--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys4/posix/GeneralRegexBound.thy	Mon Aug 29 23:16:28 2022 +0100
@@ -0,0 +1,239 @@
+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
+