ya" in allE)
+by auto
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about stream_finite                                            *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "stream_finite"
+
+lemma stream_finite_UU [simp]: "stream_finite UU"
+by (simp add: stream.finite_def)
+
+lemma stream_finite_UU_rev: "~  stream_finite s ==> s ~= UU"
+by (auto simp add: stream.finite_def)
+
+lemma stream_finite_lemma1: "stream_finite xs ==> stream_finite (x && xs)"
+apply (simp add: stream.finite_def,auto)
+apply (rule_tac x="Suc n" in exI)
+by (simp add: stream_take_lemma4)
+
+lemma stream_finite_lemma2: "[| x ~= UU; stream_finite (x && xs) |] ==> stream_finite xs"
+apply (simp add: stream.finite_def, auto)
+apply (rule_tac x="n" in exI)
+by (erule stream_take_lemma3,simp)
+
+lemma stream_finite_rt_eq: "stream_finite (rt$s) = stream_finite s"
+apply (rule stream.casedist [of s], auto)
+apply (rule stream_finite_lemma1, simp)
+by (rule stream_finite_lemma2,simp)
+
+lemma stream_finite_less: "stream_finite s ==> !t. t<<s --> stream_finite t"
+apply (erule stream_finite_ind [of s])
+apply (clarsimp, drule eq_UU_iff [THEN iffD2], auto)
+apply (case_tac "t=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1],auto)
+apply (drule stream.inverts, auto)
+apply (erule_tac x="y" in allE, simp)
+by (rule stream_finite_lemma1, simp)
+
+lemma stream_take_finite [simp]: "stream_finite (stream_take n$s)"
+apply (simp add: stream.finite_def)
+by (rule_tac x="n" in exI,simp)
+
+lemma adm_not_stream_finite: "adm (%x. ~ stream_finite x)"
+apply (rule admI2, auto)
+apply (drule stream_finite_less,drule is_ub_thelub)
+by auto
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about stream length                                            *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "slen"
+
+lemma slen_empty [simp]: "#\<bottom> = 0"
+apply (simp add: slen_def stream.finite_def)
+by (simp add: inat_defs Least_equality)
+
+lemma slen_scons [simp]: "x ~= \<bottom> ==> #(x&&xs) = iSuc (#xs)"
+apply (case_tac "stream_finite (x && xs)")
+apply (simp add: slen_def, auto)
+apply (simp add: stream.finite_def, auto)
+apply (rule Least_Suc2,auto)
+apply (drule sym)
+apply (drule sym scons_eq_UU [THEN iffD1],simp)
+apply (erule stream_finite_lemma2, simp)
+apply (simp add: slen_def, auto)
+by (drule stream_finite_lemma1,auto)
+
+lemma slen_less_1_eq: "(#x < Fin (Suc 0)) = (x = \<bottom>)"
+by (rule stream.casedist [of x], auto simp del: iSuc_Fin 
+    simp add: Fin_0 iSuc_Fin[THEN sym] i0_iless_iSuc iSuc_mono)
+
+lemma slen_empty_eq: "(#x = 0) = (x = \<bottom>)"
+by (rule stream.casedist [of x], auto)
+
+lemma slen_scons_eq: "(Fin (Suc n) < #x) = (? a y. x = a && y &  a ~= \<bottom> &  Fin n < #y)"
+apply (auto, case_tac "x=UU",auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (rule_tac x="a" in exI)
+apply (rule_tac x="y" in exI, simp)
+by (simp add: inat_defs split:inat_splits)+
+
+lemma slen_iSuc: "#x = iSuc n --> (? a y. x = a&&y &  a ~= \<bottom> &  #y = n)"
+by (rule stream.casedist [of x], auto)
+
+lemma slen_stream_take_finite [simp]: "#(stream_take n$s) ~= \<infinity>"
+by (simp add: slen_def)
+
+lemma slen_scons_eq_rev: "(#x < Fin (Suc (Suc n))) = (!a y. x ~= a && y |  a = \<bottom> |  #y < Fin (Suc n))"
+apply (rule stream.casedist [of x], auto)
