diff -r cef2893f353b -r c64241fa4dff Myhill.thy
--- a/Myhill.thy	Fri Dec 31 13:47:53 2010 +0000
+++ b/Myhill.thy	Fri Jan 07 14:25:23 2011 +0000
@@ -3,8 +3,7 @@
 begin
 
 text {* sequential composition of languages *}
-definition
-  Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
+definition Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
 where 
   "L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
 
@@ -19,25 +18,22 @@
   "(A \<union> B) ;; C = (A ;; C) \<union> (B ;; C)"
 by (auto simp:Seq_def)
 
+lemma seq_intro:
+  "\<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x @ y \<in> A ;; B "
+by (auto simp:Seq_def)
+
 lemma seq_assoc:
   "(A ;; B) ;; C = A ;; (B ;; C)"
-unfolding Seq_def
-apply(auto)
-apply(metis)
+apply(auto simp:Seq_def)
+apply blast
 by (metis append_assoc)
 
-lemma union_seq:
-  "\<Union> {f x y ;; z| x y. P x y } = (\<Union> {f x y|x y. P x y });; z"
-apply (auto simp add:Seq_def)
-apply metis
-done
-
 theorem ardens_revised:
   assumes nemp: "[] \<notin> A"
   shows "(X = X ;; A \<union> B) \<longleftrightarrow> (X = B ;; A\<star>)"
 proof
   assume eq: "X = B ;; A\<star>"
-  have "A\<star> =  {[]} \<union> A\<star> ;; A" sorry
+  have "A\<star> =  {[]} \<union> A\<star> ;; A" sorry 
   then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)" unfolding Seq_def by simp
   also have "\<dots> = B \<union> B ;; (A\<star> ;; A)"  unfolding Seq_def by auto
   also have "\<dots> = B \<union> (B ;; A\<star>) ;; A"  unfolding Seq_def
@@ -46,7 +42,7 @@
 next
   assume "X = X ;; A \<union> B"
   then have "B \<subseteq> X" "X ;; A \<subseteq> X" by auto
-  show "X = B ;; A\<star>" sorry
+  thus "X = B ;; A\<star>" sorry
 qed
 
 datatype rexp =
@@ -778,286 +774,71 @@
   "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
 proof
   fix x
-  assume h: "x \<in> UNIV // \<approx>{[]}"
-  show "x \<in> {{[]}, UNIV - {[]}}"
-    
- 
-  have "\<And> s. s \<approx>{[]} [] \<Longrightarrow> s = []"
-    apply (auto simp add:str_eq_def)
-    apply blast
-  
-  hence "False"
-    apply (simp add:quotient_def)
-
-
-lemma other_direction:
-  "Lang = L (r::rexp) \<Longrightarrow> finite (UNIV // (\<approx>Lang))"
-proof (induct arbitrary:Lang rule:rexp.induct)
-  case NULL
-  have "UNIV // (\<approx>{}) \<subseteq> {UNIV} "
-    by (auto simp:quotient_def str_eq_rel_def str_eq_def)
-  with prems show "?case" by (auto intro:finite_subset)
-next
-  case EMPTY
-  have "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
-    sorry
-  with prems show ?case by (auto intro:finite_subset)
-next
-  case (CHAR c)
-  have "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
-    sorry
-  with prems show ?case by (auto intro:finite_subset)
-next
-  case (SEQ r1 r2)
-  show ?case sorry
-next
-  case (ALT r1 r1)
-  show ?case sorry
-next
-  case (STAR r)
-  show ?case sorry
-qed
-    
-
-      
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-apply (induct arbitrary:Lang rule:rexp.induct)
-apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)  
-
-(* Alternative definition for Quo *)
-definition 
-  QUOT :: "string set \<Rightarrow> (string set) set"  
-where
-  "QUOT Lang \<equiv> (\<Union>x. {\<lbrakk>x\<rbrakk>Lang})"
-
-lemma test: 
-  "UNIV Quo Lang = QUOT Lang"
-by (auto simp add: quot_def QUOT_def)
-
-lemma test1:
-  assumes finite_CS: "finite (QUOT Lang)"
-  shows "reg Lang"
-using finite_CS
-unfolding test[symmetric]
-by (auto dest: myhill_nerode)
-
-lemma cons_one: "x @ y \<in> {z} \<Longrightarrow> x @ y = z"
-by simp
-
-lemma quot_lambda: "QUOT {[]} = {{[]}, UNIV - {[]}}"
-proof 
-  show "QUOT {[]} \<subseteq> {{[]}, UNIV - {[]}}"
-  proof 
-    fix x 
-    assume in_quot: "x \<in> QUOT {[]}"
-    show "x \<in> {{[]}, UNIV - {[]}}"
-    proof -
-      from in_quot 
-      have "x = {[]} \<or> x = UNIV - {[]}"
-        unfolding QUOT_def equiv_class_def
-      proof 
-        fix xa
-        assume in_univ: "xa \<in> UNIV"
-           and in_eqiv: "x \<in> {{y. xa \<equiv>{[]}\<equiv> y}}"
-        show "x = {[]} \<or> x = UNIV - {[]}"
-        proof(cases "xa = []")
-          case True
-          hence "{y. xa \<equiv>{[]}\<equiv> y} = {[]}" using in_eqiv
-            by (auto simp add:equiv_str_def)
-          thus ?thesis using in_eqiv by (rule_tac disjI1, simp)
-        next
-          case False
-          hence "{y. xa \<equiv>{[]}\<equiv> y} = UNIV - {[]}" using in_eqiv
-            by (auto simp:equiv_str_def)
-          thus ?thesis using in_eqiv by (rule_tac disjI2, simp)
-        qed
-      qed
-      thus ?thesis by simp
-    qed
-  qed
-next
-  show "{{[]}, UNIV - {[]}} \<subseteq> QUOT {[]}"
-  proof
-    fix x
-    assume in_res: "x \<in> {{[]}, (UNIV::string set) - {[]}}"
-    show "x \<in> (QUOT {[]})"
-    proof -
-      have "x = {[]} \<Longrightarrow> x \<in> QUOT {[]}"
-        apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-        by (rule_tac x = "[]" in exI, auto)
-      moreover have "x = UNIV - {[]} \<Longrightarrow> x \<in> QUOT {[]}"
-        apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-        by (rule_tac x = "''a''" in exI, auto)
-      ultimately show ?thesis using in_res by blast
-    qed
+  assume "x \<in> UNIV // \<approx>{[]}"
+  then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[]}}" unfolding quotient_def Image_def by blast
+  show "x \<in> {{[]}, UNIV - {[]}}" 
+  proof (cases "y = []")
+    case True with h
+    have "x = {[]}" by (auto simp:str_eq_rel_def str_eq_def)
+    thus ?thesis by simp
+  next
+    case False with h
+    have "x = UNIV - {[]}" by (auto simp:str_eq_rel_def str_eq_def)
+    thus ?thesis by simp
   qed
 qed
 
