# HG changeset patch
# User Chengsong
# Date 1645396796 0
# Node ID 210df4cd512be3d40de091a485f9dffc76d0dc34
# Parent  994403dbbed53f46d901c87baddcaec04899edc0
hi

diff -r 994403dbbed5 -r 210df4cd512b thys2/SizeBound6CT.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/SizeBound6CT.thy	Sun Feb 20 22:39:56 2022 +0000
@@ -0,0 +1,1083 @@
+
+theory SizeBound6CT
+  imports "Lexer" "PDerivs" 
+begin
+
+section \<open>Bit-Encodings\<close>
+
+fun orderedSufAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
+  where
+ "orderedSufAux (Suc i) ss = (drop i ss) # (orderedSufAux i ss)"
+|"orderedSufAux 0 ss = Nil"
+
+fun 
+orderedSuf :: "char list \<Rightarrow> char list list"
+where
+"orderedSuf s = orderedSufAux (length s) s"
+
+fun orderedPrefAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
+  where
+"orderedPrefAux (Suc i) ss = (take i ss) # (orderedPrefAux i ss)"
+|"orderedPrefAux 0 ss = Nil"
+
+
+fun orderedPref :: "char list \<Rightarrow> char list list"
+  where
+"orderedPref s = orderedPrefAux (length s) s"
+
+lemma shape_of_pref_1list:
+  shows "orderedPref [c] = [[]]"
+  apply auto
+  done
+
+lemma shape_of_suf_1list:
+  shows "orderedSuf [c] = [[c]]"
+  by auto
+
+lemma shape_of_suf_2list:
+  shows "orderedSuf [c2, c3] = [[c3], [c2,c3]]"
+  by auto
+
+lemma shape_of_prf_2list:
+  shows "orderedPref [c1, c2] = [[c1], []]"
+  by auto
+
+
+lemma shape_of_suf_3list:
+  shows "orderedSuf [c1, c2, c3] = [[c3], [c2, c3], [c1, c2, c3]]"
+  by auto
+
+lemma throwing_elem_around:
+  shows "orderedSuf (s1 @ [a] @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf ( s1 @ [a]) ))"
+and "orderedSuf (s1 @ [a] @ s) = (orderedSuf ([a] @ s) @ (map (\<lambda>s11. s11 @ ([a] @ s))) (orderedSuf s1) )"
+  sorry
+
+
+lemma suf_cons:
+  shows "orderedSuf (s1 @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf s1))"
+  apply(induct s arbitrary: s1)
+   apply simp
+  apply(subgoal_tac "s1 @ a # s = (s1 @ [a]) @ s")
+  prefer 2
+   apply simp
+  apply(subgoal_tac "orderedSuf (s1 @ a # s) = orderedSuf ((s1 @ [a]) @ s)")
+  prefer 2
+   apply presburger
+  apply(drule_tac x="s1 @ [a]" in meta_spec)
+  sorry
+
+
+
+lemma shape_of_prf_3list:
+  shows "orderedPref [c1, c2, c3] = [[c1, c2], [c1], []]"
+  by auto
+
+fun zip_concat :: "char list list \<Rightarrow> char list list \<Rightarrow> char list list"
+  where 
+    "zip_concat (s1#ss1) (s2#ss2) = (s1@s2) # (zip_concat ss1 ss2)"
+  |   "zip_concat [] [] = []"
+  | "zip_concat [] (s2#ss2) = s2 # (zip_concat [] ss2)"
+  | "zip_concat (s1#ss1) [] = s1 # (zip_concat ss1 [])"
+
+
+
+lemma compliment_pref_suf:
+  shows "zip_concat (orderedPref s) (orderedSuf s) = replicate (length s) s "
+  apply(induct s)
+   apply auto[1]
+  sorry
+
+
+
+
+datatype rrexp = 
+  RZERO
+| RONE 
+| RCHAR char
+| RSEQ rrexp rrexp
+| RALTS "rrexp list"
+| RSTAR rrexp
+
+abbreviation
+  "RALT r1 r2 \<equiv> RALTS [r1, r2]"
+
+
+
+fun
+ rnullable :: "rrexp \<Rightarrow> bool"
+where
+  "rnullable (RZERO) = False"
+| "rnullable (RONE  ) = True"
+| "rnullable (RCHAR   c) = False"
+| "rnullable (RALTS   rs) = (\<exists>r \<in> set rs. rnullable r)"
+| "rnullable (RSEQ  r1 r2) = (rnullable r1 \<and> rnullable r2)"
+| "rnullable (RSTAR   r) = True"
+
+
+fun
+ rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
+where
+  "rder c (RZERO) = RZERO"
+| "rder c (RONE) = 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
+      then RALT   (RSEQ (rder c r1) r2) (rder c r2)
+      else RSEQ   (rder c r1) r2)"
+| "rder c (RSTAR r) = RSEQ  (rder c r) (RSTAR r)"
+
+
+fun 
+  rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
+where
+  "rders r [] = r"
+| "rders r (c#s) = rders (rder c r) s"
+
+fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
+  where
+  "rdistinct [] acc = []"
+| "rdistinct (x#xs)  acc = 
+     (if x \<in> acc then rdistinct xs  acc 
+      else x # (rdistinct xs  ({x} \<union> acc)))"
+
+lemma rdistinct_idem:
+  shows "rdistinct (x # (rdistinct rs {x})) {} = x # (rdistinct rs {x})"
+  
+  sorry
+
+
+
+
+
+fun rflts :: "rrexp list \<Rightarrow> rrexp list"
+  where 
+  "rflts [] = []"
+| "rflts (RZERO # rs) = rflts rs"
+| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
+| "rflts (r1 # rs) = r1 # rflts rs"
+
+
+fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
+  where
