--- a/thys2/SizeBound6CT.thy	Wed Mar 02 23:53:11 2022 +0000
+++ b/thys2/SizeBound6CT.thy	Sat Mar 05 11:31:59 2022 +0000
@@ -116,16 +116,11 @@
 
 *)
 
-datatype cchar = 
-Achar
-|Bchar
-|Cchar
-|Dchar
 
 datatype rrexp = 
   RZERO
 | RONE 
-| RCHAR cchar
+| RCHAR char
 | RSEQ rrexp rrexp
 | RALTS "rrexp list"
 | RSTAR rrexp
@@ -145,19 +140,13 @@
 | "rnullable (RSEQ  r1 r2) = (rnullable r1 \<and> rnullable r2)"
 | "rnullable (RSTAR   r) = True"
 
-fun convert_cchar_char :: "cchar \<Rightarrow> char"
-  where
-"convert_cchar_char Achar = (CHR 0x41) "
-| "convert_cchar_char Bchar = (CHR 0x42) "
-| "convert_cchar_char Cchar = (CHR 0x43) "
-| "convert_cchar_char Dchar = (CHR 0x44) "
 
 fun
  rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
 where
   "rder c (RZERO) = RZERO"
 | "rder c (RONE) = RZERO"
-| "rder c (RCHAR d) = (if c = (convert_cchar_char d) then RONE else RZERO)"
+| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
 | "rder c (RALTS rs) = RALTS (map (rder c) rs)"
 | "rder c (RSEQ r1 r2) = 
      (if rnullable r1
@@ -879,11 +868,17 @@
 |"rexp_enum (Suc n) = [(RSEQ r1 r2). r1 \<in> set (rexp_enum i) \<and> r2 \<in> set (rexp_enum j) \<and> i + j = n]"
 
 *)
+definition SEQ_set where
+  "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 ALT_set where
+"ALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> sum_list (map rsize rs) \<le> n}"
+
 context notes rev_conj_cong[fundef_cong] begin
 function (sequential) rexp_enum :: "nat \<Rightarrow> rrexp set"
   where 
 "rexp_enum 0 = {}"
-|"rexp_enum (Suc 0) =  {RALTS []} \<union> {RZERO} \<union> {RONE} \<union> { (RCHAR c) |c. c \<in>{Achar, Bchar, Cchar, Dchar} }"
+|"rexp_enum (Suc 0) =  {RALTS []} \<union> {RZERO} \<union> {RONE} \<union> { (RCHAR c) |c::char. True }"
 |"rexp_enum (Suc n) = {(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = n} \<union>
 {(RALTS (a # rs)) | a rs i j. a \<in> (rexp_enum i) \<and> (RALTS rs) \<in> (rexp_enum j) \<and> i + j = Suc n \<and> i \<le> n \<and> j \<le> n} \<union>
 {RSTAR r0 | r0. r0 \<in> (rexp_enum n)} \<union>
@@ -904,16 +899,146 @@
   shows "n \<le> m \<Longrightarrow> (rexp_enum n) \<subseteq> (rexp_enum m)"
   by (simp add: lift_Suc_mono_le rexp_enum_inclusion)
 
+lemma zero_in_Suc0:
+  shows "RZERO \<in> rexp_enum (Suc 0)"
+and "RZERO \<in> rexp_enum 1"
+  apply simp
+  by simp
+
+lemma one_in_Suc0:
+  shows "RONE \<in> rexp_enum (Suc 0)"
+and "RONE \<in> rexp_enum 1"
+   apply simp
+  by simp
+
+lemma char_in_Suc0:
+  shows "RCHAR c \<in> rexp_enum (Suc 0)"
+  apply simp
+  done
+
+
+lemma char_in1:
+  shows "RCHAR c \<in> rexp_enum 1"
+  using One_nat_def char_in_Suc0 by presburger
+
+lemma alts_nil_in_Suc0:
+  shows "RALTS [] \<in> rexp_enum (Suc 0)"
+  and "RALTS [] \<in> rexp_enum 1"
+  apply simp
+  by simp
+
+
+lemma zero_in_positive:
+  shows "RZERO \<in> rexp_enum (Suc N)"
+  by (metis le_add1 plus_1_eq_Suc rexp_enum_mono subsetD zero_in_Suc0(2))
+
+lemma one_in_positive:
+  shows "RONE \<in> rexp_enum (Suc N)"
+  by (metis le_add1 plus_1_eq_Suc rexp_enum_mono subsetD one_in_Suc0(2))
+
+lemma alts_in_positive:
+  shows "RALTS [] \<in> rexp_enum (Suc N)"
+  by (metis One_nat_def alts_nil_in_Suc0(1) le_add_same_cancel1 less_Suc_eq_le plus_1_eq_Suc rexp_enum_mono subsetD zero_less_Suc)
+
+lemma char_in_positive:
+  shows "RCHAR c \<in> rexp_enum (Suc N)"
+  apply(cases c)
+     apply (metis Suc_eq_plus1 char_in1 le_add2 rexp_enum_mono subsetD)+
+  done
+
 lemma enum_inductive_cases:
   shows "rsize (RSEQ r1 r2) = Suc n \<Longrightarrow> \<exists>i j. rsize r1 = i \<and> rsize r2 = j\<and> i + j = n"
   by (metis Suc_inject rsize.simps(5))
-thm rsize.simps(5)
+
 