+apply (drule sym, drule scons_eq_UU [THEN iffD1],auto)
+apply (simp add: inat_defs split:inat_splits)
+apply (subgoal_tac "s=y & aa=a",simp);
+apply (simp add: inat_defs split:inat_splits)
+apply (case_tac "aa=UU",auto)
+apply (erule_tac x="a" in allE, simp)
+by (simp add: inat_defs split:inat_splits)
+
+lemma slen_take_lemma4 [rule_format]: 
+  "!s. stream_take n$s ~= s --> #(stream_take n$s) = Fin n"
+apply (induct_tac n,auto simp add: Fin_0)
+apply (case_tac "s=UU",simp)
+by (drule stream_exhaust_eq [THEN iffD1], auto)
+
+(*
+lemma stream_take_idempotent [simp]: 
+ "stream_take n$(stream_take n$s) = stream_take n$s"
+apply (case_tac "stream_take n$s = s")
+apply (auto,insert slen_take_lemma4 [of n s]);
+by (auto,insert slen_take_lemma1 [of "stream_take n$s" n],simp)
+
+lemma stream_take_take_Suc [simp]: "stream_take n$(stream_take (Suc n)$s) = 
+                                    stream_take n$s"
+apply (simp add: po_eq_conv,auto)
+ apply (simp add: stream_take_take_less)
+apply (subgoal_tac "stream_take n$s = stream_take n$(stream_take n$s)")
+ apply (erule ssubst)
+ apply (rule_tac monofun_cfun_arg)
+ apply (insert chain_stream_take [of s])
+by (simp add: chain_def,simp)
+*)
+
+lemma slen_take_eq: "ALL x. (Fin n < #x) = (stream_take n\<cdot>x ~= x)"
+apply (induct_tac n, auto)
+apply (simp add: Fin_0, clarsimp)
+apply (drule not_sym)
+apply (drule slen_empty_eq [THEN iffD1], simp)
+apply (case_tac "x=UU", simp)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (erule_tac x="y" in allE, auto)
+apply (simp add: inat_defs split:inat_splits)
+apply (case_tac "x=UU", simp)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (erule_tac x="y" in allE, simp)
+by (simp add: inat_defs split:inat_splits)
+
+lemma slen_take_eq_rev: "(#x <= Fin n) = (stream_take n\<cdot>x = x)"
+by (simp add: ile_def slen_take_eq)
+
+lemma slen_take_lemma1: "#x = Fin n ==> stream_take n\<cdot>x = x"
+by (rule slen_take_eq_rev [THEN iffD1], auto)
+
+lemma slen_rt_mono: "#s2 <= #s1 ==> #(rt$s2) <= #(rt$s1)"
+apply (rule stream.casedist [of s1])
+ by (rule stream.casedist [of s2],simp+)+
+
+lemma slen_take_lemma5: "#(stream_take n$s) <= Fin n"; 
+apply (case_tac "stream_take n$s = s")
+ apply (simp add: slen_take_eq_rev)
+by (simp add: slen_take_lemma4)
+
+lemma slen_take_lemma2: "!x. ~stream_finite x --> #(stream_take i\<cdot>x) = Fin i"
+apply (simp add: stream.finite_def, auto)
+by (simp add: slen_take_lemma4)
+
+lemma slen_infinite: "stream_finite x = (#x ~= Infty)"
+by (simp add: slen_def)
+
+lemma slen_mono_lemma: "stream_finite s ==> ALL t. s << t --> #s <= #t"
+apply (erule stream_finite_ind [of s], auto)
+apply (case_tac "t=UU", auto)
+apply (drule eq_UU_iff [THEN iffD2])
+apply (drule scons_eq_UU [THEN iffD2], simp)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (erule_tac x="y" in allE, auto)
+by (drule stream.inverts, auto)
+
+lemma slen_mono: "s << t ==> #s <= #t"
+apply (case_tac "stream_finite t")
+apply (frule stream_finite_less) 
+apply (erule_tac x="s" in allE, simp)
+apply (drule slen_mono_lemma, auto)
+by (simp add: slen_def)
+
+lemma iterate_lemma: "F$(iterate n F x) = iterate n F (F$x)" 
+by (insert iterate_Suc2 [of n F x], auto)
+
+lemma slen_rt_mult [rule_format]: "!x. Fin (i + j) <= #x --> Fin j <= #(iterate i rt x)"
+apply (induct_tac i, auto)
+apply (case_tac "x=UU", auto)
