--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Myhill.thy Tue Dec 14 14:31:31 2010 +0000
@@ -0,0 +1,1421 @@
+theory MyhillNerode
+ imports "Main" "List_Prefix"
+begin
+
+text {* sequential composition of languages *}
+definition
+ Seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100)
+where
+ "L1 ;; L2 = {s1 @ s2 | s1 s2. s1 \<in> L1 \<and> s2 \<in> L2}"
+
+inductive_set
+ Star :: "string set \<Rightarrow> string set" ("_\<star>" [101] 102)
+ for L :: "string set"
+where
+ start[intro]: "[] \<in> L\<star>"
+| step[intro]: "\<lbrakk>s1 \<in> L; s2 \<in> L\<star>\<rbrakk> \<Longrightarrow> s1@s2 \<in> L\<star>"
+
+theorem ardens_revised:
+ assumes nemp: "[] \<notin> A"
+ shows "(X = X ;; A \<union> B) \<longleftrightarrow> (X = B ;; A\<star>)"
+proof
+ assume eq: "X = B ;; A\<star>"
+ have "A\<star> = {[]} \<union> A\<star> ;; A" sorry
+ then have "B ;; A\<star> = B ;; ({[]} \<union> A\<star> ;; A)" unfolding Seq_def by simp
+ also have "\<dots> = B \<union> B ;; (A\<star> ;; A)" unfolding Seq_def by auto
+ also have "\<dots> = B \<union> (B ;; A\<star>) ;; A" unfolding Seq_def
+ by (auto) (metis append_assoc)+
+ finally show "X = X ;; A \<union> B" using eq by auto
+next
+ assume "X = X ;; A \<union> B"
+ then have "B \<subseteq> X" "X ;; A \<subseteq> X" by auto
+ show "X = B ;; A\<star>" sorry
+qed
+
+datatype rexp =
+ NULL
+| EMPTY
+| CHAR char
+| SEQ rexp rexp
+| ALT rexp rexp
+| STAR rexp
+
+consts L:: "'a \<Rightarrow> string set"
+
+overloading L_rexp \<equiv> "L:: rexp \<Rightarrow> string set"
+begin
+
+fun
+ L_rexp :: "rexp \<Rightarrow> string set"
+where
+ "L_rexp (NULL) = {}"
+ | "L_rexp (EMPTY) = {[]}"
+ | "L_rexp (CHAR c) = {[c]}"
+ | "L_rexp (SEQ r1 r2) = (L_rexp r1) ;; (L_rexp r2)"
+ | "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+ | "L_rexp (STAR r) = (L_rexp r)\<star>"
+end
+
+definition
+ folds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b"
+where
+ "folds f z S \<equiv> SOME x. fold_graph f z S x"
+
+lemma folds_simp_null [simp]:
+ "finite rs \<Longrightarrow> x \<in> L (folds ALT NULL rs) \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> L r)"
+apply (simp add: folds_def)
+apply (rule someI2_ex)
+apply (erule finite_imp_fold_graph)
+apply (erule fold_graph.induct)
+apply (auto)
+done
+
+lemma [simp]:
+ shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+by simp
+
+definition
+ str_eq ("_ \<approx>_ _")
+where
+ "x \<approx>Lang y \<equiv> (\<forall>z. x @ z \<in> Lang \<longleftrightarrow> y @ z \<in> Lang)"
+
+definition
+ str_eq_rel ("\<approx>_")
+where
+ "\<approx>Lang \<equiv> {(x, y). x \<approx>Lang y}"
+
+
+
+section {* finite \<Rightarrow> regular *}
+
+definition
+ transitions :: "string set \<Rightarrow> string set \<Rightarrow> rexp set" ("_\<Turnstile>\<Rightarrow>_")
+where
+ "Y \<Turnstile>\<Rightarrow> X \<equiv> {CHAR c | c. Y ;; {[c]} \<subseteq> X}"
+
+definition
+ transitions_rexp ("_ \<turnstile>\<rightarrow> _")
+where
+ "Y \<turnstile>\<rightarrow> X \<equiv> folds ALT NULL (Y \<Turnstile>\<Rightarrow>X)"
+
+definition
+ "init_rhs CS X \<equiv> if X = {[]}
+ then {({[]}, EMPTY)}
+ else if ([] \<in> X)
+ then insert ({[]}, EMPTY) {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}
+ else {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}"
+
+overloading L_rhs \<equiv> "L:: (string set \<times> rexp) set \<Rightarrow> string set"
+begin
+fun L_rhs:: "(string set \<times> rexp) set \<Rightarrow> string set"
+where
+ "L_rhs rhs = \<Union> {(Y;; L r) | Y r . (Y, r) \<in> rhs}"
+end
+
+definition
+ "eqs CS \<equiv> (\<Union>X \<in> CS. {(X, init_rhs CS X)})"
+
+lemma [simp]:
+ shows "finite (Y \<Turnstile>\<Rightarrow> X)"
+unfolding transitions_def
+by auto
+
+lemma defined_by_str:
+ assumes "s \<in> X"
+ and "X \<in> UNIV // (\<approx>Lang)"
+ shows "X = (\<approx>Lang) `` {s}"
+using assms
+unfolding quotient_def Image_def
+unfolding str_eq_rel_def str_eq_def
+by auto
+
+
+
+(************ arden's lemma variation ********************)
+definition
+ "rexp_of rhs X \<equiv> folds ALT NULL {r. (X, r) \<in> rhs}"
+
+definition
+ "arden_variate X rhs \<equiv> {(Y, SEQ r (STAR (rexp_of rhs X)))| Y r. (Y, r) \<in> rhs \<and> Y \<noteq> X}"
+
+(************* rhs/equations property **************)
+
+definition
+ "distinct_equas ES \<equiv> \<forall> X rhs rhs'. (X, rhs) \<in> ES \<and> (X, rhs') \<in> ES \<longrightarrow> rhs = rhs'"
+
+(*********** substitution of ES *************)
+
+text {* rhs_subst rhs X xrhs: substitude all occurence of X in rhs with xrhs *}
+definition
+ "rhs_subst rhs X xrhs \<equiv> {(Y, r) | Y r. Y \<noteq> X \<and> (Y, r) \<in> rhs} \<union>
+ {(X, SEQ r\<^isub>1 r\<^isub>2 ) | r\<^isub>1 r\<^isub>2. (X, r\<^isub>1) \<in> xrhs \<and> (X, r\<^isub>2) \<in> rhs}"
+
+definition
+ "eqs_subst ES X xrhs \<equiv> {(Y, rhs_subst yrhs X xrhs) | Y yrhs. (Y, yrhs) \<in> ES}"
+
+text {*
+ Inv: Invairance of the equation-system, during the decrease of the equation-system, Inv holds.
+*}
+
+definition
+ "ardenable ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> ([] \<notin> L (rexp_of rhs X)) \<and> X = L rhs"
+
+definition
+ "non_empty ES \<equiv> \<forall> X rhs. (X, rhs) \<in> ES \<longrightarrow> X \<noteq> {}"
+
+definition
+ "self_contained ES \<equiv> \<forall> X xrhs. (X, xrhs) \<in> ES
+ \<longrightarrow> (\<forall> Y r.(Y, r) \<in> xrhs \<and> Y \<noteq> {[]} \<longrightarrow> (\<exists> yrhs. (Y, yrhs) \<in> ES))"
+
+definition
+ "Inv ES \<equiv> finite ES \<and> distinct_equas ES \<and> ardenable ES \<and> non_empty ES \<and> self_contained ES"
+
+lemma wf_iter [rule_format]:
+ fixes f
+ assumes step: "\<And> e. \<lbrakk>P e; \<not> Q e\<rbrakk> \<Longrightarrow> (\<exists> e'. P e' \<and> (f(e'), f(e)) \<in> less_than)"
+ shows pe: "P e \<longrightarrow> (\<exists> e'. P e' \<and> Q e')"
+proof(induct e rule: wf_induct
+ [OF wf_inv_image[OF wf_less_than, where f = "f"]], clarify)
+ fix x
+ assume h [rule_format]:
+ "\<forall>y. (y, x) \<in> inv_image less_than f \<longrightarrow> P y \<longrightarrow> (\<exists>e'. P e' \<and> Q e')"
+ and px: "P x"
+ show "\<exists>e'. P e' \<and> Q e'"
+ proof(cases "Q x")
+ assume "Q x" with px show ?thesis by blast
+ next
+ assume nq: "\<not> Q x"
+ from step [OF px nq]
+ obtain e' where pe': "P e'" and ltf: "(f e', f x) \<in> less_than" by auto
+ show ?thesis
+ proof(rule h)
+ from ltf show "(e', x) \<in> inv_image less_than f"
+ by (simp add:inv_image_def)
+ next
+ from pe' show "P e'" .