 lemma enumeration_finite:
   shows "\<exists>Nn. card (rexp_enum n) < Nn"
   apply(simp add:no_top_class.gt_ex)
   done
 
+
+lemma s1:
+"{r::rexp . size r = 0} = ({ZERO, ONE} \<union> {CH c| c. True})"
+apply(auto)
+apply(case_tac x)
+apply(simp_all)
+done
+
+
+
+
+lemma enum_Suc0:
+  shows " rexp_enum (Suc 0) = {RZERO} \<union> {RONE} \<union> {RCHAR c | c. True} \<union> {RALTS []}"
+  by auto
+
+lemma enumeration_chars_finite:
+  shows "finite {RCHAR c |c. True}"
+  apply(subgoal_tac "finite (RCHAR ` (UNIV::char set))")
+  prefer 2
+  using finite_code apply blast
+  by (simp add: full_SetCompr_eq)
+
+lemma enum_Suc0_finite:
+  shows "finite (rexp_enum (Suc 0))"
+  apply(subgoal_tac "finite ( {RZERO} \<union> {RONE} \<union> {RCHAR c | c. True} \<union> {RALTS []})")
+  using enum_Suc0 apply presburger
+  using enumeration_chars_finite by blast
+
+lemma enum_1_finite:
+  shows "finite (rexp_enum 1)"
+  using enum_Suc0_finite by force
+
+lemma enum_stars_finite:
+  shows " finite (rexp_enum n) \<Longrightarrow> finite {RSTAR r0 | r0. r0 \<in> (rexp_enum n)}"
+  apply(induct n)
+   apply simp
+  apply simp
+  done
+
+definition RSEQ_set
+  where
+  "RSEQ_set A B \<equiv> (\<lambda>(r1, r2) . (RSEQ r1 r2 )) ` (A \<times> B)"
+
+
+lemma enum_seq_finite:
+  shows "(\<forall>k. k < n \<longrightarrow> finite (rexp_enum k)) \<Longrightarrow> finite  
+{(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = n}"
+  apply(induct n)
+   apply simp
+  apply(subgoal_tac "{(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = Suc n}
+\<subseteq> RSEQ_set (rexp_enum n) (rexp_enum n)")
+   apply(subgoal_tac "finite ( RSEQ_set (rexp_enum n) (rexp_enum n))")
+  using rev_finite_subset
+    apply fastforce
+
+  sorry
+
+
+
+lemma enum_induct_finite:
+  shows " finite ( {(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = n} \<union>
+{(RALTS (a # rs)) | a rs i j. a \<in> (rexp_enum i) \<and> (RALTS rs) \<in> (rexp_enum j) \<and> i + j = Suc n \<and> i \<le> n \<and> j \<le> n} \<union>
+{RSTAR r0 | r0. r0 \<in> (rexp_enum n)} \<union>
+(rexp_enum n))"
+  apply(induct n)
+  apply simp
+  sorry
+
+lemma enumeration_finite2:
+  shows "finite (rexp_enum n)"
+  apply(cases n)
+  apply auto[1]
+  apply(case_tac nat)
+  using enum_Suc0_finite apply blast
+  apply(subgoal_tac "rexp_enum ( Suc n) =  {(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = n} \<union>
+{(RALTS (a # rs)) | a rs i j. a \<in> (rexp_enum i) \<and> (RALTS rs) \<in> (rexp_enum j) \<and> i + j = Suc n \<and> i \<le> n \<and> j \<le> n} \<union>
+{RSTAR r0 | r0. r0 \<in> (rexp_enum n)} \<union>
+(rexp_enum n)")
+  prefer 2
+  using rexp_enum.simps(3) apply presburger
+  using enum_induct_finite by auto
+
+
 lemma size1_rexps:
   shows "RCHAR x \<in> rexp_enum 1"
   apply(cases x)
@@ -978,15 +1103,14 @@
    apply auto[1]
   apply(subgoal_tac "RALTS (a # list) \<in>  {uu.
       \<exists>a rs i j. uu = RALTS (a # rs) \<and> a \<in> rexp_enum i \<and> RALTS rs \<in> rexp_enum j \<and> i + j = Suc N \<and> i \<le> N \<and> j \<le> N}")
-
-  
    apply fastforce
   apply(subgoal_tac "a \<in> rexp_enum i")
   prefer 2
-  
    apply linarith
   by blast
 