-lemma quot_single_aux: "\<lbrakk>x \<noteq> []; x \<noteq> [c]\<rbrakk> \<Longrightarrow> x @ z \<noteq> [c]"
-by (induct x, auto)
-
-lemma quot_single: "\<And> (c::char). QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
-proof - 
-  fix c::"char" 
-  have exist_another: "\<exists> a. a \<noteq> c" 
-    apply (case_tac "c = CHR ''a''")
-    apply (rule_tac x = "CHR ''b''" in exI, simp)
-    by (rule_tac x = "CHR ''a''" in exI, simp)  
-  show "QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
-  proof
-    show "QUOT {[c]} \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
-    proof 
-      fix x 
-      assume in_quot: "x \<in> QUOT {[c]}"
-      show "x \<in> {{[]}, {[c]}, UNIV - {[],[c]}}"
-      proof -
-        from in_quot 
-        have "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[],[c]}"
-          unfolding QUOT_def equiv_class_def
-        proof 
-          fix xa
-          assume in_eqiv: "x \<in> {{y. xa \<equiv>{[c]}\<equiv> y}}"
-          show "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[], [c]}"
-          proof-
-            have "xa = [] \<Longrightarrow> x = {[]}" using in_eqiv 
-              by (auto simp add:equiv_str_def)
-            moreover have "xa = [c] \<Longrightarrow> x = {[c]}"
-            proof -
-              have "xa = [c] \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = {[c]}" using in_eqiv
-                apply (simp add:equiv_str_def)
-                apply (rule set_ext, rule iffI, simp)
-                apply (drule_tac x = "[]" in spec, auto)
-                done   
-              thus "xa = [c] \<Longrightarrow> x = {[c]}" using in_eqiv by simp 
-            qed
-            moreover have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}"
-            proof -
-              have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = UNIV - {[],[c]}" 
-                apply (clarsimp simp add:equiv_str_def)
-                apply (rule set_ext, rule iffI, simp)
-                apply (rule conjI)
-                apply (drule_tac x = "[c]" in spec, simp)
-                apply (drule_tac x = "[]" in spec, simp)
-                by (auto dest:quot_single_aux)
-              thus "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}" using in_eqiv by simp
-            qed
-            ultimately show ?thesis by blast
-          qed
-        qed
-        thus ?thesis by simp
-      qed
-    qed
-  next
-    show "{{[]}, {[c]}, UNIV - {[],[c]}} \<subseteq> QUOT {[c]}"
-    proof
-      fix x
-      assume in_res: "x \<in> {{[]},{[c]}, (UNIV::string set) - {[],[c]}}"
-      show "x \<in> (QUOT {[c]})"
-      proof -
-        have "x = {[]} \<Longrightarrow> x \<in> QUOT {[c]}"
-          apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-          by (rule_tac x = "[]" in exI, auto)
-        moreover have "x = {[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
-          apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-          apply (rule_tac x = "[c]" in exI, simp)
-          apply (rule set_ext, rule iffI, simp+)
-          by (drule_tac x = "[]" in spec, simp)
-        moreover have "x = UNIV - {[],[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
-          using exist_another
-          apply (clarsimp simp add:QUOT_def equiv_class_def equiv_str_def)        
-          apply (rule_tac x = "[a]" in exI, simp)
-          apply (rule set_ext, rule iffI, simp)
-          apply (clarsimp simp:quot_single_aux, simp)
-          apply (rule conjI)
-          apply (drule_tac x = "[c]" in spec, simp)
-          by (drule_tac x = "[]" in spec, simp)     
-        ultimately show ?thesis using in_res by blast
-      qed
-    qed
+lemma quot_char_subset:
+  "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
+proof 
+  fix x 
+  assume "x \<in> UNIV // \<approx>{[c]}"
+  then obtain y where h: "x = {z. (y, z) \<in> \<approx>{[c]}}" unfolding quotient_def Image_def by blast
+  show "x \<in> {{[]},{[c]}, UNIV - {[], [c]}}"
+  proof -
+    { assume "y = []" hence "x = {[]}" using h by (auto simp:str_eq_rel_def str_eq_def)
+    } moreover {
+      assume "y = [c]" hence "x = {[c]}" using h 
+        by (auto dest!:spec[where x = "[]"] simp:str_eq_rel_def str_eq_def)
+    } moreover {
+      assume "y \<noteq> []" and "y \<noteq> [c]"
+      hence "\<forall> z. (y @ z) \<noteq> [c]" by (case_tac y, auto)
+      moreover have "\<And> p. (p \<noteq> [] \<and> p \<noteq> [c]) = (\<forall> q. p @ q \<noteq> [c])" by (case_tac p, auto)
+      ultimately have "x = UNIV - {[],[c]}" using h
+        by (auto simp add:str_eq_rel_def str_eq_def)
+    } ultimately show ?thesis by blast
   qed
 qed
 
-lemma eq_class_imp_eq_str:
-  "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang \<Longrightarrow> x \<equiv>lang\<equiv> y"
-by (auto simp:equiv_class_def equiv_str_def)
+text {* *************** Some common lemmas for following ALT, SEQ & STAR cases ******************* *}
 
-lemma finite_tag_image: 
+lemma finite_tag_imageI: 
   "finite (range tag) \<Longrightarrow> finite (((op `) tag) ` S)"
 apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
 by (auto simp add:image_def Pow_def)
 
