diff -r a8a442ba0dbf -r e93760534354 Myhill_2.thy
--- a/Myhill_2.thy	Thu May 12 05:55:05 2011 +0000
+++ b/Myhill_2.thy	Wed May 18 19:54:43 2011 +0000
@@ -1,64 +1,15 @@
 theory Myhill_2
-  imports Myhill_1 
-          Prefix_subtract
+  imports Myhill_1 Prefix_subtract
           "~~/src/HOL/Library/List_Prefix"
 begin
 
 section {* Direction @{text "regular language \<Rightarrow>finite partition"} *}
 
-subsection {* The scheme*}
-
-text {* 
-  The following convenient notation @{text "x \<approx>A y"} means:
-  string @{text "x"} and @{text "y"} are equivalent with respect to 
-  language @{text "A"}.
-  *}
-
 definition
   str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _")
 where
   "x \<approx>A y \<equiv> (x, y) \<in> (\<approx>A)"
 
-text {*
-  The main lemma (@{text "rexp_imp_finite"}) is proved by a structural
-  induction over regular expressions.  where base cases (cases for @{const
-  "NULL"}, @{const "EMPTY"}, @{const "CHAR"}) are quite straightforward to
-  proof. Real difficulty lies in inductive cases.  By inductive hypothesis,
-  languages defined by sub-expressions induce finite partitiions. Under such
-  hypothsis, we need to prove that the language defined by the composite
-  regular expression gives rise to finite partion.  The basic idea is to
-  attach a tag @{text "tag(x)"} to every string @{text "x"}. The tagging
-  fuction @{text "tag"} is carefully devised, which returns tags made of
-  equivalent classes of the partitions induced by subexpressoins, and
-  therefore has a finite range. Let @{text "Lang"} be the composite language,
-  it is proved that:
-  \begin{quote}
-  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.
-  \end{quote}
-  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 
    tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=")
 where
@@ -69,7 +20,6 @@
   shows "finite (UNIV // =tag=)"
 proof -
   let "?f" =  "\<lambda>X. tag ` X" and ?A = "(UNIV // =tag=)"
-    -- {* The finiteness of @{text "f"}-image is a consequence of @{text "rng_fnt"} *}
   have "finite (?f ` ?A)" 
   proof -
     have "range ?f \<subseteq> (Pow (range tag))" unfolding Pow_def by auto
@@ -82,25 +32,23 @@
     ultimately show "finite (?f ` ?A)" by (rule rev_finite_subset) 
   qed
   moreover
-    -- {* The injectivity of @{text "f"}-image follows from the definition of @{text "(=tag=)"} *}
   have "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"
-      then
-      obtain x y 
+      then obtain x y 
         where "x \<in> X" "y \<in> Y" "tag x = tag y"
         unfolding quotient_def Image_def image_def tag_eq_rel_def
         by (simp) (blast)
       with X_in Y_in 
       have "X = Y"
 	unfolding quotient_def tag_eq_rel_def by auto
-    } then show "inj_on ?f ?A" unfolding inj_on_def by auto
+    } 
+    then show "inj_on ?f ?A" unfolding inj_on_def by auto
   qed
-  ultimately 
-  show "finite (UNIV // =tag=)" by (rule finite_imageD)
+  ultimately show "finite (UNIV // =tag=)" by (rule finite_imageD)
 qed
 
 lemma refined_partition_finite:
@@ -142,7 +90,7 @@
 
 lemma tag_finite_imageD:
   assumes rng_fnt: "finite (range tag)" 
-  and same_tag_eqvt: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<approx>A n"
+  and same_tag_eqvt: "\<And>m n. tag m = tag n \<Longrightarrow> m \<approx>A n"
   shows "finite (UNIV // \<approx>A)"
 proof (rule_tac refined_partition_finite [of "=tag="])
   show "finite (UNIV // =tag=)" by (rule finite_eq_tag_rel[OF rng_fnt])
@@ -161,48 +109,23 @@
 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.
-  *}
+subsection {* The proof *}
 