+  "rsimp_ALTs  [] = RZERO"
+| "rsimp_ALTs [r] =  r"
+| "rsimp_ALTs rs = RALTS rs"
+
+fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
+  where
+  "rsimp_SEQ  RZERO _ = RZERO"
+| "rsimp_SEQ  _ RZERO = RZERO"
+| "rsimp_SEQ RONE r2 = r2"
+| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
+
+
+fun rsimp :: "rrexp \<Rightarrow> rrexp" 
+  where
+  "rsimp (RSEQ r1 r2) = rsimp_SEQ  (rsimp r1) (rsimp r2)"
+| "rsimp (RALTS rs) = rsimp_ALTs  (rdistinct (rflts (map rsimp rs))  {}) "
+| "rsimp r = r"
+
+
+fun 
+  rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
+where
+  "rders_simp r [] = r"
+| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
+
+fun rsize :: "rrexp \<Rightarrow> nat" where
+  "rsize RZERO = 1"
+| "rsize (RONE) = 1" 
+| "rsize (RCHAR  c) = 1"
+| "rsize (RALTS  rs) = Suc (sum_list (map rsize rs))"
+| "rsize (RSEQ  r1 r2) = Suc (rsize r1 + rsize r2)"
+| "rsize (RSTAR  r) = Suc (rsize r)"
+
+
+fun rlist_size :: "rrexp list \<Rightarrow> nat" where
+"rlist_size (r # rs) = rsize r + rlist_size rs" 
+| "rlist_size [] = 0"
+
+thm neq_Nil_conv
+
+
+lemma rsimp_aalts_smaller:
+  shows "rsize (rsimp_ALTs  rs) \<le> rsize (RALTS rs)"
+  apply(induct rs)
+   apply simp
+  apply simp
+  apply(case_tac "rs = []")
+   apply simp
+  apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
+   apply(erule exE)+
+   apply simp
+  apply simp
+  by(meson neq_Nil_conv)
+  
+
+lemma finite_list_of_ders:
+  shows"\<exists>dersset. ( (finite dersset) \<and> (\<forall>s. (rders_simp r s) \<in> dersset) )"
+  sorry
+
+
+
+
+
+
+(*
+lemma rders_shape:
+  shows "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s = 
+         rsimp (RALTS  ((RSEQ (rders r1 s) r2) #
+           (map (rders r2) (orderedSuf s))) )"
+  apply(induct s arbitrary: r1 r2 rule: rev_induct)
+   apply simp
+  apply simp
+
+  sorry
+*)
+
+
+
+fun rders_cond_list :: "rrexp \<Rightarrow> bool list \<Rightarrow> char list list \<Rightarrow> rrexp list"
+  where
+"rders_cond_list r2 (True # bs) (s # strs) = (rders_simp r2 s) # (rders_cond_list r2 bs strs)"
+| "rders_cond_list r2 (False # bs) (s # strs) = rders_cond_list r2 bs strs"
+| "rders_cond_list r2 [] s = []"
+| "rders_cond_list r2 bs [] = []"
+
+fun nullable_bools :: "rrexp \<Rightarrow> char list list \<Rightarrow> bool list"
+  where
+"nullable_bools r (s#strs) = (rnullable (rders r s)) # (nullable_bools r strs) "
+|"nullable_bools r [] = []"
+
+fun cond_list :: "rrexp \<Rightarrow> rrexp \<Rightarrow> char list \<Rightarrow> rrexp list"
+  where
+"cond_list r1 r2 s = rders_cond_list r2 (nullable_bools r1 (orderedPref s) ) (orderedSuf s)"
+
+thm rsimp_SEQ.simps
+lemma rSEQ_mono:
+  shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"
+  apply auto
+  apply(induct r1)
+       apply auto
+      apply(case_tac "r2")
+       apply simp_all
+     apply(case_tac r2)
+          apply simp_all
+     apply(case_tac r2)
+         apply simp_all
+     apply(case_tac r2)
+        apply simp_all
+     apply(case_tac r2)
+  apply simp_all
+  done
+
+lemma rsimp_mono:
+  shows "rsize (rsimp r) \<le> rsize r"
+  apply(induct r)
+       apply simp_all
+  apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
+  
+    apply force
+  using rSEQ_mono
+  apply presburger
+  sorry
+
+lemma idiot:
+  shows "rsimp_SEQ RONE r = r"
+  apply(case_tac r)
+       apply simp_all
+  done
+
+lemma no_dup_after_simp:
+  shows "RALTS rs = rsimp r \<Longrightarrow> distinct rs"
+  sorry
+
+lemma no_further_dB_after_simp:
+  shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
+  sorry
+
+lemma longlist_withstands_rsimp_alts:
+  shows "length rs \<ge> 2 \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
+  sorry
+
+lemma no_alt_short_list_after_simp:
+  shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
+  sorry
+
+lemma idiot2:
+  shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
+    \<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
+  apply(case_tac r1)
+       apply(case_tac r2)
+  apply simp_all
+     apply(case_tac r2)
+  apply simp_all
+     apply(case_tac r2)
+  apply simp_all
+   apply(case_tac r2)
+  apply simp_all
+  apply(case_tac r2)
+       apply simp_all
+  done
+
+lemma rsimp_aalts_another:
+  shows "\<forall>r \<in> (set  (map rsimp  ((RSEQ (rders r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))  )) ). (rsize r) < N "
