
diff -r b794db0b79db -r b1258b7d2789 MyhillNerode.thy
--- a/MyhillNerode.thy	Fri Jun 03 13:59:21 2011 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1816 +0,0 @@
-theory MyhillNerode
-  imports "Main" "List_Prefix"
-begin
-
-text {* sequential composition of languages *}
-
-definition
-  lang_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}"
-
-lemma lang_seq_empty:
-  shows "{[]} ; L = L"
-  and   "L ; {[]} = L"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_null:
-  shows "{} ; L = {}"
-  and   "L ; {} = {}"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_append:
-  assumes a: "s1 \<in> L1"
-  and     b: "s2 \<in> L2"
-  shows "s1@s2 \<in> L1 ; L2"
-unfolding lang_seq_def
-using a b by auto 
-
-lemma lang_seq_union:
-  shows "(L1 \<union> L2); L3 = (L1; L3) \<union> (L2; L3)"
-  and   "L1; (L2 \<union> L3) = (L1; L2) \<union> (L1; L3)"
-unfolding lang_seq_def by auto
-
-lemma lang_seq_assoc:
-  shows "(L1 ; L2) ; L3 = L1 ; (L2 ; L3)"
-unfolding lang_seq_def
-apply(auto)
-apply(metis)
-by (metis append_assoc)
-
-
-section {* Kleene star for languages defined as least fixed point *}
-
-inductive_set
-  Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
-  for L :: "string set"
-where
-  start[intro]: "[] \<in> L\<star>"
-| step[intro]:  "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
-
-lemma lang_star_empty:
-  shows "{}\<star> = {[]}"
-by (auto elim: Star.cases)
-
-lemma lang_star_cases:
-  shows "L\<star> =  {[]} \<union> L ; L\<star>"
-proof
-  { fix x
-    have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L ; L\<star>"
-      unfolding lang_seq_def
-    by (induct rule: Star.induct) (auto)
-  }
-  then show "L\<star> \<subseteq> {[]} \<union> L ; L\<star>" by auto
-next
-  show "{[]} \<union> L ; L\<star> \<subseteq> L\<star>" 
-    unfolding lang_seq_def by auto
-qed
-
-lemma lang_star_cases':
-  shows "L\<star> =  {[]} \<union> L\<star> ; L"
-proof
-  { fix x
-    have "x \<in> L\<star> \<Longrightarrow> x \<in> {[]} \<union> L\<star> ; L"
-      unfolding lang_seq_def
-    apply (induct rule: Star.induct)
-    apply simp
-    apply simp
-    apply (erule disjE)
-    apply (auto)[1]
-    apply (erule exE | erule conjE)+
-    apply (rule disjI2)
-    apply (rule_tac x = "s1 @ s1a" in exI)
-    by auto
-  }
-  then show "L\<star> \<subseteq> {[]} \<union> L\<star> ; L" by auto
-next
-  show "{[]} \<union> L\<star> ; L \<subseteq> L\<star>" 
-    unfolding lang_seq_def
-    apply auto
-    apply (erule Star.induct)
-    apply auto
-    apply (drule step[of _ _ "[]"])
-    by (auto intro:start)
-qed
-
-lemma lang_star_simple:
-  shows "L \<subseteq> L\<star>"
-by (subst lang_star_cases)
-   (auto simp only: lang_seq_def)
-
-lemma lang_star_prop0_aux:
-  "s2 \<in> L\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L \<longrightarrow> (\<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4)" 
-apply (erule Star.induct)
-apply (clarify, rule_tac x = "[]" in exI, rule_tac x = s1 in exI, simp add:start)
-apply (clarify, drule_tac x = s1 in spec)
-apply (drule mp, simp, clarify)
-apply (rule_tac x = "s1a @ s3" in exI, rule_tac x = s4 in exI)
-by auto
-
-lemma lang_star_prop0:
-  "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> \<exists> s3 s4. s3 \<in> L\<star> \<and> s4 \<in> L \<and> s1 @ s2 = s3 @ s4" 
-by (auto dest:lang_star_prop0_aux)
-
-lemma lang_star_prop1:
-  assumes asm: "L1; L2 \<subseteq> L2" 
-  shows "L1\<star>; L2 \<subseteq> L2"
-proof -
-  { fix s1 s2
-    assume minor: "s2 \<in> L2"
-    assume major: "s1 \<in> L1\<star>"
-    then have "s1@s2 \<in> L2"
-    proof(induct rule: Star.induct)
-      case start
-      show "[]@s2 \<in> L2" using minor by simp
-    next
-      case (step s1 s1')
-      have "s1 \<in> L1" by fact
-      moreover
-      have "s1'@s2 \<in> L2" by fact
-      ultimately have "s1@(s1'@s2) \<in> L1; L2" by (auto simp add: lang_seq_def)
-      with asm have "s1@(s1'@s2) \<in> L2" by auto
-      then show "(s1@s1')@s2 \<in> L2" by simp
-    qed
-  } 
-  then show "L1\<star>; L2 \<subseteq> L2" by (auto simp add: lang_seq_def)
-qed
-
-lemma lang_star_prop2_aux:
-  "s2 \<in> L2\<star> \<Longrightarrow> \<forall> s1. s1 \<in> L1 \<and> L1 ; L2 \<subseteq> L1 \<longrightarrow> s1 @ s2 \<in> L1"
-apply (erule Star.induct, simp)
-apply (clarify, drule_tac x = "s1a @ s1" in spec)
-by (auto simp:lang_seq_def)
-
-lemma lang_star_prop2:
-  "L1; L2 \<subseteq> L1 \<Longrightarrow> L1 ; L2\<star> \<subseteq> L1"
-by (auto dest!:lang_star_prop2_aux simp:lang_seq_def)
-
-lemma lang_star_seq_subseteq: 
-  shows "L ; L\<star> \<subseteq> L\<star>"
-using lang_star_cases by blast
-
-lemma lang_star_double:
-  shows "L\<star>; L\<star> = L\<star>"
-proof
-  show "L\<star> ; L\<star> \<subseteq> L\<star>" 
-    using lang_star_prop1 lang_star_seq_subseteq by blast
-next
-  have "L\<star> \<subseteq> L\<star> \<union> L\<star>; (L; L\<star>)" by auto
-  also have "\<dots> = L\<star>;{[]} \<union> L\<star>; (L; L\<star>)" by (simp add: lang_seq_empty)