-lemma str_inj_imps:
-  assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<equiv>lang\<equiv> n"
-  shows "inj_on ((op `) tag) (QUOT lang)"
-proof (clarsimp simp add:inj_on_def QUOT_def)
-  fix x y
-  assume eq_tag: "tag ` \<lbrakk>x\<rbrakk>lang = tag ` \<lbrakk>y\<rbrakk>lang"
-  show "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang"
-  proof -
-    have aux1:"\<And>a b. a \<in> (\<lbrakk>b\<rbrakk>lang) \<Longrightarrow> (a \<equiv>lang\<equiv> b)"
-      by (simp add:equiv_class_def equiv_str_def)
-    have aux2: "\<And> A B f. \<lbrakk>f ` A = f ` B; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> a b. a \<in> A \<and> b \<in> B \<and> f a = f b"
-      by auto
-    have aux3: "\<And> a l. \<lbrakk>a\<rbrakk>l \<noteq> {}" 
-      by (auto simp:equiv_class_def equiv_str_def)
-    show ?thesis using eq_tag
-      apply (drule_tac aux2, simp add:aux3, clarsimp)
-      apply (drule_tac str_inj, (drule_tac aux1)+)
-      by (simp add:equiv_str_def equiv_class_def)
-  qed
+lemma eq_class_equalI:
+  "\<lbrakk>X \<in> UNIV // \<approx>lang; Y \<in> UNIV // \<approx>lang; x \<in> X; y \<in> Y; x \<approx>lang y\<rbrakk> \<Longrightarrow> X = Y"
+by (auto simp:quotient_def str_eq_rel_def str_eq_def)
+
+lemma tag_image_injI:
+  assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<approx>lang n"
+  shows "inj_on ((op `) tag) (UNIV // \<approx>lang)"
+proof-
+  { fix X Y
+    assume X_in: "X \<in> UNIV // \<approx>lang"
+      and  Y_in: "Y \<in> UNIV // \<approx>lang"
+      and  tag_eq: "tag ` X = tag ` Y"
+    then obtain x y where "x \<in> X" and "y \<in> Y" and "tag x = tag y"
+      unfolding quotient_def Image_def str_eq_rel_def str_eq_def image_def
+      apply simp by blast
+    with X_in Y_in str_inj
+    have "X = Y" by (rule_tac eq_class_equalI, simp+)
+  }
+  thus ?thesis unfolding inj_on_def by auto
 qed
 
-definition tag_str_ALT :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
-where
-  "tag_str_ALT L\<^isub>1 L\<^isub>2 x \<equiv> (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)"
-
-lemma tag_str_alt_range_finite:
-  assumes finite1: "finite (QUOT L\<^isub>1)"
-  and finite2: "finite (QUOT L\<^isub>2)"
-  shows "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
-proof -
-  have "{y. \<exists>x. y = (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)} \<subseteq> (QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"
-    by (auto simp:QUOT_def)
-  thus ?thesis using finite1 finite2
-    by (auto simp: image_def tag_str_ALT_def UNION_def 
-            intro: finite_subset[where B = "(QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"])
-qed
-
-lemma tag_str_alt_inj:
-  "tag_str_ALT L\<^isub>1 L\<^isub>2 x = tag_str_ALT L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<equiv>(L\<^isub>1 \<union> L\<^isub>2)\<equiv> y"
-apply (simp add:tag_str_ALT_def equiv_class_def equiv_str_def)
-by blast
-  
-lemma quot_alt:
-  assumes finite1: "finite (QUOT L\<^isub>1)"
-  and finite2: "finite (QUOT L\<^isub>2)"
-  shows "finite (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
-proof (rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
-  show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 \<union> L\<^isub>2))"
-    using finite_tag_image tag_str_alt_range_finite finite1 finite2
-    by auto
-next
-  show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
-    apply (rule_tac str_inj_imps)
-    by (erule_tac tag_str_alt_inj)
-qed
+text {* **************** the SEQ case ************************ *}
 
 (* list_diff:: list substract, once different return tailer *)
 fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
@@ -1079,273 +860,297 @@
 lemma [simp]: "x - [] = x"
 by (induct x, auto)
 
-lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
+lemma [simp]: "(x - y = []) \<Longrightarrow> (x \<le> y)"
+proof-   
+  have "\<exists>xa. x = xa @ (x - y) \<and> xa \<le> y"
+    apply (rule list_diff.induct[of _ x y], simp+)
+    by (clarsimp, rule_tac x = "y # xa" in exI, simp+)
+  thus "(x - y = []) \<Longrightarrow> (x \<le> y)" by simp
+qed
+
+lemma diff_prefix:
+  "\<lbrakk>c \<le> a - b; b \<le> a\<rbrakk> \<Longrightarrow> b @ c \<le> a"
 by (auto elim:prefixE)
 
-definition tag_str_SEQ:: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set set)"
-where
-  "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> if (\<exists> xa \<le> x. xa \<in> L\<^isub>1)
-                         then (\<lbrakk>x\<rbrakk>L\<^isub>1, {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa.  xa \<le> x \<and> xa \<in> L\<^isub>1})
-                         else (\<lbrakk>x\<rbrakk>L\<^isub>1, {})"
+lemma diff_diff_appd: 
+  "\<lbrakk>c < a - b; b < a\<rbrakk> \<Longrightarrow> (a - b) - c = a - (b @ c)"
+apply (clarsimp simp:strict_prefix_def)
+by (drule diff_prefix, auto elim:prefixE)
 
-lemma tag_seq_eq_E:
-  "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y \<Longrightarrow>
-   ((\<exists> xa \<le> x. xa \<in> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1 \<and> 
-    {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1} ) \<or>
-   ((\<forall> xa \<le> x. xa \<notin> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1)"
-by (simp add:tag_str_SEQ_def split:if_splits, blast)
+lemma app_eq_cases[rule_format]:
+  "\<forall> x . x @ y = m @ n \<longrightarrow> (x \<le> m \<or> m \<le> x)"
+apply (induct y, simp)
+apply (clarify, drule_tac x = "x @ [a]" in spec)
+by (clarsimp, auto simp:prefix_def)
+
+lemma app_eq_dest:
+  "x @ y = m @ n \<Longrightarrow> (x \<le> m \<and> (m - x) @ n = y) \<or> (m \<le> x \<and> (x - m) @ y = n)"
+by (frule_tac app_eq_cases, auto elim:prefixE)
+
+definition 
+  "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> ((\<approx>L\<^isub>1) `` {x}, {(\<approx>L\<^isub>2) `` {x - xa}| xa.  xa \<le> x \<and> xa \<in> L\<^isub>1})"
 