+thm rsize.elims
+
 lemma rexp_enum_covers:
   shows " rsize r \<le> N \<Longrightarrow> r \<in> rexp_enum N \<and> r \<in> rexp_enum (rsize r)"
   apply(induct N arbitrary : r)
@@ -1016,14 +1140,77 @@
          apply(subgoal_tac "RALTS list \<in> rexp_enum N")
   prefer 2
           apply presburger
+  using def_enum_alts rexp_size_induct apply presburger
+  using rexp_size_induct apply presburger
+  using rexp_size_induct apply presburger  
+  using rexp_size_induct apply presburger
+      apply(subgoal_tac "r \<in> rexp_enum 1")
+  apply (metis rsize.simps(1))
+  apply(subgoal_tac "rsize r = Suc 0")
+  prefer 2
+  using rsize.simps(1) apply presburger
+      apply(subgoal_tac "r \<in> rexp_enum (Suc 0)")
+       apply force
+  using zero_in_Suc0 apply blast
+  apply simp
   
-  sorry
+  using one_in_positive apply auto[1]
+  
+  apply (metis char_in_positive)
+   apply(subgoal_tac "rsize x41 \<le> N")
+    apply(subgoal_tac "rsize x42 \<le> N")
+  prefer 2
+     apply auto[1]
+  prefer 2
+  using enum_inductive_cases nat_le_iff_add apply blast
+   apply(subgoal_tac "x41 \<in> rexp_enum (rsize x41)")
+    prefer 2
+    apply blast
+   apply(subgoal_tac "x42 \<in> rexp_enum (rsize x42)")
+  prefer 2
+  apply blast
+   apply(subgoal_tac "rsize x42 + rsize x41 = N")
+  prefer 2
+  using add.commute enum_inductive_cases apply blast
+  apply(subgoal_tac "rexp_enum (Suc N) =  {(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = N} \<union>
+{(RALTS (a # rs)) | a rs i j. a \<in> (rexp_enum i) \<and> (RALTS rs) \<in> (rexp_enum j) \<and> i + j = Suc  N \<and> i \<le> N \<and> j \<le> N} \<union>
+{RSTAR r0 | r0. r0 \<in> (rexp_enum N)} \<union>
+(rexp_enum N)")
+    apply (smt (verit, del_insts) UnCI mem_Collect_eq old.nat.inject rsize.simps(5))
+   apply (smt (verit, ccfv_threshold) One_nat_def nle_le not_less_eq_eq rexp_enum_case3 size_geq1)
+  apply(subgoal_tac "x6 \<in> rexp_enum N")
+  prefer 2
+
+   apply force
+  apply(subgoal_tac "N \<ge> Suc 0")
+  prefer 2
+  apply (metis less_Suc_eq_le non_zero_size rsize.simps(6))
+  apply(subgoal_tac "rexp_enum (Suc N) =  {(RSEQ r1 r2)|r1 r2 i j. r1 \<in>  (rexp_enum i) \<and> r2 \<in>  (rexp_enum j) \<and> i + j = N} \<union>
+{(RALTS (a # rs)) | a rs i j. a \<in> (rexp_enum i) \<and> (RALTS rs) \<in> (rexp_enum j) \<and> i + j = Suc  N \<and> i \<le> N \<and> j \<le> N} \<union>
+{RSTAR r0 | r0. r0 \<in> (rexp_enum N)} \<union>
+(rexp_enum N)")
+  prefer 2
+  using rexp_enum_case3 apply presburger
+  by (metis (mono_tags, lifting) Un_iff mem_Collect_eq)
 
 
-lemma finite_size_finite_regx:
-  shows " \<exists>l. \<forall>rs. ((\<forall>r \<in> (set rs). rsize r < N) \<and> (distinct rs) \<longrightarrow> (length rs) < l) "
+
+
+
+definition
+  "sizeNregex N \<equiv> {r. rsize r \<le> N}"
+
+
 
-  sorry
+lemma sizeNregex_covered:
+  shows "sizeNregex N \<subseteq> rexp_enum N"
+  using rexp_enum_covers sizeNregex_def by auto
+
+lemma finiteness_of_sizeN_regex:
+  shows "finite (sizeNregex N)"
+  by (meson enumeration_finite2 rev_finite_subset sizeNregex_covered)
+
+
 
 (*below  probably needs proved concurrently*)