-  also have "\<dots> = L\<star>; ({[]} \<union> L; L\<star>)" by (simp only: lang_seq_union)
-  also have "\<dots> = L\<star>; L\<star>" using lang_star_cases by simp 
-  finally show "L\<star> \<subseteq> L\<star> ; L\<star>" by simp
-qed
-
-lemma lang_star_seq_subseteq': 
-  shows "L\<star>; L \<subseteq> L\<star>"
-proof -
-  have "L \<subseteq> L\<star>" by (rule lang_star_simple)
-  then have "L\<star>; L \<subseteq> L\<star>; L\<star>" by (auto simp add: lang_seq_def)
-  then show "L\<star>; L \<subseteq> L\<star>" using lang_star_double by blast
-qed
-
-lemma
-  shows "L\<star> \<subseteq> L\<star>\<star>"
-by (rule lang_star_simple)
-
-
-section {* regular expressions *}
-
-datatype rexp =
-  NULL
-| EMPTY
-| CHAR char
-| SEQ rexp rexp
-| ALT rexp rexp
-| STAR rexp
-
-
-consts L:: "'a \<Rightarrow> string set"
-
-overloading L_rexp \<equiv> "L::  rexp \<Rightarrow> string set"
-begin
-fun
-  L_rexp :: "rexp \<Rightarrow> string set"
-where
-    "L_rexp (NULL) = {}"
-  | "L_rexp (EMPTY) = {[]}"
-  | "L_rexp (CHAR c) = {[c]}"
-  | "L_rexp (SEQ r1 r2) = (L_rexp r1) ; (L_rexp r2)"
-  | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
-  | "L_rexp (STAR r) = (L_rexp r)\<star>"
-end
-
-
-text{* ************ now is the codes writen by chunhan ************************************* *}
-
-section {* Arden's Lemma revised *}
-
-lemma arden_aux1:
-  assumes a: "X \<subseteq> X ; A \<union> B"
-  and     b: "[] \<notin> A"
-  shows      "x \<in> X \<Longrightarrow> x \<in> B ; A\<star>"
-apply (induct x taking:length rule:measure_induct)
-apply (subgoal_tac "x \<in> X ; A \<union> B")
-defer
-using a
-apply (auto)[1]
-apply simp
-apply (erule disjE)
-defer
-apply (auto simp add:lang_seq_def) [1]
-apply (subgoal_tac "\<exists> x1 x2. x = x1 @ x2 \<and> x1 \<in> X \<and> x2 \<in> A")
-defer
-apply (auto simp add:lang_seq_def) [1]
-apply (erule exE | erule conjE)+
-apply simp
-apply (drule_tac x = x1 in spec)
-apply (simp)
-using b
-apply -
-apply (auto)[1]
-apply (subgoal_tac "x1 @ x2 \<in> (B ; A\<star>) ; A")
-defer
-apply (auto simp add:lang_seq_def)[1]
-by (metis Un_absorb1 lang_seq_assoc lang_seq_union(2) lang_star_double lang_star_simple mem_def sup1CI)
-
-theorem ardens_revised:
-  assumes nemp: "[] \<notin> A"
-  shows "(X = X ; A \<union> B) \<longleftrightarrow> (X = B ; A\<star>)"
-apply(rule iffI)
-defer
-apply(simp)
-apply(subst lang_star_cases')
-apply(subst lang_seq_union)
-apply(simp add: lang_seq_empty)
-apply(simp add: lang_seq_assoc)
-apply(auto)[1]
-proof -
-  assume "X = X ; A \<union> B"
-  then have as1: "X ; A \<union> B \<subseteq> X" and as2: "X \<subseteq> X ; A \<union> B" by simp_all
-  from as1 have a: "X ; A \<subseteq> X" and b: "B \<subseteq> X" by simp_all
-  from b have "B; A\<star> \<subseteq> X ; A\<star>" by (auto simp add: lang_seq_def)
-  moreover
-  from a have "X ; A\<star> \<subseteq> X" 
-
-by (rule lang_star_prop2)
-  ultimately have f1: "B ; A\<star> \<subseteq> X" by simp
-  from as2 nemp
-  have f2: "X \<subseteq> B; A\<star>" using arden_aux1 by auto
-  from f1 f2 show "X = B; A\<star>" by auto
-qed
-
-
-
-section {* equiv class' definition *}
-
-definition 
-  equiv_str :: "string \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> bool" ("_ \<equiv>_\<equiv> _" [100, 100, 100] 100)
-where
-  "x \<equiv>Lang\<equiv> y \<longleftrightarrow> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
-
-definition
-  equiv_class :: "string \<Rightarrow> (string set) \<Rightarrow> string set" ("\<lbrakk>_\<rbrakk>_" [100, 100] 100)
-where
-  "\<lbrakk>x\<rbrakk>Lang \<equiv> {y. x \<equiv>Lang\<equiv> y}"
-
-text {* Chunhan modifies Q to Quo *}
-
-definition  
-  quot :: "string set \<Rightarrow> string set \<Rightarrow> (string set) set" ("_ Quo _" [100, 100] 100)
-where
-  "L1 Quo L2 \<equiv> { \<lbrakk>x\<rbrakk>L2 | x. x \<in> L1}" 
-
-
-lemma lang_eqs_union_of_eqcls: 
-  "Lang = \<Union> {X. X \<in> (UNIV Quo Lang) \<and> (\<forall> x \<in> X. x \<in> Lang)}"
-proof
-  show "Lang \<subseteq> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
-  proof
-    fix x
-    assume "x \<in> Lang"
-    thus "x \<in> \<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}"
-    proof (simp add:quot_def)
-      assume "(1)": "x \<in> Lang"
-      show "\<exists>xa. (\<exists>x. xa = \<lbrakk>x\<rbrakk>Lang) \<and> (\<forall>x\<in>xa. x \<in> Lang) \<and> x \<in> xa" (is "\<exists>xa.?P xa")
-      proof -
-        have "?P (\<lbrakk>x\<rbrakk>Lang)" using "(1)"
-          by (auto simp:equiv_class_def equiv_str_def dest: spec[where  x = "[]"])
-        thus ?thesis by blast
-      qed
-    qed
-  qed   
-next
-  show "\<Union>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<subseteq> Lang"
-    by auto
-qed
-
-lemma empty_notin_CS: "{} \<notin> UNIV Quo Lang"
-apply (clarsimp simp:quot_def equiv_class_def)
-by (rule_tac x = x in exI, auto simp:equiv_str_def)
-
-lemma no_two_cls_inters:
-  "\<lbrakk>X \<in> UNIV Quo Lang; Y \<in> UNIV Quo Lang; X \<noteq> Y\<rbrakk> \<Longrightarrow> X \<inter> Y = {}"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-text {* equiv_class transition *}
-definition 
-  CT :: "string set \<Rightarrow> char \<Rightarrow> string set \<Rightarrow> bool" ("_-_\<rightarrow>_" [99,99]99)
-where
-  "X-c\<rightarrow>Y \<equiv> ((X;{[c]}) \<subseteq> Y)"
-
-types t_equa_rhs = "(string set \<times> rexp) set"
-
-types t_equa = "(string set \<times> t_equa_rhs)"
-
-types t_equas = "t_equa set"
-
-text {* 
-  "empty_rhs" generates "\<lambda>" for init-state, just like "\<lambda>" for final states 
-  in Brzozowski method. But if the init-state is "{[]}" ("\<lambda>" itself) then 
-  empty set is returned, see definition of "equation_rhs" 
-*} 
-
-definition 
-  empty_rhs :: "string set \<Rightarrow> t_equa_rhs"
-where
-  "empty_rhs X \<equiv> if ([] \<in> X) then {({[]}, EMPTY)} else {}"
-
-definition 
-  folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
-where
-  "folds f z S \<equiv> SOME x. fold_graph f z S x"
-
-definition 
-  equation_rhs :: "(string set) set \<Rightarrow> string set \<Rightarrow> t_equa_rhs"
-where
-  "equation_rhs CS X \<equiv> if (X = {[]}) then {({[]}, EMPTY)}
-                         else {(S, folds ALT NULL {CHAR c| c. S-c\<rightarrow>X} )| S. S \<in> CS} \<union> empty_rhs X"
-
-definition 
-  equations :: "(string set) set \<Rightarrow> t_equas"
-where
-  "equations CS \<equiv> {(X, equation_rhs CS X) | X. X \<in> CS}"
-
-overloading L_rhs \<equiv> "L:: t_equa_rhs \<Rightarrow> string set"
-begin
-fun L_rhs:: "t_equa_rhs \<Rightarrow> string set"
-where
-  "L_rhs rhs = {x. \<exists> X r. (X, r) \<in> rhs \<and> x \<in> X;(L r)}"
-end
-
-definition 
-  distinct_rhs :: "t_equa_rhs \<Rightarrow> bool"
-where
-  "distinct_rhs rhs \<equiv> \<forall> X reg\<^isub>1 reg\<^isub>2. (X, reg\<^isub>1) \<in> rhs \<and> (X, reg\<^isub>2) \<in> rhs \<longrightarrow> reg\<^isub>1 = reg\<^isub>2"
-
-definition
-  distinct_equas_rhs :: "t_equas \<Rightarrow> bool"
-where
-  "distinct_equas_rhs equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> distinct_rhs rhs"
-
-definition 
-  distinct_equas :: "t_equas \<Rightarrow> bool"
-where
-  "distinct_equas equas \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> equas \<and> (X, rhs') \<in> equas \<longrightarrow> rhs = rhs'"
-
-definition 
-  seq_rhs_r :: "t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
-where
-  "seq_rhs_r rhs r \<equiv> (\<lambda>(X, reg). (X, SEQ reg r)) ` rhs"
-
-definition 
-  del_x_paired :: "('a \<times> 'b) set \<Rightarrow> 'a \<Rightarrow> ('a \<times> 'b) set"
-where
-  "del_x_paired S x \<equiv> S - {X. X \<in> S \<and> fst X = x}"
-
-definition
-  arden_variate :: "string set \<Rightarrow> rexp \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
-where
-  "arden_variate X r rhs \<equiv> seq_rhs_r (del_x_paired rhs X) (STAR r)"
-
-definition
-  no_EMPTY_rhs :: "t_equa_rhs \<Rightarrow> bool"
-where
-  "no_EMPTY_rhs rhs \<equiv> \<forall> X r. (X, r) \<in> rhs \<and> X \<noteq> {[]} \<longrightarrow> [] \<notin> L r"
-
-definition 
-  no_EMPTY_equas :: "t_equas \<Rightarrow> bool"
-where
-  "no_EMPTY_equas equas \<equiv> \<forall> X rhs. (X, rhs) \<in> equas \<longrightarrow> no_EMPTY_rhs rhs"
-
-lemma fold_alt_null_eqs:
-  "finite rS \<Longrightarrow> x \<in> L (folds ALT NULL rS) = (\<exists> r \<in> rS. x \<in> L r)"
-apply (simp add:folds_def)
-apply (rule someI2_ex)
-apply (erule finite_imp_fold_graph)
-apply (erule fold_graph.induct)
-by auto (*??? how do this be in Isar ?? *)
-
-lemma seq_rhs_r_prop1:
-  "L (seq_rhs_r rhs r) = (L rhs);(L r)"
-apply (auto simp:seq_rhs_r_def image_def lang_seq_def)
-apply (rule_tac x = "s1 @ s1a" in exI, rule_tac x = "s2a" in exI, simp)
-apply (rule_tac x = a in exI, rule_tac x = b in exI, simp)
-apply (rule_tac x = s1 in exI, rule_tac x = s1a in exI, simp)
-apply (rule_tac x = X in exI, rule_tac x  = "SEQ ra r" in exI, simp)
-apply (rule conjI)
-apply (rule_tac x = "(X, ra)" in bexI, simp+) 
-apply (rule_tac x = s1a in exI, rule_tac x = "s2a @ s2" in exI, simp)
-apply (simp add:lang_seq_def)
-by (rule_tac x = s2a in exI, rule_tac x = s2 in exI, simp)
-
-lemma del_x_paired_prop1:  
-  "\<lbrakk>distinct_rhs rhs; (X, r) \<in> rhs\<rbrakk> \<Longrightarrow> X ; L r \<union> L (del_x_paired rhs X) = L rhs"
-  apply (simp add:del_x_paired_def)
-  apply (simp add: distinct_rhs_def)
-  apply(auto simp add: lang_seq_def)
-  apply(metis)
-  done
-
-lemma arden_variate_prop:
-  assumes "(X, rx) \<in> rhs"
-  shows "(\<forall> Y. Y \<noteq> X \<longrightarrow> (\<exists> r. (Y, r) \<in> rhs) = (\<exists> r. (Y, r) \<in> (arden_variate X rx rhs)))"
-proof (rule allI, rule impI)
-  fix Y
-  assume "(1)": "Y \<noteq> X"
-  show "(\<exists>r. (Y, r) \<in> rhs) = (\<exists>r. (Y, r) \<in> arden_variate X rx rhs)"
-  proof
-    assume "(1_1)": "\<exists>r. (Y, r) \<in> rhs"
-    show "\<exists>r. (Y, r) \<in> arden_variate X rx rhs" (is "\<exists>r. ?P r")
-    proof -
-      from "(1_1)" obtain r where "(Y, r) \<in> rhs" ..
