diff -r 070f543e2560 -r d94209ad2880 Myhill.thy
--- a/Myhill.thy	Thu Feb 03 09:54:19 2011 +0000
+++ b/Myhill.thy	Thu Feb 03 12:00:06 2011 +0000
@@ -1,900 +1,11 @@
 theory Myhill
-  imports Myhill_1
+  imports Myhill_2
 begin
 
 section {* Direction @{text "regular language \<Rightarrow>finite partition"} *}
 
-subsection {* The scheme*}
-
-text {* 
-  The following convenient notation @{text "x \<approx>Lang y"} means:
-  string @{text "x"} and @{text "y"} are equivalent with respect to 
-  language @{text "Lang"}.
-  *}
-
-definition
-  str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _")
-where
-  "x \<approx>Lang y \<equiv> (x, y) \<in> (\<approx>Lang)"
-
 text {*
-  The main lemma (@{text "rexp_imp_finite"}) is proved by a structural induction over regular expressions.
-  While base cases (cases for @{const "NULL"}, @{const "EMPTY"}, @{const "CHAR"}) are quite straight forward,
-  the inductive cases are rather involved. What we have when starting to prove these inductive caes is that
-  the partitions induced by the componet language are finite. The basic idea to show the finiteness of the 
-  partition induced by the composite language is to attach a tag @{text "tag(x)"} to every string 
-  @{text "x"}. The tags are made of equivalent classes from the component partitions. Let @{text "tag"}
-  be the tagging function and @{text "Lang"} be the composite language, it can be proved that
-  if strings with the same tag are equivalent with respect to @{text "Lang"}, expressed as:
-  \[
-  @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"}
-  \]
-  then the partition induced by @{text "Lang"} must be finite. There are two arguments for this. 
-  The first goes as the following:
-  \begin{enumerate}
-    \item First, the tagging function @{text "tag"} induces an equivalent relation @{text "(=tag=)"} 
-          (defiintion of @{text "f_eq_rel"} and lemma @{text "equiv_f_eq_rel"}).
-    \item It is shown that: if the range of @{text "tag"} (denoted @{text "range(tag)"}) is finite, 
-           the partition given rise by @{text "(=tag=)"} is finite (lemma @{text "finite_eq_f_rel"}).
-           Since tags are made from equivalent classes from component partitions, and the inductive
-           hypothesis ensures the finiteness of these partitions, it is not difficult to prove
-           the finiteness of @{text "range(tag)"}.
-    \item It is proved that if equivalent relation @{text "R1"} is more refined than @{text "R2"}
-           (expressed as @{text "R1 \<subseteq> R2"}),
-           and the partition induced by @{text "R1"} is finite, then the partition induced by @{text "R2"}
-           is finite as well (lemma @{text "refined_partition_finite"}).
-    \item The injectivity assumption @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"} implies that
-            @{text "(=tag=)"} is more refined than @{text "(\<approx>Lang)"}.
-    \item Combining the points above, we have: the partition induced by language @{text "Lang"}
-          is finite (lemma @{text "tag_finite_imageD"}).
-  \end{enumerate}
-*}
-
-definition 
-   f_eq_rel ("=_=")
-where
-   "(=f=) = {(x, y) | x y. f x = f y}"
-
-lemma equiv_f_eq_rel:"equiv UNIV (=f=)"
-  by (auto simp:equiv_def f_eq_rel_def refl_on_def sym_def trans_def)
-
-lemma finite_range_image: "finite (range f) \<Longrightarrow> finite (f ` A)"
-  by (rule_tac B = "{y. \<exists>x. y = f x}" in finite_subset, auto simp:image_def)
-
-lemma finite_eq_f_rel:
-  assumes rng_fnt: "finite (range tag)"
-  shows "finite (UNIV // (=tag=))"
-proof -
-  let "?f" =  "op ` tag" and ?A = "(UNIV // (=tag=))"
-  show ?thesis
-  proof (rule_tac f = "?f" and A = ?A in finite_imageD) 
-    -- {* 
-      The finiteness of @{text "f"}-image is a simple consequence of assumption @{text "rng_fnt"}:
-      *}
-    show "finite (?f ` ?A)" 
-    proof -
-      have "\<forall> X. ?f X \<in> (Pow (range tag))" by (auto simp:image_def Pow_def)
-      moreover from rng_fnt have "finite (Pow (range tag))" by simp
-      ultimately have "finite (range ?f)"
-        by (auto simp only:image_def intro:finite_subset)
-      from finite_range_image [OF this] show ?thesis .