+ qed
+ qed
+qed
+
+text {* ******BEGIN: proving the initial equation-system satisfies Inv ****** *}
+
+lemma init_ES_satisfy_Inv:
+ assumes finite_CS: "finite (UNIV // (\<approx>Lang))"
+ and X_in_eq_cls: "X \<in> (UNIV // (\<approx>Lang))"
+ shows "Inv (eqs (UNIV // (\<approx>Lang)))"
+proof -
+ have "finite (eqs (UNIV // (\<approx>Lang)))" using finite_CS
+ by (auto simp add:eqs_def)
+ moreover have "distinct_equas (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:distinct_equas_def eqs_def)
+ moreover have "ardenable (eqs (UNIV // (\<approx>Lang)))"
+ proof-
+ have "\<And> X rhs. (X, rhs) \<in> (eqs (UNIV // (\<approx>Lang))) \<Longrightarrow> ([] \<notin> L (rexp_of rhs X))"
+ proof
+ apply (auto simp:eqs_def rexp_of_def)
+ sorry
+ moreover have "\<forall> X rhs. (X, rhs) \<in> (eqs (UNIV // (\<approx>Lang))) \<longrightarrow> X = L rhs"
+ sorry
+ ultimately show ?thesis by (simp add:ardenable_def)
+ qed
+ moreover have "non_empty (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:non_empty_def eqs_def quotient_def Image_def str_eq_rel_def str_eq_def)
+ moreover have "self_contained (eqs (UNIV // (\<approx>Lang)))"
+ by (auto simp:self_contained_def eqs_def init_rhs_def)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+
+text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
+
+lemma iteration_step:
+ assumes Inv_ES: "Inv ES"
+ and X_in_ES: "\<exists> xrhs. (X, xrhs) \<in> ES"
+ and not_T: "card ES > 1"
+ shows "(\<exists> ES' xrhs'. Inv ES' \<and> (card ES', card ES) \<in> less_than \<and> (X, xrhs') \<in> ES')"
+proof -
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+lemma distinct_rhs_equations:
+ "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> distinct_rhs xrhs"
+by (auto simp: equations_def equation_rhs_def distinct_rhs_def empty_rhs_def dest:no_two_cls_inters)
+
+lemma every_nonempty_eqclass_has_strings:
+ "\<lbrakk>X \<in> (UNIV Quo Lang); X \<noteq> {[]}\<rbrakk> \<Longrightarrow> \<exists> clist. clist \<in> X \<and> clist \<noteq> []"
+by (auto simp:quot_def equiv_class_def equiv_str_def)
+
+lemma every_eqclass_is_derived_from_empty:
+ assumes not_empty: "X \<noteq> {[]}"
+ shows "X \<in> (UNIV Quo Lang) \<Longrightarrow> \<exists> clist. {[]};{clist} \<subseteq> X \<and> clist \<noteq> []"
+using not_empty
+apply (drule_tac every_nonempty_eqclass_has_strings, simp)
+by (auto intro:exI[where x = clist] simp:lang_seq_def)
+
+lemma equiv_str_in_CS:
+ "\<lbrakk>clist\<rbrakk>Lang \<in> (UNIV Quo Lang)"
+by (auto simp:quot_def)
+
+lemma has_str_imp_defined_by_str:
+ "\<lbrakk>str \<in> X; X \<in> UNIV Quo Lang\<rbrakk> \<Longrightarrow> X = \<lbrakk>str\<rbrakk>Lang"
+by (auto simp:quot_def equiv_class_def equiv_str_def)
+
+lemma every_eqclass_has_ascendent:
+ assumes has_str: "clist @ [c] \<in> X"
+ and in_CS: "X \<in> UNIV Quo Lang"
+ shows "\<exists> Y. Y \<in> UNIV Quo Lang \<and> Y-c\<rightarrow>X \<and> clist \<in> Y" (is "\<exists> Y. ?P Y")
+proof -
+ have "?P (\<lbrakk>clist\<rbrakk>Lang)"
+ proof -
+ have "\<lbrakk>clist\<rbrakk>Lang \<in> UNIV Quo Lang"
+ by (simp add:quot_def, rule_tac x = clist in exI, simp)
+ moreover have "\<lbrakk>clist\<rbrakk>Lang-c\<rightarrow>X"
+ proof -
+ have "X = \<lbrakk>(clist @ [c])\<rbrakk>Lang" using has_str in_CS
+ by (auto intro!:has_str_imp_defined_by_str)
+ moreover have "\<forall> sl. sl \<in> \<lbrakk>clist\<rbrakk>Lang \<longrightarrow> sl @ [c] \<in> \<lbrakk>(clist @ [c])\<rbrakk>Lang"
+ by (auto simp:equiv_class_def equiv_str_def)
+ ultimately show ?thesis unfolding CT_def lang_seq_def
+ by auto
+ qed
+ moreover have "clist \<in> \<lbrakk>clist\<rbrakk>Lang"
+ by (auto simp:equiv_str_def equiv_class_def)
+ ultimately show "?P (\<lbrakk>clist\<rbrakk>Lang)" by simp
+ qed
+ thus ?thesis by blast
+qed
+
+lemma finite_charset_rS:
+ "finite {CHAR c |c. Y-c\<rightarrow>X}"
+by (rule_tac A = UNIV and f = CHAR in finite_surj, auto)
+
+lemma l_eq_r_in_equations:
+ assumes X_in_equas: "(X, xrhs) \<in> equations (UNIV Quo Lang)"
+ shows "X = L xrhs"
+proof (cases "X = {[]}")
+ case True
+ thus ?thesis using X_in_equas
+ by (simp add:equations_def equation_rhs_def lang_seq_def)
+next
+ case False
+ show ?thesis
+ proof
+ show "X \<subseteq> L xrhs"
+ proof
+ fix x
+ assume "(1)": "x \<in> X"
+ show "x \<in> L xrhs"
+ proof (cases "x = []")
+ assume empty: "x = []"
+ hence "x \<in> L (empty_rhs X)" using "(1)"
+ by (auto simp:empty_rhs_def lang_seq_def)
+ thus ?thesis using X_in_equas False empty "(1)"
+ unfolding equations_def equation_rhs_def by auto
+ next
+ assume not_empty: "x \<noteq> []"
+ hence "\<exists> clist c. x = clist @ [c]" by (case_tac x rule:rev_cases, auto)
+ then obtain clist c where decom: "x = clist @ [c]" by blast
+ moreover have "\<And> Y. \<lbrakk>Y \<in> UNIV Quo Lang; Y-c\<rightarrow>X; clist \<in> Y\<rbrakk>
+ \<Longrightarrow> [c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
+ proof -
+ fix Y
+ assume Y_is_eq_cl: "Y \<in> UNIV Quo Lang"
+ and Y_CT_X: "Y-c\<rightarrow>X"
+ and clist_in_Y: "clist \<in> Y"
+ with finite_charset_rS
+ show "[c] \<in> L (folds ALT NULL {CHAR c |c. Y-c\<rightarrow>X})"
+ by (auto simp :fold_alt_null_eqs)
+ qed
+ hence "\<exists>Xa. Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})"
+ using X_in_equas False not_empty "(1)" decom
+ by (auto dest!:every_eqclass_has_ascendent simp:equations_def equation_rhs_def lang_seq_def)
+ then obtain Xa where
+ "Xa \<in> UNIV Quo Lang \<and> clist @ [c] \<in> Xa ; L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})" by blast
+ hence "x \<in> L {(S, folds ALT NULL {CHAR c |c. S-c\<rightarrow>X}) |S. S \<in> UNIV Quo Lang}"
+ using X_in_equas "(1)" decom
+ by (auto simp add:equations_def equation_rhs_def intro!:exI[where x = Xa])
+ thus "x \<in> L xrhs" using X_in_equas False not_empty unfolding equations_def equation_rhs_def
+ by auto
+ qed
+ qed
+ next
+ show "L xrhs \<subseteq> X"
+ proof
+ fix x
+ assume "(2)": "x \<in> L xrhs"
+ have "(2_1)": "\<And> s1 s2 r Xa. \<lbrakk>s1 \<in> Xa; s2 \<in> L (folds ALT NULL {CHAR c |c. Xa-c\<rightarrow>X})\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
+ using finite_charset_rS
+ by (auto simp:CT_def lang_seq_def fold_alt_null_eqs)
+ have "(2_2)": "\<And> s1 s2 Xa r.\<lbrakk>s1 \<in> Xa; s2 \<in> L r; (Xa, r) \<in> empty_rhs X\<rbrakk> \<Longrightarrow> s1 @ s2 \<in> X"
+ by (simp add:empty_rhs_def split:if_splits)
+ show "x \<in> X" using X_in_equas False "(2)"
+ by (auto intro:"(2_1)" "(2_2)" simp:equations_def equation_rhs_def lang_seq_def)
+ qed
+ qed
+qed
+
+
+
+lemma no_EMPTY_equations:
+ "(X, xrhs) \<in> equations CS \<Longrightarrow> no_EMPTY_rhs xrhs"
+apply (clarsimp simp add:equations_def equation_rhs_def)
+apply (simp add:no_EMPTY_rhs_def empty_rhs_def, auto)
+apply (subgoal_tac "finite {CHAR c |c. Xa-c\<rightarrow>X}", drule_tac x = "[]" in fold_alt_null_eqs, clarsimp, rule finite_charset_rS)+
+done
+
+lemma init_ES_satisfy_ardenable:
+ "(X, xrhs) \<in> equations (UNIV Quo Lang) \<Longrightarrow> ardenable (X, xrhs)"
+ unfolding ardenable_def
+ by (auto intro:distinct_rhs_equations no_EMPTY_equations simp:l_eq_r_in_equations simp del:L_rhs.simps)
+
+lemma init_ES_satisfy_Inv:
+ assumes finite_CS: "finite (UNIV Quo Lang)"
+ and X_in_eq_cls: "X \<in> UNIV Quo Lang"
+ shows "Inv X (equations (UNIV Quo Lang))"
+proof -
+ have "finite (equations (UNIV Quo Lang))" using finite_CS
+ by (auto simp:equations_def)
+ moreover have "\<exists>rhs. (X, rhs) \<in> equations (UNIV Quo Lang)" using X_in_eq_cls
+ by (simp add:equations_def)
+ moreover have "distinct_equas (equations (UNIV Quo Lang))"
+ by (auto simp:distinct_equas_def equations_def)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow>
+ rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equations (UNIV Quo Lang)))"
+ apply (simp add:left_eq_cls_def equations_def rhs_eq_cls_def equation_rhs_def)
+ by (auto simp:empty_rhs_def split:if_splits)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> X \<noteq> {}"
