theory Myhill_2 imports Myhill_1 Prefix_subtract "~~/src/HOL/Library/List_Prefix"beginsection {* 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)"lemma str_eq_def2: shows "\<approx>A = {(x, y) | x y. x \<approx>A y}"unfolding str_eq_defby simpdefinition tag_eq_rel :: "(string \<Rightarrow> 'b) \<Rightarrow> (string \<times> string) set" ("=_=")where "=tag= \<equiv> {(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)qedlemma 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)qedlemma 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 autonext show "equiv UNIV =tag=" unfolding equiv_def tag_eq_rel_def refl_on_def sym_def trans_def by autonext show "equiv UNIV (\<approx>A)" unfolding equiv_def str_eq_rel_def sym_def refl_on_def trans_def by blastqedsubsection {* 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 autolemma quot_null_finiteI [intro]: shows "finite (UNIV // \<approx>{})"unfolding quot_null_eq by simpsubsubsection {* 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 qedqedlemma 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 qedqedlemma 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]: 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 autoqedsubsubsection {* 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 \<cdot> 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 assmsunfolding Seq_def prefix_defby (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 \<cdot> B) y"proof - { fix x y z assume xz_in_seq: "x @ z \<in> A \<cdot> B" and tag_xy: "tag_str_SEQ A B x = tag_str_SEQ A B y" have"y @ z \<in> A \<cdot> 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 \<cdot> 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 \<cdot> B" unfolding prefix_def Seq_def by (auto) (metis append_assoc) } ultimately show "y @ z \<in> A \<cdot> 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 \<cdot> B) y" unfolding str_eq_def str_eq_rel_def by blastqed 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 \<cdot> 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 \<cdot> 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 autoqedsubsubsection {* 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 simpnext 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 qedqedtext {* 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 blastqedlemma quot_star_finiteI [intro]: assumes finite1: "finite (UNIV // \<approx>A)" shows "finite (UNIV // \<approx>(A\<star>))"proof (rule_tac tag = "tag_str_STAR A" in tag_finite_imageD) show "\<And>x y. tag_str_STAR A x = tag_str_STAR A y \<Longrightarrow> x \<approx>(A\<star>) y" by (rule tag_str_STAR_injI)next have *: "finite (Pow (UNIV // \<approx>A))" using finite1 by auto show "finite (range (tag_str_STAR A))" unfolding tag_str_STAR_def apply(rule finite_subset[OF _ *]) unfolding quotient_def by autoqedsubsubsection{* The conclusion *}lemma Myhill_Nerode2: shows "finite (UNIV // \<approx>(L_rexp r))"by (induct r) (auto)theorem Myhill_Nerode: shows "(\<exists>r. A = L_rexp r) \<longleftrightarrow> finite (UNIV // \<approx>A)"using Myhill_Nerode1 Myhill_Nerode2 by autoend