-      hence "?P (SEQ r (STAR rx))"
-      proof (simp add:arden_variate_def image_def)
-        have "(Y, r) \<in> rhs \<Longrightarrow> (Y, r) \<in> del_x_paired rhs X"
-          by (auto simp:del_x_paired_def "(1)")
-        thus "(Y, r) \<in> rhs \<Longrightarrow> (Y, SEQ r (STAR rx)) \<in> seq_rhs_r (del_x_paired rhs X) (STAR rx)"
-          by (auto simp:seq_rhs_r_def)
-      qed
-      thus ?thesis by blast
-    qed
-  next
-    assume "(2_1)": "\<exists>r. (Y, r) \<in> arden_variate X rx rhs"
-    thus "\<exists>r. (Y, r) \<in> rhs" (is "\<exists> r. ?P r")
-      by (auto simp:arden_variate_def del_x_paired_def seq_rhs_r_def image_def)
-  qed
-qed
-
-text {*
-  arden_variate_valid:  proves variation from 
-  
-   "X = X;r + Y;ry + \<dots>" to "X = Y;(SEQ ry (STAR r)) + \<dots>" 
-
-  holds the law of "language of left equiv language of right" 
-*}
-lemma arden_variate_valid:
-  assumes X_not_empty: "X \<noteq> {[]}"
-  and     l_eq_r:   "X = L rhs"
-  and     dist: "distinct_rhs rhs"
-  and     no_empty: "no_EMPTY_rhs rhs"
-  and     self_contained: "(X, r) \<in> rhs"
-  shows   "X = L (arden_variate X r rhs)" 
-proof -
-  have "[] \<notin> L r" using no_empty X_not_empty self_contained
-    by (auto simp:no_EMPTY_rhs_def)
-  hence ardens: "X = X;(L r) \<union> (L (del_x_paired rhs X)) \<longleftrightarrow> X = (L (del_x_paired rhs X)) ; (L r)\<star>" 
-    by (rule ardens_revised)
-  have del_x: "X = X ; L r \<union> L (del_x_paired rhs X) \<longleftrightarrow> X = L rhs" using dist l_eq_r self_contained
-    by (auto dest!:del_x_paired_prop1)
-  show ?thesis
-  proof
-    show "X \<subseteq> L (arden_variate X r rhs)"
-    proof
-      fix x
-      assume "(1_1)": "x \<in> X" with l_eq_r ardens del_x
-      show "x \<in> L (arden_variate X r rhs)" 
-        by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
-    qed
-  next
-    show "L (arden_variate X r rhs) \<subseteq> X"
-    proof
-      fix x
-      assume "(2_1)": "x \<in> L (arden_variate X r rhs)" with ardens del_x l_eq_r
-      show "x \<in> X" 
-        by (simp add:arden_variate_def seq_rhs_r_prop1 del:L_rhs.simps)
-    qed
-  qed
-qed
-
-text {* 
-  merge_rhs {(x1, r1), (x2, r2}, (x4, r4), \<dots>} {(x1, r1'), (x3, r3'), \<dots>} = 
-     {(x1, ALT r1 r1'}, (x2, r2), (x3, r3'), (x4, r4), \<dots>} *}  
-definition 
-  merge_rhs :: "t_equa_rhs \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa_rhs"
-where
-  "merge_rhs rhs rhs' \<equiv> {(X, r). (\<exists> r1 r2. ((X,r1) \<in> rhs \<and> (X, r2) \<in> rhs') \<and> r = ALT r1 r2)  \<or>
-                                 (\<exists> r1. (X, r1) \<in> rhs \<and> (\<not> (\<exists> r2. (X, r2) \<in> rhs')) \<and> r = r1) \<or>
-                                 (\<exists> r2. (X, r2) \<in> rhs' \<and> (\<not> (\<exists> r1. (X, r1) \<in> rhs)) \<and> r = r2)    }"                                  
-
-
-text {* rhs_subst rhs X=xrhs r: substitude all occurence of X in rhs((X,r) \<in> rhs) with xrhs *}
-definition 
-  rhs_subst :: "t_equa_rhs \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> rexp \<Rightarrow> t_equa_rhs"
-where
-  "rhs_subst rhs X xrhs r \<equiv> merge_rhs (del_x_paired rhs X) (seq_rhs_r xrhs r)"
-
-definition 
-  equas_subst_f :: "string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equa \<Rightarrow> t_equa"
-where
-  "equas_subst_f X xrhs equa \<equiv> let (Y, rhs) = equa in
-                                 if (\<exists> r. (X, r) \<in> rhs)
-                                 then (Y, rhs_subst rhs X xrhs (SOME r. (X, r) \<in> rhs))
-                                 else equa"
-
-definition
-  equas_subst :: "t_equas \<Rightarrow> string set \<Rightarrow> t_equa_rhs \<Rightarrow> t_equas"
-where
-  "equas_subst ES X xrhs \<equiv> del_x_paired (equas_subst_f X xrhs ` ES) X"
-
-lemma lang_seq_prop1:
- "x \<in> X ; L r \<Longrightarrow> x \<in> X ; (L r \<union> L r')"
-by (auto simp:lang_seq_def)
-
-lemma lang_seq_prop1':
-  "x \<in> X; L r \<Longrightarrow> x \<in> X ; (L r' \<union> L r)"
-by (auto simp:lang_seq_def)
-
-lemma lang_seq_prop2:
-  "x \<in> X; (L r \<union> L r') \<Longrightarrow> x \<in> X;L r \<or> x \<in> X;L r'"
-by (auto simp:lang_seq_def)
-
-lemma merge_rhs_prop1:
-  shows "L (merge_rhs rhs rhs') = L rhs \<union> L rhs' "
-apply (auto simp add:merge_rhs_def dest!:lang_seq_prop2 intro:lang_seq_prop1)
-apply (rule_tac x = X in exI, rule_tac x = r1 in exI, simp)
-apply (case_tac "\<exists> r2. (X, r2) \<in> rhs'")
-apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r r2" in exI, simp add:lang_seq_prop1)
-apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
-apply (case_tac "\<exists> r1. (X, r1) \<in> rhs")
-apply (clarify, rule_tac x = X in exI, rule_tac x = "ALT r1 r" in exI, simp add:lang_seq_prop1')
-apply (rule_tac x = X in exI, rule_tac x = r in exI, simp)
-done
-
-lemma no_EMPTY_rhss_imp_merge_no_EMPTY:
-  "\<lbrakk>no_EMPTY_rhs rhs; no_EMPTY_rhs rhs'\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (merge_rhs rhs rhs')"
-apply (simp add:no_EMPTY_rhs_def merge_rhs_def)
-apply (clarify, (erule conjE | erule exE | erule disjE)+)
-by auto
-
-lemma distinct_rhs_prop:
-  "\<lbrakk>distinct_rhs rhs; (X, r1) \<in> rhs; (X, r2) \<in> rhs\<rbrakk> \<Longrightarrow> r1 = r2"
-by (auto simp:distinct_rhs_def)
-
-lemma merge_rhs_prop2:
-  assumes dist_rhs: "distinct_rhs rhs"
-  and     dist_rhs':"distinct_rhs rhs'"
-  shows "distinct_rhs (merge_rhs rhs rhs')"
-apply (auto simp:merge_rhs_def distinct_rhs_def)
-using dist_rhs
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs'
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs
-apply (drule distinct_rhs_prop, simp+)
-using dist_rhs'
-apply (drule distinct_rhs_prop, simp+)
-done
-
-lemma seq_rhs_r_holds_distinct: 
-  "distinct_rhs rhs \<Longrightarrow> distinct_rhs (seq_rhs_r rhs r)"
-by (auto simp:distinct_rhs_def seq_rhs_r_def)
-
-lemma seq_rhs_r_prop0:
-  assumes l_eq_r: "X = L xrhs"
-  shows "L (seq_rhs_r xrhs r) = X ; L r "
-using l_eq_r
-by (auto simp only:seq_rhs_r_prop1)
-
-lemma rhs_subst_prop1:
-  assumes l_eq_r: "X = L xrhs"
-  and     dist:  "distinct_rhs rhs"
-  shows "(X, r) \<in> rhs \<Longrightarrow> L rhs = L (rhs_subst rhs X xrhs r)"
-apply (simp add:rhs_subst_def merge_rhs_prop1 del:L_rhs.simps)
-using l_eq_r 
-apply (drule_tac r = r in seq_rhs_r_prop0, simp del:L_rhs.simps)
-using dist
-by (auto dest!:del_x_paired_prop1 simp del:L_rhs.simps)
-
-lemma del_x_paired_holds_distinct_rhs:
-  "distinct_rhs rhs \<Longrightarrow> distinct_rhs (del_x_paired rhs x)"
-by (auto simp:distinct_rhs_def del_x_paired_def)
-
-lemma rhs_subst_holds_distinct_rhs:
-  "\<lbrakk>distinct_rhs rhs; distinct_rhs xrhs\<rbrakk> \<Longrightarrow> distinct_rhs (rhs_subst rhs X xrhs r)"
-apply (drule_tac r = r and rhs = xrhs in seq_rhs_r_holds_distinct)
-apply (drule_tac x = X in del_x_paired_holds_distinct_rhs)
-by (auto dest:merge_rhs_prop2[where rhs = "del_x_paired rhs X"] simp:rhs_subst_def)
-
-section {* myhill-nerode theorem *}
-
-definition left_eq_cls :: "t_equas \<Rightarrow> (string set) set"
-where
-  "left_eq_cls ES \<equiv> {X. \<exists> rhs. (X, rhs) \<in> ES} "
-
-definition right_eq_cls :: "t_equas \<Rightarrow> (string set) set"
-where
-  "right_eq_cls ES \<equiv> {Y. \<exists> X rhs r. (X, rhs) \<in> ES \<and> (Y, r) \<in> rhs }"
-
-definition rhs_eq_cls :: "t_equa_rhs \<Rightarrow> (string set) set"