-    qed
-  next
-    -- {* 
-      The injectivity of @{text "f"}-image is a consequence of the definition of @{text "(=tag=)"}:
-      *}
-    show "inj_on ?f ?A" 
-    proof-
-      { fix X Y
-        assume X_in: "X \<in> ?A"
-          and  Y_in: "Y \<in> ?A"
-          and  tag_eq: "?f X = ?f Y"
-        have "X = Y"
-        proof -
-          from X_in Y_in tag_eq 
-          obtain x y 
-            where x_in: "x \<in> X" and y_in: "y \<in> Y" and eq_tg: "tag x = tag y"
-            unfolding quotient_def Image_def str_eq_rel_def 
-                                   str_eq_def image_def f_eq_rel_def
-            apply simp by blast
-          with X_in Y_in show ?thesis 
-            by (auto simp:quotient_def str_eq_rel_def str_eq_def f_eq_rel_def) 
-        qed
-      } thus ?thesis unfolding inj_on_def by auto
-    qed
-  qed
-qed
-
-lemma finite_image_finite: "\<lbrakk>\<forall> x \<in> A. f x \<in> B; finite B\<rbrakk> \<Longrightarrow> finite (f ` A)"
-  by (rule finite_subset [of _ B], auto)
-
-lemma refined_partition_finite:
-  fixes R1 R2 A
-  assumes fnt: "finite (A // R1)"
-  and refined: "R1 \<subseteq> R2"
-  and eq1: "equiv A R1" and eq2: "equiv A R2"
-  shows "finite (A // R2)"
-proof -
-  let ?f = "\<lambda> X. {R1 `` {x} | x. x \<in> X}" 
-    and ?A = "(A // R2)" and ?B = "(A // R1)"
-  show ?thesis
-  proof(rule_tac f = ?f and A = ?A in finite_imageD)
-    show "finite (?f ` ?A)"
-    proof(rule finite_subset [of _ "Pow ?B"])
-      from fnt show "finite (Pow (A // R1))" by simp
-    next
-      from eq2
-      show " ?f ` A // R2 \<subseteq> Pow ?B"
-        unfolding image_def Pow_def quotient_def
-        apply auto
-        by (rule_tac x = xb in bexI, simp, 
-                 unfold equiv_def sym_def refl_on_def, blast)
-    qed
-  next
-    show "inj_on ?f ?A"
-    proof -
-      { fix X Y
-        assume X_in: "X \<in> ?A" and Y_in: "Y \<in> ?A" 
-          and eq_f: "?f X = ?f Y" (is "?L = ?R")
-        have "X = Y" using X_in
-        proof(rule quotientE)
-          fix x
-          assume "X = R2 `` {x}" and "x \<in> A" with eq2
-          have x_in: "x \<in> X" 
-            unfolding equiv_def quotient_def refl_on_def by auto
-          with eq_f have "R1 `` {x} \<in> ?R" by auto
-          then obtain y where 
-            y_in: "y \<in> Y" and eq_r: "R1 `` {x} = R1 ``{y}" by auto
-          have "(x, y) \<in> R1"
-          proof -
-            from x_in X_in y_in Y_in eq2
-            have "x \<in> A" and "y \<in> A" 
-              unfolding equiv_def quotient_def refl_on_def by auto
-            from eq_equiv_class_iff [OF eq1 this] and eq_r
-            show ?thesis by simp
-          qed
-          with refined have xy_r2: "(x, y) \<in> R2" by auto
-          from quotient_eqI [OF eq2 X_in Y_in x_in y_in this]
-          show ?thesis .
-        qed
-      } thus ?thesis by (auto simp:inj_on_def)
-    qed
-  qed
-qed
-
-lemma equiv_lang_eq: "equiv UNIV (\<approx>Lang)"
-  unfolding equiv_def str_eq_rel_def sym_def refl_on_def trans_def
-  by blast
-
-lemma tag_finite_imageD:
-  fixes tag
-  assumes rng_fnt: "finite (range tag)" 
-  -- {* Suppose the rang of tagging fucntion @{text "tag"} is finite. *}
-  and same_tag_eqvt: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<approx>Lang n"
-  -- {* And strings with same tag are equivalent *}
-  shows "finite (UNIV // (\<approx>Lang))"
-proof -
-  let ?R1 = "(=tag=)"
-  show ?thesis
-  proof(rule_tac refined_partition_finite [of _ ?R1])
-    from finite_eq_f_rel [OF rng_fnt]
-     show "finite (UNIV // =tag=)" . 