+ by (clarsimp simp:equations_def empty_notin_CS intro:classical)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equations (UNIV Quo Lang) \<longrightarrow> ardenable (X, xrhs)"
+ by (auto intro!:init_ES_satisfy_ardenable)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+
+text {* *********** END: proving the initial equation-system satisfies Inv ******* *}
+
+
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+
+text {* ****** BEGIN: proving every equation-system's iteration step satisfies Inv ***** *}
+
+lemma not_T_aux: "\<lbrakk>\<not> TCon (insert a A); x = a\<rbrakk>
+ \<Longrightarrow> \<exists>y. x \<noteq> y \<and> y \<in> insert a A "
+apply (case_tac "insert a A = {a}")
+by (auto simp:TCon_def)
+
+lemma not_T_atleast_2[rule_format]:
+ "finite S \<Longrightarrow> \<forall> x. x \<in> S \<and> (\<not> TCon S)\<longrightarrow> (\<exists> y. x \<noteq> y \<and> y \<in> S)"
+apply (erule finite.induct, simp)
+apply (clarify, case_tac "x = a")
+by (erule not_T_aux, auto)
+
+lemma exist_another_equa:
+ "\<lbrakk>\<not> TCon ES; finite ES; distinct_equas ES; (X, rhl) \<in> ES\<rbrakk> \<Longrightarrow> \<exists> Y yrhl. (Y, yrhl) \<in> ES \<and> X \<noteq> Y"
+apply (drule not_T_atleast_2, simp)
+apply (clarsimp simp:distinct_equas_def)
+apply (drule_tac x= X in spec, drule_tac x = rhl in spec, drule_tac x = b in spec)
+by auto
+
+lemma Inv_mono_with_lambda:
+ assumes Inv_ES: "Inv X ES"
+ and X_noteq_Y: "X \<noteq> {[]}"
+ shows "Inv X (ES - {({[]}, yrhs)})"
+proof -
+ have "finite (ES - {({[]}, yrhs)})" using Inv_ES
+ by (simp add:Inv_def)
+ moreover have "\<exists>rhs. (X, rhs) \<in> ES - {({[]}, yrhs)}" using Inv_ES X_noteq_Y
+ by (simp add:Inv_def)
+ moreover have "distinct_equas (ES - {({[]}, yrhs)})" using Inv_ES X_noteq_Y
+ apply (clarsimp simp:Inv_def distinct_equas_def)
+ by (drule_tac x = Xa in spec, simp)
+ moreover have "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ ardenable (X, xrhs) \<and> X \<noteq> {}" using Inv_ES
+ by (clarify, simp add:Inv_def)
+ moreover
+ have "insert {[]} (left_eq_cls (ES - {({[]}, yrhs)})) = insert {[]} (left_eq_cls ES)"
+ by (auto simp:left_eq_cls_def)
+ hence "\<forall>X xrhs.(X, xrhs) \<in> ES - {({[]}, yrhs)} \<longrightarrow>
+ rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (ES - {({[]}, yrhs)}))"
+ using Inv_ES by (auto simp:Inv_def)
+ ultimately show ?thesis by (simp add:Inv_def)
+qed
+
+lemma non_empty_card_prop:
+ "finite ES \<Longrightarrow> \<forall>e. e \<in> ES \<longrightarrow> card ES - Suc 0 < card ES"
+apply (erule finite.induct, simp)
+apply (case_tac[!] "a \<in> A")
+by (auto simp:insert_absorb)
+
+lemma ardenable_prop:
+ assumes not_lambda: "Y \<noteq> {[]}"
+ and ardable: "ardenable (Y, yrhs)"
+ shows "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" (is "\<exists> yrhs'. ?P yrhs'")
+proof (cases "(\<exists> reg. (Y, reg) \<in> yrhs)")
+ case True
+ thus ?thesis
+ proof
+ fix reg
+ assume self_contained: "(Y, reg) \<in> yrhs"
+ show ?thesis
+ proof -
+ have "?P (arden_variate Y reg yrhs)"
+ proof -
+ have "Y = L (arden_variate Y reg yrhs)"
+ using self_contained not_lambda ardable
+ by (rule_tac arden_variate_valid, simp_all add:ardenable_def)
+ moreover have "distinct_rhs (arden_variate Y reg yrhs)"
+ using ardable
+ by (auto simp:distinct_rhs_def arden_variate_def seq_rhs_r_def del_x_paired_def ardenable_def)
+ moreover have "rhs_eq_cls (arden_variate Y reg yrhs) = rhs_eq_cls yrhs - {Y}"
+ proof -
+ have "\<And> rhs r. rhs_eq_cls (seq_rhs_r rhs r) = rhs_eq_cls rhs"
+ apply (auto simp:rhs_eq_cls_def seq_rhs_r_def image_def)
+ by (rule_tac x = "SEQ ra r" in exI, rule_tac x = "(x, ra)" in bexI, simp+)
+ moreover have "\<And> rhs X. rhs_eq_cls (del_x_paired rhs X) = rhs_eq_cls rhs - {X}"
+ by (auto simp:rhs_eq_cls_def del_x_paired_def)
+ ultimately show ?thesis by (simp add:arden_variate_def)
+ qed
+ ultimately show ?thesis by simp
+ qed
+ thus ?thesis by (rule_tac x= "arden_variate Y reg yrhs" in exI, simp)
+ qed
+ qed
+next
+ case False
+ hence "(2)": "rhs_eq_cls yrhs - {Y} = rhs_eq_cls yrhs"
+ by (auto simp:rhs_eq_cls_def)
+ show ?thesis
+ proof -
+ have "?P yrhs" using False ardable "(2)"
+ by (simp add:ardenable_def)
+ thus ?thesis by blast
+ qed
+qed
+
+lemma equas_subst_f_del_no_other:
+ assumes self_contained: "(Y, rhs) \<in> ES"
+ shows "\<exists> rhs'. (Y, rhs') \<in> (equas_subst_f X xrhs ` ES)" (is "\<exists> rhs'. ?P rhs'")
+proof -
+ have "\<exists> rhs'. equas_subst_f X xrhs (Y, rhs) = (Y, rhs')"
+ by (auto simp:equas_subst_f_def)
+ then obtain rhs' where "equas_subst_f X xrhs (Y, rhs) = (Y, rhs')" by blast
+ hence "?P rhs'" unfolding image_def using self_contained
+ by (auto intro:bexI[where x = "(Y, rhs)"])
+ thus ?thesis by blast
+qed
+
+lemma del_x_paired_del_only_x:
+ "\<lbrakk>X \<noteq> Y; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> (X, rhs) \<in> del_x_paired ES Y"
+by (auto simp:del_x_paired_def)
+
+lemma equas_subst_del_no_other:
+ "\<lbrakk>(X, rhs) \<in> ES; X \<noteq> Y\<rbrakk> \<Longrightarrow> (\<exists>rhs. (X, rhs) \<in> equas_subst ES Y rhs')"
+unfolding equas_subst_def
+apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other)
+by (erule exE, drule del_x_paired_del_only_x, auto)
+
+lemma equas_subst_holds_distinct:
+ "distinct_equas ES \<Longrightarrow> distinct_equas (equas_subst ES Y rhs')"
+apply (clarsimp simp add:equas_subst_def distinct_equas_def del_x_paired_def equas_subst_f_def)
+by (auto split:if_splits)
+
+lemma del_x_paired_dels:
+ "(X, rhs) \<in> ES \<Longrightarrow> {Y. Y \<in> ES \<and> fst Y = X} \<inter> ES \<noteq> {}"
+by (auto)
+
+lemma del_x_paired_subset:
+ "(X, rhs) \<in> ES \<Longrightarrow> ES - {Y. Y \<in> ES \<and> fst Y = X} \<subset> ES"
+apply (drule del_x_paired_dels)
+by auto
+
+lemma del_x_paired_card_less:
+ "\<lbrakk>finite ES; (X, rhs) \<in> ES\<rbrakk> \<Longrightarrow> card (del_x_paired ES X) < card ES"
+apply (simp add:del_x_paired_def)
+apply (drule del_x_paired_subset)
+by (auto intro:psubset_card_mono)
+
+lemma equas_subst_card_less:
+ "\<lbrakk>finite ES; (Y, yrhs) \<in> ES\<rbrakk> \<Longrightarrow> card (equas_subst ES Y rhs') < card ES"
+apply (simp add:equas_subst_def)
+apply (frule_tac h = "equas_subst_f Y rhs'" in finite_imageI)
+apply (drule_tac f = "equas_subst_f Y rhs'" in Finite_Set.card_image_le)
+apply (drule_tac X = Y and xrhs = rhs' in equas_subst_f_del_no_other,erule exE)
+by (drule del_x_paired_card_less, auto)
+
+lemma equas_subst_holds_distinct_rhs:
+ assumes dist': "distinct_rhs yrhs'"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ shows "distinct_rhs xrhs"
+using X_in history
+apply (clarsimp simp:equas_subst_def del_x_paired_def)
+apply (drule_tac x = a in spec, drule_tac x = b in spec)
+apply (simp add:ardenable_def equas_subst_f_def)
+by (auto intro:rhs_subst_holds_distinct_rhs simp:dist' split:if_splits)
+
+lemma r_no_EMPTY_imp_seq_rhs_r_no_EMPTY:
+ "[] \<notin> L r \<Longrightarrow> no_EMPTY_rhs (seq_rhs_r rhs r)"
+by (auto simp:no_EMPTY_rhs_def seq_rhs_r_def lang_seq_def)
+
+lemma del_x_paired_holds_no_EMPTY:
+ "no_EMPTY_rhs yrhs \<Longrightarrow> no_EMPTY_rhs (del_x_paired yrhs Y)"
+by (auto simp:no_EMPTY_rhs_def del_x_paired_def)
+
+lemma rhs_subst_holds_no_EMPTY:
+ "\<lbrakk>no_EMPTY_rhs yrhs; (Y, r) \<in> yrhs; Y \<noteq> {[]}\<rbrakk> \<Longrightarrow> no_EMPTY_rhs (rhs_subst yrhs Y rhs' r)"
+apply (auto simp:rhs_subst_def intro!:no_EMPTY_rhss_imp_merge_no_EMPTY r_no_EMPTY_imp_seq_rhs_r_no_EMPTY del_x_paired_holds_no_EMPTY)
+by (auto simp:no_EMPTY_rhs_def)
+
+lemma equas_subst_holds_no_EMPTY:
+ assumes substor: "Y \<noteq> {[]}"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ shows "no_EMPTY_rhs xrhs"
+proof-
+ from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ by (auto simp add:equas_subst_def del_x_paired_def)
+ then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ hence dist_zrhs: "distinct_rhs zrhs" using history
+ by (auto simp:ardenable_def)
+ show ?thesis
+ proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ case True
+ then obtain r where Y_in_zrhs: "(Y, r) \<in> zrhs" ..