-where
-  "rhs_eq_cls rhs \<equiv> {Y. \<exists> r. (Y, r) \<in> rhs}"
-
-definition ardenable :: "t_equa \<Rightarrow> bool"
-where
-  "ardenable equa \<equiv> let (X, rhs) = equa in 
-                      distinct_rhs rhs \<and> no_EMPTY_rhs rhs \<and> X = L rhs"
-
-text {*
-  Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
-*}
-definition Inv :: "string set \<Rightarrow> t_equas \<Rightarrow> bool"
-where
-  "Inv X ES \<equiv> finite ES \<and> (\<exists> rhs. (X, rhs) \<in> ES) \<and> distinct_equas ES \<and> 
-            (\<forall> X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs) \<and> X \<noteq> {} \<and> rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls ES))"
-
-text {*
-  TCon: Termination Condition of the equation-system decreasion.
-*}
-definition TCon:: "'a set \<Rightarrow> bool"
-where
-  "TCon ES \<equiv> card ES = 1"
-
-
-text {* 
-  The following is a iteration principle, and is the main framework for the proof:
-   1: We can form the initial equation-system using "equations" defined above, and prove it has invariance Inv by lemma "init_ES_satisfy_Inv";
-   2: We can decrease the number of the equation-system using ardens_lemma_revised and substitution ("equas_subst", defined above), 
-        and prove it holds the property "step" in "wf_iter" by lemma "iteration_step"
-   and finally using property Inv and TCon to prove the myhill-nerode theorem
-  
-*}
-lemma wf_iter [rule_format]: 
-  fixes f
-  assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and>  (f(e'), f(e)) \<in> less_than)"
-  shows pe:     "P e \<longrightarrow> (\<exists> e'. P e' \<and>  Q e')"
-proof(induct e rule: wf_induct 
-           [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
-  fix x 
-  assume h [rule_format]: 
-    "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
-    and px: "P x"
-  show "\<exists>e'. P e' \<and> Q e'"
-  proof(cases "Q x")
-    assume "Q x" with px show ?thesis by blast
-  next
-    assume nq: "\<not> Q x"
-    from step [OF px nq]
-    obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
-    show ?thesis
-    proof(rule h)
-      from ltf show "(e', x) \<in> inv_image less_than f" 
-	by (simp add:inv_image_def)
-    next
-      from pe' show "P e'" .
-    qed
-  qed
-qed
-
-
-text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
-
-lemma distinct_rhs_equations:
-  "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
-by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
-
-lemma every_nonempty_eqclass_has_strings:
-  "\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-lemma every_eqclass_is_derived_from_empty:
-  assumes not_empty: "X \<noteq> {[]}"
-  shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
-using not_empty
-apply (drule_tac every_nonempty_eqclass_has_strings, simp)
-by (auto intro:exI[where x = clist] simp:lang_seq_def)
-
-lemma equiv_str_in_CS:
-  "\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
-by (auto simp:quot_def)
-
-lemma has_str_imp_defined_by_str:
-  "\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
-by (auto simp:quot_def equiv_class_def equiv_str_def)
-
-lemma every_eqclass_has_ascendent:
-  assumes has_str: "clist @ [c] \<in> X"
-  and     in_CS:   "X \<in> UNIV Quo Lang"
-  shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
-proof -
-  have "?P (\<lbrakk>clist\<rbrakk>Lang)" 
-  proof -
-    have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"       
-      by (simp add:quot_def, rule_tac x = clist in exI, simp)
-    moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X" 
-    proof -
-      have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
-        by (auto intro!:has_str_imp_defined_by_str)
-      moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
-        by (auto simp:equiv_class_def equiv_str_def)
-      ultimately show ?thesis unfolding CT_def lang_seq_def
-        by auto
-    qed
-    moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang" 
-      by (auto simp:equiv_str_def equiv_class_def)
-    ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
-  qed
-  thus ?thesis by blast
-qed
-
-lemma finite_charset_rS:
-  "finite {CHAR c |c. Y-c\<rightarrow>X}"
-by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
-
-lemma l_eq_r_in_equations:
-  assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
-  shows "X = L xrhs"    
-proof (cases "X = {[]}")
-  case True
-  thus ?thesis using X_in_equas 
-    by (simp add:equations_def equation_rhs_def lang_seq_def)
-next
-  case False 
-  show ?thesis
-  proof 
-    show "X \<subseteq> L xrhs"
-    proof
-      fix x
-      assume "(1)": "x \<in> X"
-      show "x \<in> L xrhs"          
-      proof (cases "x = []")
-        assume empty: "x = []"
-        hence "x \<in> L (empty_rhs X)" using "(1)"
-          by (auto simp:empty_rhs_def lang_seq_def)
-        thus ?thesis using X_in_equas False empty "(1)" 
-          unfolding equations_def equation_rhs_def by auto
-      next
-        assume not_empty: "x \<noteq> []"
-        hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
-        then obtain clist c where decom: "x = clist @ [c]" by blast
-        moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>
-          \<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
-        proof -
-          fix Y
-          assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
-            and Y_CT_X: "Y-c\<rightarrow>X"
-            and clist_in_Y: "clist \<in> Y"
-          with finite_charset_rS 
-          show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
-            by (auto simp :fold_alt_null_eqs)
-        qed
-        hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" 
-          using X_in_equas False not_empty "(1)" decom
-          by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def lang_seq_def)
-        then obtain Xa where 
-          "Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
-        hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}" 
-          using X_in_equas "(1)" decom
-          by (auto simp add:equations_def equation_rhs_def intro!:exI[where x = Xa])
-        thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
-          by auto
-      qed
-    qed
-  next
-    show "L xrhs \<subseteq> X"
-    proof
-      fix x 
-      assume "(2)": "x \<in> L xrhs"
-      have "(2_1)": "\<And> s1 s2 r Xa. \<lbrakk>s1 \<in> Xa; s2 \<in> L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
-        using finite_charset_rS
-        by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
-      have "(2_2)": "\<And> s1 s2 Xa r.\<lbrakk>s1 \<in> Xa; s2 \<in> L r; (Xa, r) \<in> empty_rhs X\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
-        by (simp add:empty_rhs_def split:if_splits)
-      show "x \<in> X" using X_in_equas False "(2)"         
-        by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def lang_seq_def)
-    qed
-  qed
-qed
-
-
-
-lemma no_EMPTY_equations:
-  "(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
-apply (clarsimp simp add:equations_def equation_rhs_def)
-apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
-apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_charset_rS)+
-done
-
-lemma init_ES_satisfy_ardenable:
-  "(X, xrhs) \<in> equations (UNIV Quo Lang)  \<Longrightarrow> ardenable (X, xrhs)"  
-  unfolding ardenable_def