-   next
-     from same_tag_eqvt
-     show "(=tag=) \<subseteq> (\<approx>Lang)"
-       by (auto simp:f_eq_rel_def str_eq_def)
-   next
-     from equiv_f_eq_rel
-     show "equiv UNIV (=tag=)" by blast
-   next
-     from equiv_lang_eq
-     show "equiv UNIV (\<approx>Lang)" by blast
-  qed
-qed
-
-text {*
-  A more concise, but less intelligible argument for @{text "tag_finite_imageD"} 
-  is given as the following. The basic idea is still using standard library 
-  lemma @{thm [source] "finite_imageD"}:
-  \[
-  @{thm "finite_imageD" [no_vars]}
-  \]
-  which says: if the image of injective function @{text "f"} over set @{text "A"} is 
-  finite, then @{text "A"} must be finte, as we did in the lemmas above.
-  *}
-
-lemma 
-  fixes tag
-  assumes rng_fnt: "finite (range tag)" 
-  -- {* Suppose the rang of tagging fucntion @{text "tag"} is finite. *}
-  and same_tag_eqvt: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<approx>Lang n"
-  -- {* And strings with same tag are equivalent *}
-  shows "finite (UNIV // (\<approx>Lang))"
-  -- {* Then the partition generated by @{text "(\<approx>Lang)"} is finite. *}
-proof -
-  -- {* The particular @{text "f"} and @{text "A"} used in @{thm [source] "finite_imageD"} are:*}
-  let "?f" =  "op ` tag" and ?A = "(UNIV // \<approx>Lang)"
-  show ?thesis
-  proof (rule_tac f = "?f" and A = ?A in finite_imageD) 
-    -- {* 
-      The finiteness of @{text "f"}-image is a simple consequence of assumption @{text "rng_fnt"}:
-      *}
-    show "finite (?f ` ?A)" 
-    proof -
-      have "\<forall> X. ?f X \<in> (Pow (range tag))" by (auto simp:image_def Pow_def)
-      moreover from rng_fnt have "finite (Pow (range tag))" by simp
-      ultimately have "finite (range ?f)"
-        by (auto simp only:image_def intro:finite_subset)
-      from finite_range_image [OF this] show ?thesis .
-    qed
-  next
-    -- {* 
-      The injectivity of @{text "f"} is the consequence of assumption @{text "same_tag_eqvt"}:
-      *}
-    show "inj_on ?f ?A" 
-    proof-
-      { fix X Y
-        assume X_in: "X \<in> ?A"
-          and  Y_in: "Y \<in> ?A"
-          and  tag_eq: "?f X = ?f Y"
-        have "X = Y"
-        proof -
-          from X_in Y_in tag_eq 
-          obtain x y where x_in: "x \<in> X" and y_in: "y \<in> Y" and eq_tg: "tag x = tag y"
-            unfolding quotient_def Image_def str_eq_rel_def str_eq_def image_def
-            apply simp by blast 
-          from same_tag_eqvt [OF eq_tg] have "x \<approx>Lang y" .
-          with X_in Y_in x_in y_in
-          show ?thesis by (auto simp:quotient_def str_eq_rel_def str_eq_def) 
-        qed
-      } thus ?thesis unfolding inj_on_def by auto
-    qed
-  qed
-qed
-
-subsection {* The proof*}
-
-text {*
-  Each case is given in a separate section, as well as the final main lemma. Detailed explainations accompanied by
-  illustrations are given for non-trivial cases.
-
-  For ever inductive case, there are two tasks, the easier one is to show the range finiteness of
-  of the tagging function based on the finiteness of component partitions, the
-  difficult one is to show that strings with the same tag are equivalent with respect to the 
-  composite language. Suppose the composite language be @{text "Lang"}, tagging function be 
-  @{text "tag"}, it amounts to show:
-  \[
-  @{text "tag(x) = tag(y) \<Longrightarrow> x \<approx>Lang y"}
-  \]
-  expanding the definition of @{text "\<approx>Lang"}, it amounts to show:
-  \[
-  @{text "tag(x) = tag(y) \<Longrightarrow> (\<forall> z. x@z \<in> Lang \<longleftrightarrow> y@z \<in> Lang)"}
-  \]
-  Because the assumed tag equlity @{text "tag(x) = tag(y)"} is symmetric,
-  it is suffcient to show just one direction:
-  \[
-  @{text "\<And> x y z. \<lbrakk>tag(x) = tag(y); x@z \<in> Lang\<rbrakk> \<Longrightarrow> y@z \<in> Lang"}
-  \]
-  This is the pattern followed by every inductive case.
