regexp: comparison My.thy

equal deleted inserted replaced

-:e31b733ace44
+:f72c82bf59e5
 theory My
-imports Main
+imports Main Infinite_Set
 begin
 definition
-lang_seq :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ; _" [100,100] 100)
+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}"
+"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>"
+lemma lang_star_cases:
+shows "L\<star> =  {[]} \<union> L ;; L\<star>"
+unfolding Seq_def
+by (auto) (metis Star.simps)
+lemma lang_star_cases2:
+shows "L ;; L\<star>  = L\<star> ;; L"
+sorry
+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
 fun
-L_rexp :: "rexp \<Rightarrow> string set"
+Sem :: "rexp \<Rightarrow> string set" ("\<lparr>_\<rparr>" [0] 1000)
 where
-"L_rexp (NULL) = {}"
+"\<lparr>NULL\<rparr> = {}"
-| "L_rexp (EMPTY) = {[]}"
+| "\<lparr>EMPTY\<rparr> = {[]}"
-| "L_rexp (CHAR c) = {[c]}"
+| "\<lparr>CHAR c\<rparr> = {[c]}"
-| "L_rexp (SEQ r1 r2) = (L_rexp r1) ; (L_rexp r2)"
+| "\<lparr>SEQ r1 r2\<rparr> = \<lparr>r1\<rparr> ;; \<lparr>r2\<rparr>"
-| "L_rexp (ALT r1 r2) = (L_rexp r1) \<union> (L_rexp r2)"
+| "\<lparr>ALT r1 r2\<rparr> = \<lparr>r1\<rparr> \<union> \<lparr>r2\<rparr>"
-| "L_rexp (STAR r) = (L_rexp r)\<star>"
+| "\<lparr>STAR r\<rparr> = \<lparr>r\<rparr>\<star>"
 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_rexp (folds ALT NULL rs) = (\<exists>r \<in> rs. x \<in> L_rexp r)"
+"finite rs \<Longrightarrow> x \<in> \<lparr>folds ALT NULL rs\<rparr> \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> \<lparr>r\<rparr>)"
 apply (simp add: folds_def)
 apply (rule someI2_ex)
 apply (erule finite_imp_fold_graph)
 apply (erule fold_graph.induct)
 apply (auto)
 done
 lemma folds_simp_empty [simp]:
-"finite rs \<Longrightarrow> x \<in> L_rexp (folds ALT EMPTY rs) = ((\<exists>r \<in> rs. x \<in> L_rexp r) \<or> x = [])"
+"finite rs \<Longrightarrow> x \<in> \<lparr>folds ALT EMPTY rs\<rparr> \<longleftrightarrow> (\<exists>r \<in> rs. x \<in> \<lparr>r\<rparr>) \<or> x = []"
 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]:
-"(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
+shows "(x, y) \<in> {(x, y). P x y} \<longleftrightarrow> P x y"
 by simp
 definition
 str_eq ("_ \<approx>_ _")
 where
 show "Lang = \<Union> {X. final X Lang}"
 by blast
 qed
 lemma all_rexp:
-"\<lbrakk>finite (UNIV // \<approx>Lang); X \<in> (UNIV // \<approx>Lang)\<rbrakk> \<Longrightarrow> \<exists>r. X = L_rexp r"
+"\<lbrakk>finite (UNIV // \<approx>Lang); X \<in> (UNIV // \<approx>Lang)\<rbrakk> \<Longrightarrow> \<exists>r. X = \<lparr>r\<rparr>"
 sorry
 lemma final_rexp:
-"\<lbrakk>finite (UNIV // (\<approx>Lang)); final X Lang\<rbrakk> \<Longrightarrow> \<exists>r. X = L_rexp r"
+"\<lbrakk>finite (UNIV // (\<approx>Lang)); final X Lang\<rbrakk> \<Longrightarrow> \<exists>r. X = \<lparr>r\<rparr>"
 unfolding final_def
 using all_rexp by blast
 lemma finite_f_one_to_one:
 assumes "finite B"
 qed
 lemma finite_regular_aux:
 fixes Lang :: "string set"
 assumes "finite (UNIV // (\<approx>Lang))"
-shows "\<exists>rs. Lang =  L_rexp (folds ALT NULL rs)"
+shows "\<exists>rs. Lang = \<lparr>folds ALT NULL rs\<rparr>"
 apply(subst lang_is_union_of_finals)
 using assms
 apply -
 apply(drule finite_final)
-apply(drule_tac f="L_rexp" in finite_f_one_to_one)
+apply(drule_tac f="Sem" in finite_f_one_to_one)
 apply(clarify)
 apply(drule final_rexp[OF assms])
 apply(auto)[1]
 apply(clarify)
 apply(rule_tac x="rs" in exI)
 done
 lemma finite_regular:
 fixes Lang :: "string set"
 assumes "finite (UNIV // (\<approx>Lang))"
-shows "\<exists>r. Lang =  L_rexp r"
