diff -r dabd25e8e4e9 -r 7a016eeb118d thys2/GeneralRegexBound.thy
--- a/thys2/GeneralRegexBound.thy	Sat Mar 12 14:33:54 2022 +0000
+++ b/thys2/GeneralRegexBound.thy	Tue Mar 15 16:37:41 2022 +0000
@@ -1,5 +1,5 @@
 theory GeneralRegexBound imports 
-"BasicIdentities"
+"BasicIdentities" 
 begin
 
 
@@ -17,19 +17,257 @@
   "SEQ_set A n \<equiv> {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A \<and> rsize r1 + rsize r2 \<le> n}"
 
 definition SEQ_set_cartesian where
-"SEQ_set_cartesian A n  = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"
+"SEQ_set_cartesian A  = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"
 
 definition ALT_set where
 "ALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> sum_list (map rsize rs) \<le> n}"
 
+definition ALTs_set
+  where
+  "ALTs_set A n \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A \<and> sum_list (map rsize rs) \<le> n}"
+
+
+
+lemma alts_set_2defs:
+  shows "ALT_set A n = ALTs_set A n"
+  apply(subgoal_tac "ALT_set A n \<subseteq> ALTs_set A n")
+   apply(subgoal_tac "ALTs_set A n \<subseteq> ALT_set A n")
+    apply auto[1]
+  prefer 2
+  using ALT_set_def ALTs_set_def apply fastforce
+  apply(subgoal_tac "\<forall>r \<in> ALTs_set A n. r \<in> ALT_set A n")
+   apply blast
+  apply(rule ballI)
+  apply(subgoal_tac "\<exists>rs. r = RALTS rs \<and> sum_list (map rsize rs) \<le> n")
+   prefer 2
+  using ALTs_set_def apply fastforce
+  apply(erule exE)
+  apply(subgoal_tac "set rs \<subseteq> A")
+  prefer 2
+  apply (simp add: ALTs_set_def subsetI)
+  using ALT_set_def by blast
+
+
 