-  by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations simp del:L_rhs.simps)
-
-lemma init_ES_satisfy_Inv:
-  assumes finite_CS: "finite (UNIV Quo Lang)"
-  and X_in_eq_cls: "X \<in> UNIV Quo Lang"
-  shows "Inv X (equations (UNIV Quo Lang))"
-proof -
-  have "finite (equations (UNIV Quo Lang))" using finite_CS
-    by (auto simp:equations_def)    
-  moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls 
-    by (simp add:equations_def)
-  moreover have "distinct_equas (equations (UNIV Quo Lang))" 
-    by (auto simp:distinct_equas_def equations_def)
-  moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
-                 rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))" 
-    apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
-    by (auto simp:empty_rhs_def split:if_splits)
-  moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
-    by (clarsimp simp:equations_def empty_notin_CS intro:classical)
-  moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
-    by (auto intro!:init_ES_satisfy_ardenable)
-  ultimately show ?thesis by (simp add:Inv_def)
-qed
-
-
-text {* *********** END: proving the initial equation-system satisfies Inv ******* *}
-
-
-text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
-
-lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
-       \<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
-apply (case_tac "insert a A = {a}")
-by (auto simp:TCon_def)
-
-lemma not_T_atleast_2[rule_format]:
-  "finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
-apply (erule finite.induct, simp)
-apply (clarify, case_tac "x = a")
-by (erule not_T_aux, auto)
-
-lemma exist_another_equa: 
-  "\<lbrakk>\<not> TCon ES; finite ES; distinct_equas ES; (X, rhl) \<in> ES\<rbrakk> \<Longrightarrow> \<exists> Y yrhl. (Y, yrhl) \<in> ES \<and> X \<noteq> Y"
-apply (drule not_T_atleast_2, simp)
-apply (clarsimp simp:distinct_equas_def)
-apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
-by auto
-
-lemma Inv_mono_with_lambda:
-  assumes Inv_ES: "Inv X ES"
-  and X_noteq_Y:  "X \<noteq> {[]}"
-  shows "Inv X (ES - {({[]}, yrhs)})"
-proof -
-  have "finite (ES - {({[]}, yrhs)})" using Inv_ES
-    by (simp add:Inv_def)
-  moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
-    by (simp add:Inv_def)
-  moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
-    apply (clarsimp simp:Inv_def distinct_equas_def)
-    by (drule_tac x = Xa in spec, simp)    
-  moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
-                          ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
-    by (clarify, simp add:Inv_def)
-  moreover 
-  have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
-    by (auto simp:left_eq_cls_def)
-  hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
-                          rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
-    using Inv_ES by (auto simp:Inv_def)
-  ultimately show ?thesis by (simp add:Inv_def)
-qed
-
-lemma non_empty_card_prop:
-  "finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
-apply (erule finite.induct, simp)
-apply (case_tac[!] "a \<in> A")
-by (auto simp:insert_absorb)
-
-lemma ardenable_prop:
-  assumes not_lambda: "Y \<noteq> {[]}"
-  and ardable: "ardenable (Y, yrhs)"
-  shows "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" (is "\<exists> yrhs'. ?P yrhs'")
-proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
-  case True
-  thus ?thesis 
-  proof 
-    fix reg
-    assume self_contained: "(Y, reg) \<in> yrhs"
-    show ?thesis 
-    proof -
-      have "?P (arden_variate Y reg yrhs)"
-      proof -
-        have "Y = L (arden_variate Y reg yrhs)" 
-          using self_contained not_lambda ardable
-          by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
-        moreover have "distinct_rhs (arden_variate Y reg yrhs)" 
-          using ardable 
-          by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
-        moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
-        proof -
-          have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
-            apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
-            by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
-          moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
-            by (auto simp:rhs_eq_cls_def del_x_paired_def)
-          ultimately show ?thesis by (simp add:arden_variate_def)
-        qed
-        ultimately show ?thesis by simp
-      qed
-      thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
-    qed
-  qed
-next
-  case False
-  hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
-    by (auto simp:rhs_eq_cls_def)
-  show ?thesis 
-  proof -
-    have "?P yrhs" using False ardable "(2)" 
-      by (simp add:ardenable_def)      
-    thus ?thesis by blast
-  qed
-qed
-
-lemma equas_subst_f_del_no_other:
-  assumes self_contained: "(Y, rhs) \<in> ES"
-  shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
-proof -
-  have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
-    by (auto simp:equas_subst_f_def)
-  then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
-  hence "?P rhs'" unfolding image_def using self_contained
-    by (auto intro:bexI[where x = "(Y, rhs)"])
-  thus ?thesis by blast
-qed
-
-lemma del_x_paired_del_only_x: 
-  "\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
-by (auto simp:del_x_paired_def)
-
-lemma equas_subst_del_no_other:
- "\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
-unfolding equas_subst_def
-apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
-by (erule exE, drule del_x_paired_del_only_x, auto)
-
-lemma equas_subst_holds_distinct:
-  "distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
-apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
-by (auto split:if_splits)
-
-lemma del_x_paired_dels: 
-  "(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
-by (auto)
-
-lemma del_x_paired_subset:
-  "(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
-apply (drule del_x_paired_dels)
-by auto
-
-lemma del_x_paired_card_less: 
-  "\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
-apply (simp add:del_x_paired_def)
-apply (drule del_x_paired_subset)
-by (auto intro:psubset_card_mono)
-
-lemma equas_subst_card_less:
-  "\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
-apply (simp add:equas_subst_def)
-apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
-apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
-apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
-by (drule del_x_paired_card_less, auto)
-
-lemma equas_subst_holds_distinct_rhs:
-  assumes   dist': "distinct_rhs yrhs'"
-  and     history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
-  and     X_in :  "(X, xrhs) \<in> equas_subst ES Y yrhs'"
-  shows "distinct_rhs xrhs"
-using X_in history
-apply (clarsimp simp:equas_subst_def del_x_paired_def)
-apply (drule_tac x = a in spec, drule_tac x = b in spec)
-apply (simp add:ardenable_def equas_subst_f_def)
-by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
-
-lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
-  "[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
-by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
-
-lemma del_x_paired_holds_no_EMPTY:
-  "no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