+shows "\<exists>r. Lang =  \<lparr>r\<rparr>"
 using assms finite_regular_aux
 by auto
 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}"
+"Y \<Turnstile>\<Rightarrow> X \<equiv> {CHAR c | c. Y ;; {[c]} \<subseteq> X}"
 definition
 transitions_rexp ("_ \<turnstile>\<rightarrow> _")
 where
 "Y \<turnstile>\<rightarrow> X \<equiv> if [] \<in> X then folds ALT EMPTY (Y \<Turnstile>\<Rightarrow>X) else folds ALT NULL (Y \<Turnstile>\<Rightarrow>X)"
 definition
 "rhs CS X \<equiv> if X = {[]} then {({[]}, EMPTY)} else {(Y, Y \<turnstile>\<rightarrow>X) | Y. Y \<in> CS}"
 definition
-"rhs_sem CS X \<equiv> \<Union> {(Y; L_rexp r) | Y r . (Y, r) \<in> rhs CS X}"
+"rhs_sem CS X \<equiv> \<Union> {(Y;; \<lparr>r\<rparr>) | Y r . (Y, r) \<in> rhs CS X}"
 definition
 "eqs CS \<equiv> (\<Union>X \<in> CS. {(X, rhs CS X)})"
 definition
 by auto
 lemma every_eqclass_has_transition:
 assumes has_str: "s @ [c] \<in> X"
 and     in_CS:   "X \<in> UNIV // (\<approx>Lang)"
-obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ; {[c]} \<subseteq> X" and "s \<in> Y"
+obtains Y where "Y \<in> UNIV // (\<approx>Lang)" and "Y ;; {[c]} \<subseteq> X" and "s \<in> Y"
 proof -
 def Y \<equiv> "(\<approx>Lang) `` {s}"
 have "Y \<in> UNIV // (\<approx>Lang)"
 unfolding Y_def quotient_def by auto
 moreover
 have "X = (\<approx>Lang) `` {s @ [c]}"
 using has_str in_CS defined_by_str by blast
-then have "Y ; {[c]} \<subseteq> X"
+then have "Y ;; {[c]} \<subseteq> X"
-unfolding Y_def Image_def lang_seq_def
+unfolding Y_def Image_def Seq_def
 unfolding str_eq_rel_def
 by (auto) (simp add: str_eq_def)
 moreover
 have "s \<in> Y" unfolding Y_def
 unfolding Image_def str_eq_rel_def str_eq_def by simp
-moreover
+(*moreover
-have "True" by simp (* FIXME *)
+have "True" by simp FIXME *)
-note that
+ultimately show thesis by (blast intro: that)
-ultimately show thesis by blast
 qed
 lemma test:
 assumes "[] \<in> X"
-shows "[] \<in> L_rexp (Y \<turnstile>\<rightarrow> X)"
+shows "[] \<in> \<lparr>Y \<turnstile>\<rightarrow> X\<rparr>"
 using assms
 by (simp add: transitions_rexp_def)
 lemma rhs_sem:
 assumes "X \<in> (UNIV // (\<approx>Lang))"
 shows "X \<subseteq> rhs_sem (UNIV // (\<approx>Lang)) X"
 apply(case_tac "X = {[]}")
 apply(simp)
-apply(simp add: rhs_sem_def rhs_def lang_seq_def)
+apply(simp add: rhs_sem_def rhs_def Seq_def)
 apply(rule subsetI)
 apply(case_tac "x = []")
 apply(simp add: rhs_sem_def rhs_def)
 apply(rule_tac x = "X" in exI)
 apply(simp)
 apply(rule_tac x = "X" in exI)
 apply(simp add: assms)
 apply(simp add: transitions_rexp_def)
 oops
+(*
+fun
+power :: "string \<Rightarrow> nat \<Rightarrow> string" (infixr "\<Up>" 100)
+where
+"s \<Up> 0 = s"
+| "s \<Up> (Suc n) = s @ (s \<Up> n)"
+definition
+"Lone = {(''0'' \<Up> n) @ (''1'' \<Up> n) | n. True }"
+lemma
+"infinite (UNIV // (\<approx>Lone))"
+unfolding infinite_iff_countable_subset
+apply(rule_tac x="\<lambda>n. {(''0'' \<Up> n) @ (''1'' \<Up> i) | i. i \<in> {..n} }" in exI)
+apply(auto)
+prefer 2
+unfolding Lone_def
+unfolding quotient_def
+unfolding Image_def
+apply(simp)
+unfolding str_eq_rel_def
+unfolding str_eq_def
+apply(auto)
+apply(rule_tac x="''0'' \<Up> n" in exI)
+apply(auto)
+unfolding infinite_nat_iff_unbounded
+unfolding Lone_def
+*)
+text {* Derivatives *}
+definition
+DERS :: "string \<Rightarrow> string set \<Rightarrow> string set"
+where
+"DERS s L \<equiv> {s'. s @ s' \<in> L}"
+lemma
+shows "x \<approx>L y \<longleftrightarrow> DERS x L = DERS y L"
+unfolding DERS_def str_eq_def
+by auto

changeset 24	f72c82bf59e5
parent 22	0792821035b6