+apply (simp add: inat_defs)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (erule_tac x="y" in allE, auto)
+apply (simp add: inat_defs split:inat_splits)
+by (simp add: iterate_lemma)
+
+lemma slen_take_lemma3 [rule_format]: 
+  "!(x::'a::flat stream) y. Fin n <= #x --> x << y --> stream_take n\<cdot>x = stream_take n\<cdot>y"
+apply (induct_tac n, auto)
+apply (case_tac "x=UU", auto)
+apply (simp add: inat_defs)
+apply (simp add: Suc_ile_eq)
+apply (case_tac "y=UU", clarsimp)
+apply (drule eq_UU_iff [THEN iffD2],simp)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)+
+apply (erule_tac x="ya" in allE, simp)
+apply (drule stream.inverts,auto)
+by (drule ax_flat [rule_format], simp)
+
+lemma slen_strict_mono_lemma: 
+  "stream_finite t ==> !s. #(s::'a::flat stream) = #t &  s << t --> s = t"
+apply (erule stream_finite_ind, auto)
+apply (drule eq_UU_iff [THEN iffD2], simp)
+apply (case_tac "sa=UU", auto)
+apply (drule stream_exhaust_eq [THEN iffD1], clarsimp)
+apply (drule stream.inverts, simp, simp, clarsimp)
+by (drule ax_flat [rule_format], simp)
+
+lemma slen_strict_mono: "[|stream_finite t; s ~= t; s << (t::'a::flat stream) |] ==> #s < #t"
+apply (intro ilessI1, auto)
+apply (simp add: slen_mono)
+by (drule slen_strict_mono_lemma, auto)
+
+lemma stream_take_Suc_neq: "stream_take (Suc n)$s ~=s ==> 
+                     stream_take n$s ~= stream_take (Suc n)$s"
+apply auto
+apply (subgoal_tac "stream_take n$s ~=s")
+ apply (insert slen_take_lemma4 [of n s],auto)
+apply (rule stream.casedist [of s],simp)
+apply (simp add: inat_defs split:inat_splits)
+by (simp add: slen_take_lemma4)
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about smap                                                     *)
+(* ----------------------------------------------------------------------- *)
+
+
+section "smap"
+
+lemma smap_unfold: "smap = (\<Lambda> f t. case t of x&&xs \<Rightarrow> f$x && smap$f$xs)"
+by (insert smap_def [THEN fix_eq2], auto)
+
+lemma smap_empty [simp]: "smap\<cdot>f\<cdot>\<bottom> = \<bottom>"
+by (subst smap_unfold, simp)
+
+lemma smap_scons [simp]: "x~=\<bottom> ==> smap\<cdot>f\<cdot>(x&&xs) = (f\<cdot>x)&&(smap\<cdot>f\<cdot>xs)"
+by (subst smap_unfold, force)
+
+
+
+(* ----------------------------------------------------------------------- *)
+(* theorems about sfilter                                                  *)
+(* ----------------------------------------------------------------------- *)
+
+section "sfilter"
+
+lemma sfilter_unfold: 
+ "sfilter = (\<Lambda> p s. case s of x && xs \<Rightarrow>
+  If p\<cdot>x then x && sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs fi)"
+by (insert sfilter_def [THEN fix_eq2], auto)
+
+lemma strict_sfilter: "sfilter\<cdot>\<bottom> = \<bottom>"
+apply (rule ext_cfun)
+apply (subst sfilter_unfold, auto)
+apply (case_tac "x=UU", auto)
+by (drule stream_exhaust_eq [THEN iffD1], auto)
+
+lemma sfilter_empty [simp]: "sfilter\<cdot>f\<cdot>\<bottom> = \<bottom>"
+by (subst sfilter_unfold, force)
+
+lemma sfilter_scons [simp]: 
+  "x ~= \<bottom> ==> sfilter\<cdot>f\<cdot>(x && xs) = 
+                           If f\<cdot>x then x && sfilter\<cdot>f\<cdot>xs else sfilter\<cdot>f\<cdot>xs fi" 
+by (subst sfilter_unfold, force)
+
+
+(* ----------------------------------------------------------------------- *)