 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>
+                         (SEQ_set (sizeNregex n) n)
+                       \<union>  (ALTs_set (sizeNregex n) n))"
+  apply(auto)
+        apply(case_tac x)
+             apply(auto simp add: SEQ_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: ALTs_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: ALTs_set_def)
+  apply(simp add: sizeNregex_def)
+  apply(auto)
+  using ex_in_conv by fastforce
+
+lemma sizeN_inclusion:
+  shows "sizeNregex n   \<subseteq> sizeNregex (Suc n)"
+  by (simp add: Collect_mono sizeNregex_def)
+
+lemma ralts_nil_in_altset:
+  shows " RALTS [] \<in> ALT_set (sizeNregex n) n "
+  using ALT_set_def by auto
+
+
 lemma sizenregex_induct:
   shows "sizeNregex (Suc n) = sizeNregex n \<union> {RZERO, RONE, RALTS []} \<union> {RCHAR c| c. True} \<union>
 SEQ_set ( sizeNregex n) n \<union> ALT_set (sizeNregex n) n \<union> (RSTAR ` (sizeNregex n))"
-  sorry
+  apply(subgoal_tac "sizeNregex (Suc n) =  {RZERO, RONE, RALTS []} \<union> {RCHAR c| c. True} \<union>
+SEQ_set ( sizeNregex n) n \<union> ALT_set (sizeNregex n) n \<union> (RSTAR ` (sizeNregex n))")
+  using sizeN_inclusion apply blast
+  apply(subgoal_tac " {RZERO, RONE, RALTS []} \<union> {RCHAR c |c. True} \<union> SEQ_set (sizeNregex n) n \<union>
+    ALT_set (sizeNregex n) n \<union>
+    RSTAR ` sizeNregex n =  (({RZERO, RONE} \<union> {RCHAR c| c. True}) 
+                       \<union> (RSTAR ` sizeNregex n) \<union>
+                         (SEQ_set (sizeNregex n) n)
+                       \<union>  (ALTs_set (sizeNregex n) n))")
+  using sizenregex_induct1 apply presburger
+  apply(subgoal_tac "{RZERO, RONE} \<union> {RCHAR c |c. True} \<union> SEQ_set (sizeNregex n) n \<union>
+    ALT_set (sizeNregex n) n \<union>
+    RSTAR ` sizeNregex n =
+    {RZERO, RONE} \<union> {RCHAR c |c. True} \<union> RSTAR ` sizeNregex n \<union> SEQ_set (sizeNregex n) n \<union>
+    ALTs_set (sizeNregex n) n ")
+  prefer 2
+  using alts_set_2defs apply auto[1]
+  apply(subgoal_tac " {RZERO, RONE, RALTS []} \<union> {RCHAR c |c. True} \<union> SEQ_set (sizeNregex n) n \<union>
+    ALT_set (sizeNregex n) n \<union>
+    RSTAR ` sizeNregex n = 
+ {RZERO, RONE} \<union> {RCHAR c |c. True} \<union> SEQ_set (sizeNregex n) n \<union>
+    (insert (RALTS []) (ALT_set (sizeNregex n) n)) \<union>
+    RSTAR ` sizeNregex n")
+  prefer 2
+  apply fastforce
+  by (simp add: insert_absorb ralts_nil_in_altset)
+
+
+
+lemma s4:
+  "SEQ_set A n \<subseteq> SEQ_set_cartesian A"
+  using SEQ_set_cartesian_def SEQ_set_def by fastforce
+
+lemma s5:
+  "finite A \<Longrightarrow> finite (SEQ_set_cartesian A)"
+  apply(subgoal_tac "SEQ_set_cartesian A = (\<lambda>(x1, x2). RSEQ x1 x2) ` (A \<times> A)")
+  apply simp
+  unfolding SEQ_set_cartesian_def
+  apply(auto)
+  done
+
+thm size_list_def
+
+definition ALTs_set_length
+  where
+  "ALTs_set_length A n l \<equiv> {RALTS rs | rs. \<forall>r \<in> set rs. r \<in> A 
+                                      \<and> sum_list (map rsize rs) \<le> n 
+                                      \<and> length rs \<le> l}"
+
+
+definition ALTs_set_length2
+  where
+  "ALTs_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 "ALTs_set_length A n l \<subseteq> ALTs_set_length2 A l"
+  apply(auto simp add: ALTs_set_length2_def ALTs_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 "sum_list (map rsize 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 " sum_list (map rsize rs) \<le> n \<Longrightarrow> length rs \<le> n"
+  by (meson order.trans size_sum_more_than_len)
+
+  
+
+
+lemma t2:
+  shows "ALTs_set A n \<subseteq> ALTs_set_length A n n"
+  unfolding ALTs_set_length_def ALTs_set_def
+  apply(auto)
+  using sum_list_len by blast
+
+  
+thm ALTs_set_def
+
+lemma s8_aux:
+  assumes "finite A" 
+  shows "finite (ALTs_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)) = ALTs_set_length2 A n"
+    unfolding ALTs_set_length2_def set_length2_def
+    by (auto)
+  ultimately have "finite (ALTs_set_length2 A n)"
+    by (metis finite_imageI)
+  then show ?thesis
+    by (metis infinite_super r000)
+qed
+
+lemma s1:
+  shows "{r::rrexp . rsize r = 1} = ({RZERO, RONE, RALTS []} \<union> {RCHAR c| c. True})"
+  apply(auto)
+  apply(case_tac x)
+  apply(simp_all)
+    apply (metis One_nat_def Suc_n_not_le_n size_geq1)
+  apply (metis One_nat_def Suc_n_not_le_n ex_in_conv set_empty2 size_geq1)
+  by (metis not_one_le_zero size_geq1)
+
+
+
+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 chars_finite:
@@ -53,42 +291,33 @@
    apply(simp add: full_SetCompr_eq)
   using non_zero_size not_less_eq_eq sizeNregex_def by fastforce
 
-lemma seq_included_in_cart:
-  shows "SEQ_set A n \<subseteq> SEQ_set_cartesian A n"
-  using SEQ_set_cartesian_def SEQ_set_def by fastforce
-
-lemma finite_seq:
-  shows " finite (sizeNregex n) \<Longrightarrow> finite (SEQ_set (sizeNregex n) n)"
-  apply(rule finite_subset)
-  sorry
-
-
-lemma finite_size_n:
-  shows "finite (sizeNregex n)"
-  apply(induct n)
-  apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
-  apply(subst sizenregex_induct)
-  apply(subst finite_Un)+
-  apply(rule conjI)+
-       apply simp
-      apply simp
-     apply (simp add: full_SetCompr_eq)
-
-  sorry
 
 lemma three_easy_cases0: shows 
 "rsize (rders_simp RZERO s) \<le> Suc 0"
-  sorry
+  apply(induct s)
+   apply simp
+  apply simp
+  done
 
 
 lemma three_easy_cases1: shows 
 "rsize (rders_simp RONE s) \<le> Suc 0"
-  sorry
+    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"
-
-  sorry
+  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
+