(*  Author: Tobias Nipkow, Alex Krauss  *)

header "Regular sets"

theory Regular_Set

imports Main

begin

type_synonym 'a lang = "'a list set"

definition conc :: "'a lang \<Rightarrow> 'a lang \<Rightarrow> 'a lang" (infixr "@@" 75) where

"A @@ B = {xs@ys | xs ys. xs:A & ys:B}"

overloading lang_pow == "compow :: nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang"

begin

  primrec lang_pow :: "nat \<Rightarrow> 'a lang \<Rightarrow> 'a lang" where

  "lang_pow 0 A = {[]}" |

  "lang_pow (Suc n) A = A @@ (lang_pow n A)"

end

definition star :: "'a lang \<Rightarrow> 'a lang" where

"star A = (\<Union>n. A ^^ n)"

subsection{* @{term "op @@"} *}

lemma concI[simp,intro]: "u : A \<Longrightarrow> v : B \<Longrightarrow> u@v : A @@ B"

by (auto simp add: conc_def)

lemma concE[elim]: 

assumes "w \<in> A @@ B"

obtains u v where "u \<in> A" "v \<in> B" "w = u@v"

using assms by (auto simp: conc_def)

lemma conc_mono: "A \<subseteq> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> A @@ B \<subseteq> C @@ D"

by (auto simp: conc_def) 

lemma conc_empty[simp]: shows "{} @@ A = {}" and "A @@ {} = {}"

by auto

lemma conc_epsilon[simp]: shows "{[]} @@ A = A" and "A @@ {[]} = A"

by (simp_all add:conc_def)

lemma conc_assoc: "(A @@ B) @@ C = A @@ (B @@ C)"

by (auto elim!: concE) (simp only: append_assoc[symmetric] concI)

lemma conc_Un_distrib:

shows "A @@ (B \<union> C) = A @@ B \<union> A @@ C"

and   "(A \<union> B) @@ C = A @@ C \<union> B @@ C"

by auto

lemma conc_UNION_distrib:

shows "A @@ UNION I M = UNION I (%i. A @@ M i)"

and   "UNION I M @@ A = UNION I (%i. M i @@ A)"

by auto

subsection{* @{term "A ^^ n"} *}

lemma lang_pow_add: "A ^^ (n + m) = A ^^ n @@ A ^^ m"

by (induct n) (auto simp: conc_assoc)

lemma lang_pow_empty: "{} ^^ n = (if n = 0 then {[]} else {})"

by (induct n) auto

lemma lang_pow_empty_Suc[simp]: "({}::'a lang) ^^ Suc n = {}"

by (simp add: lang_pow_empty)

lemma conc_pow_comm:

  shows "A @@ (A ^^ n) = (A ^^ n) @@ A"

by (induct n) (simp_all add: conc_assoc[symmetric])

lemma length_lang_pow_ub:

  "ALL w : A. length w \<le> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<le> k*n"

by(induct n arbitrary: w) (fastsimp simp: conc_def)+

lemma length_lang_pow_lb:

  "ALL w : A. length w \<ge> k \<Longrightarrow> w : A^^n \<Longrightarrow> length w \<ge> k*n"

by(induct n arbitrary: w) (fastsimp simp: conc_def)+

subsection{* @{const star} *}

lemma star_if_lang_pow[simp]: "w : A ^^ n \<Longrightarrow> w : star A"

by (auto simp: star_def)

lemma Nil_in_star[iff]: "[] : star A"

proof (rule star_if_lang_pow)

  show "[] : A ^^ 0" by simp

qed

lemma star_if_lang[simp]: assumes "w : A" shows "w : star A"

proof (rule star_if_lang_pow)

  show "w : A ^^ 1" using `w : A` by simp

qed

lemma append_in_starI[simp]:

assumes "u : star A" and "v : star A" shows "u@v : star A"

proof -

  from `u : star A` obtain m where "u : A ^^ m" by (auto simp: star_def)

  moreover

  from `v : star A` obtain n where "v : A ^^ n" by (auto simp: star_def)

  ultimately have "u@v : A ^^ (m+n)" by (simp add: lang_pow_add)

  thus ?thesis by simp

qed

lemma conc_star_star: "star A @@ star A = star A"

by (auto simp: conc_def)

lemma conc_star_comm:

  shows "A @@ star A = star A @@ A"

unfolding star_def conc_pow_comm conc_UNION_distrib

by simp

lemma star_induct[consumes 1, case_names Nil append, induct set: star]:

assumes "w : star A"

  and "P []"

  and step: "!!u v. u : A \<Longrightarrow> v : star A \<Longrightarrow> P v \<Longrightarrow> P (u@v)"

shows "P w"

proof -

  { fix n have "w : A ^^ n \<Longrightarrow> P w"

    by (induct n arbitrary: w) (auto intro: `P []` step star_if_lang_pow) }

  with `w : star A` show "P w" by (auto simp: star_def)

qed

lemma star_empty[simp]: "star {} = {[]}"

by (auto elim: star_induct)

lemma star_epsilon[simp]: "star {[]} = {[]}"

by (auto elim: star_induct)

lemma star_idemp[simp]: "star (star A) = star A"

by (auto elim: star_induct)

lemma star_unfold_left: "star A = A @@ star A \<union> {[]}" (is "?L = ?R")

proof

  show "?L \<subseteq> ?R" by (rule, erule star_induct) auto

qed auto

lemma concat_in_star: "set ws \<subseteq> A \<Longrightarrow> concat ws : star A"

by (induct ws) simp_all

lemma in_star_iff_concat:

  "w : star A = (EX ws. set ws \<subseteq> A & w = concat ws)"

  (is "_ = (EX ws. ?R w ws)")

proof

  assume "w : star A" thus "EX ws. ?R w ws"

  proof induct

    case Nil have "?R [] []" by simp

    thus ?case ..

  next

    case (append u v)

    moreover

    then obtain ws where "set ws \<subseteq> A \<and> v = concat ws" by blast

    ultimately have "?R (u@v) (u#ws)" by auto

    thus ?case ..

  qed

next

  assume "EX us. ?R w us" thus "w : star A"

  by (auto simp: concat_in_star)

qed

lemma star_conv_concat: "star A = {concat ws|ws. set ws \<subseteq> A}"

by (fastsimp simp: in_star_iff_concat)

lemma star_insert_eps[simp]: "star (insert [] A) = star(A)"

proof-

  { fix us

    have "set us \<subseteq> insert [] A \<Longrightarrow> EX vs. concat us = concat vs \<and> set vs \<subseteq> A"

      (is "?P \<Longrightarrow> EX vs. ?Q vs")

    proof

      let ?vs = "filter (%u. u \<noteq> []) us"

      show "?P \<Longrightarrow> ?Q ?vs" by (induct us) auto

    qed

  } thus ?thesis by (auto simp: star_conv_concat)

qed

lemma star_decom: 

  assumes a: "x \<in> star A" "x \<noteq> []"

  shows "\<exists>a b. x = a @ b \<and> a \<noteq> [] \<and> a \<in> A \<and> b \<in> star A"

using a by (induct rule: star_induct) (blast)+

subsection {* Arden's Lemma *}

lemma arden_helper:

  assumes eq: "X = A @@ X \<union> B"

  shows "X = (A ^^ Suc n) @@ X \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)"

proof (induct n)

  case 0 

  show "X = (A ^^ Suc 0) @@ X \<union> (\<Union>m\<le>0. (A ^^ m) @@ B)"

    using eq by simp

next

  case (Suc n)

  have ih: "X = (A ^^ Suc n) @@ X \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)" by fact

  also have "\<dots> = (A ^^ Suc n) @@ (A @@ X \<union> B) \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)" using eq by simp

  also have "\<dots> = (A ^^ Suc (Suc n)) @@ X \<union> ((A ^^ Suc n) @@ B) \<union> (\<Union>m\<le>n. (A ^^ m) @@ B)"

    by (simp add: conc_Un_distrib conc_assoc[symmetric] conc_pow_comm)

  also have "\<dots> = (A ^^ Suc (Suc n)) @@ X \<union> (\<Union>m\<le>Suc n. (A ^^ m) @@ B)"

    by (auto simp add: le_Suc_eq)

  finally show "X = (A ^^ Suc (Suc n)) @@ X \<union> (\<Union>m\<le>Suc n. (A ^^ m) @@ B)" .

qed

lemma Arden:

  assumes "[] \<notin> A" 

  shows "X = A @@ X \<union> B \<longleftrightarrow> X = star A @@ B"

proof

  assume eq: "X = A @@ X \<union> B"