-  *}
-
-subsubsection {* The base case for @{const "NULL"} *}
-
-lemma quot_null_eq:
-  shows "(UNIV // \<approx>{}) = ({UNIV}::lang set)"
-  unfolding quotient_def Image_def str_eq_rel_def by auto
-
-lemma quot_null_finiteI [intro]:
-  shows "finite ((UNIV // \<approx>{})::lang set)"
-unfolding quot_null_eq by simp
-
-
-subsubsection {* The base case for @{const "EMPTY"} *}
-
-
-lemma quot_empty_subset:
-  "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
-proof
-  fix x
-  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)
-    thus ?thesis by simp
-  next
-    case False with h
-    have "x = UNIV - {[]}" by (auto simp: str_eq_rel_def)
-    thus ?thesis by simp
-  qed
-qed
-
-lemma quot_empty_finiteI [intro]:
-  shows "finite (UNIV // (\<approx>{[]}))"
-by (rule finite_subset[OF quot_empty_subset]) (simp)
-
-
-subsubsection {* The base case for @{const "CHAR"} *}
-
-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)
-    } moreover {
-      assume "y = [c]" hence "x = {[c]}" using h 
-        by (auto dest!:spec[where x = "[]"] simp:str_eq_rel_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)
-    } ultimately show ?thesis by blast
-  qed
-qed
-
-lemma quot_char_finiteI [intro]:
-  shows "finite (UNIV // (\<approx>{[c]}))"
-by (rule finite_subset[OF quot_char_subset]) (simp)
-
-
-subsubsection {* The inductive case for @{const ALT} *}
-
-definition 
-  tag_str_ALT :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang)"
-where
-  "tag_str_ALT L1 L2 = (\<lambda>x. (\<approx>L1 `` {x}, \<approx>L2 `` {x}))"
-
-lemma quot_union_finiteI [intro]:
-  fixes L1 L2::"lang"
-  assumes finite1: "finite (UNIV // \<approx>L1)"
-  and     finite2: "finite (UNIV // \<approx>L2)"
-  shows "finite (UNIV // \<approx>(L1 \<union> L2))"
-proof (rule_tac tag = "tag_str_ALT L1 L2" in tag_finite_imageD)
-  show "\<And>x y. tag_str_ALT L1 L2 x = tag_str_ALT L1 L2 y \<Longrightarrow> x \<approx>(L1 \<union> L2) y"
-    unfolding tag_str_ALT_def 
-    unfolding str_eq_def
-    unfolding Image_def 
-    unfolding str_eq_rel_def
-    by auto
-next
-  have *: "finite ((UNIV // \<approx>L1) \<times> (UNIV // \<approx>L2))" 
-    using finite1 finite2 by auto
-  show "finite (range (tag_str_ALT L1 L2))"
-    unfolding tag_str_ALT_def
-    apply(rule finite_subset[OF _ *])
-    unfolding quotient_def
-    by auto
-qed
-
-subsubsection {* The inductive case for @{text "SEQ"}*}
-
-text {*
-  For case @{const "SEQ"}, the language @{text "L"} is @{text "L\<^isub>1 ;; L\<^isub>2"}.
-  Given @{text "x @ z \<in> L\<^isub>1 ;; L\<^isub>2"}, according to the defintion of @{text " L\<^isub>1 ;; L\<^isub>2"},
-  string @{text "x @ z"} can be splitted with the prefix in @{text "L\<^isub>1"} and suffix in @{text "L\<^isub>2"}.
-  The split point can either be in @{text "x"} (as shown in Fig. \ref{seq_first_split}),
-  or in @{text "z"} (as shown in Fig. \ref{seq_snd_split}). Whichever way it goes, the structure
-  on @{text "x @ z"} cn be transfered faithfully onto @{text "y @ z"} 
-  (as shown in Fig. \ref{seq_trans_first_split} and \ref{seq_trans_snd_split}) with the the help of the assumed 
-  tag equality. The following tag function @{text "tag_str_SEQ"} is such designed to facilitate
-  such transfers and lemma @{text "tag_str_SEQ_injI"} formalizes the informal argument above. The details 
-  of structure transfer will be given their.
-\input{fig_seq}
-
+  It is now the time for use to discuss further about the way.
   *}
 
-definition 
-  tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)"
-where
-  "tag_str_SEQ L1 L2 = 
-     (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa.  xa \<le> x \<and> xa \<in> L1}))"
-
-text {* The following is a techical lemma which helps to split the @{text "x @ z \<in> L\<^isub>1 ;; L\<^isub>2"} mentioned above.*}
-
-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 eq_xys: "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)
-  from app_eq_dest [OF eq_xys]
-  have
-    "(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)" 
-               (is "?Split1 \<or> ?Split2") .
-  moreover have "?Split1 \<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 "?Split2 \<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 tag_str_SEQ_injI:
-  fixes v w 
-  assumes eq_tag: "tag_str_SEQ L\<^isub>1 L\<^isub>2 v = tag_str_SEQ L\<^isub>1 L\<^isub>2 w" 
-  shows "v \<approx>(L\<^isub>1 ;; L\<^isub>2) w"
-proof-
-    -- {* As explained before, a pattern for just one direction needs to be dealt with:*}
-  { 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-
-      -- {* There are two ways to split @{text "x@z"}: *}
-      from append_seq_elim [OF xz_in_seq]
-      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)" .