 lemma tag_str_seq_range_finite:
-  assumes finite1: "finite (QUOT L\<^isub>1)"
-  and finite2: "finite (QUOT L\<^isub>2)"
-  shows "finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
-proof -
-  have "(range (tag_str_SEQ L\<^isub>1 L\<^isub>2)) \<subseteq> (QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))"
-    by (auto simp:image_def tag_str_SEQ_def QUOT_def)
-  thus ?thesis using finite1 finite2 
-    by (rule_tac B = "(QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))" in finite_subset, auto)
-qed
-  
-lemma app_in_seq_decom[rule_format]:
-  "\<forall> x. x @ z \<in> L\<^isub>1 ; L\<^isub>2 \<longrightarrow> (\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2) \<or> 
-                            (\<exists> za \<le> z. (x @ za) \<in> L\<^isub>1 \<and> (z - za) \<in> L\<^isub>2)"
-apply (induct z)
-apply (simp, rule allI, rule impI, rule disjI1)
-apply (clarsimp simp add:lang_seq_def)
-apply (rule_tac x = s1 in exI, simp)
-apply (rule allI | rule impI)+
-apply (drule_tac x = "x @ [a]" in spec, simp)
-apply (erule exE | erule conjE | erule disjE)+
-apply (rule disjI2, rule_tac x = "[a]" in exI, simp)
-apply (rule disjI1, rule_tac x = xa in exI, simp) 
-apply (erule exE | erule conjE)+
-apply (rule disjI2, rule_tac x = "a # za" in exI, simp)
-done
+  "\<lbrakk>finite (UNIV // \<approx>L\<^isub>1); finite (UNIV // \<approx>L\<^isub>2)\<rbrakk> \<Longrightarrow> finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
+apply (rule_tac B = "(UNIV // \<approx>L\<^isub>1) \<times> (Pow (UNIV // \<approx>L\<^isub>2))" in finite_subset)
+by (auto simp:tag_str_SEQ_def Image_def quotient_def split:if_splits)
 
-lemma tag_str_seq_inj:
-  assumes tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
-  shows "(x::string) \<equiv>(L\<^isub>1 ; L\<^isub>2)\<equiv> y"
-proof -
-  have aux: "\<And> x y z. \<lbrakk>tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y; x @ z \<in> L\<^isub>1 ; L\<^isub>2\<rbrakk> 
-                       \<Longrightarrow> y @ z \<in> L\<^isub>1 ; L\<^isub>2"
-  proof (drule app_in_seq_decom, erule disjE)
-    fix x y z
-    assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
-      and x_gets_l2: "\<exists>xa\<le>x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2"
-    from x_gets_l2 have "\<exists> xa \<le> x. xa \<in> L\<^isub>1" by blast
-    hence xy_l2:"{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1}"
-      using tag_eq tag_seq_eq_E by blast
-    from x_gets_l2 obtain xa where xa_le_x: "xa \<le> x"
-                               and xa_in_l1: "xa \<in> L\<^isub>1"
-                               and rest_in_l2: "(x - xa) @ z \<in> L\<^isub>2" by blast
-    hence "\<exists> ya. \<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 \<and> ya \<le> y \<and> ya \<in> L\<^isub>1" using xy_l2 by auto
-    then obtain ya where ya_le_x: "ya \<le> y"
-                     and ya_in_l1: "ya \<in> L\<^isub>1"
-                     and rest_eq: "\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2" by blast
-    from rest_eq rest_in_l2 have "(y - ya) @ z \<in> L\<^isub>2" 
-      by (auto simp:equiv_class_def equiv_str_def)
-    hence "ya @ ((y - ya) @ z) \<in> L\<^isub>1 ; L\<^isub>2" using ya_in_l1
-      by (auto simp:lang_seq_def)
-    thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using ya_le_x 
-      by (erule_tac prefixE, simp)
-  next
-    fix x y z
-    assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
-      and x_gets_l1: "\<exists>za\<le>z. x @ za \<in> L\<^isub>1 \<and> z - za \<in> L\<^isub>2"
-    from tag_eq tag_seq_eq_E have x_y_eq: "\<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1" by blast
-    from x_gets_l1 obtain za where za_le_z: "za \<le> z"
-                               and x_za_in_l1: "(x @ za) \<in> L\<^isub>1"
-                               and rest_in_l2: "z - za \<in> L\<^isub>2" by blast
-    from x_y_eq x_za_in_l1 have y_za_in_l1: "y @ za \<in> L\<^isub>1"
-      by (auto simp:equiv_class_def equiv_str_def)
-    hence "(y @ za) @ (z - za) \<in> L\<^isub>1 ; L\<^isub>2" using rest_in_l2
-      apply (simp add:lang_seq_def)
-      by (rule_tac x = "y @ za" in exI, rule_tac x = "z - za" in exI, simp)
-    thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using za_le_z
-      by (erule_tac prefixE, simp)
-  qed
-  show ?thesis using tag_eq
-    apply (simp add:equiv_str_def)
-    by (auto intro:aux)
+lemma append_seq_elim:
+  assumes "x @ y \<in> L\<^isub>1 ;; L\<^isub>2"
+  shows "(\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ y \<in> L\<^isub>2) \<or> (\<exists> ya \<le> y. (x @ ya) \<in> L\<^isub>1 \<and> (y - ya) \<in> L\<^isub>2)"
+proof-
+  from assms obtain s\<^isub>1 s\<^isub>2 where "x @ y = s\<^isub>1 @ s\<^isub>2" and in_seq: "s\<^isub>1 \<in> L\<^isub>1 \<and> s\<^isub>2 \<in> L\<^isub>2" 
+    by (auto simp:Seq_def)
+  hence "(x \<le> s\<^isub>1 \<and> (s\<^isub>1 - x) @ s\<^isub>2 = y) \<or> (s\<^isub>1 \<le> x \<and> (x - s\<^isub>1) @ y = s\<^isub>2)"
+    using app_eq_dest by auto
+  moreover have "\<lbrakk>x \<le> s\<^isub>1; (s\<^isub>1 - x) @ s\<^isub>2 = y\<rbrakk> \<Longrightarrow> \<exists> ya \<le> y. (x @ ya) \<in> L\<^isub>1 \<and> (y - ya) \<in> L\<^isub>2" using in_seq
+    by (rule_tac x = "s\<^isub>1 - x" in exI, auto elim:prefixE)
+  moreover have "\<lbrakk>s\<^isub>1 \<le> x; (x - s\<^isub>1) @ y = s\<^isub>2\<rbrakk> \<Longrightarrow> \<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ y \<in> L\<^isub>2" using in_seq
+    by (rule_tac x = s\<^isub>1 in exI, auto)
+  ultimately show ?thesis by blast
 qed
 