+  sorry
+
+
+
+lemma shape_derssimpseq_onechar:
+  shows "   (rders_simp r [c]) =  (rsimp (rders r [c]))"
+and "rders_simp (RSEQ r1 r2) [c] = 
+         rsimp (RALTS  ((RSEQ (rders r1 [c]) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref [c]))  (orderedSuf [c]))) )"
+   apply simp
+  apply(simp add: rders.simps)
+  apply(case_tac "rsimp (rder c r1) = RZERO")
+   apply auto
+  apply(case_tac "rsimp (rder c r1) = RONE")
+   apply auto
+   apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp r2")
+  prefer 2
+  using idiot
+    apply simp
+   apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp r2]) {})")
+    prefer 2
+    apply auto
+   apply(case_tac "rsimp r2")
+        apply auto
+   apply(subgoal_tac "rdistinct x5 {} = x5")
+  prefer 2
+  using no_further_dB_after_simp
+    apply metis
+   apply(subgoal_tac "rsimp_ALTs (rdistinct x5 {}) = rsimp_ALTs x5")
+    prefer 2
+    apply fastforce  
+   apply auto
+   apply (metis no_alt_short_list_after_simp)
+  apply (case_tac "rsimp r2 = RZERO")
+   apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RZERO")
+    prefer 2
+    apply(case_tac "rsimp ( rder c r1)")
+         apply auto
+  apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RSEQ (rsimp (rder c r1)) (rsimp r2)")
+   prefer 2
+   apply auto
+  sorry
+
+
+
+lemma shape_derssimpseq_onechar2:
+  shows "rders_simp (RSEQ r1 r2) [c] = 
+         rsimp (RALTS  ((RSEQ (rders_simp r1 [c]) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref [c]))  (orderedSuf [c]))) )"
+  sorry
+
+
+lemma rders__onechar:
+  shows " (rders_simp r [c]) =  (rsimp (rders r [c]))"
+  by simp
+
+lemma rders_append:
+  "rders c (s1 @ s2) = rders (rders c s1) s2"
+  apply(induct s1 arbitrary: c s2)
+  apply(simp_all)
+  done
+
+lemma rders_simp_append:
+  "rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
+  apply(induct s1 arbitrary: c s2)
+  apply(simp_all)
+  done
+
+lemma inside_simp_removal:
+  shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
+  
+  sorry
+
+lemma set_related_list:
+  shows "distinct rs  \<Longrightarrow> length rs = card (set rs)"
+  by (simp add: distinct_card)
+(*this section deals with the property of distinctBy: creates a list without duplicates*)
+lemma rdistinct_never_added_twice:
+  shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
+  by force
+
+
+lemma rdistinct_does_the_job:
+  shows "distinct (rdistinct rs s)"
+  apply(induct rs arbitrary: s)
+   apply simp
+  apply simp
+  sorry
+
+
+lemma simp_helps_der_pierce:
+  shows " rsimp
+            (rder x
+              (rsimp_ALTs rs)) = 
+          rsimp 
+            (rsimp_ALTs 
+              (map (rder x )
+                rs
+              )
+            )"
+  sorry
+
+lemma simp_helps_der_pierce_dB:
+  shows " rsimp
+            (rsimp_ALTs
+              (map (rder x)
+                (rdistinct rs {}))) = 
+          rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
+
+  sorry
+
+lemma simp_helps_der_pierce_flts:
+  shows " rsimp
+            (rsimp_ALTs
+             (rdistinct 
+                (map (rder x)
+                  (rflts rs )
+                ) {}
+             )
+            ) = 
+          rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {})  )"
+
+  sorry
+
+
+lemma unfold_ders_simp_inside_only: 
+  shows "    (rders_simp (RSEQ r1 r2) xs =
+           rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))
+       \<Longrightarrow> rsimp (rder x (rders_simp (RSEQ r1 r2) xs)) = rsimp (rder x (rsimp (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))))"
+ by presburger
+
+
+
+lemma unfold_ders_simp_inside_only_nosimp: 
+  shows "    (rders_simp (RSEQ r1 r2) xs =
+           rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))
+       \<Longrightarrow> rsimp (rder x (rders_simp (RSEQ r1 r2) xs)) = rsimp (rder x (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))"
+  using inside_simp_removal by presburger
+
+
+
+
+lemma rders_simp_one_char:
+  shows "rders_simp r [c] = rsimp (rder c r)"
+  apply auto
+  done
+
+lemma rsimp_idem:
+  shows "rsimp (rsimp r) = rsimp r"
+  sorry
+
+lemma head_one_more_simp:
+  shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
+  by (simp add: rsimp_idem)
+
