--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Myhill_2.thy Thu Feb 03 12:44:46 2011 +0000
@@ -0,0 +1,900 @@
+theory Myhill_2
+ imports Myhill_1
+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}
+
+ *}
+
+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