--- 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