+ hence some: "(SOME r. (Y, r) \<in> zrhs) = r" using Z_in dist_zrhs
+ by (auto simp:distinct_rhs_def)
+ hence "no_EMPTY_rhs (rhs_subst zrhs Y yrhs' r)"
+ using substor Y_in_zrhs history Z_in
+ by (rule_tac rhs_subst_holds_no_EMPTY, auto simp:ardenable_def)
+ thus ?thesis using X_Z True some
+ by (simp add:equas_subst_def equas_subst_f_def)
+ next
+ case False
+ hence "(X, xrhs) = (Z, zrhs)" using Z_in X_Z
+ by (simp add:equas_subst_f_def)
+ thus ?thesis using history Z_in
+ by (auto simp:ardenable_def)
+ qed
+qed
+
+lemma equas_subst_f_holds_left_eq_right:
+ assumes substor: "Y = L rhs'"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> distinct_rhs xrhs \<and> X = L xrhs"
+ and subst: "(X, xrhs) = equas_subst_f Y rhs' (Z, zrhs)"
+ and self_contained: "(Z, zrhs) \<in> ES"
+ shows "X = L xrhs"
+proof (cases "\<exists> r. (Y, r) \<in> zrhs")
+ case True
+ from True obtain r where "(1)":"(Y, r) \<in> zrhs" ..
+ show ?thesis
+ proof -
+ from history self_contained
+ have dist: "distinct_rhs zrhs" by auto
+ hence "(SOME r. (Y, r) \<in> zrhs) = r" using self_contained "(1)"
+ using distinct_rhs_def by (auto intro:some_equality)
+ moreover have "L zrhs = L (rhs_subst zrhs Y rhs' r)" using substor dist "(1)" self_contained
+ by (rule_tac rhs_subst_prop1, auto simp:distinct_equas_rhs_def)
+ ultimately show ?thesis using subst history self_contained
+ by (auto simp:equas_subst_f_def split:if_splits)
+ qed
+next
+ case False
+ thus ?thesis using history subst self_contained
+ by (auto simp:equas_subst_f_def)
+qed
+
+lemma equas_subst_holds_left_eq_right:
+ assumes history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and substor: "Y = L rhs'"
+ and X_in : "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ shows "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y rhs' \<longrightarrow> X = L xrhs"
+apply (clarsimp simp add:equas_subst_def del_x_paired_def)
+using substor
+apply (drule_tac equas_subst_f_holds_left_eq_right)
+using history
+by (auto simp:ardenable_def)
+
+lemma equas_subst_holds_ardenable:
+ assumes substor: "Y = L yrhs'"
+ and history: "\<forall>X xrhs. (X, xrhs) \<in> ES \<longrightarrow> ardenable (X, xrhs)"
+ and X_in:"(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ and dist': "distinct_rhs yrhs'"
+ and not_lambda: "Y \<noteq> {[]}"
+ shows "ardenable (X, xrhs)"
+proof -
+ have "distinct_rhs xrhs" using history X_in dist'
+ by (auto dest:equas_subst_holds_distinct_rhs)
+ moreover have "no_EMPTY_rhs xrhs" using history X_in not_lambda
+ by (auto intro:equas_subst_holds_no_EMPTY)
+ moreover have "X = L xrhs" using history substor X_in
+ by (auto dest: equas_subst_holds_left_eq_right simp del:L_rhs.simps)
+ ultimately show ?thesis using ardenable_def by simp
+qed
+
+lemma equas_subst_holds_cls_defined:
+ assumes X_in: "(X, xrhs) \<in> equas_subst ES Y yrhs'"
+ and Inv_ES: "Inv X' ES"
+ and subst: "(Y, yrhs) \<in> ES"
+ and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ shows "rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+proof-
+ have tac: "\<lbrakk> A \<subseteq> B; C \<subseteq> D; E \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> E \<subseteq> B \<union> D" by auto
+ from X_in have "\<exists> (Z, zrhs) \<in> ES. (X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)"
+ by (auto simp add:equas_subst_def del_x_paired_def)
+ then obtain Z zrhs where Z_in: "(Z, zrhs) \<in> ES"
+ and X_Z: "(X, xrhs) = equas_subst_f Y yrhs' (Z, zrhs)" by blast
+ hence "rhs_eq_cls zrhs \<subseteq> insert {[]} (left_eq_cls ES)" using Inv_ES
+ by (auto simp:Inv_def)
+ moreover have "rhs_eq_cls yrhs' \<subseteq> insert {[]} (left_eq_cls ES) - {Y}"
+ using Inv_ES subst cls_holds_but_Y
+ by (auto simp:Inv_def)
+ moreover have "rhs_eq_cls xrhs \<subseteq> rhs_eq_cls zrhs \<union> rhs_eq_cls yrhs' - {Y}"
+ using X_Z cls_holds_but_Y
+ apply (clarsimp simp add:equas_subst_f_def rhs_subst_def split:if_splits)
+ by (auto simp:rhs_eq_cls_def merge_rhs_def del_x_paired_def seq_rhs_r_def)
+ moreover have "left_eq_cls (equas_subst ES Y yrhs') = left_eq_cls ES - {Y}" using subst
+ by (auto simp: left_eq_cls_def equas_subst_def del_x_paired_def equas_subst_f_def
+ dest: equas_subst_f_del_no_other
+ split: if_splits)
+ ultimately show ?thesis by blast
+qed
+
+lemma iteration_step:
+ assumes Inv_ES: "Inv X ES"
+ and not_T: "\<not> TCon ES"
+ shows "(\<exists> ES'. Inv X ES' \<and> (card ES', card ES) \<in> less_than)"
+proof -
+ from Inv_ES not_T have another: "\<exists>Y yrhs. (Y, yrhs) \<in> ES \<and> X \<noteq> Y" unfolding Inv_def
+ by (clarify, rule_tac exist_another_equa[where X = X], auto)
+ then obtain Y yrhs where subst: "(Y, yrhs) \<in> ES" and not_X: " X \<noteq> Y" by blast
+ show ?thesis (is "\<exists> ES'. ?P ES'")
+ proof (cases "Y = {[]}")
+ case True
+ --"in this situation, we pick a \"\<lambda>\" equation, thus directly remove it from the equation-system"
+ have "?P (ES - {(Y, yrhs)})"
+ proof
+ show "Inv X (ES - {(Y, yrhs)})" using True not_X
+ by (simp add:Inv_ES Inv_mono_with_lambda)
+ next
+ show "(card (ES - {(Y, yrhs)}), card ES) \<in> less_than" using Inv_ES subst
+ by (auto elim:non_empty_card_prop[rule_format] simp:Inv_def)
+ qed
+ thus ?thesis by blast
+ next
+ case False
+ --"in this situation, we pick a equation and using ardenable to get a
