theory My

imports Main Infinite_Set

begin

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>"

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

  Sem :: "rexp \<Rightarrow> string set" ("\<lparr>_\<rparr>" [0] 1000)

where

    "\<lparr>NULL\<rparr> = {}"

  | "\<lparr>EMPTY\<rparr> = {[]}"

  | "\<lparr>CHAR c\<rparr> = {[c]}"

  | "\<lparr>SEQ r1 r2\<rparr> = \<lparr>r1\<rparr> ;; \<lparr>r2\<rparr>"

  | "\<lparr>ALT r1 r2\<rparr> = \<lparr>r1\<rparr> \<union> \<lparr>r2\<rparr>"

  | "\<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> \<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> \<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]:

  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}"

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 = \<lparr>r\<rparr>"

sorry

lemma final_rexp:

  "\<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"

  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 = \<lparr>folds ALT NULL rs\<rparr>"

apply(subst lang_is_union_of_finals)

using assms

apply -

apply(drule finite_final)

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)

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 =  \<lparr>r\<rparr>"

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;; \<lparr>r\<rparr>) | 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 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 *) 

  ultimately show thesis by (blast intro: that)

qed

lemma test:

  assumes "[] \<in> 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 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