--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Theories/Myhill_2.thy Wed Mar 23 12:17:30 2011 +0000
@@ -0,0 +1,470 @@
+theory Myhill_2
+ imports Myhill_1 Prefix_subtract
+ "~~/src/HOL/Library/List_Prefix"
+begin
+
+section {* Direction @{text "regular language \<Rightarrow>finite partition"} *}
+
+definition
+ str_eq :: "string \<Rightarrow> lang \<Rightarrow> string \<Rightarrow> bool" ("_ \<approx>_ _")
+where
+ "x \<approx>A y \<equiv> (x, y) \<in> (\<approx>A)"
+
+definition
+ tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=")
+where
+ "=tag= \<equiv> {(x, y) | x y. tag x = tag y}"
+
+lemma finite_eq_tag_rel:
+ assumes rng_fnt: "finite (range tag)"
+ shows "finite (UNIV // =tag=)"
+proof -
+ let "?f" = "\<lambda>X. tag ` X" and ?A = "(UNIV // =tag=)"
+ have "finite (?f ` ?A)"
+ proof -
+ have "range ?f \<subseteq> (Pow (range tag))" unfolding Pow_def by auto
+ moreover
+ have "finite (Pow (range tag))" using rng_fnt by simp
+ ultimately
+ have "finite (range ?f)" unfolding image_def by (blast intro: finite_subset)
+ moreover
+ have "?f ` ?A \<subseteq> range ?f" by auto
+ ultimately show "finite (?f ` ?A)" by (rule rev_finite_subset)
+ qed
+ moreover
+ 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
+ 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
+ qed
+ ultimately show "finite (UNIV // =tag=)" by (rule finite_imageD)
+qed
+
+lemma refined_partition_finite:
+ assumes fnt: "finite (UNIV // R1)"
+ and refined: "R1 \<subseteq> R2"
+ and eq1: "equiv UNIV R1" and eq2: "equiv UNIV R2"
+ shows "finite (UNIV // R2)"
+proof -
+ let ?f = "\<lambda>X. {R1 `` {x} | x. x \<in> X}"
+ and ?A = "UNIV // R2" and ?B = "UNIV // R1"
+ have "?f ` ?A \<subseteq> Pow ?B"
+ unfolding image_def Pow_def quotient_def by auto
+ moreover
+ have "finite (Pow ?B)" using fnt by simp
+ ultimately
+ have "finite (?f ` ?A)" by (rule finite_subset)
+ moreover
+ have "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"
+ from quotientE [OF X_in]
+ obtain x where "X = R2 `` {x}" by blast
+ with equiv_class_self[OF eq2] have x_in: "x \<in> X" by simp
+ then have "R1 ``{x} \<in> ?f X" by auto
+ with eq_f have "R1 `` {x} \<in> ?f Y" by simp
+ then obtain y
+ where y_in: "y \<in> Y" and eq_r1_xy: "R1 `` {x} = R1 `` {y}" by auto
+ with eq_equiv_class[OF _ eq1]
+ have "(x, y) \<in> R1" by blast
+ with refined have "(x, y) \<in> R2" by auto
+ with quotient_eqI [OF eq2 X_in Y_in x_in y_in]
+ have "X = Y" .
+ }
+ then show "inj_on ?f ?A" unfolding inj_on_def by blast
+ qed
+ ultimately show "finite (UNIV // R2)" by (rule finite_imageD)
+qed
+
+lemma tag_finite_imageD:
+ assumes rng_fnt: "finite (range tag)"
+ 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])
+next
+ from same_tag_eqvt
+ show "=tag= \<subseteq> \<approx>A" unfolding tag_eq_rel_def str_eq_def
+ by auto
+next
+ show "equiv UNIV =tag="
+ unfolding equiv_def tag_eq_rel_def refl_on_def sym_def trans_def
+ by auto
+next
+ show "equiv UNIV (\<approx>A)"
+ unfolding equiv_def str_eq_rel_def sym_def refl_on_def trans_def
+ by blast
+qed
+
+
+subsection {* The proof *}
+
+subsubsection {* The base case for @{const "NULL"} *}
+
+lemma quot_null_eq:
+ shows "UNIV // \<approx>{} = {UNIV}"
+unfolding quotient_def Image_def str_eq_rel_def by auto
+
+lemma quot_null_finiteI [intro]:
+ shows "finite (UNIV // \<approx>{})"
+unfolding quot_null_eq by simp
+
+
+subsubsection {* The base case for @{const "EMPTY"} *}
+
+lemma quot_empty_subset:
+ shows "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 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))"
+proof (rule_tac tag = "tag_str_ALT A B" in tag_finite_imageD)
+ have "finite ((UNIV // \<approx>A) \<times> (UNIV // \<approx>B))"
+ using finite1 finite2 by auto
+ then show "finite (range (tag_str_ALT A B))"
+ unfolding tag_str_ALT_def quotient_def
+ by (rule rev_finite_subset) (auto)
+next
+ show "\<And>x y. tag_str_ALT A B x = tag_str_ALT A B y \<Longrightarrow> x \<approx>(A \<union> B) y"
+ unfolding tag_str_ALT_def
+ unfolding str_eq_def
+ unfolding str_eq_rel_def
+ by auto
+qed
+
+
+subsubsection {* The inductive case for @{text "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 - xa}) | xa. xa \<le> x \<and> xa \<in> L1}))"
+
+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:
+ 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> 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 -
+ 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
+ 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
+
+ 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
+ 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
+ with h1 h3 have "y @ z \<in> A ;; B"
+ unfolding prefix_def Seq_def
+ by (auto) (metis append_assoc)
+ }
+ ultimately show "y @ z \<in> A ;; B"
+ using Seq_in_cases [OF xz_in_seq] by blast
+ qed
+ }
+ from this [OF _ eq_tag] and this [OF _ eq_tag [THEN sym]]
+ show "x \<approx>(A ;; B) y" 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"} *}
+
+definition
+ tag_str_STAR :: "lang \<Rightarrow> string \<Rightarrow> lang set"
+where
+ "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>
+ (\<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:
+ assumes eq_tag: "tag_str_STAR L\<^isub>1 v = tag_str_STAR L\<^isub>1 w"
+ shows "v \<approx>(L\<^isub>1\<star>) w"
+proof-
+ { 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 = []")
+ 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
+ case False
+ 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> 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
+ 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
+ have "(y - ya) @ z \<in> L\<^isub>1\<star>"
+ proof-
+ 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 -
+ 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
+ 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 -
+ 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 append_eq_dest[OF ab_max] by (auto simp:strict_prefix_def)
+ moreover {
+ 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_append)
+ moreover have "\<not> (length ?xa_max' \<le> length xa_max)"
+ using a_neq by simp
+ ultimately show ?thesis using h4 by blast
+ qed }
+ ultimately show ?P1 and ?P2 by auto
+ qed
+ hence "(x - xa_max)@?za \<in> L\<^isub>1" using a_in by (auto elim:prefixE)
+ 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
+ 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
+ }
+ 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_star_finiteI [intro]:
+ 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 Myhill_Nerode2:
+ fixes r::"rexp"
+ 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 auto
+
+end