--- a/thys3/GeneralRegexBound.thy	Sat Apr 30 00:50:08 2022 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,212 +0,0 @@
-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
-  "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))"
-  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: sizeNregex_def)
-  using sizeNregex_def apply force
-  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 by fastforce
-
-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)
-
-
-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)
-  done
-
-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
-