+lemma head_one_more_dersimp:
+  shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
+  using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
+
+thm cond_list.simps
+
+lemma suffix_plus1char:
+  shows "\<not> (rnullable (rders r1 s)) \<Longrightarrow> cond_list r1 r2 (s@[c]) = map (rder c) (cond_list r1 r2 s)"
+   apply simp
+  sorry
+
+lemma suffix_plus1charn:
+shows "rnullable (rders r1 s) \<Longrightarrow> cond_list r1 r2 (s@[c]) = (rder c r2) # (map (rder c) (cond_list r1 r2 s))"
+  sorry
+
+lemma ders_simp_nullability:
+  shows "rnullable (rders r s) = rnullable (rders_simp r s)"
+  sorry
+
+lemma  first_elem_seqder:
+  shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
+                   # rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
+  by auto
+
+lemma first_elem_seqder1:
+  shows  "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) = 
+                                          map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
+  by (simp add: rsimp_idem)
+
+lemma first_elem_seqdersimps:
+  shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) = 
+                                          map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
+  using first_elem_seqder1 rders_simp_append by auto
+
+lemma first_elem_seqder_nullable:
+  shows "rnullable (rders_simp r1 xs) \<Longrightarrow> cond_list r1 r2 (xs @ [x]) = rder x r2 # map (rder x) (cond_list r1 r2 xs)"
+  sorry
+
+
+(*nullable_seq_with_list1 related *)
+lemma LHS0_def_der_alt:
+  shows "rsimp (rder x (RALTS ( (RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) = 
+         rsimp (RALTS ((rder x (RSEQ (rders_simp r1 xs) r2)) # (map (rder x) (cond_list r1 r2 xs))))"
+  by fastforce
+
+lemma LHS1_def_der_seq:
+  shows "rnullable (rders_simp r1 xs) \<Longrightarrow> 
+rsimp (rder x (RALTS ( (RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) = 
+rsimp(RALTS ((RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) # [rder x r2]) ) # (map (rder x ) (cond_list r1 r2 xs))))"
+  by (simp add: rders_simp_append rsimp_idem)
+
+
+
+
+
+lemma cond_list_head_last:
+  shows "rnullable (rders r1 s) \<Longrightarrow> 
+        RALTS (r # (cond_list r1 r2 (s @ [c]))) = RALTS (r # ((rder c r2) # (map (rder c) (cond_list r1 r2 s))))"
+  using suffix_plus1charn by blast
+
+
+lemma simp_flatten:
+  shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
+
+  sorry
+
+lemma RHS1:
+  shows "rnullable (rders_simp r1 xs) \<Longrightarrow>
+                                    rsimp (RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) # 
+(cond_list r1 r2 (xs @[x])))) = 
+                                    rsimp (RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) # 
+( ((rder x r2) # (map (rder x) (cond_list r1 r2 xs)))))) "
+  using first_elem_seqder_nullable by presburger
+
+
+lemma nullable_seq_with_list1:
+  shows " rnullable (rders_simp r1 s) \<Longrightarrow>
+    rsimp (rder c (RALTS ( (RSEQ (rders_simp r1 s) r2) #  (cond_list r1 r2 s)) )) =
+    rsimp (RALTS ( (RSEQ (rders_simp r1 (s @ [c])) r2) # (cond_list r1 r2 (s @ [c])) ) )"
+  by (metis LHS1_def_der_seq append.left_neutral append_Cons first_elem_seqder_nullable simp_flatten)
+
+
+
+
+lemma nullable_seq_with_list:
+  shows "rnullable (rders_simp r1 xs) \<Longrightarrow> rsimp (rder x (RALTS ((RSEQ (rders_simp r1 xs) r2) # 
+      (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)) ))) = 
+                                          rsimp (RALTS ((RSEQ (rders_simp r1 (xs@[x])) r2) #
+      (rders_cond_list r2 (nullable_bools r1 (orderedPref (xs@[x]))) (orderedSuf (xs@[x]))) ) )
+    "
+  apply(subgoal_tac "rsimp (rder x (RALTS ((RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) = 
+                     rsimp (RALTS ((RSEQ (rders_simp r1 (xs@[x])) r2) # (cond_list r1 r2 (xs@[x]))))")
+   apply auto[1]
+  using nullable_seq_with_list1 by auto
+
+
+
+
+lemma r1r2ders_whole:
+"rsimp
+         (RALTS
+           (rder x (RSEQ (rders_simp r1 xs) r2) #
+            map (rder x) (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))) = 
+      rsimp(    RALTS(   ( (RSEQ (rders_simp r1 (xs@[x])) r2)
+                   #  (rders_cond_list r2 (nullable_bools r1 (orderedPref (xs @ [x]))) (orderedSuf (xs @ [x])))))) "