 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
+  shows "UNIV // \<approx>{} = {UNIV}"
+unfolding quotient_def Image_def str_eq_rel_def by auto
 
 lemma quot_null_finiteI [intro]:
-  shows "finite ((UNIV // \<approx>{})::lang set)"
+  shows "finite (UNIV // \<approx>{})"
 unfolding quot_null_eq by simp
 
 
 subsubsection {* The base case for @{const "EMPTY"} *}
 
-
 lemma quot_empty_subset:
-  "UNIV // (\<approx>{[]}) \<subseteq> {{[]}, UNIV - {[]}}"
+  shows "UNIV // \<approx>{[]} \<subseteq> {{[]}, UNIV - {[]}}"
 proof
   fix x
   assume "x \<in> UNIV // \<approx>{[]}"
@@ -221,7 +144,7 @@
 qed
 
 lemma quot_empty_finiteI [intro]:
-  shows "finite (UNIV // (\<approx>{[]}))"
+  shows "finite (UNIV // \<approx>{[]})"
 by (rule finite_subset[OF quot_empty_subset]) (simp)
 
 
@@ -237,23 +160,24 @@
   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]"
+        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
+    } 
+    ultimately show ?thesis by blast
   qed
 qed
 
 lemma quot_char_finiteI [intro]:
-  shows "finite (UNIV // (\<approx>{[c]}))"
+  shows "finite (UNIV // \<approx>{[c]})"
 by (rule finite_subset[OF quot_char_subset]) (simp)
 
 
@@ -265,7 +189,6 @@
   "tag_str_ALT A B \<equiv> (\<lambda>x. (\<approx>A `` {x}, \<approx>B `` {x}))"
 
 lemma quot_union_finiteI [intro]:
-  fixes L1 L2::"lang"
   assumes finite1: "finite (UNIV // \<approx>A)"
   and     finite2: "finite (UNIV // \<approx>B)"
   shows "finite (UNIV // \<approx>(A \<union> B))"
@@ -283,140 +206,79 @@
     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}
-
-  *}
-
 definition 
   tag_str_SEQ :: "lang \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> (lang \<times> lang set)"
 where
   "tag_str_SEQ L1 L2 \<equiv>
-     (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - x'}) | x'.  x' \<le> x \<and> x' \<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.*}
+     (\<lambda>x. (\<approx>L1 `` {x}, {(\<approx>L2 `` {x - xa}) | xa.  xa \<le> x \<and> xa \<in> L1}))"
 
-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 Seq_in_cases:
+  assumes "x @ z \<in> A ;; B"
+  shows "(\<exists> x' \<le> x. x' \<in> A \<and> (x - x') @ z \<in> B) \<or> 
+         (\<exists> z' \<le> z. (x @ z') \<in> A \<and> (z - z') \<in> B)"
+using assms
+unfolding Seq_def prefix_def
+by (auto simp add: append_eq_append_conv2)
 
 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:*}
+  assumes eq_tag: "tag_str_SEQ A B x = tag_str_SEQ A B y" 
+  shows "x \<approx>(A ;; B) y"
+proof -
   { fix x y z
-    assume xz_in_seq: "x @ z \<in> L\<^isub>1 ;; L\<^isub>2"
-    and tag_xy: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
-    have"y @ z \<in> L\<^isub>1 ;; L\<^isub>2" 
-    proof-
-      -- {* 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"
+    assume xz_in_seq: "x @ z \<in> A ;; B"
+    and tag_xy: "tag_str_SEQ A B x = tag_str_SEQ A B y"
+    have"y @ z \<in> A ;; B" 
+    proof -
+      { (* first case with x' in A and (x - x') @ z in B *)
+        fix x'
+        assume h1: "x' \<le> x" and h2: "x' \<in> A" and h3: "(x - x') @ z \<in> B"
+        obtain y' 
+          where "y' \<le> y" 
+          and "y' \<in> A" 
+          and "(y - y') @ z \<in> B"
         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")
+          have "{\<approx>B `` {x - x'} |x'. x' \<le> x \<and> x' \<in> A} = 
+                {\<approx>B `` {y - y'} |y'. y' \<le> y \<and> y' \<in> A}" (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"
+          moreover 
+	  have "\<approx>B `` {x - x'} \<in> ?Left" using h1 h2 by auto
+          ultimately 
+	  have "\<approx>B `` {x - x'} \<in> ?Right" by simp
+          then obtain y' 
+            where eq_xy': "\<approx>B `` {x - x'} = \<approx>B `` {y - y'}" 
+            and pref_y': "y' \<le> y" and y'_in: "y' \<in> A"
             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 
+	  
+	  have "(x - x')  \<approx>B (y - y')" using eq_xy'
+            unfolding Image_def str_eq_rel_def str_eq_def by auto
+          with h3 have "(y - y') @ z \<in> B" 
+	    unfolding str_eq_rel_def str_eq_def by simp
+          with pref_y' y'_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
+	then have "y @ z \<in> A ;; B" by (erule_tac prefixE) (auto simp: Seq_def)
+      } 
+      moreover 
+      { (* second case with x @ z' in A and z - z' in B *)
+        fix z'
+        assume h1: "z' \<le> z" and h2: "(x @ z') \<in> A" and h3: "z - z' \<in> B"
+	 have "\<approx>A `` {x} = \<approx>A `` {y}" 
+           using tag_xy unfolding tag_str_SEQ_def by simp
+         with h2 have "y @ z' \<in> A"
             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)
+        with h1 h3 have "y @ z \<in> A ;; B"
+	  unfolding prefix_def Seq_def
+	  by (auto) (metis append_assoc)
       }
-      ultimately show ?thesis by blast
+      ultimately show "y @ z \<in> A ;; B" 
+	using Seq_in_cases [OF xz_in_seq] 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
+    show "x \<approx>(A ;; B) y" unfolding str_eq_def str_eq_rel_def by blast
 qed 
 