-lemma quot_seq: 
-  assumes finite1: "finite (QUOT L\<^isub>1)"
-  and finite2: "finite (QUOT L\<^isub>2)"
-  shows "finite (QUOT (L\<^isub>1;L\<^isub>2))"
-proof (rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
-  show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 ; L\<^isub>2))"
-    using finite_tag_image tag_str_seq_range_finite finite1 finite2
-    by auto
+lemma tag_str_SEQ_injI:
+  "tag_str_SEQ L\<^isub>1 L\<^isub>2 m = tag_str_SEQ L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 ;; L\<^isub>2) n"
+proof-
+  { fix x y z
+    assume xz_in_seq: "x @ z \<in> L\<^isub>1 ;; L\<^isub>2"
+    and tag_xy: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+    have"y @ z \<in> L\<^isub>1 ;; L\<^isub>2" 
+    proof-
+      have "(\<exists> xa \<le> x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2) \<or> (\<exists> za \<le> z. (x @ za) \<in> L\<^isub>1 \<and> (z - za) \<in> L\<^isub>2)"
+        using xz_in_seq append_seq_elim by simp
+      moreover {
+        fix xa
+        assume h1: "xa \<le> x" and h2: "xa \<in> L\<^isub>1" and h3: "(x - xa) @ z \<in> L\<^isub>2"
+        obtain ya where "ya \<le> y" and "ya \<in> L\<^isub>1" and "(y - ya) @ z \<in> L\<^isub>2" 
+        proof -
+          have "\<exists> ya.  ya \<le> y \<and> ya \<in> L\<^isub>1 \<and> (x - xa) \<approx>L\<^isub>2 (y - ya)"
+          proof -
+            have "{\<approx>L\<^isub>2 `` {x - xa} |xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = 
+                  {\<approx>L\<^isub>2 `` {y - xa} |xa. xa \<le> y \<and> xa \<in> L\<^isub>1}" (is "?Left = ?Right") 
+              using h1 tag_xy by (auto simp:tag_str_SEQ_def)
+            moreover have "\<approx>L\<^isub>2 `` {x - xa} \<in> ?Left" using h1 h2 by auto
+            ultimately have "\<approx>L\<^isub>2 `` {x - xa} \<in> ?Right" by simp
+            thus ?thesis by (auto simp:Image_def str_eq_rel_def str_eq_def)
+          qed
+          with prems show ?thesis by (auto simp:str_eq_rel_def str_eq_def)
+        qed
+        hence "y @ z \<in> L\<^isub>1 ;; L\<^isub>2" by (erule_tac prefixE, auto simp:Seq_def)          
+      } moreover {
+        fix za
+        assume h1: "za \<le> z" and h2: "(x @ za) \<in> L\<^isub>1" and h3: "z - za \<in> L\<^isub>2"
+        hence "y @ za \<in> L\<^isub>1"
+        proof-
+          have "\<approx>L\<^isub>1 `` {x} = \<approx>L\<^isub>1 `` {y}" using h1 tag_xy by (auto simp:tag_str_SEQ_def)
+          with h2 show ?thesis by (auto simp:Image_def str_eq_rel_def str_eq_def) 
+        qed
+        with h1 h3 have "y @ z \<in> L\<^isub>1 ;; L\<^isub>2" by (drule_tac A = L\<^isub>1 in seq_intro, auto elim:prefixE)
+      }
+      ultimately show ?thesis by blast
+    qed
+  } thus "tag_str_SEQ L\<^isub>1 L\<^isub>2 m = tag_str_SEQ L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 ;; L\<^isub>2) n" 
+    by (auto simp add: str_eq_def str_eq_rel_def)
+qed 
+
+lemma quot_seq_finiteI:
+  assumes finite1: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
+  and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
+  shows "finite (UNIV // \<approx>(L\<^isub>1 ;; L\<^isub>2))"
+proof(rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
+  show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` UNIV // \<approx>L\<^isub>1 ;; L\<^isub>2)" using finite1 finite2
+    by (auto intro:finite_tag_imageI tag_str_seq_range_finite)
 next
-  show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 ; L\<^isub>2))"
-    apply (rule_tac str_inj_imps)
-    by (erule_tac tag_str_seq_inj)
+  show  "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>L\<^isub>1 ;; L\<^isub>2)"
+    apply (rule tag_image_injI)
+    apply (rule tag_str_SEQ_injI)
+    by (auto intro:tag_image_injI tag_str_SEQ_injI simp:)
 qed
 
-(****************** the STAR case ************************)
+text {* **************** the ALT case ************************ *}
+
+definition 
+  "tag_str_ALT L\<^isub>1 L\<^isub>2 (x::string) \<equiv> ((\<approx>L\<^isub>1) `` {x}, (\<approx>L\<^isub>2) `` {x})"
 
-lemma app_eq_elim[rule_format]:
-  "\<And> a. \<forall> b x y. a @ b = x @ y \<longrightarrow> (\<exists> aa ab. a = aa @ ab \<and> x = aa \<and> y = ab @ b) \<or>
-                                   (\<exists> ba bb. b = ba @ bb \<and> x = a @ ba \<and> y = bb \<and> ba \<noteq> [])"
-  apply (induct_tac a rule:List.induct, simp)
-  apply (rule allI | rule impI)+
-  by (case_tac x, auto)
+lemma tag_str_alt_range_finite:
+  "\<lbrakk>finite (UNIV // \<approx>L\<^isub>1); finite (UNIV // \<approx>L\<^isub>2)\<rbrakk> \<Longrightarrow> finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
+apply (rule_tac B = "(UNIV // \<approx>L\<^isub>1) \<times> (UNIV // \<approx>L\<^isub>2)" in finite_subset)
+by (auto simp:tag_str_ALT_def Image_def quotient_def)
 
