theory GeneralRegexBound 

  imports "BasicIdentities" 

begin

lemma size_geq1:

  shows "rsize r \<ge> 1"

  by (induct r) auto 

definition RSEQ_set where

  "RSEQ_set A n \<equiv> {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A \<and> rsize r1 + rsize r2 \<le> n}"

definition RSEQ_set_cartesian where

  "RSEQ_set_cartesian A  = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"

definition RALT_set where

  "RALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> rsizes rs \<le> n}"

definition RALTs_set where

  "RALTs_set A n \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> rsizes rs \<le> n}"

definition RNTIMES_set where

  "RNTIMES_set A n \<equiv> {RNTIMES r m | m r. r \<in> A \<and> rsize r + m \<le> n}"

definition

  "sizeNregex N \<equiv> {r. rsize r \<le> N}"

lemma sizenregex_induct1:

  "sizeNregex (Suc n) = (({RZERO, RONE} \<union> {RCHAR c| c. True}) 

                         \<union> (RSTAR ` sizeNregex n) 

                         \<union> (RSEQ_set (sizeNregex n) n)

                         \<union> (RALTs_set (sizeNregex n) n))

                         \<union> (RNTIMES_set (sizeNregex n) n)"

  apply(auto)

        apply(case_tac x)

             apply(auto simp add: RSEQ_set_def)

  using sizeNregex_def apply force

  using sizeNregex_def apply auto[1]

  apply (simp add: sizeNregex_def)

         apply (simp add: sizeNregex_def)

         apply (simp add: RALTs_set_def)

  apply (metis imageI list.set_map member_le_sum_list order_trans)

  apply (simp add: sizeNregex_def)

        apply (simp add: sizeNregex_def)

  apply (simp add: RNTIMES_set_def)

  apply (simp add: sizeNregex_def)

  using sizeNregex_def apply force

  apply (simp add: sizeNregex_def)

  apply (simp add: sizeNregex_def)

  apply (simp add: sizeNregex_def)

    apply (simp add: RALTs_set_def)

  apply(simp add: sizeNregex_def)

  apply(auto)

  using ex_in_conv apply fastforce

  apply (simp add: RNTIMES_set_def)

  apply(simp add: sizeNregex_def)

  by force

lemma s4:

  "RSEQ_set A n \<subseteq> RSEQ_set_cartesian A"

  using RSEQ_set_cartesian_def RSEQ_set_def by fastforce

lemma s5:

  assumes "finite A"

  shows "finite (RSEQ_set_cartesian A)"

  using assms

  apply(subgoal_tac "RSEQ_set_cartesian A = (\<lambda>(x1, x2). RSEQ x1 x2) ` (A \<times> A)")

  apply simp

  unfolding RSEQ_set_cartesian_def

  apply(auto)

  done

definition RALTs_set_length

  where

  "RALTs_set_length A n l \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> rsizes rs \<le> n \<and> length rs \<le> l}"

definition RALTs_set_length2

  where

  "RALTs_set_length2 A l \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> length rs \<le> l}"

definition set_length2

  where

  "set_length2 A l \<equiv> {rs. \<forall>r \<in> set rs. r \<in> A \<and> length rs \<le> l}"

lemma r000: 

  shows "RALTs_set_length A n l \<subseteq> RALTs_set_length2 A l"

  apply(auto simp add: RALTs_set_length2_def RALTs_set_length_def)

  done

lemma r02: 

  shows "set_length2 A 0 \<subseteq> {[]}"

  apply(auto simp add: set_length2_def)

  apply(case_tac x)

  apply(auto)

  done

lemma r03:

  shows "set_length2 A (Suc n) \<subseteq> 

          {[]} \<union> (\<lambda>(h, t). h # t) ` (A \<times> (set_length2 A n))"

  apply(auto simp add: set_length2_def)

  apply(case_tac x)

   apply(auto)

  done

lemma r1:

  assumes "finite A" 

  shows "finite (set_length2 A n)"

  using assms

  apply(induct n)

  apply(rule finite_subset)

    apply(rule r02)

   apply(simp)    

  apply(rule finite_subset)

   apply(rule r03)

  apply(simp)

  done

lemma size_sum_more_than_len:

  shows "rsizes rs \<ge> length rs"

  apply(induct rs)

   apply simp

  apply simp

  apply(subgoal_tac "rsize a \<ge> 1")

   apply linarith

  using size_geq1 by auto

lemma sum_list_len:

  shows "rsizes rs \<le> n \<Longrightarrow> length rs \<le> n"

  by (meson order.trans size_sum_more_than_len)

lemma t2:

  shows "RALTs_set A n \<subseteq> RALTs_set_length A n n"

  unfolding RALTs_set_length_def RALTs_set_def

  apply(auto)

  using sum_list_len by blast

lemma s8_aux:

  assumes "finite A" 

  shows "finite (RALTs_set_length A n n)"

proof -

  have "finite A" by fact

  then have "finite (set_length2 A n)"

    by (simp add: r1)

  moreover have "(RALTS ` (set_length2 A n)) = RALTs_set_length2 A n"

    unfolding RALTs_set_length2_def set_length2_def

    by (auto)

  ultimately have "finite (RALTs_set_length2 A n)"

    by (metis finite_imageI)

  then show ?thesis

    by (metis infinite_super r000)

qed

lemma char_finite:

  shows "finite  {RCHAR c |c. True}"

  apply simp

  apply(subgoal_tac "finite (RCHAR ` (UNIV::char set))")

   prefer 2

   apply simp

  by (simp add: full_SetCompr_eq)

thm RNTIMES_set_def

lemma s9_aux0:

  shows "RNTIMES_set (insert r A) n \<subseteq> RNTIMES_set A n \<union> (\<Union> i \<in> {..n}. {RNTIMES r i})"

apply(auto simp add: RNTIMES_set_def)

  done

lemma s9_aux:

  assumes "finite A"

  shows "finite (RNTIMES_set A n)"

  using assms

  apply(induct A arbitrary: n)

   apply(auto simp add: RNTIMES_set_def)[1]

  apply(subgoal_tac "finite (RNTIMES_set F n \<union> (\<Union> i \<in> {..n}. {RNTIMES x i}))")

  apply (metis finite_subset s9_aux0)

  by blast

lemma finite_size_n:

  shows "finite (sizeNregex n)"

  apply(induct n)

   apply(simp add: sizeNregex_def)

  apply (metis (mono_tags, lifting) not_finite_existsD not_one_le_zero size_geq1)

  apply(subst sizenregex_induct1)

  apply(simp only: finite_Un)

  apply(rule conjI)+

  apply(simp)

  using char_finite apply blast

    apply(simp)

   apply(rule finite_subset)

    apply(rule s4)

   apply(rule s5)

   apply(simp)

  apply(rule finite_subset)

   apply(rule t2)

  apply(rule s8_aux)

   apply(simp)

  by (simp add: s9_aux)

lemma three_easy_cases0: 

  shows "rsize (rders_simp RZERO s) \<le> Suc 0"

  apply(induct s)

   apply simp

  apply simp

  done

lemma three_easy_cases1: 

  shows "rsize (rders_simp RONE s) \<le> Suc 0"

    apply(induct s)

   apply simp

  apply simp

  using three_easy_cases0 by auto

lemma three_easy_casesC: 

  shows "rsize (rders_simp (RCHAR c) s) \<le> Suc 0"

  apply(induct s)

   apply simp

  apply simp

  apply(case_tac " a = c")

  using three_easy_cases1 apply blast

  apply simp

  using three_easy_cases0 by force

unused_thms

end