+  using ders_simp_nullability first_elem_seqdersimps nullable_seq_with_list1 suffix_plus1char by auto
+
+lemma rders_simp_same_simpders:
+  shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
+  apply(induct s rule: rev_induct)
+   apply simp
+  apply(case_tac "xs = []")
+   apply simp
+  apply(simp add: rders_append rders_simp_append)
+  using inside_simp_removal by blast
+
+lemma shape_derssimp_seq:
+  shows "\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s = 
+         rsimp (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) )"
+
+  apply(induct s arbitrary: r1 r2 rule: rev_induct)
+   apply simp
+  apply(case_tac "xs = []")
+  using shape_derssimpseq_onechar2 apply force
+  apply(simp only: rders_simp_append)
+  apply(simp only: rders_simp_one_char)
+
+  apply(subgoal_tac "rsimp (rder x (rders_simp (RSEQ r1 r2) xs))
+                   = rsimp (rder x (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))")
+   prefer 2
+  using unfold_ders_simp_inside_only_nosimp apply presburger
+  apply(subgoal_tac "rsimp (rder x (RALTS (RSEQ (rders_simp r1 xs) r2
+                   # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))) = 
+                     rsimp ( (RALTS (rder x (RSEQ (rders_simp r1 xs) r2)
+                   # (map (rder x) (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))))
+                      ")
+   prefer 2
+   apply simp
+  using r1r2ders_whole rders_simp_append rders_simp_one_char by presburger
+
+(*
+
+  apply(subgoal_tac " rsimp
+            (rder x
+              (rsimp_ALTs
+                (rdistinct
+                  (rflts
+                    (rsimp_SEQ (rsimp (rders_simp r1 xs)) (rsimp r2) #
+                     map rsimp
+                      (rders_cond_list r2 (nullable_bools r1 (orderedPrefAux (length xs) xs)) (orderedSufAux (length xs) xs))))
+                  {}))) =  
+                      rsimp
+            (
+              (rsimp_ALTs
+               (map (rder x)
+                (rdistinct
+                  (rflts
+                    (rsimp_SEQ (rsimp (rders_simp r1 xs)) (rsimp r2) #
+                     map rsimp
+                      (rders_cond_list r2 (nullable_bools r1 (orderedPrefAux (length xs) xs)) (orderedSufAux (length xs) xs))))
+                  {})
+               )
+              )
+            ) ")
+   prefer 2
+*)
+
+lemma shape_derssimp_alts:
+  shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders r s) rs))"
+  apply(case_tac "s")
+   apply simp
+  apply simp
+  sorry
+(*
+fun rexp_encode :: "rrexp \<Rightarrow> nat"
+  where
+"rexp_encode RZERO = 0"
+|"rexp_encode RONE = 1"
+|"rexp_encode (RCHAR c) = 2"
+|"rexp_encode (RSEQ r1 r2) = ( 2 ^ (rexp_encode r1)) "
+*)
+lemma finite_chars:
+  shows " \<exists>N. ( (\<forall>r \<in> (set rs). \<exists>c. r = RCHAR c) \<and> (distinct rs) \<longrightarrow> length rs < N)"
+  apply(rule_tac x = "Suc 256" in exI)
+  sorry
+
+definition all_chars :: "int \<Rightarrow> char list"
+  where "all_chars n = map char_of [0..n]"
+(*
+fun rexp_enum :: "nat \<Rightarrow> rrexp list"
+  where 
+"rexp_enum 0 = []"
+|"rexp_enum (Suc 0) =  RALTS [] # RZERO # (RONE # (map RCHAR (all_chars 255)))"
+|"rexp_enum (Suc n) = [(RSEQ r1 r2). r1 \<in> set (rexp_enum i) \<and> r2 \<in> set (rexp_enum j) \<and> i + j = n]"
+
+*)
+
+fun 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> set (all_chars 255)}"
+|"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}"
+
+
+lemma finite_sized_rexp_forms_finite_set:
+  shows " \<exists>SN. ( \<forall>r.( rsize r < N \<longrightarrow> r \<in> SN)) \<and> (finite SN)"
+  apply(induct N)
+   apply simp
+   apply auto
+ (*\<lbrakk>\<forall>r. rsize r < N \<longrightarrow> r \<in> SN; finite SN\<rbrakk> \<Longrightarrow> \<exists>SN. (\<forall>r. rsize r < Suc N \<longrightarrow> r \<in> SN) \<and> finite SN*)
+ (* \<And>N. \<exists>SN. (\<forall>r. rsize r < N \<longrightarrow> r \<in> SN) \<and> finite SN \<Longrightarrow> \<exists>SN. (\<forall>r. rsize r < Suc N \<longrightarrow> r \<in> SN) \<and> finite SN*)
+  sorry
+
+
+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) "
+  sorry
+
+(*below  probably needs proved concurrently*)
+
+lemma finite_r1r2_ders_list:
+  shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
+           \<Longrightarrow>  \<exists>l. \<forall>s.