-      -- {* It can be shown that @{text "?thesis"} holds in either case: *}
-      moreover {
-        -- {* The case for the first split:*}
-        fix xa
-        assume h1: "xa \<le> x" and h2: "xa \<in> L\<^isub>1" and h3: "(x - xa) @ z \<in> L\<^isub>2"
-        -- {* The following subgoal implements the structure transfer:*}
-        obtain ya 
-          where "ya \<le> y" 
-          and "ya \<in> L\<^isub>1" 
-          and "(y - ya) @ z \<in> L\<^isub>2"
-        proof -
-        -- {*
-            \begin{minipage}{0.8\textwidth}
-            By expanding the definition of 
-            @{thm [display] "tag_xy"}
-            and extracting the second compoent, we get:
-            \end{minipage}
-            *}
-          have "{\<approx>L\<^isub>2 `` {x - xa} |xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = 
-                   {\<approx>L\<^isub>2 `` {y - ya} |ya. ya \<le> y \<and> ya \<in> L\<^isub>1}" (is "?Left = ?Right")
-            using tag_xy unfolding tag_str_SEQ_def by simp
-            -- {* Since @{thm "h1"} and @{thm "h2"} hold, it is not difficult to show: *}
-          moreover have "\<approx>L\<^isub>2 `` {x - xa} \<in> ?Left" using h1 h2 by auto
-            -- {* 
-            \begin{minipage}{0.7\textwidth}
-            Through tag equality, equivalent class @{term "\<approx>L\<^isub>2 `` {x - xa}"} also 
-                  belongs to the @{text "?Right"}:
-            \end{minipage}
-            *}
-          ultimately have "\<approx>L\<^isub>2 `` {x - xa} \<in> ?Right" by simp
-            -- {* From this, the counterpart of @{text "xa"} in @{text "y"} is obtained:*}
-          then obtain ya 
-            where eq_xya: "\<approx>L\<^isub>2 `` {x - xa} = \<approx>L\<^isub>2 `` {y - ya}" 
-            and pref_ya: "ya \<le> y" and ya_in: "ya \<in> L\<^isub>1"
-            by simp blast
-          -- {* It can be proved that @{text "ya"} has the desired property:*}
-          have "(y - ya)@z \<in> L\<^isub>2" 
-          proof -
-            from eq_xya have "(x - xa)  \<approx>L\<^isub>2 (y - ya)" 
-              unfolding Image_def str_eq_rel_def str_eq_def by auto
-            with h3 show ?thesis unfolding str_eq_rel_def str_eq_def by simp
-          qed
-          -- {* Now, @{text "ya"} has all properties to be a qualified candidate:*}
-          with pref_ya ya_in 
-          show ?thesis using that by blast
-        qed
-          -- {* From the properties of @{text "ya"}, @{text "y @ z \<in> L\<^isub>1 ;; L\<^isub>2"} is derived easily.*}
-        hence "y @ z \<in> L\<^isub>1 ;; L\<^isub>2" by (erule_tac prefixE, auto simp:Seq_def)
-      } moreover {
-        -- {* The other case is even more simpler: *}
-        fix za
-        assume h1: "za \<le> z" and h2: "(x @ za) \<in> L\<^isub>1" and h3: "z - za \<in> L\<^isub>2"
-        have "y @ za \<in> L\<^isub>1"
-        proof-
-          have "\<approx>L\<^isub>1 `` {x} = \<approx>L\<^isub>1 `` {y}" 
-            using tag_xy unfolding tag_str_SEQ_def by simp
-          with h2 show ?thesis
-            unfolding Image_def str_eq_rel_def str_eq_def by auto
-        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
-  } 
-  -- {* 
-      \begin{minipage}{0.8\textwidth}
-      @{text "?thesis"} is proved by exploiting the symmetry of 
-      @{thm [source] "eq_tag"}:
-      \end{minipage}
-      *}
-  from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
-    show ?thesis unfolding str_eq_def str_eq_rel_def by blast
-qed 
-
-lemma quot_seq_finiteI [intro]:
-  fixes L1 L2::"lang"
-  assumes fin1: "finite (UNIV // \<approx>L1)" 
-  and     fin2: "finite (UNIV // \<approx>L2)" 
-  shows "finite (UNIV // \<approx>(L1 ;; L2))"
-proof (rule_tac tag = "tag_str_SEQ L1 L2" in tag_finite_imageD)
-  show "\<And>x y. tag_str_SEQ L1 L2 x = tag_str_SEQ L1 L2 y \<Longrightarrow> x \<approx>(L1 ;; L2) y"
-    by (rule tag_str_SEQ_injI)
-next
-  have *: "finite ((UNIV // \<approx>L1) \<times> (Pow (UNIV // \<approx>L2)))" 
-    using fin1 fin2 by auto
-  show "finite (range (tag_str_SEQ L1 L2))" 
-    unfolding tag_str_SEQ_def
-    apply(rule finite_subset[OF _ *])
-    unfolding quotient_def
-    by auto
-qed
-
-subsubsection {* The inductive case for @{const "STAR"} *}
-
-text {* 
-  This turned out to be the trickiest case. The essential goal is 
-  to proved @{text "y @ z \<in>  L\<^isub>1*"} under the assumptions that @{text "x @ z \<in>  L\<^isub>1*"}
-  and that @{text "x"} and @{text "y"} have the same tag. The reasoning goes as the following:
-  \begin{enumerate}
-    \item Since @{text "x @ z \<in>  L\<^isub>1*"} holds, a prefix @{text "xa"} of @{text "x"} can be found
-          such that @{text "xa \<in> L\<^isub>1*"} and @{text "(x - xa)@z \<in> L\<^isub>1*"}, as shown in Fig. \ref{first_split}.