 lemma quot_seq_finiteI [intro]:
@@ -437,53 +299,13 @@
     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 \<equiv> (\<lambda>x. {\<approx>L1 `` {x - x'} | x'. x' < x \<and> x' \<in> L1\<star>})"
+  "tag_str_STAR L1 \<equiv> (\<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> 
@@ -513,7 +335,8 @@
 qed
 
 
-text {* The following is a technical lemma.which helps to show the range finiteness of tag function. *}
+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}")
@@ -521,46 +344,26 @@
 
 
 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"
+  shows "v \<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 
@@ -570,12 +373,6 @@
         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)"
@@ -588,47 +385,25 @@
         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)
+              using append_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"
@@ -639,24 +414,19 @@
                 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)
+                  using b_eqs b_in np h1 by (simp add:diff_diff_append)
                 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
@@ -664,123 +434,11 @@
         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)
@@ -803,76 +461,9 @@
   shows "finite (UNIV // \<approx>(L r))"
 by (induct r) (auto)
 
+
 theorem Myhill_Nerode:
   shows "(\<exists>r::rexp. A = L r) \<longleftrightarrow> finite (UNIV // \<approx>A)"
-using Myhill_Nerode1 Myhill_Nerode2 by metis
-
-(*
-section {* Closure properties *}
-
-abbreviation
-  reg :: "lang \<Rightarrow> bool"
-where
-  "reg A \<equiv> \<exists>r::rexp. A = L r"
-
-
-
-lemma closure_union[intro]:
-  assumes "reg A" "reg B" 
-  shows "reg (A \<union> B)"
-using assms
-apply(auto)
-apply(rule_tac x="ALT r ra" in exI)
-apply(auto)
-done
-
-lemma closure_seq[intro]:
-  assumes "reg A" "reg B" 
-  shows "reg (A ;; B)"
-using assms
-apply(auto)
-apply(rule_tac x="SEQ r ra" in exI)
-apply(auto)
-done
-
-lemma closure_star[intro]:
-  assumes "reg A"
-  shows "reg (A\<star>)"
-using assms
-apply(auto)
-apply(rule_tac x="STAR r" in exI)
-apply(auto)
-done
-
-lemma closure_complement[intro]:
-  assumes "reg A"
-  shows "reg (- A)"
-using assms
-unfolding Myhill_Nerode
-unfolding str_eq_rel_def
-by auto
-
-lemma closure_difference[intro]:
-  assumes "reg A" "reg B" 
-  shows "reg (A - B)"
-proof -
-  have "A - B = - ((- A) \<union> B)" by blast
-  moreover
-  have "reg (- ((- A) \<union> B))" 
-    using assms by blast
-  ultimately show "reg (A - B)" by simp
-qed
-
-lemma closure_intersection[intro]:
-  assumes "reg A" "reg B" 
-  shows "reg (A \<inter> B)"
-proof -
-  have "A \<inter> B = - ((- A) \<union> (- B))" by blast
-  moreover
-  have "reg (- ((- A) \<union> (- B)))" 
-    using assms by blast
-  ultimately show "reg (A \<inter> B)" by simp
-qed
-*)
+using Myhill_Nerode1 Myhill_Nerode2 by auto
 
 end