theory My
imports Main
begin
definition
lang_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>"
datatype rexp =
NULL
| EMPTY
| CHAR char
| SEQ rexp rexp
| ALT rexp rexp
| STAR rexp
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>"
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)"
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 = [])"
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"
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}"
definition
final :: "string set \<Rightarrow> string set \<Rightarrow> bool"
where
"final X Lang \<equiv> (X \<in> UNIV // \<approx>Lang) \<and> (\<forall>s \<in> X. s \<in> Lang)"
lemma lang_is_union_of_finals:
"Lang = \<Union> {X. final X Lang}"
proof -
have "Lang \<subseteq> \<Union> {X. final X Lang}"
unfolding final_def
unfolding quotient_def Image_def
unfolding str_eq_rel_def
apply(simp)
apply(auto)
apply(rule_tac x="(\<approx>Lang) `` {x}" in exI)
unfolding Image_def
unfolding str_eq_rel_def
apply(auto)
unfolding str_eq_def
apply(auto)
apply(drule_tac x="[]" in spec)
apply(simp)
done
moreover
have "\<Union>{X. final X Lang} \<subseteq> Lang"
unfolding final_def by auto
ultimately
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"
sorry
lemma final_rexp:
"\<lbrakk>finite (UNIV // (\<approx>Lang)); final X Lang\<rbrakk> \<Longrightarrow> \<exists>r. X = L_rexp r"
unfolding final_def
using all_rexp by blast
lemma finite_f_one_to_one:
assumes "finite B"
and "\<forall>x \<in> B. \<exists>y. f y = x"
shows "\<exists>rs. finite rs \<and> (B = {f y | y . y \<in> rs})"
using assms
by (induct) (auto)
lemma finite_final:
assumes "finite (UNIV // (\<approx>Lang))"
shows "finite {X. final X Lang}"
using assms
proof -
have "{X. final X Lang} \<subseteq> (UNIV // (\<approx>Lang))"
unfolding final_def by auto
with assms show "finite {X. final X Lang}"
using finite_subset by auto
qed
lemma finite_regular_aux:
fixes Lang :: "string set"
assumes "finite (UNIV // (\<approx>Lang))"
shows "\<exists>rs. Lang = L_rexp (folds ALT NULL rs)"
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(clarify)
apply(drule final_rexp[OF assms])
apply(auto)[1]
apply(clarify)
apply(rule_tac x="rs" in exI)
apply(simp)
apply(rule set_eqI)
apply(auto)
done
lemma finite_regular:
fixes Lang :: "string set"
assumes "finite (UNIV // (\<approx>Lang))"
shows "\<exists>r. Lang = L_rexp r"
using assms finite_regular_aux
by auto
section {* other direction *}
lemma inj_image_lang:
fixes f::"string \<Rightarrow> 'a"
assumes str_inj: "\<And>x y. f x = f y \<Longrightarrow> x \<approx>Lang y"
shows "inj_on (image f) (UNIV // (\<approx>Lang))"
proof -
{ fix x y::string
assume eq_tag: "f ` {z. x \<approx>Lang z} = f ` {z. y \<approx>Lang z}"
moreover
have "{z. x \<approx>Lang z} \<noteq> {}" unfolding str_eq_def by auto
ultimately obtain a b where "x \<approx>Lang a" "y \<approx>Lang b" "f a = f b" by blast
then have "x \<approx>Lang a" "y \<approx>Lang b" "a \<approx>Lang b" using str_inj by auto
then have "x \<approx>Lang y" unfolding str_eq_def by simp
then have "{z. x \<approx>Lang z} = {z. y \<approx>Lang z}" unfolding str_eq_def by simp
}
then have "\<forall>x\<in>UNIV // \<approx>Lang. \<forall>y\<in>UNIV // \<approx>Lang. f ` x = f ` y \<longrightarrow> x = y"
unfolding quotient_def Image_def str_eq_rel_def by simp
then show "inj_on (image f) (UNIV // (\<approx>Lang))"
unfolding inj_on_def by simp
qed
lemma finite_range_image:
assumes fin: "finite (range f)"
shows "finite ((image f) ` X)"
proof -
from fin have "finite (Pow (f ` UNIV))" by auto
moreover
have "(image f) ` X \<subseteq> Pow (f ` UNIV)" by auto
ultimately show "finite ((image f) ` X)" using finite_subset by auto
qed
definition
tag1 :: "string set \<Rightarrow> string set \<Rightarrow> string \<Rightarrow> (string set \<times> string set)"
where
"tag1 L\<^isub>1 L\<^isub>2 \<equiv> \<lambda>x. ((\<approx>L\<^isub>1) `` {x}, (\<approx>L\<^isub>2) `` {x})"
lemma tag1_range_finite:
assumes finite1: "finite (UNIV // \<approx>L\<^isub>1)"
and finite2: "finite (UNIV // \<approx>L\<^isub>2)"
shows "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
proof -
have "finite (UNIV // \<approx>L\<^isub>1 \<times> UNIV // \<approx>L\<^isub>2)" using finite1 finite2 by auto
moreover
have "range (tag1 L\<^isub>1 L\<^isub>2) \<subseteq> (UNIV // \<approx>L\<^isub>1) \<times> (UNIV // \<approx>L\<^isub>2)"
unfolding tag1_def quotient_def by auto
ultimately show "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
using finite_subset by blast
qed
lemma tag1_inj:
"tag1 L\<^isub>1 L\<^isub>2 x = tag1 L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<approx>(L\<^isub>1 \<union> L\<^isub>2) y"
unfolding tag1_def Image_def str_eq_rel_def str_eq_def
by auto
lemma quot_alt_cu:
fixes L\<^isub>1 L\<^isub>2::"string set"
assumes fin1: "finite (UNIV // \<approx>L\<^isub>1)"
and fin2: "finite (UNIV // \<approx>L\<^isub>2)"
shows "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
proof -
have "finite (range (tag1 L\<^isub>1 L\<^isub>2))"
using fin1 fin2 tag1_range_finite by simp
then have "finite (image (tag1 L\<^isub>1 L\<^isub>2) ` (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2)))"
using finite_range_image by blast
moreover
have "\<And>x y. tag1 L\<^isub>1 L\<^isub>2 x = tag1 L\<^isub>1 L\<^isub>2 y \<Longrightarrow> x \<approx>(L\<^isub>1 \<union> L\<^isub>2) y"
using tag1_inj by simp
then have "inj_on (image (tag1 L\<^isub>1 L\<^isub>2)) (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))"
using inj_image_lang by blast
ultimately
show "finite (UNIV // \<approx>(L\<^isub>1 \<union> L\<^isub>2))" by (rule finite_imageD)
qed
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> 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}"
definition
"eqs CS \<equiv> (\<Union>X \<in> CS. {(X, rhs CS X)})"
definition
"eqs_sem CS \<equiv> (\<Union>X \<in> CS. {(X, rhs_sem 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
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"
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"
unfolding Y_def Image_def lang_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
have "True" by simp (* FIXME *)
note that
ultimately show thesis by blast
qed
lemma test:
assumes "[] \<in> X"
shows "[] \<in> L_rexp (Y \<turnstile>\<rightarrow> X)"
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(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