  { fix w assume "w : X"

    let ?n = "size w"

    from `[] \<notin> A` have "ALL u : A. length u \<ge> 1"

      by (metis Suc_eq_plus1 add_leD2 le_0_eq length_0_conv not_less_eq_eq)

    hence "ALL u : A^^(?n+1). length u \<ge> ?n+1"

      by (metis length_lang_pow_lb nat_mult_1)

    hence "ALL u : A^^(?n+1)@@X. length u \<ge> ?n+1"

      by(auto simp only: conc_def length_append)

    hence "w \<notin> A^^(?n+1)@@X" by auto

    hence "w : star A @@ B" using `w : X` using arden_helper[OF eq, where n="?n"]

      by (auto simp add: star_def conc_UNION_distrib)

  } moreover

  { fix w assume "w : star A @@ B"

    hence "EX n. w : A^^n @@ B" by(auto simp: conc_def star_def)

    hence "w : X" using arden_helper[OF eq] by blast

  } ultimately show "X = star A @@ B" by blast 

next

  assume eq: "X = star A @@ B"

  have "star A = A @@ star A \<union> {[]}"

    by (rule star_unfold_left)

  then have "star A @@ B = (A @@ star A \<union> {[]}) @@ B"

    by metis

  also have "\<dots> = (A @@ star A) @@ B \<union> B"

    unfolding conc_Un_distrib by simp

  also have "\<dots> = A @@ (star A @@ B) \<union> B" 

    by (simp only: conc_assoc)

  finally show "X = A @@ X \<union> B" 

    using eq by blast 

qed

lemma reversed_arden_helper:

  assumes eq: "X = X @@ A \<union> B"

  shows "X = X @@ (A ^^ Suc n) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))"

proof (induct n)

  case 0 

  show "X = X @@ (A ^^ Suc 0) \<union> (\<Union>m\<le>0. B @@ (A ^^ m))"

    using eq by simp

next

  case (Suc n)

  have ih: "X = X @@ (A ^^ Suc n) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))" by fact

  also have "\<dots> = (X @@ A \<union> B) @@ (A ^^ Suc n) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))" using eq by simp

  also have "\<dots> = X @@ (A ^^ Suc (Suc n)) \<union> (B @@ (A ^^ Suc n)) \<union> (\<Union>m\<le>n. B @@ (A ^^ m))"

    by (simp add: conc_Un_distrib conc_assoc)

  also have "\<dots> = X @@ (A ^^ Suc (Suc n)) \<union> (\<Union>m\<le>Suc n. B @@ (A ^^ m))"

    by (auto simp add: le_Suc_eq)

  finally show "X = X @@ (A ^^ Suc (Suc n)) \<union> (\<Union>m\<le>Suc n. B @@ (A ^^ m))" .

qed

theorem reversed_Arden:

  assumes nemp: "[] \<notin> A"

  shows "X = X @@ A \<union> B \<longleftrightarrow> X = B @@ star A"

proof

 assume eq: "X = X @@ A \<union> B"

  { fix w assume "w : X"

    let ?n = "size w"

    from `[] \<notin> A` have "ALL u : A. length u \<ge> 1"

      by (metis Suc_eq_plus1 add_leD2 le_0_eq length_0_conv not_less_eq_eq)

    hence "ALL u : A^^(?n+1). length u \<ge> ?n+1"

      by (metis length_lang_pow_lb nat_mult_1)

    hence "ALL u : X @@ A^^(?n+1). length u \<ge> ?n+1"

      by(auto simp only: conc_def length_append)

    hence "w \<notin> X @@ A^^(?n+1)" by auto

    hence "w : B @@ star A" using `w : X` using reversed_arden_helper[OF eq, where n="?n"]

      by (auto simp add: star_def conc_UNION_distrib)

  } moreover

  { fix w assume "w : B @@ star A"

    hence "EX n. w : B @@ A^^n" by (auto simp: conc_def star_def)

    hence "w : X" using reversed_arden_helper[OF eq] by blast

  } ultimately show "X = B @@ star A" by blast 

next 

  assume eq: "X = B @@ star A"

  have "star A = {[]} \<union> star A @@ A" 

    unfolding conc_star_comm[symmetric]

    by(metis Un_commute star_unfold_left)

  then have "B @@ star A = B @@ ({[]} \<union> star A @@ A)"

    by metis

  also have "\<dots> = B \<union> B @@ (star A @@ A)"

    unfolding conc_Un_distrib by simp

  also have "\<dots> = B \<union> (B @@ star A) @@ A" 

    by (simp only: conc_assoc)

  finally show "X = X @@ A \<union> B" 

    using eq by blast 

qed

end