+   section "i_rt"
+(* ----------------------------------------------------------------------- *)
+
+lemma i_rt_UU [simp]: "i_rt n UU = UU"
+apply (simp add: i_rt_def)
+by (rule iterate.induct,auto)
+
+lemma i_rt_0 [simp]: "i_rt 0 s = s"
+by (simp add: i_rt_def)
+
+lemma i_rt_Suc [simp]: "a ~= UU ==> i_rt (Suc n) (a&&s) = i_rt n s"
+by (simp add: i_rt_def iterate_Suc2 del: iterate_Suc)
+
+lemma i_rt_Suc_forw: "i_rt (Suc n) s = i_rt n (rt$s)"
+by (simp only: i_rt_def iterate_Suc2)
+
+lemma i_rt_Suc_back:"i_rt (Suc n) s = rt$(i_rt n s)"
+by (simp only: i_rt_def,auto)
+
+lemma i_rt_mono: "x << s ==> i_rt n x  << i_rt n s"
+by (simp add: i_rt_def monofun_rt_mult)
+
+lemma i_rt_ij_lemma: "Fin (i + j) <= #x ==> Fin j <= #(i_rt i x)"
+by (simp add: i_rt_def slen_rt_mult)
+
+lemma slen_i_rt_mono: "#s2 <= #s1 ==> #(i_rt n s2) <= #(i_rt n s1)"
+apply (induct_tac n,auto)
+apply (simp add: i_rt_Suc_back)
+by (drule slen_rt_mono,simp)
+
+lemma i_rt_take_lemma1 [rule_format]: "ALL s. i_rt n (stream_take n$s) = UU"
+apply (induct_tac n); 
+ apply (simp add: i_rt_Suc_back,auto)
+apply (case_tac "s=UU",auto)
+by (drule stream_exhaust_eq [THEN iffD1],auto)
+
+lemma i_rt_slen: "(i_rt n s = UU) = (stream_take n$s = s)"
+apply auto
+ apply (insert i_rt_ij_lemma [of n "Suc 0" s]);
+ apply (subgoal_tac "#(i_rt n s)=0")
+  apply (case_tac "stream_take n$s = s",simp+)
+  apply (insert slen_take_eq [rule_format,of n s],simp)
+  apply (simp add: inat_defs split:inat_splits)
+ apply (simp add: slen_take_eq )
+by (simp, insert i_rt_take_lemma1 [of n s],simp)
+
+lemma i_rt_lemma_slen: "#s=Fin n ==> i_rt n s = UU"
+by (simp add: i_rt_slen slen_take_lemma1)
+
+lemma stream_finite_i_rt [simp]: "stream_finite (i_rt n s) = stream_finite s"
+apply (induct_tac n, auto)
+ apply (rule stream.casedist [of "s"], auto simp del: i_rt_Suc)
+by (simp add: i_rt_Suc_back stream_finite_rt_eq)+
+
+lemma take_i_rt_len_lemma: "ALL sl x j t. Fin sl = #x & n <= sl &
+                            #(stream_take n$x) = Fin t & #(i_rt n x)= Fin j 
+                                              --> Fin (j + t) = #x"
+apply (induct_tac n,auto)
+ apply (simp add: inat_defs)
+apply (case_tac "x=UU",auto)
+ apply (simp add: inat_defs)
+apply (drule stream_exhaust_eq [THEN iffD1],clarsimp)
+apply (subgoal_tac "EX k. Fin k = #y",clarify)
+ apply (erule_tac x="k" in allE)
+ apply (erule_tac x="y" in allE,auto)
+ apply (erule_tac x="THE p. Suc p = t" in allE,auto)
+   apply (simp add: inat_defs split:inat_splits)
+  apply (simp add: inat_defs split:inat_splits)
+  apply (simp only: the_equality)
+ apply (simp add: inat_defs split:inat_splits)
+ apply force
+by (simp add: inat_defs split:inat_splits)
+
+lemma take_i_rt_len: 
+"[| Fin sl = #x; n <= sl; #(stream_take n$x) = Fin t; #(i_rt n x) = Fin j |] ==>
+    Fin (j + t) = #x"
+by (blast intro: take_i_rt_len_lemma [rule_format])
+
+
+(* ----------------------------------------------------------------------- *)
+   section "i_th"
+(* ----------------------------------------------------------------------- *)
+
+lemma i_th_i_rt_step:
+"[| i_th n s1 << i_th n s2; i_rt (Suc n) s1 << i_rt (Suc n) s2 |] ==> 
+   i_rt n s1 << i_rt n s2"
+apply (simp add: i_th_def i_rt_Suc_back)
+apply (rule stream.casedist [of "i_rt n s1"],simp)