-definition tag_str_STAR:: "string set \<Rightarrow> string \<Rightarrow> string set set"
-where
-  "tag_str_STAR L\<^isub>1 x \<equiv> if (x = []) 
-                       then {}
-                       else {\<lbrakk>x\<^isub>2\<rbrakk>L\<^isub>1 | x\<^isub>1 x\<^isub>2. x =  x\<^isub>1 @ x\<^isub>2 \<and> x\<^isub>1 \<in> L\<^isub>1\<star> \<and> x\<^isub>2 \<noteq> []}"
-
-lemma tag_str_star_range_finite:
-  assumes finite1: "finite (QUOT L\<^isub>1)"
-  shows "finite (range (tag_str_STAR L\<^isub>1))"
-proof -
-  have "range (tag_str_STAR L\<^isub>1) \<subseteq> Pow (QUOT L\<^isub>1)"
-    by (auto simp:image_def tag_str_STAR_def QUOT_def)
-  thus ?thesis using finite1
-    by (rule_tac B = "Pow (QUOT L\<^isub>1)" in finite_subset, auto)
+lemma quot_union_finiteI:
+  assumes finite1: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
+  and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
+  shows "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
+proof(rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
+  show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` UNIV // \<approx>L\<^isub>1 \<union> L\<^isub>2)" using finite1 finite2
+    by (auto intro:finite_tag_imageI tag_str_alt_range_finite)
+next
+  show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>L\<^isub>1 \<union> L\<^isub>2)"
+  proof-
+    have "\<And>m n. tag_str_ALT L\<^isub>1 L\<^isub>2 m = tag_str_ALT L\<^isub>1 L\<^isub>2 n \<Longrightarrow> m \<approx>(L\<^isub>1 \<union> L\<^isub>2) n"
+      unfolding tag_str_ALT_def str_eq_def Image_def str_eq_rel_def by auto
+    thus ?thesis by (auto intro:tag_image_injI)
+  qed
 qed
 
-lemma star_prop[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
+text {* **************** the Star case ****************** *}
+
+lemma finite_set_has_max: "\<lbrakk>finite A; A \<noteq> {}\<rbrakk> \<Longrightarrow> (\<exists> max \<in> A. \<forall> a \<in> A. f a <= (f max :: nat))"
+proof (induct rule:finite.induct)
+  case emptyI thus ?case by simp
+next
+  case (insertI A a)
+  show ?case
+  proof (cases "A = {}")
+    case True thus ?thesis by (rule_tac x = a in bexI, auto)
+  next
+    case False
+    with prems obtain max where h1: "max \<in> A" and h2: "\<forall>a\<in>A. f a \<le> f max" by blast
+    show ?thesis
+    proof (cases "f a \<le> f max")
+      assume "f a \<le> f max"
+      with h1 h2 show ?thesis by (rule_tac x = max in bexI, auto)
+    next
+      assume "\<not> (f a \<le> f max)"
+      thus ?thesis using h2 by (rule_tac x = a in bexI, auto)
+    qed
+  qed
+qed
+
+lemma star_intro1[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
 by (erule Star.induct, auto)
 
-lemma star_prop2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
+lemma star_intro2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
 by (drule step[of y lang "[]"], auto simp:start)
 
-lemma star_prop3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
-by (erule Star.induct, auto intro:star_prop2)
+lemma star_intro3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto intro:star_intro2)
+
+lemma star_decom: "\<lbrakk>x \<in> lang\<star>; x \<noteq> []\<rbrakk> \<Longrightarrow>(\<exists> a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> lang \<and> b \<in> lang\<star>)"
+by (induct x rule: Star.induct, simp, blast)
 
-lemma postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
-by (erule postfixE, induct x arbitrary:y, auto)
+lemma finite_strict_prefix_set: "finite {xa. xa < (x::string)}"
+apply (induct x rule:rev_induct, simp)
+apply (subgoal_tac "{xa. xa < xs @ [x]} = {xa. xa < xs} \<union> {xs}")
+by (auto simp:strict_prefix_def)
+
+definition 
+  "tag_str_STAR L\<^isub>1 x \<equiv> {(\<approx>L\<^isub>1) `` {x - xa} | xa. xa < x \<and> xa \<in> L\<^isub>1\<star>}"
+
+lemma tag_str_star_range_finite:
+  "finite (UNIV // \<approx>L\<^isub>1) \<Longrightarrow> finite (range (tag_str_STAR L\<^isub>1))"
+apply (rule_tac B = "Pow (UNIV // \<approx>L\<^isub>1)" in finite_subset)
+by (auto simp:tag_str_STAR_def Image_def quotient_def split:if_splits)
 