-          Such a prefix always exists, @{text "xa = []"}, for example, is one. 
-    \item There could be many but fintie many of such @{text "xa"}, from which we can find the longest
-          and name it @{text "xa_max"}, as shown in Fig. \ref{max_split}.
-    \item The next step is to split @{text "z"} into @{text "za"} and @{text "zb"} such that
-           @{text "(x - xa_max) @ za \<in> L\<^isub>1"} and @{text "zb \<in> L\<^isub>1*"}  as shown in Fig. \ref{last_split}.
-          Such a split always exists because:
-          \begin{enumerate}
-            \item Because @{text "(x - x_max) @ z \<in> L\<^isub>1*"}, it can always be splitted into prefix @{text "a"}
-              and suffix @{text "b"}, such that @{text "a \<in> L\<^isub>1"} and @{text "b \<in> L\<^isub>1*"},
-              as shown in Fig. \ref{ab_split}.
-            \item But the prefix @{text "a"} CANNOT be shorter than @{text "x - xa_max"} 
-              (as shown in Fig. \ref{ab_split_wrong}), becasue otherwise,
-                   @{text "ma_max@a"} would be in the same kind as @{text "xa_max"} but with 
-                   a larger size, conflicting with the fact that @{text "xa_max"} is the longest.
-          \end{enumerate}
-    \item  \label{tansfer_step} 
-         By the assumption that @{text "x"} and @{text "y"} have the same tag, the structure on @{text "x @ z"}
-          can be transferred to @{text "y @ z"} as shown in Fig. \ref{trans_split}. The detailed steps are:
-          \begin{enumerate}
-            \item A @{text "y"}-prefix @{text "ya"} corresponding to @{text "xa"} can be found, 
-                  which satisfies conditions: @{text "ya \<in> L\<^isub>1*"} and @{text "(y - ya)@za \<in> L\<^isub>1"}.
-            \item Since we already know @{text "zb \<in> L\<^isub>1*"}, we get @{text "(y - ya)@za@zb \<in> L\<^isub>1*"},
-                  and this is just @{text "(y - ya)@z \<in> L\<^isub>1*"}.
-            \item With fact @{text "ya \<in> L\<^isub>1*"}, we finally get @{text "y@z \<in> L\<^isub>1*"}.
-          \end{enumerate}
-  \end{enumerate}
-
-  The formal proof of lemma @{text "tag_str_STAR_injI"} faithfully follows this informal argument 
-  while the tagging function @{text "tag_str_STAR"} is defined to make the transfer in step
-  \ref{ansfer_step} feasible.