+apply (rule stream.casedist [of "i_rt n s2"],auto)
+apply (drule eq_UU_iff [THEN iffD2], simp add: scons_eq_UU)
+by (intro monofun_cfun, auto)
+
+lemma i_th_stream_take_Suc [rule_format]: 
+ "ALL s. i_th n (stream_take (Suc n)$s) = i_th n s"
+apply (induct_tac n,auto)
+ apply (simp add: i_th_def)
+ apply (case_tac "s=UU",auto)
+ apply (drule stream_exhaust_eq [THEN iffD1],auto)
+apply (case_tac "s=UU",simp add: i_th_def)
+apply (drule stream_exhaust_eq [THEN iffD1],auto)
+by (simp add: i_th_def i_rt_Suc_forw)
+
+lemma i_th_last: "i_th n s && UU = i_rt n (stream_take (Suc n)$s)"
+apply (insert surjectiv_scons [of "i_rt n (stream_take (Suc n)$s)"])
+apply (rule i_th_stream_take_Suc [THEN subst])
+apply (simp add: i_th_def  i_rt_Suc_back [symmetric])
+by (simp add: i_rt_take_lemma1)
+
+lemma i_th_last_eq: 
+"i_th n s1 = i_th n s2 ==> i_rt n (stream_take (Suc n)$s1) = i_rt n (stream_take (Suc n)$s2)"
+apply (insert i_th_last [of n s1])
+apply (insert i_th_last [of n s2])
+by auto
+
+lemma i_th_prefix_lemma:
+"[| k <= n; stream_take (Suc n)$s1 << stream_take (Suc n)$s2 |] ==> 
+    i_th k s1 << i_th k s2"
+apply (insert i_th_stream_take_Suc [of k s1, THEN sym])
+apply (insert i_th_stream_take_Suc [of k s2, THEN sym],auto)
+apply (simp add: i_th_def)
+apply (rule monofun_cfun, auto)
+apply (rule i_rt_mono)
+by (blast intro: stream_take_lemma10)
+
+lemma take_i_rt_prefix_lemma1: 
+  "stream_take (Suc n)$s1 << stream_take (Suc n)$s2 ==>
+   i_rt (Suc n) s1 << i_rt (Suc n) s2 ==> 
+   i_rt n s1 << i_rt n s2 & stream_take n$s1 << stream_take n$s2"
+apply auto
+ apply (insert i_th_prefix_lemma [of n n s1 s2])
+ apply (rule i_th_i_rt_step,auto)
+by (drule mono_stream_take_pred,simp)
+
+lemma take_i_rt_prefix_lemma: 
+"[| stream_take n$s1 << stream_take n$s2; i_rt n s1 << i_rt n s2 |] ==> s1 << s2"
+apply (case_tac "n=0",simp)
+apply (insert neq0_conv [of n])
+apply (insert not0_implies_Suc [of n],auto)
+apply (subgoal_tac "stream_take 0$s1 << stream_take 0$s2 & 
+                    i_rt 0 s1 << i_rt 0 s2")
+ defer 1
+ apply (rule zero_induct,blast)
+ apply (blast dest: take_i_rt_prefix_lemma1)
+by simp
+
+lemma streams_prefix_lemma: "(s1 << s2) = 
+  (stream_take n$s1 << stream_take n$s2 & i_rt n s1 << i_rt n s2)"; 
+apply auto
+  apply (simp add: monofun_cfun_arg)
+ apply (simp add: i_rt_mono)
+by (erule take_i_rt_prefix_lemma,simp)
+
+lemma streams_prefix_lemma1:
+ "[| stream_take n$s1 = stream_take n$s2; i_rt n s1 = i_rt n s2 |] ==> s1 = s2"
+apply (simp add: po_eq_conv,auto)
+ apply (insert streams_prefix_lemma)
+ by blast+
+
+
+(* ----------------------------------------------------------------------- *)
+   section "sconc"
+(* ----------------------------------------------------------------------- *)
+
+lemma UU_sconc [simp]: " UU ooo s = s "
+by (simp add: sconc_def inat_defs)
+
+lemma scons_neq_UU: "a~=UU ==> a && s ~=UU"
+by auto
+
+lemma singleton_sconc [rule_format, simp]: "x~=UU --> (x && UU) ooo y = x && y"
+apply (simp add: sconc_def inat_defs split:inat_splits,auto)
+apply (rule someI2_ex,auto)
+ apply (rule_tac x="x && y" in exI,auto)
+apply (simp add: i_rt_Suc_forw)
+apply (case_tac "xa=UU",simp)
+by (drule stream_exhaust_eq [THEN iffD1],auto)
+
+lemma ex_sconc [rule_format]: 
+  "ALL k y. #x = Fin k --> (EX w. stream_take k$w = x & i_rt k w = y)"
+apply (case_tac "#x")