-lemma inj_aux:
-  "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
-    \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk> 
-  \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>"
-proof- 
-  have "\<And>s. s \<in> L\<^isub>1\<star> \<Longrightarrow> \<forall> m z yb. (s = m @ z \<and> m \<equiv>L\<^isub>1\<equiv> yb \<and> x = xa @ m \<and> xa \<in> L\<^isub>1\<star> \<and> m \<noteq> [] \<and>  
-    (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m) \<longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>)"    
-    apply (erule Star.induct, simp)
-    apply (rule allI | rule impI | erule conjE)+
-    apply (drule app_eq_elim)
-    apply (erule disjE | erule exE | erule conjE)+
-    apply simp
-    apply (simp (no_asm) only:append_assoc[THEN sym])
-    apply (rule step) 
-    apply (simp add:equiv_str_def)
-    apply simp
-
-    apply (erule exE | erule conjE)+    
-    apply (rotate_tac 3)
-    apply (frule_tac x = "xa @ s1" in spec)
-    apply (rotate_tac 12)
-    apply (drule_tac x = ba in spec)
-    apply (erule impE)
-    apply ( simp add:star_prop3)
-    apply (simp)
-    apply (drule postfix_prop)
-    apply simp
-    done
-  thus "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
-         \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk> 
-        \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>" by blast
+lemma tag_str_STAR_injI:
+  "tag_str_STAR L\<^isub>1 m = tag_str_STAR L\<^isub>1 n \<Longrightarrow> m \<approx>(L\<^isub>1\<star>) n"
+proof-
+  { fix x y z
+    assume xz_in_star: "x @ z \<in> L\<^isub>1\<star>"
+    and tag_xy: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
+    have "y @ z \<in> L\<^isub>1\<star>"
+    proof(cases "x = []")
+      case True
+      with tag_xy have "y = []" by (auto simp:tag_str_STAR_def strict_prefix_def)
+      thus ?thesis using xz_in_star True by simp
+    next
+      case False
+      obtain x_max where h1: "x_max < x" and h2: "x_max \<in> L\<^isub>1\<star>" and h3: "(x - x_max) @ z \<in> L\<^isub>1\<star>" 
+        and h4:"\<forall> xa < x. xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star> \<longrightarrow> length xa \<le> length x_max"
+      proof-
+        let ?S = "{xa. xa < x \<and> xa \<in> L\<^isub>1\<star> \<and> (x - xa) @ z \<in> L\<^isub>1\<star>}"
+        have "finite ?S"
+          by (rule_tac B = "{xa. xa < x}" in finite_subset, auto simp:finite_strict_prefix_set)
+        moreover have "?S \<noteq> {}" using False xz_in_star
+          by (simp, rule_tac x = "[]" in exI, auto simp:strict_prefix_def)
+        ultimately have "\<exists> max \<in> ?S. \<forall> a \<in> ?S. length a \<le> length max" using finite_set_has_max by blast
+        with prems show ?thesis by blast
+      qed
+      obtain ya where h5: "ya < y" and h6: "ya \<in> L\<^isub>1\<star>" and h7: "(x - x_max) \<approx>L\<^isub>1 (y - ya)"
+      proof-
+        from tag_xy have "{\<approx>L\<^isub>1 `` {x - xa} |xa. xa < x \<and> xa \<in> L\<^isub>1\<star>} = 
+          {\<approx>L\<^isub>1 `` {y - xa} |xa. xa < y \<and> xa \<in> L\<^isub>1\<star>}" (is "?left = ?right")
+          by (auto simp:tag_str_STAR_def)
+        moreover have "\<approx>L\<^isub>1 `` {x - x_max} \<in> ?left" using h1 h2 by auto
+        ultimately have "\<approx>L\<^isub>1 `` {x - x_max} \<in> ?right" by simp
+        with prems show ?thesis apply (simp add:Image_def str_eq_rel_def str_eq_def) by blast
+      qed      
+      have "(y - ya) @ z \<in> L\<^isub>1\<star>" 
+      proof-
+        from h3 h1 obtain a b where a_in: "a \<in> L\<^isub>1" and a_neq: "a \<noteq> []" and b_in: "b \<in> L\<^isub>1\<star>" 
+          and ab_max: "(x - x_max) @ z = a @ b" 
+          by (drule_tac star_decom, auto simp:strict_prefix_def elim:prefixE)
+        have "(x - x_max) \<le> a \<and> (a - (x - x_max)) @ b = z" 
+        proof -
+          have "((x - x_max) \<le> a \<and> (a - (x - x_max)) @ b = z) \<or> (a < (x - x_max) \<and> ((x - x_max) - a) @ z = b)" 
+            using app_eq_dest[OF ab_max] by (auto simp:strict_prefix_def)
+          moreover { 
+            assume np: "a < (x - x_max)" and b_eqs: " ((x - x_max) - a) @ z = b"
+            have "False"
+            proof -
+              let ?x_max' = "x_max @ a"
+              have "?x_max' < x" using np h1 by (clarsimp simp:strict_prefix_def diff_prefix) 
+              moreover have "?x_max' \<in> L\<^isub>1\<star>" using a_in h2 by (simp add:star_intro3) 
+              moreover have "(x - ?x_max') @ z \<in> L\<^isub>1\<star>" using b_eqs b_in np h1 by (simp add:diff_diff_appd)
+              moreover have "\<not> (length ?x_max' \<le> length x_max)" using a_neq by simp
+              ultimately show ?thesis using h4 by blast
+            qed 
+          } ultimately show ?thesis by blast
+        qed
+        then obtain za where z_decom: "z = za @ b" and x_za: "(x - x_max) @ za \<in> L\<^isub>1" 
+          using a_in by (auto elim:prefixE)        
+        from x_za h7 have "(y - ya) @ za \<in> L\<^isub>1" by (auto simp:str_eq_def)
+        with z_decom b_in show ?thesis by (auto dest!:step[of "(y - ya) @ za"])
+      qed
+      with h5 h6 show ?thesis by (drule_tac star_intro1, auto simp:strict_prefix_def elim:prefixE)
+    qed      
+  } thus "tag_str_STAR L\<^isub>1 m = tag_str_STAR L\<^isub>1 n \<Longrightarrow> m \<approx>(L\<^isub>1\<star>) n"
+    by (auto simp add:str_eq_def str_eq_rel_def)
 qed
 