+ rhs without itself in it, then use equas_subst to form a new equation-system"
+ hence "\<exists> yrhs'. Y = L yrhs' \<and> distinct_rhs yrhs' \<and> rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}"
+ using subst Inv_ES
+ by (auto intro:ardenable_prop simp add:Inv_def simp del:L_rhs.simps)
+ then obtain yrhs' where Y'_l_eq_r: "Y = L yrhs'"
+ and dist_Y': "distinct_rhs yrhs'"
+ and cls_holds_but_Y: "rhs_eq_cls yrhs' = rhs_eq_cls yrhs - {Y}" by blast
+ hence "?P (equas_subst ES Y yrhs')"
+ proof -
+ have finite_del: "\<And> S x. finite S \<Longrightarrow> finite (del_x_paired S x)"
+ apply (rule_tac A = "del_x_paired S x" in finite_subset)
+ by (auto simp:del_x_paired_def)
+ have "finite (equas_subst ES Y yrhs')" using Inv_ES
+ by (auto intro!:finite_del simp:equas_subst_def Inv_def)
+ moreover have "\<exists>rhs. (X, rhs) \<in> equas_subst ES Y yrhs'" using Inv_ES not_X
+ by (auto intro:equas_subst_del_no_other simp:Inv_def)
+ moreover have "distinct_equas (equas_subst ES Y yrhs')" using Inv_ES
+ by (auto intro:equas_subst_holds_distinct simp:Inv_def)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> ardenable (X, xrhs)"
+ using Inv_ES dist_Y' False Y'_l_eq_r
+ apply (clarsimp simp:Inv_def)
+ by (rule equas_subst_holds_ardenable, simp_all)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow> X \<noteq> {}" using Inv_ES
+ by (clarsimp simp:equas_subst_def Inv_def del_x_paired_def equas_subst_f_def split:if_splits, auto)
+ moreover have "\<forall>X xrhs. (X, xrhs) \<in> equas_subst ES Y yrhs' \<longrightarrow>
+ rhs_eq_cls xrhs \<subseteq> insert {[]} (left_eq_cls (equas_subst ES Y yrhs'))"
+ using Inv_ES subst cls_holds_but_Y
+ apply (rule_tac impI | rule_tac allI)+
+ by (erule equas_subst_holds_cls_defined, auto)
+ moreover have "(card (equas_subst ES Y yrhs'), card ES) \<in> less_than"using Inv_ES subst
+ by (simp add:equas_subst_card_less Inv_def)
+ ultimately show "?P (equas_subst ES Y yrhs')" by (auto simp:Inv_def)
+ qed
+ thus ?thesis by blast
+ qed
+qed
+
+text {* ***** END: proving every equation-system's iteration step satisfies Inv ************** *}
+
+lemma iteration_conc:
+ assumes history: "Inv X ES"
+ shows "\<exists> ES'. Inv X ES' \<and> TCon ES'" (is "\<exists> ES'. ?P ES'")
+proof (cases "TCon ES")
+ case True
+ hence "?P ES" using history by simp
+ thus ?thesis by blast
+next
+ case False
+ thus ?thesis using history iteration_step
+ by (rule_tac f = card in wf_iter, simp_all)
+qed
+
+lemma eqset_imp_iff': "A = B \<Longrightarrow> \<forall> x. x \<in> A \<longleftrightarrow> x \<in> B"
+apply (auto simp:mem_def)
+done
+
+lemma set_cases2:
+ "\<lbrakk>(A = {} \<Longrightarrow> R A); \<And> x. (A = {x}) \<Longrightarrow> R A; \<And> x y. \<lbrakk>x \<noteq> y; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> R A\<rbrakk> \<Longrightarrow> R A"
+apply (case_tac "A = {}", simp)
+by (case_tac "\<exists> x. A = {x}", clarsimp, blast)
+
+lemma rhs_aux:"\<lbrakk>distinct_rhs rhs; {Y. \<exists>r. (Y, r) \<in> rhs} = {X}\<rbrakk> \<Longrightarrow> (\<exists>r. rhs = {(X, r)})"
+apply (rule_tac A = rhs in set_cases2, simp)
+apply (drule_tac x = X in eqset_imp_iff, clarsimp)
+apply (drule eqset_imp_iff',clarsimp)
+apply (frule_tac x = a in spec, drule_tac x = aa in spec)
+by (auto simp:distinct_rhs_def)
+
+lemma every_eqcl_has_reg:
+ assumes finite_CS: "finite (UNIV Quo Lang)"
+ and X_in_CS: "X \<in> (UNIV Quo Lang)"
+ shows "\<exists> (reg::rexp). L reg = X" (is "\<exists> r. ?E r")
+proof-
+ have "\<exists>ES'. Inv X ES' \<and> TCon ES'" using finite_CS X_in_CS
+ by (auto intro:init_ES_satisfy_Inv iteration_conc)
+ then obtain ES' where Inv_ES': "Inv X ES'"
+ and TCon_ES': "TCon ES'" by blast
+ from Inv_ES' TCon_ES'
+ have "\<exists> rhs. ES' = {(X, rhs)}"
+ apply (clarsimp simp:Inv_def TCon_def)
+ apply (rule_tac x = rhs in exI)
+ by (auto dest!:card_Suc_Diff1 simp:card_eq_0_iff)
+ then obtain rhs where ES'_single_equa: "ES' = {(X, rhs)}" ..
+ hence X_ardenable: "ardenable (X, rhs)" using Inv_ES'
+ by (simp add:Inv_def)
+
+ from X_ardenable have X_l_eq_r: "X = L rhs"
+ by (simp add:ardenable_def)
+ hence rhs_not_empty: "rhs \<noteq> {}" using Inv_ES' ES'_single_equa
+ by (auto simp:Inv_def ardenable_def)
+ have rhs_eq_cls: "rhs_eq_cls rhs \<subseteq> {X, {[]}}"
+ using Inv_ES' ES'_single_equa
+ by (auto simp:Inv_def ardenable_def left_eq_cls_def)
+ have X_not_empty: "X \<noteq> {}" using Inv_ES' ES'_single_equa
+ by (auto simp:Inv_def)
+ show ?thesis
+ proof (cases "X = {[]}")
+ case True
+ hence "?E EMPTY" by (simp)
+ thus ?thesis by blast
+ next
+ case False with X_ardenable
+ have "\<exists> rhs'. X = L rhs' \<and> rhs_eq_cls rhs' = rhs_eq_cls rhs - {X} \<and> distinct_rhs rhs'"
+ by (drule_tac ardenable_prop, auto)
+ then obtain rhs' where X_eq_rhs': "X = L rhs'"
+ and rhs'_eq_cls: "rhs_eq_cls rhs' = rhs_eq_cls rhs - {X}"
+ and rhs'_dist : "distinct_rhs rhs'" by blast
+ have "rhs_eq_cls rhs' \<subseteq> {{[]}}" using rhs_eq_cls False rhs'_eq_cls rhs_not_empty
+ by blast
+ hence "rhs_eq_cls rhs' = {{[]}}" using X_not_empty X_eq_rhs'
+ by (auto simp:rhs_eq_cls_def)
+ hence "\<exists> r. rhs' = {({[]}, r)}" using rhs'_dist
+ by (auto intro:rhs_aux simp:rhs_eq_cls_def)
+ then obtain r where "rhs' = {({[]}, r)}" ..