-
-  \input{fig_star}
-*} 
-
-definition 
-  tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set"
-where
-  "tag_str_STAR L1 = (\<lambda>x. {\<approx>L1 `` {x - xa} | xa. xa < x \<and> xa \<in> L1\<star>})"
-
-text {* A technical lemma. *}
-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 insertI.hyps and False  
-    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
-
-
-text {* The following is a technical lemma.which helps to show the range finiteness of tag function. *}
-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)
-
-
-lemma tag_str_STAR_injI:
-  fixes v w
-  assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w"
-  shows "(v::string) \<approx>(L\<^isub>1\<star>) w"
-proof-
-    -- {* As explained before, a pattern for just one direction needs to be dealt with:*}
-  { 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 = []")
-      -- {* 
-        The degenerated case when @{text "x"} is a null string is easy to prove:
-        *}
-      case True
-      with tag_xy have "y = []" 
-        by (auto simp add: tag_str_STAR_def strict_prefix_def)
-      thus ?thesis using xz_in_star True by simp
-    next
-        -- {* The nontrival case:
-        *}
-      case False
-      -- {* 
-        \begin{minipage}{0.8\textwidth}
-        Since @{text "x @ z \<in> L\<^isub>1\<star>"}, @{text "x"} can always be splitted
-        by a prefix @{text "xa"} together with its suffix @{text "x - xa"}, such
-        that both @{text "xa"} and @{text "(x - xa) @ z"} are in @{text "L\<^isub>1\<star>"},
-        and there could be many such splittings.Therefore, the following set @{text "?S"} 
-        is nonempty, and finite as well:
-        \end{minipage}
-        *}
-      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)
-      -- {* \begin{minipage}{0.7\textwidth} 
-            Since @{text "?S"} is finite, we can always single out the longest and name it @{text "xa_max"}: 
-            \end{minipage}
-          *}
-      ultimately have "\<exists> xa_max \<in> ?S. \<forall> xa \<in> ?S. length xa \<le> length xa_max" 
-        using finite_set_has_max by blast
-      then obtain xa_max 
-        where h1: "xa_max < x" 
-        and h2: "xa_max \<in> L\<^isub>1\<star>" 
-        and h3: "(x - xa_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 xa_max"
-        by blast
-      -- {*
-          \begin{minipage}{0.8\textwidth}
-          By the equality of tags, the counterpart of @{text "xa_max"} among 
-          @{text "y"}-prefixes, named @{text "ya"}, can be found:
-          \end{minipage}
-          *}
-      obtain ya 
-        where h5: "ya < y" and h6: "ya \<in> L\<^isub>1\<star>" 
-        and eq_xya: "(x - xa_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 - xa_max} \<in> ?left" using h1 h2 by auto
-        ultimately have "\<approx>L\<^isub>1 `` {x - xa_max} \<in> ?right" by simp
-        thus ?thesis using that 
-          apply (simp add:Image_def str_eq_rel_def str_eq_def) by blast
-      qed 
-      -- {*
-          \begin{minipage}{0.8\textwidth}
-          The @{text "?thesis"}, @{prop "y @ z \<in> L\<^isub>1\<star>"}, is a simple consequence
-          of the following proposition:
-          \end{minipage}
-          *}
-      have "(y - ya) @ z \<in> L\<^isub>1\<star>" 
-      proof-
-        -- {* The idea is to split the suffix @{text "z"} into @{text "za"} and @{text "zb"}, 
-          such that: *}
-        obtain za zb where eq_zab: "z = za @ zb" 
-          and l_za: "(y - ya)@za \<in> L\<^isub>1" and ls_zb: "zb \<in> L\<^isub>1\<star>"
-        proof -
-          -- {* 
-            \begin{minipage}{0.8\textwidth}
-            Since @{thm "h1"}, @{text "x"} can be splitted into
-            @{text "a"} and @{text "b"} such that:
-            \end{minipage}
-            *}
-          from h1 have "(x - xa_max) @ z \<noteq> []" 
-            by (auto simp:strict_prefix_def elim:prefixE)
-          from star_decom [OF h3 this]
-          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 - xa_max) @ z = a @ b" by blast
-          -- {* Now the candiates for @{text "za"} and @{text "zb"} are found:*}
-          let ?za = "a - (x - xa_max)" and ?zb = "b"
-          have pfx: "(x - xa_max) \<le> a" (is "?P1") 
-            and eq_z: "z = ?za @ ?zb" (is "?P2")
-          proof -
-            -- {* 
-              \begin{minipage}{0.8\textwidth}
-              Since @{text "(x - xa_max) @ z = a @ b"}, string @{text "(x - xa_max) @ z"}
-              can be splitted in two ways:
-              \end{minipage}
-              *}
-            have "((x - xa_max) \<le> a \<and> (a - (x - xa_max)) @ b = z) \<or> 
-              (a < (x - xa_max) \<and> ((x - xa_max) - a) @ z = b)" 