+ apply (rule stream_finite_ind [of x],auto)
+  apply (simp add: stream.finite_def)
+  apply (drule slen_take_lemma1,blast)
+ apply (simp add: inat_defs split:inat_splits)+
+apply (erule_tac x="y" in allE,auto)
+by (rule_tac x="a && w" in exI,auto)
+
+lemma rt_sconc1: "Fin n = #x ==> i_rt n (x ooo y) = y"; 
+apply (simp add: sconc_def inat_defs split:inat_splits, arith?,auto)
+apply (rule someI2_ex,auto)
+by (drule ex_sconc,simp)
+
+lemma sconc_inj2: "\<lbrakk>Fin n = #x; x ooo y = x ooo z\<rbrakk> \<Longrightarrow> y = z"
+apply (frule_tac y=y in rt_sconc1)
+by (auto elim: rt_sconc1)
+
+lemma sconc_UU [simp]:"s ooo UU = s"
+apply (case_tac "#s")
+ apply (simp add: sconc_def inat_defs)
+ apply (rule someI2_ex)
+  apply (rule_tac x="s" in exI)
+  apply auto
+   apply (drule slen_take_lemma1,auto)
+  apply (simp add: i_rt_lemma_slen)
+ apply (drule slen_take_lemma1,auto)
+ apply (simp add: i_rt_slen)
+by (simp add: sconc_def inat_defs)
+
+lemma stream_take_sconc [simp]: "Fin n = #x ==> stream_take n$(x ooo y) = x"
+apply (simp add: sconc_def)
+apply (simp add: inat_defs split:inat_splits,auto)
+apply (rule someI2_ex,auto)
+by (drule ex_sconc,simp)
+
+lemma scons_sconc [rule_format,simp]: "a~=UU --> (a && x) ooo y = a && x ooo y"
+apply (case_tac "#x",auto)
+ apply (simp add: sconc_def) 
+ apply (rule someI2_ex)
+  apply (drule ex_sconc,simp)
+ apply (rule someI2_ex,auto)
+  apply (simp add: i_rt_Suc_forw)
+  apply (rule_tac x="a && x" in exI,auto)
+ apply (case_tac "xa=UU",auto)
+  apply (drule_tac s="stream_take nat$x" in scons_neq_UU)
+  apply (simp add: i_rt_Suc_forw)
+ apply (drule stream_exhaust_eq [THEN iffD1],auto)
+ apply (drule streams_prefix_lemma1,simp+)
+by (simp add: sconc_def)
+
+lemma ft_sconc: "x ~= UU ==> ft$(x ooo y) = ft$x"
+by (rule stream.casedist [of x],auto)
+
+lemma sconc_assoc: "(x ooo y) ooo z = x ooo y ooo z"
+apply (case_tac "#x")
+ apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)
+  apply (simp add: stream.finite_def del: scons_sconc)
+  apply (drule slen_take_lemma1,auto simp del: scons_sconc)
+ apply (case_tac "a = UU", auto)
+by (simp add: sconc_def)
+
+
+(* ----------------------------------------------------------------------- *)
+
+lemma sconc_mono: "y << y' ==> x ooo y << x ooo y'"
+apply (case_tac "#x")
+ apply (rule stream_finite_ind [of "x"])
+   apply (auto simp add: stream.finite_def)
+  apply (drule slen_take_lemma1,blast)
+ by (simp add: stream_prefix',auto simp add: sconc_def)
+
+lemma sconc_mono1 [simp]: "x << x ooo y"
+by (rule sconc_mono [of UU, simplified])
+
+(* ----------------------------------------------------------------------- *)
+
+lemma empty_sconc [simp]: "(x ooo y = UU) = (x = UU & y = UU)"
+apply (case_tac "#x",auto)
+   apply (insert sconc_mono1 [of x y]);
+   by (insert eq_UU_iff [THEN iffD2, of x],auto)
+
+(* ----------------------------------------------------------------------- *)
+
+lemma rt_sconc [rule_format, simp]: "s~=UU --> rt$(s ooo x) = rt$s ooo x"
+by (rule stream.casedist,auto)
+
+lemma i_th_sconc_lemma [rule_format]: 
+  "ALL x y. Fin n < #x --> i_th n (x ooo y) = i_th n x"
+apply (induct_tac n, auto)
+apply (simp add: Fin_0 i_th_def)
+apply (simp add: slen_empty_eq ft_sconc)
+apply (simp add: i_th_def)
+apply (case_tac "x=UU",auto)
+apply (drule stream_exhaust_eq [THEN iffD1], auto)