-by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
-
-lemma rhs_subst_holds_no_EMPTY:
-  "\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
-apply (auto simp:rhs_subst_def intro!:no_EMPTY_rhss_imp_merge_no_EMPTY r_no_EMPTY_imp_seq_rhs_r_no_EMPTY del_x_paired_holds_no_EMPTY)
-by (auto simp:no_EMPTY_rhs_def)
-
-lemma equas_subst_holds_no_EMPTY:
-  assumes substor: "Y \<noteq> {[]}"
-  and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
-  and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
-  shows "no_EMPTY_rhs xrhs"
-proof-
-  from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
-    by (auto simp add:equas_subst_def del_x_paired_def)
-  then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
-                       and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
-  hence dist_zrhs: "distinct_rhs zrhs" using history
-    by (auto simp:ardenable_def)
-  show ?thesis
-  proof (cases "\<exists> r. (Y, r) \<in> zrhs")
-    case True
-    then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
-    hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
-      by (auto simp:distinct_rhs_def)
-    hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
-      using substor Y_in_zrhs history Z_in
-      by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
-    thus ?thesis using X_Z True some
-      by (simp add:equas_subst_def equas_subst_f_def)
-  next
-    case False
-    hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
-      by (simp add:equas_subst_f_def)
-    thus ?thesis using history Z_in
-      by (auto simp:ardenable_def)
-  qed
-qed
-
-lemma equas_subst_f_holds_left_eq_right:
-  assumes substor: "Y = L rhs'"
-  and     history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
-  and       subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
-  and     self_contained: "(Z, zrhs) \<in> ES"
-  shows "X = L xrhs"
-proof (cases "\<exists> r. (Y, r) \<in> zrhs")
-  case True
-  from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
-  show ?thesis
-  proof -
-    from history self_contained
-    have dist: "distinct_rhs zrhs" by auto
-    hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
-      using distinct_rhs_def by (auto intro:some_equality)
-    moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
-      by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
-    ultimately show ?thesis using subst history self_contained
-      by (auto simp:equas_subst_f_def split:if_splits)
-  qed
-next
-  case False
-  thus ?thesis using history subst self_contained
-    by (auto simp:equas_subst_f_def)
-qed
-
-lemma equas_subst_holds_left_eq_right:
-  assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
-  and     substor: "Y = L rhs'"
-  and     X_in :  "(X, xrhs) \<in> equas_subst ES Y yrhs'"
-  shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
-apply (clarsimp simp add:equas_subst_def del_x_paired_def)
-using substor
-apply (drule_tac equas_subst_f_holds_left_eq_right)
-using history
-by (auto simp:ardenable_def)
-
-lemma equas_subst_holds_ardenable:
-  assumes substor: "Y = L yrhs'"
-  and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
-  and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
-  and dist': "distinct_rhs yrhs'"
-  and not_lambda: "Y \<noteq> {[]}"
-  shows "ardenable (X, xrhs)"
-proof -
-  have "distinct_rhs xrhs" using history X_in dist' 
-    by (auto dest:equas_subst_holds_distinct_rhs)
-  moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
-    by (auto intro:equas_subst_holds_no_EMPTY)
-  moreover have "X = L xrhs" using history substor X_in
-  by (auto dest: equas_subst_holds_left_eq_right simp del:L_rhs.simps)
-  ultimately show ?thesis using ardenable_def by simp
-qed
-
-lemma equas_subst_holds_cls_defined:
-  assumes         X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
-  and           Inv_ES: "Inv X' ES"
-  and            subst: "(Y, yrhs) \<in> ES"
-  and  cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
-  shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
-proof-
-  have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
-  from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
-    by (auto simp add:equas_subst_def del_x_paired_def)
-  then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
-                       and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
-  hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
-    by (auto simp:Inv_def)
-  moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}" 
-    using Inv_ES subst cls_holds_but_Y
-    by (auto simp:Inv_def)
-  moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
-    using X_Z cls_holds_but_Y
-    apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
-    by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
-  moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
-    by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
-             dest: equas_subst_f_del_no_other
-             split: if_splits)
-  ultimately show ?thesis by blast
-qed
-
-lemma iteration_step: 
-  assumes Inv_ES: "Inv X ES"
-  and    not_T: "\<not> TCon ES"
-  shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)" 
-proof -
-  from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
-    by (clarify, rule_tac exist_another_equa[where X = X], auto)
-  then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
-  show ?thesis (is "\<exists> ES'. ?P ES'")
-  proof (cases "Y = {[]}") 
-    case True
-      --"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
-    have "?P (ES - {(Y, yrhs)})" 
-    proof 
-      show "Inv X (ES - {(Y, yrhs)})" using True not_X
-        by (simp add:Inv_ES Inv_mono_with_lambda)
-    next 
-      show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
-        by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
-    qed
-    thus ?thesis by blast
-  next
-    case False
-      --"in this situation, we pick a equation and using ardenable to get a 
-        rhs without itself in it, then use equas_subst to form a new equation-system"
-    hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" 
-      using subst Inv_ES
-      by (auto intro:ardenable_prop simp add:Inv_def simp del:L_rhs.simps)
-    then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
-      and dist_Y': "distinct_rhs yrhs'"
-      and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
-    hence "?P (equas_subst ES Y yrhs')"
-    proof -
-      have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)" 
-        apply (rule_tac A = "del_x_paired S x" in finite_subset)
-        by (auto simp:del_x_paired_def)
-      have "finite (equas_subst ES Y yrhs')" using Inv_ES 
-        by (auto intro!:finite_del simp:equas_subst_def Inv_def)
-      moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
-        by (auto intro:equas_subst_del_no_other simp:Inv_def)
-      moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
-        by (auto intro:equas_subst_holds_distinct simp:Inv_def)
-      moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
-        using Inv_ES dist_Y' False Y'_l_eq_r
-        apply (clarsimp simp:Inv_def)
-        by (rule equas_subst_holds_ardenable, simp_all)
-      moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
-        by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
-      moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