-              using app_eq_dest[OF ab_max] by (auto simp:strict_prefix_def)
-            moreover {
-              -- {* However, the undsired way can be refuted by absurdity: *}
-              assume np: "a < (x - xa_max)" 
-                and b_eqs: "((x - xa_max) - a) @ z = b"
-              have "False"
-              proof -
-                let ?xa_max' = "xa_max @ a"
-                have "?xa_max' < x" 
-                  using np h1 by (clarsimp simp:strict_prefix_def diff_prefix) 
-                moreover have "?xa_max' \<in> L\<^isub>1\<star>" 
-                  using a_in h2 by (simp add:star_intro3) 
-                moreover have "(x - ?xa_max') @ z \<in> L\<^isub>1\<star>" 
-                  using b_eqs b_in np h1 by (simp add:diff_diff_appd)
-                moreover have "\<not> (length ?xa_max' \<le> length xa_max)" 
-                  using a_neq by simp
-                ultimately show ?thesis using h4 by blast
-              qed }
-            -- {* Now it can be shown that the splitting goes the way we desired. *}
-            ultimately show ?P1 and ?P2 by auto
-          qed
-          hence "(x - xa_max)@?za \<in> L\<^isub>1" using a_in by (auto elim:prefixE)
-          -- {* Now candidates @{text "?za"} and @{text "?zb"} have all the requred properteis. *}
-          with eq_xya have "(y - ya) @ ?za \<in> L\<^isub>1" 
-            by (auto simp:str_eq_def str_eq_rel_def)
-           with eq_z and b_in 
-          show ?thesis using that by blast
-        qed
-        -- {* 
-           @{text "?thesis"} can easily be shown using properties of @{text "za"} and @{text "zb"}:
-            *}
-        have "((y - ya) @ za) @ zb \<in> L\<^isub>1\<star>" using  l_za ls_zb by blast
-        with eq_zab show ?thesis by simp
-      qed
-      with h5 h6 show ?thesis 
-        by (drule_tac star_intro1, auto simp:strict_prefix_def elim:prefixE)
-    qed
-  } 
-  -- {* By instantiating the reasoning pattern just derived for both directions:*} 
-  from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
-  -- {* The thesis is proved as a trival consequence: *} 
-    show  ?thesis unfolding str_eq_def str_eq_rel_def by blast
-qed
-
-lemma -- {* The oringal version with less explicit details. *}
-  fixes v w
-  assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w"
-  shows "(v::string) \<approx>(L\<^isub>1\<star>) w"
-proof-
-    -- {* 
-    \begin{minipage}{0.8\textwidth}
-    According to the definition of @{text "\<approx>Lang"}, 
-    proving @{text "v \<approx>(L\<^isub>1\<star>) w"} amounts to
-    showing: for any string @{text "u"},
-    if @{text "v @ u \<in> (L\<^isub>1\<star>)"} then @{text "w @ u \<in> (L\<^isub>1\<star>)"} and vice versa.
-    The reasoning pattern for both directions are the same, as derived
-    in the following:
-    \end{minipage}
-    *}
-  { 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 = []")
-      -- {* 
-        The degenerated case when @{text "x"} is a null string is easy to prove:
-        *}
-      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
-        -- {*
-        \begin{minipage}{0.8\textwidth}
-        The case when @{text "x"} is not null, and
-        @{text "x @ z"} is in @{text "L\<^isub>1\<star>"}, 
-        \end{minipage}
-        *}
-      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
-        thus ?thesis using that 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 that 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 str_eq_rel_def)
-	with b_in have "((y - ya) @ za) @ b \<in> L\<^isub>1\<star>" by blast
-        with z_decom show ?thesis by auto 
-      qed
-      with h5 h6 show ?thesis 
-        by (drule_tac star_intro1, auto simp:strict_prefix_def elim:prefixE)
-    qed
-  } 
-  -- {* By instantiating the reasoning pattern just derived for both directions:*} 
-  from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
-  -- {* The thesis is proved as a trival consequence: *} 
-    show  ?thesis unfolding str_eq_def str_eq_rel_def by blast
-qed
-
-lemma quot_star_finiteI [intro]:
-  fixes L1::"lang"
-  assumes finite1: "finite (UNIV // \<approx>L1)"
-  shows "finite (UNIV // \<approx>(L1\<star>))"
-proof (rule_tac tag = "tag_str_STAR L1" in tag_finite_imageD)
-  show "\<And>x y. tag_str_STAR L1 x = tag_str_STAR L1 y \<Longrightarrow> x \<approx>(L1\<star>) y"
-    by (rule tag_str_STAR_injI)
-next
-  have *: "finite (Pow (UNIV // \<approx>L1))" 
-    using finite1 by auto
-  show "finite (range (tag_str_STAR L1))"
-    unfolding tag_str_STAR_def
-    apply(rule finite_subset[OF _ *])
-    unfolding quotient_def
-    by auto
-qed
-
-subsubsection{* The conclusion *}
-
-lemma rexp_imp_finite:
-  fixes r::"rexp"
-  shows "finite (UNIV // \<approx>(L r))"
-by (induct r) (auto)
-
 end