+apply (erule_tac x="ya" in allE)
+by (simp add: inat_defs split:inat_splits)
+
+
+
+(* ----------------------------------------------------------------------- *)
+
+lemma sconc_lemma [rule_format, simp]: "ALL s. stream_take n$s ooo i_rt n s = s"
+apply (induct_tac n,auto)
+apply (case_tac "s=UU",auto)
+by (drule stream_exhaust_eq [THEN iffD1],auto)
+
+(* ----------------------------------------------------------------------- *)
+   subsection "pointwise equality"
+(* ----------------------------------------------------------------------- *)
+
+lemma ex_last_stream_take_scons: "stream_take (Suc n)$s = 
+                     stream_take n$s ooo i_rt n (stream_take (Suc n)$s)"
+by (insert sconc_lemma [of n "stream_take (Suc n)$s"],simp)
+
+lemma i_th_stream_take_eq: 
+"!!n. ALL n. i_th n s1 = i_th n s2 ==> stream_take n$s1 = stream_take n$s2"
+apply (induct_tac n,auto)
+apply (subgoal_tac "stream_take (Suc na)$s1 =
+                    stream_take na$s1 ooo i_rt na (stream_take (Suc na)$s1)")
+ apply (subgoal_tac "i_rt na (stream_take (Suc na)$s1) = 
+                    i_rt na (stream_take (Suc na)$s2)")
+  apply (subgoal_tac "stream_take (Suc na)$s2 = 
+                    stream_take na$s2 ooo i_rt na (stream_take (Suc na)$s2)")
+   apply (insert ex_last_stream_take_scons,simp)
+  apply blast
+ apply (erule_tac x="na" in allE)
+ apply (insert i_th_last_eq [of _ s1 s2])
+by blast+
+
+lemma pointwise_eq_lemma[rule_format]: "ALL n. i_th n s1 = i_th n s2 ==> s1 = s2"
+by (insert i_th_stream_take_eq [THEN stream.take_lemmas],blast)
+
+(* ----------------------------------------------------------------------- *)
+   subsection "finiteness"
+(* ----------------------------------------------------------------------- *)
+
+lemma slen_sconc_finite1:
+  "[| #(x ooo y) = Infty; Fin n = #x |] ==> #y = Infty"
+apply (case_tac "#y ~= Infty",auto)
+apply (simp only: slen_infinite [symmetric])
+apply (drule_tac y=y in rt_sconc1)
+apply (insert stream_finite_i_rt [of n "x ooo y"])
+by (simp add: slen_infinite)
+
+lemma slen_sconc_infinite1: "#x=Infty ==> #(x ooo y) = Infty"
+by (simp add: sconc_def)
+
+lemma slen_sconc_infinite2: "#y=Infty ==> #(x ooo y) = Infty"
+apply (case_tac "#x")
+ apply (simp add: sconc_def)
+ apply (rule someI2_ex)
+  apply (drule ex_sconc,auto)
+ apply (erule contrapos_pp)
+ apply (insert stream_finite_i_rt)
+ apply (simp add: slen_infinite,auto)
+by (simp add: sconc_def)
+
+lemma sconc_finite: "(#x~=Infty & #y~=Infty) = (#(x ooo y)~=Infty)"
+apply auto
+  apply (case_tac "#x",auto)
+  apply (erule contrapos_pp,simp)
+  apply (erule slen_sconc_finite1,simp)
+ apply (drule slen_sconc_infinite1 [of _ y],simp)
+by (drule slen_sconc_infinite2 [of _ x],simp)
+
+(* ----------------------------------------------------------------------- *)
+
+lemma slen_sconc_mono3: "[| Fin n = #x; Fin k = #(x ooo y) |] ==> n <= k"
+apply (insert slen_mono [of "x" "x ooo y"])
+by (simp add: inat_defs split: inat_splits)
+
+(* ----------------------------------------------------------------------- *)
+   subsection "finite slen"
+(* ----------------------------------------------------------------------- *)
+
+lemma slen_sconc: "[| Fin n = #x; Fin m = #y |] ==> #(x ooo y) = Fin (n + m)"
+apply (case_tac "#(x ooo y)")
+ apply (frule_tac y=y in rt_sconc1)