+ hence "?E r" using X_eq_rhs' by (auto simp add:lang_seq_def)
+ thus ?thesis by blast
+ qed
+qed
+
+text {* definition of a regular language *}
+
+abbreviation
+ reg :: "string set => bool"
+where
+ "reg L1 \<equiv> (\<exists>r::rexp. L r = L1)"
+
+theorem myhill_nerode:
+ assumes finite_CS: "finite (UNIV Quo Lang)"
+ shows "\<exists> (reg::rexp). Lang = L reg" (is "\<exists> r. ?P r")
+proof -
+ have has_r_each: "\<forall>C\<in>{X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists>(r::rexp). C = L r" using finite_CS
+ by (auto dest:every_eqcl_has_reg)
+ have "\<exists> (rS::rexp set). finite rS \<and>
+ (\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> rS. C = L r) \<and>
+ (\<forall> r \<in> rS. \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r)"
+ (is "\<exists> rS. ?Q rS")
+ proof-
+ have "\<And> C. C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang} \<Longrightarrow> C = L (SOME (ra::rexp). C = L ra)"
+ using has_r_each
+ apply (erule_tac x = C in ballE, erule_tac exE)
+ by (rule_tac a = r in someI2, simp+)
+ hence "?Q ((\<lambda> C. SOME r. C = L r) ` {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang})" using has_r_each
+ using finite_CS by auto
+ thus ?thesis by blast
+ qed
+ then obtain rS where finite_rS : "finite rS"
+ and has_r_each': "\<forall> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. \<exists> r \<in> (rS::rexp set). C = L r"
+ and has_cl_each: "\<forall> r \<in> (rS::rexp set). \<exists> C \<in> {X \<in> UNIV Quo Lang. \<forall>x\<in>X. x \<in> Lang}. C = L r" by blast
+ have "?P (folds ALT NULL rS)"
+ proof
+ show "Lang \<subseteq> L (folds ALT NULL rS)" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_r_each'
+ apply (clarsimp simp:fold_alt_null_eqs) by blast
+ next
+ show "L (folds ALT NULL rS) \<subseteq> Lang" using finite_rS lang_eqs_union_of_eqcls[of Lang] has_cl_each
+ by (clarsimp simp:fold_alt_null_eqs)
+ qed
+ thus ?thesis by blast
+qed
+
+
+text {* tests by Christian *}
+
+(* Alternative definition for Quo *)
+definition
+ QUOT :: "string set \<Rightarrow> (string set) set"
+where
+ "QUOT Lang \<equiv> (\<Union>x. {\<lbrakk>x\<rbrakk>Lang})"
+
+lemma test:
+ "UNIV Quo Lang = QUOT Lang"
+by (auto simp add: quot_def QUOT_def)
+
+lemma test1:
+ assumes finite_CS: "finite (QUOT Lang)"
+ shows "reg Lang"
+using finite_CS
+unfolding test[symmetric]
+by (auto dest: myhill_nerode)
+
+lemma cons_one: "x @ y \<in> {z} \<Longrightarrow> x @ y = z"
+by simp
+
+lemma quot_lambda: "QUOT {[]} = {{[]}, UNIV - {[]}}"
+proof
+ show "QUOT {[]} \<subseteq> {{[]}, UNIV - {[]}}"
+ proof
+ fix x
+ assume in_quot: "x \<in> QUOT {[]}"
+ show "x \<in> {{[]}, UNIV - {[]}}"
+ proof -
+ from in_quot
+ have "x = {[]} \<or> x = UNIV - {[]}"
+ unfolding QUOT_def equiv_class_def
+ proof
+ fix xa
+ assume in_univ: "xa \<in> UNIV"
+ and in_eqiv: "x \<in> {{y. xa \<equiv>{[]}\<equiv> y}}"
+ show "x = {[]} \<or> x = UNIV - {[]}"
+ proof(cases "xa = []")
+ case True
+ hence "{y. xa \<equiv>{[]}\<equiv> y} = {[]}" using in_eqiv
+ by (auto simp add:equiv_str_def)
+ thus ?thesis using in_eqiv by (rule_tac disjI1, simp)
+ next
+ case False
+ hence "{y. xa \<equiv>{[]}\<equiv> y} = UNIV - {[]}" using in_eqiv
+ by (auto simp:equiv_str_def)
+ thus ?thesis using in_eqiv by (rule_tac disjI2, simp)
+ qed
+ qed
+ thus ?thesis by simp
+ qed
+ qed
+next
+ show "{{[]}, UNIV - {[]}} \<subseteq> QUOT {[]}"
+ proof
+ fix x
+ assume in_res: "x \<in> {{[]}, (UNIV::string set) - {[]}}"
+ show "x \<in> (QUOT {[]})"
+ proof -
+ have "x = {[]} \<Longrightarrow> x \<in> QUOT {[]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ by (rule_tac x = "[]" in exI, auto)
+ moreover have "x = UNIV - {[]} \<Longrightarrow> x \<in> QUOT {[]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ by (rule_tac x = "''a''" in exI, auto)
+ ultimately show ?thesis using in_res by blast
+ qed
+ qed
+qed
+
+lemma quot_single_aux: "\<lbrakk>x \<noteq> []; x \<noteq> [c]\<rbrakk> \<Longrightarrow> x @ z \<noteq> [c]"
+by (induct x, auto)
+
+lemma quot_single: "\<And> (c::char). QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
+proof -
+ fix c::"char"
+ have exist_another: "\<exists> a. a \<noteq> c"
+ apply (case_tac "c = CHR ''a''")
+ apply (rule_tac x = "CHR ''b''" in exI, simp)
+ by (rule_tac x = "CHR ''a''" in exI, simp)
+ show "QUOT {[c]} = {{[]}, {[c]}, UNIV - {[], [c]}}"
+ proof
+ show "QUOT {[c]} \<subseteq> {{[]},{[c]}, UNIV - {[], [c]}}"
+ proof
+ fix x
+ assume in_quot: "x \<in> QUOT {[c]}"
+ show "x \<in> {{[]}, {[c]}, UNIV - {[],[c]}}"
+ proof -
+ from in_quot
+ have "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[],[c]}"
+ unfolding QUOT_def equiv_class_def
+ proof
+ fix xa
+ assume in_eqiv: "x \<in> {{y. xa \<equiv>{[c]}\<equiv> y}}"
+ show "x = {[]} \<or> x = {[c]} \<or> x = UNIV - {[], [c]}"
+ proof-
+ have "xa = [] \<Longrightarrow> x = {[]}" using in_eqiv
+ by (auto simp add:equiv_str_def)
+ moreover have "xa = [c] \<Longrightarrow> x = {[c]}"
+ proof -
+ have "xa = [c] \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = {[c]}" using in_eqiv
+ apply (simp add:equiv_str_def)
+ apply (rule set_ext, rule iffI, simp)
+ apply (drule_tac x = "[]" in spec, auto)
+ done
+ thus "xa = [c] \<Longrightarrow> x = {[c]}" using in_eqiv by simp
+ qed
+ moreover have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}"
+ proof -
+ have "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> {y. xa \<equiv>{[c]}\<equiv> y} = UNIV - {[],[c]}"
+ apply (clarsimp simp add:equiv_str_def)
+ apply (rule set_ext, rule iffI, simp)
+ apply (rule conjI)
+ apply (drule_tac x = "[c]" in spec, simp)
+ apply (drule_tac x = "[]" in spec, simp)
+ by (auto dest:quot_single_aux)
+ thus "\<lbrakk>xa \<noteq> []; xa \<noteq> [c]\<rbrakk> \<Longrightarrow> x = UNIV - {[],[c]}" using in_eqiv by simp
+ qed
+ ultimately show ?thesis by blast
+ qed
+ qed
+ thus ?thesis by simp
+ qed
+ qed
+ next
+ show "{{[]}, {[c]}, UNIV - {[],[c]}} \<subseteq> QUOT {[c]}"
+ proof
+ fix x
+ assume in_res: "x \<in> {{[]},{[c]}, (UNIV::string set) - {[],[c]}}"
+ show "x \<in> (QUOT {[c]})"
+ proof -
+ have "x = {[]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ by (rule_tac x = "[]" in exI, auto)
+ moreover have "x = {[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+ apply (rule_tac x = "[c]" in exI, simp)
+ apply (rule set_ext, rule iffI, simp+)
+ by (drule_tac x = "[]" in spec, simp)
+ moreover have "x = UNIV - {[],[c]} \<Longrightarrow> x \<in> QUOT {[c]}"
+ using exist_another
+ apply (clarsimp simp add:QUOT_def equiv_class_def equiv_str_def)
+ apply (rule_tac x = "[a]" in exI, simp)
+ apply (rule set_ext, rule iffI, simp)
+ apply (clarsimp simp:quot_single_aux, simp)
+ apply (rule conjI)
+ apply (drule_tac x = "[c]" in spec, simp)
+ by (drule_tac x = "[]" in spec, simp)
+ ultimately show ?thesis using in_res by blast
+ qed
+ qed
+ qed
+qed
+
+lemma eq_class_imp_eq_str:
+ "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang \<Longrightarrow> x \<equiv>lang\<equiv> y"
+by (auto simp:equiv_class_def equiv_str_def)
+
+lemma finite_tag_image:
+ "finite (range tag) \<Longrightarrow> finite (((op `) tag) ` S)"
+apply (rule_tac B = "Pow (tag ` UNIV)" in finite_subset)
+by (auto simp add:image_def Pow_def)
+
+lemma str_inj_imps:
+ assumes str_inj: "\<And> m n. tag m = tag (n::string) \<Longrightarrow> m \<equiv>lang\<equiv> n"
+ shows "inj_on ((op `) tag) (QUOT lang)"
+proof (clarsimp simp add:inj_on_def QUOT_def)
+ fix x y
+ assume eq_tag: "tag ` \<lbrakk>x\<rbrakk>lang = tag ` \<lbrakk>y\<rbrakk>lang"
+ show "\<lbrakk>x\<rbrakk>lang = \<lbrakk>y\<rbrakk>lang"
+ proof -
+ have aux1:"\<And>a b. a \<in> (\<lbrakk>b\<rbrakk>lang) \<Longrightarrow> (a \<equiv>lang\<equiv> b)"
