(* Author: Christian Urban, Xingyuan Zhang, Chunhan Wu *)

theory Closures

imports Derivatives

begin

section {* Closure properties of regular languages *}

abbreviation

  regular :: "'a lang \<Rightarrow> bool"

where

  "regular A \<equiv> \<exists>r. A = lang r"

subsection {* Closure under set operations *}

lemma closure_union [intro]:

  assumes "regular A" "regular B" 

  shows "regular (A \<union> B)"

proof -

  from assms obtain r1 r2::"'a rexp" where "lang r1 = A" "lang r2 = B" by auto

  then have "A \<union> B = lang (Plus r1 r2)" by simp

  then show "regular (A \<union> B)" by blast

qed

lemma closure_seq [intro]:

  assumes "regular A" "regular B" 

  shows "regular (A \<cdot> B)"

proof -

  from assms obtain r1 r2::"'a rexp" where "lang r1 = A" "lang r2 = B" by auto

  then have "A \<cdot> B = lang (Times r1 r2)" by simp

  then show "regular (A \<cdot> B)" by blast

qed

lemma closure_star [intro]:

  assumes "regular A"

  shows "regular (A\<star>)"

proof -

  from assms obtain r::"'a rexp" where "lang r = A" by auto

  then have "A\<star> = lang (Star r)" by simp

  then show "regular (A\<star>)" by blast

qed

text {* Closure under complementation is proved via the 

  Myhill-Nerode theorem *}

lemma closure_complement [intro]:

  fixes A::"('a::finite) lang"

  assumes "regular A"

  shows "regular (- A)"

proof -

  from assms have "finite (UNIV // \<approx>A)" by (simp add: Myhill_Nerode)

  then have "finite (UNIV // \<approx>(-A))" by (simp add: str_eq_def)

  then show "regular (- A)" by (simp add: Myhill_Nerode)

qed

lemma closure_difference [intro]:

  fixes A::"('a::finite) lang"

  assumes "regular A" "regular B" 

  shows "regular (A - B)"

proof -

  have "A - B = - (- A \<union> B)" by blast

  moreover

  have "regular (- (- A \<union> B))" 

    using assms by blast

  ultimately show "regular (A - B)" by simp

qed

lemma closure_intersection [intro]:

  fixes A::"('a::finite) lang"

  assumes "regular A" "regular B" 

  shows "regular (A \<inter> B)"

proof -

  have "A \<inter> B = - (- A \<union> - B)" by blast

  moreover

  have "regular (- (- A \<union> - B))" 

    using assms by blast

  ultimately show "regular (A \<inter> B)" by simp

qed

subsection {* Closure under string reversal *}

fun

  Rev :: "'a rexp \<Rightarrow> 'a rexp"

where

  "Rev Zero = Zero"

| "Rev One = One"

| "Rev (Atom c) = Atom c"

| "Rev (Plus r1 r2) = Plus (Rev r1) (Rev r2)"

| "Rev (Times r1 r2) = Times (Rev r2) (Rev r1)"

| "Rev (Star r) = Star (Rev r)"

lemma rev_seq[simp]:

  shows "rev ` (B \<cdot> A) = (rev ` A) \<cdot> (rev ` B)"

unfolding conc_def image_def

by (auto) (metis rev_append)+

lemma rev_star1:

  assumes a: "s \<in> (rev ` A)\<star>"

  shows "s \<in> rev ` (A\<star>)"

using a

proof(induct rule: star_induct)

  case (append s1 s2)

  have inj: "inj (rev::'a list \<Rightarrow> 'a list)" unfolding inj_on_def by auto

  have "s1 \<in> rev ` A" "s2 \<in> rev ` (A\<star>)" by fact+

  then obtain x1 x2 where "x1 \<in> A" "x2 \<in> A\<star>" and eqs: "s1 = rev x1" "s2 = rev x2" by auto

  then have "x1 \<in> A\<star>" "x2 \<in> A\<star>" by (auto)

  then have "x2 @ x1 \<in> A\<star>" by (auto)

  then have "rev (x2 @ x1) \<in> rev ` A\<star>" using inj by (simp only: inj_image_mem_iff)

  then show "s1 @ s2 \<in>  rev ` A\<star>" using eqs by simp

qed (auto)

lemma rev_star2:

  assumes a: "s \<in> A\<star>"

  shows "rev s \<in> (rev ` A)\<star>"

using a

proof(induct rule: star_induct)

  case (append s1 s2)

  have inj: "inj (rev::'a list \<Rightarrow> 'a list)" unfolding inj_on_def by auto

  have "s1 \<in> A"by fact

  then have "rev s1 \<in> rev ` A" using inj by (simp only: inj_image_mem_iff)

  then have "rev s1 \<in> (rev ` A)\<star>" by (auto)

  moreover

  have "rev s2 \<in> (rev ` A)\<star>" by fact

  ultimately show "rev (s1 @ s2) \<in>  (rev ` A)\<star>" by (auto)

qed (auto)

lemma rev_star [simp]:

  shows " rev ` (A\<star>) = (rev ` A)\<star>"

using rev_star1 rev_star2 by auto

lemma rev_lang:

  shows "rev ` (lang r) = lang (Rev r)"

by (induct r) (simp_all add: image_Un)

lemma closure_reversal [intro]:

  assumes "regular A"

  shows "regular (rev ` A)"

proof -

  from assms obtain r::"'a rexp" where "A = lang r" by auto

  then have "lang (Rev r) = rev ` A" by (simp add: rev_lang)

  then show "regular (rev` A)" by blast

qed

subsection {* Closure under left-quotients *}

abbreviation

  "Ders_lang A B \<equiv> \<Union>x \<in> A. Ders x B"

lemma closure_left_quotient:

  assumes "regular A"

  shows "regular (Ders_lang B A)"

proof -

  from assms obtain r::"'a rexp" where eq: "lang r = A" by auto

  have fin: "finite (pders_lang B r)" by (rule finite_pders_lang)

  have "Ders_lang B (lang r) = (\<Union> lang ` (pders_lang B r))"

    by (simp add: Ders_pders pders_lang_def)

  also have "\<dots> = lang (\<Uplus>(pders_lang B r))" using fin by simp

  finally have "Ders_lang B A = lang (\<Uplus>(pders_lang B r))" using eq

    by simp

  then show "regular (Ders_lang B A)" by auto

qed

end