+ apply (insert take_i_rt_len [of "THE j. Fin j = #(x ooo y)" "x ooo y" n n m],simp)
+ apply (insert slen_sconc_mono3 [of n x _ y],simp)
+by (insert sconc_finite [of x y],auto)
+
+(* ----------------------------------------------------------------------- *)
+   subsection "flat prefix"
+(* ----------------------------------------------------------------------- *)
+
+lemma sconc_prefix: "(s1::'a::flat stream) << s2 ==> EX t. s1 ooo t = s2"
+apply (case_tac "#s1")
+ apply (subgoal_tac "stream_take nat$s1 = stream_take nat$s2");
+  apply (rule_tac x="i_rt nat s2" in exI)
+  apply (simp add: sconc_def)
+  apply (rule someI2_ex)
+   apply (drule ex_sconc)
+   apply (simp,clarsimp,drule streams_prefix_lemma1)
+   apply (simp+,rule slen_take_lemma3 [of _ s1 s2]);
+  apply (simp+,rule_tac x="UU" in exI)
+apply (insert slen_take_lemma3 [of _ s1 s2]);
+by (rule stream.take_lemmas,simp)
+
+(* ----------------------------------------------------------------------- *)
+   subsection "continuity"
+(* ----------------------------------------------------------------------- *)
+
+lemma chain_sconc: "chain S ==> chain (%i. (x ooo S i))"
+by (simp add: chain_def,auto simp add: sconc_mono)
+
+lemma chain_scons: "chain S ==> chain (%i. a && S i)"
+apply (simp add: chain_def,auto)
+by (rule monofun_cfun_arg,simp)
+
+lemma contlub_scons: "contlub (%x. a && x)"
+by (simp add: contlub_Rep_CFun2)
+
+lemma contlub_scons_lemma: "chain S ==> (LUB i. a && S i) = a && (LUB i. S i)"
+apply (insert contlub_scons [of a])
+by (simp only: contlub)
+
+lemma finite_lub_sconc: "chain Y ==> (stream_finite x) ==> 
+                        (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
+apply (rule stream_finite_ind [of x])
+ apply (auto)
+apply (subgoal_tac "(LUB i. a && (s ooo Y i)) = a && (LUB i. s ooo Y i)")
+ by (force,blast dest: contlub_scons_lemma chain_sconc)
+
+lemma contlub_sconc_lemma: 
+  "chain Y ==> (LUB i. x ooo Y i) = (x ooo (LUB i. Y i))"
+apply (case_tac "#x=Infty")
+ apply (simp add: sconc_def)
+ prefer 2
+ apply (drule finite_lub_sconc,auto simp add: slen_infinite)
+apply (simp add: slen_def)
+apply (insert lub_const [of x] unique_lub [of _ x _])
+by (auto simp add: lub)
+
+lemma contlub_sconc: "contlub (%y. x ooo y)"; 
+by (rule contlubI, insert contlub_sconc_lemma [of _ x], simp);
+
+lemma monofun_sconc: "monofun (%y. x ooo y)"
+by (simp add: monofun sconc_mono)
+
+lemma cont_sconc: "cont (%y. x ooo y)"
+apply (rule  monocontlub2cont)
+ apply (rule monofunI, simp add: sconc_mono)
+by (rule contlub_sconc);
+
+
+(* ----------------------------------------------------------------------- *)
+   section "constr_sconc"
+(* ----------------------------------------------------------------------- *)
+
+lemma constr_sconc_UUs [simp]: "constr_sconc UU s = s"
+by (simp add: constr_sconc_def inat_defs)
+
+lemma "x ooo y = constr_sconc x y"
+apply (case_tac "#x")
+ apply (rule stream_finite_ind [of x],auto simp del: scons_sconc)
+  defer 1
+  apply (simp add: constr_sconc_def del: scons_sconc)
+  apply (case_tac "#s")
+   apply (simp add: inat_defs)
+   apply (case_tac "a=UU",auto simp del: scons_sconc)
+   apply (simp)
+  apply (simp add: sconc_def)
+ apply (simp add: constr_sconc_def)
+apply (simp add: stream.finite_def)
+by (drule slen_take_lemma1,auto)
+
+declare eq_UU_iff [THEN sym, simp add]
 
 end
-
-