+ by (simp add:equiv_class_def equiv_str_def)
+ have aux2: "\<And> A B f. \<lbrakk>f ` A = f ` B; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists> a b. a \<in> A \<and> b \<in> B \<and> f a = f b"
+ by auto
+ have aux3: "\<And> a l. \<lbrakk>a\<rbrakk>l \<noteq> {}"
+ by (auto simp:equiv_class_def equiv_str_def)
+ show ?thesis using eq_tag
+ apply (drule_tac aux2, simp add:aux3, clarsimp)
+ apply (drule_tac str_inj, (drule_tac aux1)+)
+ by (simp add:equiv_str_def equiv_class_def)
+ qed
+qed
+
+definition tag_str_ALT :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
+where
+ "tag_str_ALT L\<^isub>1 L\<^isub>2 x \<equiv> (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)"
+
+lemma tag_str_alt_range_finite:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (range (tag_str_ALT L\<^isub>1 L\<^isub>2))"
+proof -
+ have "{y. \<exists>x. y = (\<lbrakk>x\<rbrakk>L\<^isub>1, \<lbrakk>x\<rbrakk>L\<^isub>2)} \<subseteq> (QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"
+ by (auto simp:QUOT_def)
+ thus ?thesis using finite1 finite2
+ by (auto simp: image_def tag_str_ALT_def UNION_def
+ intro: finite_subset[where B = "(QUOT L\<^isub>1) \<times> (QUOT L\<^isub>2)"])
+qed
+
+lemma tag_str_alt_inj:
+ "tag_str_ALT L\<^isub>1 L\<^isub>2 x = tag_str_ALT L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<equiv>(L\<^isub>1 \<union> L\<^isub>2)\<equiv> y"
+apply (simp add:tag_str_ALT_def equiv_class_def equiv_str_def)
+by blast
+
+lemma quot_alt:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
+proof (rule_tac f = "(op `) (tag_str_ALT L\<^isub>1 L\<^isub>2)" in finite_imageD)
+ show "finite (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 \<union> L\<^isub>2))"
+ using finite_tag_image tag_str_alt_range_finite finite1 finite2
+ by auto
+next
+ show "inj_on (op ` (tag_str_ALT L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 \<union> L\<^isub>2))"
+ apply (rule_tac str_inj_imps)
+ by (erule_tac tag_str_alt_inj)
+qed
+
+(* list_diff:: list substract, once different return tailer *)
+fun list_diff :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" (infix "-" 51)
+where
+ "list_diff [] xs = []" |
+ "list_diff (x#xs) [] = x#xs" |
+ "list_diff (x#xs) (y#ys) = (if x = y then list_diff xs ys else (x#xs))"
+
+lemma [simp]: "(x @ y) - x = y"
+apply (induct x)
+by (case_tac y, simp+)
+
+lemma [simp]: "x - x = []"
+by (induct x, auto)
+
+lemma [simp]: "x = xa @ y \<Longrightarrow> x - xa = y "
+by (induct x, auto)
+
+lemma [simp]: "x - [] = x"
+by (induct x, auto)
+
+lemma [simp]: "xa \<le> x \<Longrightarrow> (x @ y) - xa = (x - xa) @ y"
+by (auto elim:prefixE)
+
+definition tag_str_SEQ:: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set set)"
+where
+ "tag_str_SEQ L\<^isub>1 L\<^isub>2 x \<equiv> if (\<exists> xa \<le> x. xa \<in> L\<^isub>1)
+ then (\<lbrakk>x\<rbrakk>L\<^isub>1, {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1})
+ else (\<lbrakk>x\<rbrakk>L\<^isub>1, {})"
+
+lemma tag_seq_eq_E:
+ "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y \<Longrightarrow>
+ ((\<exists> xa \<le> x. xa \<in> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1 \<and>
+ {\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1} ) \<or>
+ ((\<forall> xa \<le> x. xa \<notin> L\<^isub>1) \<and> \<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1)"
+by (simp add:tag_str_SEQ_def split:if_splits, blast)
+
+lemma tag_str_seq_range_finite:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (range (tag_str_SEQ L\<^isub>1 L\<^isub>2))"
+proof -
+ have "(range (tag_str_SEQ L\<^isub>1 L\<^isub>2)) \<subseteq> (QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))"
+ by (auto simp:image_def tag_str_SEQ_def QUOT_def)
+ thus ?thesis using finite1 finite2
+ by (rule_tac B = "(QUOT L\<^isub>1) \<times> (Pow (QUOT L\<^isub>2))" in finite_subset, auto)
+qed
+
+lemma app_in_seq_decom[rule_format]:
+ "\<forall> x. x @ z \<in> L\<^isub>1 ; L\<^isub>2 \<longrightarrow> (\<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)"
+apply (induct z)
+apply (simp, rule allI, rule impI, rule disjI1)
+apply (clarsimp simp add:lang_seq_def)
+apply (rule_tac x = s1 in exI, simp)
+apply (rule allI | rule impI)+
+apply (drule_tac x = "x @ [a]" in spec, simp)
+apply (erule exE | erule conjE | erule disjE)+
+apply (rule disjI2, rule_tac x = "[a]" in exI, simp)
+apply (rule disjI1, rule_tac x = xa in exI, simp)
+apply (erule exE | erule conjE)+
+apply (rule disjI2, rule_tac x = "a # za" in exI, simp)
+done
+
+lemma tag_str_seq_inj:
+ assumes tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ shows "(x::string) \<equiv>(L\<^isub>1 ; L\<^isub>2)\<equiv> y"
+proof -
+ have aux: "\<And> x y z. \<lbrakk>tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y; x @ z \<in> L\<^isub>1 ; L\<^isub>2\<rbrakk>
+ \<Longrightarrow> y @ z \<in> L\<^isub>1 ; L\<^isub>2"
+ proof (drule app_in_seq_decom, erule disjE)
+ fix x y z
+ assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ and x_gets_l2: "\<exists>xa\<le>x. xa \<in> L\<^isub>1 \<and> (x - xa) @ z \<in> L\<^isub>2"
+ from x_gets_l2 have "\<exists> xa \<le> x. xa \<in> L\<^isub>1" by blast
+ hence xy_l2:"{\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 | xa. xa \<le> x \<and> xa \<in> L\<^isub>1} = {\<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 | ya. ya \<le> y \<and> ya \<in> L\<^isub>1}"
+ using tag_eq tag_seq_eq_E by blast
+ from x_gets_l2 obtain xa where xa_le_x: "xa \<le> x"
+ and xa_in_l1: "xa \<in> L\<^isub>1"
+ and rest_in_l2: "(x - xa) @ z \<in> L\<^isub>2" by blast
+ hence "\<exists> ya. \<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2 \<and> ya \<le> y \<and> ya \<in> L\<^isub>1" using xy_l2 by auto
+ then obtain ya where ya_le_x: "ya \<le> y"
+ and ya_in_l1: "ya \<in> L\<^isub>1"
+ and rest_eq: "\<lbrakk>(x - xa)\<rbrakk>L\<^isub>2 = \<lbrakk>(y - ya)\<rbrakk>L\<^isub>2" by blast
+ from rest_eq rest_in_l2 have "(y - ya) @ z \<in> L\<^isub>2"
+ by (auto simp:equiv_class_def equiv_str_def)
+ hence "ya @ ((y - ya) @ z) \<in> L\<^isub>1 ; L\<^isub>2" using ya_in_l1
+ by (auto simp:lang_seq_def)
+ thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using ya_le_x
+ by (erule_tac prefixE, simp)
+ next
+ fix x y z
+ assume tag_eq: "tag_str_SEQ L\<^isub>1 L\<^isub>2 x = tag_str_SEQ L\<^isub>1 L\<^isub>2 y"
+ and x_gets_l1: "\<exists>za\<le>z. x @ za \<in> L\<^isub>1 \<and> z - za \<in> L\<^isub>2"
+ from tag_eq tag_seq_eq_E have x_y_eq: "\<lbrakk>x\<rbrakk>L\<^isub>1 = \<lbrakk>y\<rbrakk>L\<^isub>1" by blast
+ from x_gets_l1 obtain za where za_le_z: "za \<le> z"
+ and x_za_in_l1: "(x @ za) \<in> L\<^isub>1"
+ and rest_in_l2: "z - za \<in> L\<^isub>2" by blast
+ from x_y_eq x_za_in_l1 have y_za_in_l1: "y @ za \<in> L\<^isub>1"
+ by (auto simp:equiv_class_def equiv_str_def)
+ hence "(y @ za) @ (z - za) \<in> L\<^isub>1 ; L\<^isub>2" using rest_in_l2
+ apply (simp add:lang_seq_def)
+ by (rule_tac x = "y @ za" in exI, rule_tac x = "z - za" in exI, simp)
+ thus "y @ z \<in> L\<^isub>1 ; L\<^isub>2" using za_le_z
+ by (erule_tac prefixE, simp)
+ qed
+ show ?thesis using tag_eq
+ apply (simp add:equiv_str_def)
+ by (auto intro:aux)
+qed
+
+lemma quot_seq:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ and finite2: "finite (QUOT L\<^isub>2)"
+ shows "finite (QUOT (L\<^isub>1;L\<^isub>2))"
+proof (rule_tac f = "(op `) (tag_str_SEQ L\<^isub>1 L\<^isub>2)" in finite_imageD)
+ show "finite (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2) ` QUOT (L\<^isub>1 ; L\<^isub>2))"
+ using finite_tag_image tag_str_seq_range_finite finite1 finite2
+ by auto
+next
+ show "inj_on (op ` (tag_str_SEQ L\<^isub>1 L\<^isub>2)) (QUOT (L\<^isub>1 ; L\<^isub>2))"
+ apply (rule_tac str_inj_imps)
+ by (erule_tac tag_str_seq_inj)