-
-lemma min_postfix_exists[rule_format]:
-  "finite A \<Longrightarrow> A \<noteq> {} \<and> (\<forall> a \<in> A. \<forall> b \<in> A. ((b >>= a) \<or> (a >>= b))) 
-                \<longrightarrow> (\<exists> min. (min \<in> A \<and> (\<forall> a \<in> A. a >>= min)))"
-apply (erule finite.induct)
-apply simp
-apply simp
-apply (case_tac "A = {}")
-apply simp
-apply clarsimp
-apply (case_tac "a >>= min")
-apply (rule_tac x = min in exI, simp)
-apply (rule_tac x = a in exI, simp)
-apply clarify
-apply (rotate_tac 5)
-apply (erule_tac x = aa in ballE) defer apply simp
-apply (erule_tac ys = min in postfix_trans)
-apply (erule_tac x = min in ballE) 
-by simp+
-
-lemma tag_str_star_inj:
-  "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
-proof -
-  have aux: "\<And> x y z. \<lbrakk>tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y; x @ z \<in> L\<^isub>1\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> L\<^isub>1\<star>"
-  proof-
-    fix x y z
-    assume tag_eq: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
-      and x_z: "x @ z \<in> L\<^isub>1\<star>"
-    show "y @ z \<in> L\<^isub>1\<star>"
-    proof (cases "x = []")
-      case True
-      with tag_eq have "y = []" by (simp add:tag_str_STAR_def split:if_splits, blast)
-      thus ?thesis using x_z True by simp
-    next
-      case False
-      hence not_empty: "{xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} \<noteq> {}" using x_z
-        by (simp, rule_tac x = x in exI, rule_tac x = "[]" in exI, simp add:start)
-      have finite_set: "finite {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}"
-        apply (rule_tac B = "{xb. \<exists> xa. x = xa @ xb}" in finite_subset)
-        apply auto
-        apply (induct x, simp)
-        apply (subgoal_tac "{xb. \<exists>xa. a # x = xa @ xb} = insert (a # x) {xb. \<exists>xa. x = xa @ xb}")
-        apply auto
-        by (case_tac xaa, simp+)
-      have comparable: "\<forall> a \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}. 
-                        \<forall> b \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
-                          ((b >>= a) \<or> (a >>= b))"
-        by (auto simp:postfix_def, drule app_eq_elim, blast)
-      hence "\<exists> min. min \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} 
-                \<and> (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min)"
-        using finite_set not_empty comparable
-        apply (drule_tac min_postfix_exists, simp)
-        by (erule exE, rule_tac x = min in exI, auto)
-      then obtain min xa where x_decom: "x = xa @ min \<and> xa \<in> L\<^isub>1\<star>"
-        and min_not_empty: "min \<noteq> []"
-        and min_z_in_star: "min @ z \<in> L\<^isub>1\<star>"
-        and is_min: "\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min"  by blast
-      from x_decom min_not_empty have "\<lbrakk>min\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 x"  by (auto simp:tag_str_STAR_def)
-      hence "\<exists> yb. \<lbrakk>yb\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 y \<and> \<lbrakk>min\<rbrakk>L\<^isub>1 = \<lbrakk>yb\<rbrakk>L\<^isub>1" using tag_eq by auto
-      hence "\<exists> ya yb. y = ya @ yb \<and> ya \<in> L\<^isub>1\<star> \<and> min \<equiv>L\<^isub>1\<equiv> yb \<and> yb \<noteq> [] " 
-        by (simp add:tag_str_STAR_def equiv_class_def equiv_str_def split:if_splits, blast)        
-      then obtain ya yb where y_decom: "y = ya @ yb"
-                          and ya_in_star: "ya \<in> L\<^isub>1\<star>"
-                          and yb_not_empty: "yb \<noteq> []"
-                          and min_yb_eq: "min \<equiv>L\<^isub>1\<equiv> yb"  by blast
-      from min_z_in_star min_yb_eq min_not_empty is_min x_decom
-      have "yb @ z \<in> L\<^isub>1\<star>"
-        by (rule_tac x = x and xa = xa in inj_aux, simp+)
-      thus ?thesis using ya_in_star y_decom
-        by (auto dest:star_prop)        
-    qed
-  qed
-  show "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
-    by (auto intro:aux simp:equiv_str_def)
+lemma quot_star_finiteI:
+  assumes finite: "finite (UNIV // \<approx>(L\<^isub>1::string set))"
+  shows "finite (UNIV // \<approx>(L\<^isub>1\<star>))"
+proof(rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
+  show "finite (op ` (tag_str_STAR L\<^isub>1) ` UNIV // \<approx>L\<^isub>1\<star>)" using finite
+    by (auto intro:finite_tag_imageI tag_str_star_range_finite)
+next
+  show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (UNIV // \<approx>L\<^isub>1\<star>)"
+    by (auto intro:tag_image_injI tag_str_STAR_injI)
 qed
 
-lemma quot_star:  
-  assumes finite1: "finite (QUOT L\<^isub>1)"
-  shows "finite (QUOT (L\<^isub>1\<star>))"
-proof (rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
-  show "finite (op ` (tag_str_STAR L\<^isub>1) ` QUOT (L\<^isub>1\<star>))"
-    using finite_tag_image tag_str_star_range_finite finite1
-    by auto
-next
-  show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (QUOT (L\<^isub>1\<star>))"
-    apply (rule_tac str_inj_imps)
-    by (erule_tac tag_str_star_inj)
-qed
+text {* **************** the Other Direction ************ *}
 
 lemma other_direction:
-  "Lang = L (r::rexp) \<Longrightarrow> finite (QUOT Lang)"
-apply (induct arbitrary:Lang rule:rexp.induct)
-apply (simp add:QUOT_def equiv_class_def equiv_str_def)
-by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)  
+  "Lang = L (r::rexp) \<Longrightarrow> finite (UNIV // (\<approx>Lang))"
+proof (induct arbitrary:Lang rule:rexp.induct)
+  case NULL
+  have "UNIV // (\<approx>{}) \<subseteq> {UNIV} "
+    by (auto simp:quotient_def str_eq_rel_def str_eq_def)
+  with prems show "?case" by (auto intro:finite_subset)
+next
+  case EMPTY
+  have "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}" by (rule quot_empty_subset)
+  with prems show ?case by (auto intro:finite_subset)
+next
+  case (CHAR c)
+  have "UNIV // (\<approx>{[c]}) \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}" by (rule quot_char_subset)
+  with prems show ?case by (auto intro:finite_subset)
+next
+  case (SEQ r\<^isub>1 r\<^isub>2)
+  have "\<lbrakk>finite (UNIV // \<approx>(L r\<^isub>1)); finite (UNIV // \<approx>(L r\<^isub>2))\<rbrakk> \<Longrightarrow> finite (UNIV // \<approx>(L r\<^isub>1 ;; L r\<^isub>2))"
+    by (erule quot_seq_finiteI, simp)
+  with prems show ?case by simp
+next
+  case (ALT r\<^isub>1 r\<^isub>2)
+  have "\<lbrakk>finite (UNIV // \<approx>(L r\<^isub>1)); finite (UNIV // \<approx>(L r\<^isub>2))\<rbrakk> \<Longrightarrow> finite (UNIV // \<approx>(L r\<^isub>1 \<union> L r\<^isub>2))"
+    by (erule quot_union_finiteI, simp)
+  with prems show ?case by simp  
+next
+  case (STAR r)
+  have "finite (UNIV // \<approx>(L r)) \<Longrightarrow> finite (UNIV // \<approx>((L r)\<star>))"
+    by (erule quot_star_finiteI)
+  with prems show ?case by simp
+qed 
 
-end 
+end
\ No newline at end of file