+(length (rdistinct  (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) {}) )  < l "
+  sorry
+
+(*
+\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s = 
+         rsimp (RALTS  ((RSEQ (rders r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) )
+*)
+ 
+
+lemma sum_list_size2:
+  shows "\<forall>z \<in>set rs. (rsize z ) \<le> Nr \<Longrightarrow> rlist_size rs \<le> (length rs) * Nr"
+  apply(induct rs)
+   apply simp
+  by simp
+
+lemma sum_list_size:
+  fixes rs
+  shows " \<forall>r \<in> (set rs). (rsize r) \<le> Nr \<and> (length rs) \<le> l \<Longrightarrow> rlist_size rs \<le> l * Nr"
+  by (metis dual_order.trans mult.commute mult_le_mono2 mult_zero_right sum_list_size2 zero_le)
+
+lemma seq_second_term_chain1:
+  shows " \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) \<le> 
+ rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))"
+
+  sorry
+
+
+lemma seq_second_term_chain2:
+  shows "\<forall>s.  rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) = 
+ rlist_size (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))"
+
+  oops
+
+lemma seq_second_term_bounded:
+  fixes r2 r1
+  assumes "\<forall>s. rsize (rders_simp r2 s) < N2"
+  shows "\<exists>N3. \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) < N3"
+
+  sorry
+
+
+lemma seq_first_term_bounded:
+  fixes r1 r2
+  shows "\<exists>Nr. \<forall>s. rsize (rders_simp r1 s) \<le> Nr \<Longrightarrow> \<exists>Nr'. \<forall>s. rsize (RSEQ  (rders_simp r1 s) r2) \<le> Nr'"
+  apply(erule exE)
+  apply(rule_tac x = "Nr + (rsize r2) + 1" in exI)
+  by simp
+
+
+lemma alts_triangle_inequality:
+  shows "rsize (RALTS (r # rs)) \<le> rsize r + rlist_size rs + 1 "
+  apply(subgoal_tac "rsize (RALTS (r # rs) ) =  rsize r + rlist_size rs + 1")
+   apply auto[1]
+  apply(induct rs)
+   apply simp
+  apply auto
+  done
+
+lemma seq_equal_term_nosimp_entire_bounded:
+  shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2) 
+      \<Longrightarrow> \<exists>N3. \<forall>s.  rsize  (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+          (rdistinct ((rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s)) ) {}) ) ) \<le> N3"
+  apply(subgoal_tac "\<exists>N3. \<forall>s. rsize  (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) ) \<le>
+                                        rsize (RSEQ (rders_simp r1 s) r2) + 
+            rlist_size (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) + 1")
+  prefer 2
+  using alts_triangle_inequality apply presburger
+  using seq_first_term_bounded
+  using seq_second_term_bounded
+  apply(subgoal_tac "\<exists>N3. \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) < N3")
+  prefer 2
+   apply meson
+  apply(subgoal_tac "\<exists>Nr'. \<forall>s. rsize (RSEQ  (rders_simp r1 s) r2) \<le> Nr'")
+   prefer 2
+  apply (meson order_le_less)
+  apply(erule exE)
+  apply(erule exE)
+  apply(erule exE)
+  apply(rule_tac x = "N3a + Nr'" in exI)
+  sorry
+
+lemma alts_simp_bounded_by_sloppy1_version:
+  shows "\<forall>s. rsize (rsimp (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) )) \<le> 
+rsize  (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
+                          (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))
+                         ) 
+                         {}
+              )
+        ) "
+  sorry
+
+lemma alts_simp_bounded_by_sloppy1:
+  shows "rsize  (rsimp (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
+                          (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))
+                         ) 
+                         {}
+              )
+        )) \<le>
+rsize  (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+                          (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))
+                         ) 
+        )"
+  sorry
+
+lemma hand_made_def_rlist_size:
+  shows "rlist_size rs = sum_list (map rsize rs)"
+proof (induct rs)
+  case Nil show ?case by simp
+next
+  case (Cons a rs) thus ?case
+    by simp
+qed
+
+(*this section deals with the property of distinctBy: creates a list without duplicates*)
+lemma distinct_mono:
+  shows "rlist_size (rdistinct (a # s) {}) \<le> rlist_size (a # (rdistinct s {}) )"
+  sorry
+
+lemma distinct_acc_mono:
+  shows "A \<subseteq> B \<Longrightarrow> rlist_size (rdistinct s A) \<ge> rlist_size (rdistinct s B)"
+  apply(induct s arbitrary: A B)
+   apply simp
+  apply(case_tac "a \<in> A")
+  apply(subgoal_tac "a \<in> B")
+  
+  apply simp
+  
+   apply blast
+  apply(subgoal_tac "rlist_size (rdistinct (a # s) A) = rlist_size (a # (rdistinct s (A \<union> {a})))")
+   apply(case_tac "a \<in> B")
+  apply(subgoal_tac "rlist_size (rdistinct (a # s) B) = rlist_size ( (rdistinct s B))") 
+  apply (metis Un_insert_right dual_order.trans insert_subset le_add_same_cancel2 rlist_size.simps(1) sup_bot_right zero_order(1))
+  apply simp
+  apply auto
+  by (meson insert_mono)
+
+
+lemma distinct_mono2:
+  shows " rlist_size (rdistinct s {a}) \<le> rlist_size (rdistinct s {})"
+  apply(induct s)
+   apply simp
+  apply simp
+  using distinct_acc_mono by auto
+
+
+
+lemma distinct_mono_spares_first_elem:
+  shows "rsize (RALTS (rdistinct (a # s) {})) \<le> rsize (RALTS (a # (rdistinct s {})))"
+  apply simp
+  apply (subgoal_tac "rlist_size ( (rdistinct s {a})) \<le> rlist_size ( (rdistinct s {})) ")
+  using hand_made_def_rlist_size apply auto[1]
+  using distinct_mono2 by auto
+
+lemma sloppy1_bounded_by_sloppiest:
+  shows "rsize  (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
+                          (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))
+                         ) 
+                         {}
+              )
+        ) \<le> rsize  (RALTS ((RSEQ (rders_simp r1 s) r2) #
+                          (rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s)) {})
+                         
+                         
+              )
+        )"
+  
+  sorry
+
+
+lemma alts_simp_bounded_by_sloppiest_version:
+  shows "\<forall>s. rsize (rsimp (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) )) \<le> 
+rsize  (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s)) {})  ) ) "
+  by (meson alts_simp_bounded_by_sloppy1_version order_trans sloppy1_bounded_by_sloppiest)
+
+
+lemma seq_equal_term_entire_bounded:
+  shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2) 
+      \<Longrightarrow> \<exists>N3. \<forall>s.  rsize (rsimp (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) )) \<le> N3"
+  using seq_equal_term_nosimp_entire_bounded
+  apply(subgoal_tac " \<exists>N3. \<forall>s.  rsize  (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rdistinct  (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s)) {}) ) ) \<le> N3")
+  apply(erule exE)
+   prefer 2
+  apply blast
+  apply(subgoal_tac "\<forall>s. rsize (rsimp (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s))) )) \<le> 
+rsize  (RALTS  ((RSEQ (rders_simp r1 s) r2) #
+           (rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s))  (orderedSuf s)) {})  ) ) ")
+   prefer 2
+  using alts_simp_bounded_by_sloppiest_version apply blast
+  apply(rule_tac x = "Suc N3 " in exI)
+  apply(rule allI)
+
+  apply(subgoal_tac " rsize
+             (rsimp
+               (RALTS
+                 (RSEQ (rders_simp r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))))
+            \<le> rsize
+                (RALTS
+                  (RSEQ (rders_simp r1 s) r2 #
+                   rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}))")
+   prefer 2
+   apply presburger
+  apply(subgoal_tac "  rsize
+                (RALTS
+                  (RSEQ (rders_simp r1 s) r2 #
+                   rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {})) \<le> N3")
+  
+   apply linarith
+  apply simp
+  done
+
+
+
+lemma M1seq:
+  fixes r1 r2 
+  shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
+           \<Longrightarrow> \<exists>N3.\<forall>s.(rsize (rders_simp (RSEQ r1 r2) s)) < N3"
+  apply(frule seq_equal_term_entire_bounded)
+  apply(erule exE)
+  apply(rule_tac x = "max (N3+2) (Suc (Suc (rsize r1) + (rsize r2)))" in exI)
+  apply(rule allI)
+  apply(case_tac "s = []")
+   prefer 2
+   apply (metis add_2_eq_Suc' le_imp_less_Suc less_SucI max.strict_coboundedI1 shape_derssimp_seq(2))
+  by (metis add.assoc less_Suc_eq max.strict_coboundedI2 plus_1_eq_Suc rders_simp.simps(1) rsize.simps(5))
+ (*  apply (simp add: less_SucI shape_derssimp_seq(2))
+   apply (meson less_SucI less_max_iff_disj)
+  apply simp
+  done*)
+
+(*lemma empty_diff:
+  shows "s = [] \<Longrightarrow>
+        (rsize (rders_simp (RSEQ r1 r2) s)) \<le> 
+        (max 
+        (rsize (rsimp (RALTS (RSEQ (rders r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))))
+        (Suc (rsize r1 + rsize r2)) ) "
+  apply simp
+  done*)
+(*For star related  bound*)
+
+lemma star_is_a_singleton_list_derc:
+  shows " \<exists>Ss.  rders_simp (RSTAR r) [c] = rsimp_ALTs (map (\<lambda>s1.  rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)"
+  apply simp
+  apply(rule_tac x = "[[c]]" in exI)
+  apply auto
+  done
+
+lemma rder_rsimp_ALTs_commute:
+  shows "  (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
+  apply(induct rs)
+   apply simp
+  apply(case_tac rs)
+   apply simp
+  apply auto
+  done
+
+lemma double_nested_ALTs_under_rsimp:
+  shows "rsimp (rsimp_ALTs ((RALTS rs1) # rs)) = rsimp (RALTS (rs1 @ rs))"
+  apply(case_tac rs1)
+  apply simp
+  
+   apply (metis list.simps(8) list.simps(9) neq_Nil_conv rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
+  apply(case_tac rs)
+   apply simp
+  apply auto
+  sorry
+
+fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list => char list list" where
+"star_update c r [] = []"
+|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s)) 
+                                then (s@[c]) # [c] # (star_update c r Ss) 
+                               else   s # (star_update c r Ss) )"
+
+lemma starseq_list_evolution:
+  fixes  r :: rrexp and Ss :: "char list list" and x :: char 
+  shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
+         rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))  )"   
+  apply(induct Ss)
+  apply simp
+  sorry
+
+
+lemma star_seqs_produce_star_seqs:
+  shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
+       = rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
+  by (meson comp_apply)
+
+lemma der_seqstar_res:
+  shows "rder x (RSEQ r1 r2) = RSEQ r3 r4"
+  oops
+
+lemma linearity_of_list_of_star_or_starseqs: 
+  fixes r::rrexp and Ss::"char list list" and x::char
+  shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
+                 rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)"
+  apply(simp add: rder_rsimp_ALTs_commute)
+  apply(induct Ss)
+   apply simp
+   apply (metis list.simps(8) rsimp_ALTs.simps(1))
+
+
+  sorry
+
+lemma starder_is_a_list_of_stars_or_starseqs:
+  shows "s \<noteq> [] \<Longrightarrow> \<exists>Ss.  rders_simp (RSTAR r) s = rsimp_ALTs (map (\<lambda>s1.  rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)"
+  apply(induct s rule: rev_induct)
+  apply simp
+  apply(case_tac "xs = []")
+  using star_is_a_singleton_list_derc
+  apply(simp)
+  apply auto
+  apply(simp add: rders_simp_append)
+  using linearity_of_list_of_star_or_starseqs by blast
+
+
+lemma finite_star:
+  shows "(\<forall>s. rsize (rders_simp r0 s) < N0 )
+           \<Longrightarrow> \<exists>N3. \<forall>s.(rsize (rders_simp (RSTAR r0) s)) < N3"
+
+  sorry
+
+
+lemma rderssimp_zero:
+  shows"rders_simp RZERO s = RZERO"
+  apply(induction s)
+  apply simp
+  by simp
+
+lemma rderssimp_one:
+  shows"rders_simp RONE (a # s) = RZERO"
+  apply(induction s)
+  apply simp
+  by simp
+
+lemma rderssimp_char:
+  shows "rders_simp (RCHAR c) s = RONE \<or> rders_simp (RCHAR c) s = RZERO \<or> rders_simp (RCHAR c) s = (RCHAR c)"
+  apply auto
+  by (metis rder.simps(2) rder.simps(3) rders_simp.elims rders_simp.simps(2) rderssimp_one rsimp.simps(4))
+
+lemma finite_size_ders:
+  fixes r
+  shows " \<exists>Nr. \<forall>s. rsize (rders_simp r s) < Nr"
+  apply(induct r rule: rrexp.induct)
+       apply auto
+  apply(rule_tac x = "2" in exI)
+  using rderssimp_zero rsize.simps(1) apply presburger
+      apply(rule_tac x = "2" in exI)
+      apply (metis Suc_1 lessI rders_simp.elims rderssimp_one rsize.simps(1) rsize.simps(2))
+     apply(rule_tac x = "2" in meta_spec)
+     apply (metis lessI rderssimp_char rsize.simps(1) rsize.simps(2) rsize.simps(3))
+  
+  using M1seq apply blast
+   prefer 2
+
+   apply (simp add: finite_star)
+  sorry
+
+
+unused_thms
+lemma seq_ders_shape:
+  shows "E"
+
+  oops
+
+(*rsimp (rders (RSEQ r1 r2) s) =
+         rsimp RALT [RSEQ (rders r1 s) r2, rders r2 si, ...]
+         where si is the i-th shortest suffix of s such that si \<in> L r2"
+*)
+
+
+end