+qed
+
+(****************** the STAR case ************************)
+
+lemma app_eq_elim[rule_format]:
+ "\<And> a. \<forall> b x y. a @ b = x @ y \<longrightarrow> (\<exists> aa ab. a = aa @ ab \<and> x = aa \<and> y = ab @ b) \<or>
+ (\<exists> ba bb. b = ba @ bb \<and> x = a @ ba \<and> y = bb \<and> ba \<noteq> [])"
+ apply (induct_tac a rule:List.induct, simp)
+ apply (rule allI | rule impI)+
+ by (case_tac x, auto)
+
+definition tag_str_STAR:: "string set \<Rightarrow> string \<Rightarrow> string set set"
+where
+ "tag_str_STAR L\<^isub>1 x \<equiv> if (x = [])
+ then {}
+ else {\<lbrakk>x\<^isub>2\<rbrakk>L\<^isub>1 | x\<^isub>1 x\<^isub>2. x = x\<^isub>1 @ x\<^isub>2 \<and> x\<^isub>1 \<in> L\<^isub>1\<star> \<and> x\<^isub>2 \<noteq> []}"
+
+lemma tag_str_star_range_finite:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ shows "finite (range (tag_str_STAR L\<^isub>1))"
+proof -
+ have "range (tag_str_STAR L\<^isub>1) \<subseteq> Pow (QUOT L\<^isub>1)"
+ by (auto simp:image_def tag_str_STAR_def QUOT_def)
+ thus ?thesis using finite1
+ by (rule_tac B = "Pow (QUOT L\<^isub>1)" in finite_subset, auto)
+qed
+
+lemma star_prop[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall> y. y \<in> lang\<star> \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto)
+
+lemma star_prop2: "y \<in> lang \<Longrightarrow> y \<in> lang\<star>"
+by (drule step[of y lang "[]"], auto simp:start)
+
+lemma star_prop3[rule_format]: "x \<in> lang\<star> \<Longrightarrow> \<forall>y . y \<in> lang \<longrightarrow> x @ y \<in> lang\<star>"
+by (erule Star.induct, auto intro:star_prop2)
+
+lemma postfix_prop: "y >>= (x @ y) \<Longrightarrow> x = []"
+by (erule postfixE, induct x arbitrary:y, auto)
+
+lemma inj_aux:
+ "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
+ \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
+ \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>"
+proof-
+ have "\<And>s. s \<in> L\<^isub>1\<star> \<Longrightarrow> \<forall> m z yb. (s = m @ z \<and> m \<equiv>L\<^isub>1\<equiv> yb \<and> x = xa @ m \<and> xa \<in> L\<^isub>1\<star> \<and> m \<noteq> [] \<and>
+ (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m) \<longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>)"
+ apply (erule Star.induct, simp)
+ apply (rule allI | rule impI | erule conjE)+
+ apply (drule app_eq_elim)
+ apply (erule disjE | erule exE | erule conjE)+
+ apply simp
+ apply (simp (no_asm) only:append_assoc[THEN sym])
+ apply (rule step)
+ apply (simp add:equiv_str_def)
+ apply simp
+
+ apply (erule exE | erule conjE)+
+ apply (rotate_tac 3)
+ apply (frule_tac x = "xa @ s1" in spec)
+ apply (rotate_tac 12)
+ apply (drule_tac x = ba in spec)
+ apply (erule impE)
+ apply ( simp add:star_prop3)
+ apply (simp)
+ apply (drule postfix_prop)
+ apply simp
+ done
+ thus "\<lbrakk>(m @ z) \<in> L\<^isub>1\<star>; m \<equiv>L\<^isub>1\<equiv> yb; xa @ m = x; xa \<in> L\<^isub>1\<star>; m \<noteq> [];
+ \<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= m\<rbrakk>
+ \<Longrightarrow> (yb @ z) \<in> L\<^isub>1\<star>" by blast
+qed
+
+
+lemma min_postfix_exists[rule_format]:
+ "finite A \<Longrightarrow> A \<noteq> {} \<and> (\<forall> a \<in> A. \<forall> b \<in> A. ((b >>= a) \<or> (a >>= b)))
+ \<longrightarrow> (\<exists> min. (min \<in> A \<and> (\<forall> a \<in> A. a >>= min)))"
+apply (erule finite.induct)
+apply simp
+apply simp
+apply (case_tac "A = {}")
+apply simp
+apply clarsimp
+apply (case_tac "a >>= min")
+apply (rule_tac x = min in exI, simp)
+apply (rule_tac x = a in exI, simp)
+apply clarify
+apply (rotate_tac 5)
+apply (erule_tac x = aa in ballE) defer apply simp
+apply (erule_tac ys = min in postfix_trans)
+apply (erule_tac x = min in ballE)
+by simp+
+
+lemma tag_str_star_inj:
+ "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
+proof -
+ have aux: "\<And> x y z. \<lbrakk>tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y; x @ z \<in> L\<^isub>1\<star>\<rbrakk> \<Longrightarrow> y @ z \<in> L\<^isub>1\<star>"
+ proof-
+ fix x y z
+ assume tag_eq: "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 y"
+ and x_z: "x @ z \<in> L\<^isub>1\<star>"
+ show "y @ z \<in> L\<^isub>1\<star>"
+ proof (cases "x = []")
+ case True
+ with tag_eq have "y = []" by (simp add:tag_str_STAR_def split:if_splits, blast)
+ thus ?thesis using x_z True by simp
+ next
+ case False
+ hence not_empty: "{xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>} \<noteq> {}" using x_z
+ by (simp, rule_tac x = x in exI, rule_tac x = "[]" in exI, simp add:start)
+ have finite_set: "finite {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}"
+ apply (rule_tac B = "{xb. \<exists> xa. x = xa @ xb}" in finite_subset)
+ apply auto
+ apply (induct x, simp)
+ apply (subgoal_tac "{xb. \<exists>xa. a # x = xa @ xb} = insert (a # x) {xb. \<exists>xa. x = xa @ xb}")
+ apply auto
+ by (case_tac xaa, simp+)
+ have comparable: "\<forall> a \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
+ \<forall> b \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}.
+ ((b >>= a) \<or> (a >>= b))"
+ by (auto simp:postfix_def, drule app_eq_elim, blast)
+ hence "\<exists> min. min \<in> {xb. \<exists> xa. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star>}
+ \<and> (\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min)"
+ using finite_set not_empty comparable
+ apply (drule_tac min_postfix_exists, simp)
+ by (erule exE, rule_tac x = min in exI, auto)
+ then obtain min xa where x_decom: "x = xa @ min \<and> xa \<in> L\<^isub>1\<star>"
+ and min_not_empty: "min \<noteq> []"
+ and min_z_in_star: "min @ z \<in> L\<^isub>1\<star>"
+ and is_min: "\<forall> xa xb. x = xa @ xb \<and> xa \<in> L\<^isub>1\<star> \<and> xb \<noteq> [] \<and> xb @ z \<in> L\<^isub>1\<star> \<longrightarrow> xb >>= min" by blast
+ from x_decom min_not_empty have "\<lbrakk>min\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 x" by (auto simp:tag_str_STAR_def)
+ hence "\<exists> yb. \<lbrakk>yb\<rbrakk>L\<^isub>1 \<in> tag_str_STAR L\<^isub>1 y \<and> \<lbrakk>min\<rbrakk>L\<^isub>1 = \<lbrakk>yb\<rbrakk>L\<^isub>1" using tag_eq by auto
+ hence "\<exists> ya yb. y = ya @ yb \<and> ya \<in> L\<^isub>1\<star> \<and> min \<equiv>L\<^isub>1\<equiv> yb \<and> yb \<noteq> [] "
+ by (simp add:tag_str_STAR_def equiv_class_def equiv_str_def split:if_splits, blast)
+ then obtain ya yb where y_decom: "y = ya @ yb"
+ and ya_in_star: "ya \<in> L\<^isub>1\<star>"
+ and yb_not_empty: "yb \<noteq> []"
+ and min_yb_eq: "min \<equiv>L\<^isub>1\<equiv> yb" by blast
+ from min_z_in_star min_yb_eq min_not_empty is_min x_decom
+ have "yb @ z \<in> L\<^isub>1\<star>"
+ by (rule_tac x = x and xa = xa in inj_aux, simp+)
+ thus ?thesis using ya_in_star y_decom
+ by (auto dest:star_prop)
+ qed
+ qed
+ show "tag_str_STAR L\<^isub>1 x = tag_str_STAR L\<^isub>1 (y::string) \<Longrightarrow> x \<equiv>L\<^isub>1\<star>\<equiv> y"
+ by (auto intro:aux simp:equiv_str_def)
+qed
+
+lemma quot_star:
+ assumes finite1: "finite (QUOT L\<^isub>1)"
+ shows "finite (QUOT (L\<^isub>1\<star>))"
+proof (rule_tac f = "(op `) (tag_str_STAR L\<^isub>1)" in finite_imageD)
+ show "finite (op ` (tag_str_STAR L\<^isub>1) ` QUOT (L\<^isub>1\<star>))"
+ using finite_tag_image tag_str_star_range_finite finite1
+ by auto
+next
+ show "inj_on (op ` (tag_str_STAR L\<^isub>1)) (QUOT (L\<^isub>1\<star>))"
+ apply (rule_tac str_inj_imps)
+ by (erule_tac tag_str_star_inj)
+qed
+
+lemma other_direction:
+ "Lang = L (r::rexp) \<Longrightarrow> finite (QUOT Lang)"
+apply (induct arbitrary:Lang rule:rexp.induct)
+apply (simp add:QUOT_def equiv_class_def equiv_str_def)
+by (simp_all add:quot_lambda quot_single quot_seq quot_alt quot_star)
+
+end