-                               rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
-        using Inv_ES subst cls_holds_but_Y
-        apply (rule_tac impI | rule_tac allI)+
-        by (erule equas_subst_holds_cls_defined, auto)
-      moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
-        by (simp add:equas_subst_card_less Inv_def)
-      ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)      
-    qed
-    thus ?thesis by blast
-  qed
-qed
-
-text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
-
-lemma iteration_conc: 
-  assumes history: "Inv X ES"
-  shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
-proof (cases "TCon ES")
-  case True
-  hence "?P ES" using history by simp
-  thus ?thesis by blast
-next
-  case False  
-  thus ?thesis using history iteration_step
-    by (rule_tac f = card in wf_iter, simp_all)
-qed
-
-lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
-apply (auto simp:mem_def)
-done
-
-lemma set_cases2:
-  "\<lbrakk>(A = {} \<Longrightarrow> R A); \<And> x. (A = {x}) \<Longrightarrow> R A; \<And> x y. \<lbrakk>x \<noteq> y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> R A\<rbrakk> \<Longrightarrow> R A"
-apply (case_tac "A = {}", simp)
-by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
-
-lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
-apply (rule_tac A = rhs in set_cases2, simp)
-apply (drule_tac x = X in eqset_imp_iff, clarsimp)
-apply (drule eqset_imp_iff',clarsimp)
-apply (frule_tac x = a in spec, drule_tac x = aa in spec)
-by (auto simp:distinct_rhs_def)
-
-lemma every_eqcl_has_reg: 
-  assumes finite_CS: "finite (UNIV Quo Lang)"
-  and X_in_CS: "X \<in> (UNIV Quo Lang)"
-  shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
-proof-
-  have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
-    by (auto intro:init_ES_satisfy_Inv iteration_conc)
-  then obtain ES' where Inv_ES': "Inv X ES'" 
-                   and  TCon_ES': "TCon ES'" by blast
-  from Inv_ES' TCon_ES' 
-  have "\<exists> rhs. ES' = {(X, rhs)}"
-    apply (clarsimp simp:Inv_def TCon_def)
-    apply (rule_tac x = rhs in exI)
-    by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)  
-  then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
-  hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
-    by (simp add:Inv_def)
-  
-  from X_ardenable have X_l_eq_r: "X = L rhs"
-    by (simp add:ardenable_def)
-  hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
-    by (auto simp:Inv_def ardenable_def)
-  have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
-    using Inv_ES' ES'_single_equa
-    by (auto simp:Inv_def ardenable_def left_eq_cls_def)
-  have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
-    by (auto simp:Inv_def)    
-  show ?thesis
-  proof (cases "X = {[]}")
-    case True
-    hence "?E EMPTY" by (simp)
-    thus ?thesis by blast
-  next
-    case False with  X_ardenable
-    have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
-      by (drule_tac ardenable_prop, auto)
-    then obtain rhs' where X_eq_rhs': "X = L rhs'"
-      and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}" 
-      and rhs'_dist : "distinct_rhs rhs'" by blast
-    have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty 
-      by blast
-    hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
-      by (auto simp:rhs_eq_cls_def)
-    hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
-      by (auto intro:rhs_aux simp:rhs_eq_cls_def)
-    then obtain r where "rhs' = {({[]}, r)}" ..
-    hence "?E r" using X_eq_rhs' by (auto simp add:lang_seq_def)
-    thus ?thesis by blast     
-  qed
-qed
-
-text {* definition of a regular language *}
-
-abbreviation
-  reg :: "string set => bool"
-where
-  "reg L1 \<equiv> (\<exists>r::rexp. L r = L1)"
-
-theorem myhill_nerode: 
-  assumes finite_CS: "finite (UNIV Quo Lang)"
-  shows   "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
-proof -
-  have has_r_each: "\<forall>C\<in>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists>(r::rexp). C = L r" using finite_CS
-    by (auto dest:every_eqcl_has_reg)  
-  have "\<exists> (rS::rexp set). finite rS \<and> 
-                          (\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and> 
-                          (\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)" 
-       (is "\<exists> rS. ?Q rS")
-  proof-
-    have "\<And> C. C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<Longrightarrow> C = L (SOME (ra::rexp). C = L ra)"
-      using has_r_each
-      apply (erule_tac x = C in ballE, erule_tac exE)
-      by (rule_tac a = r in someI2, simp+)
-    hence "?Q ((\<lambda> C. SOME r. C = L r) ` {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang})" using has_r_each
-      using finite_CS by auto
-    thus ?thesis by blast    
-  qed
-  then obtain rS where finite_rS : "finite rS"
-    and has_r_each': "\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> (rS::rexp set). C = L r"
-    and has_cl_each: "\<forall> r \<in> (rS::rexp set). \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r" by blast
-  have "?P (folds ALT NULL rS)"
-  proof
-    show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
-      apply (clarsimp simp:fold_alt_null_eqs) by blast
-  next 
-    show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
-      by (clarsimp simp:fold_alt_null_eqs)
-  qed
-  thus ?thesis by blast
-qed 
-
-
-text {* tests by Christian *}
-
-(* 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
-  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
-  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)
-
-lemma finite_tag_image: 
-  "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
-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
-
-(* list_diff:: list substract, once different return tailer *)
-fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
-where
-  "list_diff []  xs = []" |
-  "list_diff (x#xs) [] = x#xs" |
-  "list_diff (x#xs) (y#ys) = (if x = y then list_diff xs ys else (x#xs))"
-
-lemma [simp]: "(x @ y) - x = y"
-apply (induct x)
-by (case_tac y, simp+)
-
-lemma [simp]: "x - x = []"
-by (induct x, auto)
-
-lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
-by (induct x, auto)
-
-lemma [simp]: "x - [] = x"
-by (induct x, auto)
-
-lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
-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 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 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
-
-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)
-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
-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)
-qed
-
-(****************** the STAR case ************************)
-
-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)
-
-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)
-qed
-
-lemma star_prop[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>"
-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 postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
-by (erule postfixE, induct x arbitrary:y, auto)
-
-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
-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)
-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
-
-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)  
-
-end 




