
diff -r c6351a96bf80 -r a7e98deebb5c thys2/ClosedForms.thy
--- a/thys2/ClosedForms.thy	Tue Mar 08 00:50:40 2022 +0000
+++ b/thys2/ClosedForms.thy	Wed Mar 09 17:33:08 2022 +0000
@@ -1,117 +1,36 @@
-
-theory ClosedForms
-  imports "Lexer" "PDerivs" 
+theory ClosedForms imports
+"BasicIdentities"
 begin
 
+lemma alts_closed_form: shows 
+"rsimp (rders_simp (RALTS rs) s) = 
+rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
+  apply(induct s rule: rev_induct)
+   apply simp
+  apply simp
+  apply(subst rders_simp_append)
+  apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) = 
+ rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
+   prefer 2
+  apply (metis inside_simp_removal rders_simp_one_char)
+  apply(simp only: )
+  sorry
 
-datatype rrexp = 
-  RZERO
-| RONE 
-| RCHAR char
-| RSEQ rrexp rrexp
-| RALTS "rrexp list"
-| RSTAR rrexp
-
-abbreviation
-  "RALT r1 r2 \<equiv> RALTS [r1, r2]"
+lemma alts_closed_form_variant: shows 
+"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = 
+rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
+  sorry
 
 
 
-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 star_closed_form:
+  shows "rders_simp (RSTAR r0) (c#s) = 
+rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r0 [[c]]) ) ))"
+  apply(induct s)
+   apply simp
+  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"
-
-fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
-"vsuf [] _ = []"
-|"vsuf (c#cs) r1 = (if (rnullable r1) then  (vsuf cs (rder c r1)) @ [c # cs]
-                                      else  (vsuf cs (rder c r1))
-                   ) "
 
 lemma seq_closed_form: shows 
 "rsimp (rders_simp (RSEQ r1 r2) s) = 
@@ -124,883 +43,9 @@
   sorry
 
 
-fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> 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@[c]) # (star_update c r Ss) )"
-
-fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
-  where
-"star_updates [] r Ss = Ss"
-| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
-
-
-lemma star_closed_form:
-  shows "rders_simp (RSTAR r0) (c#s) = 
-rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r [[c]]) ) ))"
-  apply(induct s)
-   apply simp
-  sorry
-
-
-lemma star_closed_form_bounded_by_rdistinct_list_estimate:
-  shows "rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
-         (star_updates s r [[c]]) ) ))) \<le>
-        Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
-         (star_updates s r [[c]]) ) {})  ) )"
-
-  sorry
-
-lemma distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size:
-  shows "\<forall>r\<in> set rs. (rsize r ) \<le> N \<Longrightarrow> sum_list (map rsize (rdistinct rs {})) \<le>
-         (card (rexp_enum N))* N"
-  sorry
-
-
-lemma ind_hypo_on_ders_leads_to_stars_bounded:
-  shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>        
-      (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
-         (star_updates s r [[c]]) ) {})  ) ) \<le> 
-(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
-"
-  sorry
-
-lemma r0_bounded_star_bounded:
-  shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
-             \<forall>s. rsize (rders_simp (RSTAR r0) s) \<le> 
-(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))"
-
-  sorry
-
-
-(*some basic facts about rsimp*)
-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
-
-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 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 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 ralts_cap_mono:
-  shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "
-  by simp
-
-lemma rflts_def_idiot:
-  shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
-       \<Longrightarrow> rflts (a # rs) = a # rflts rs"
-  apply(case_tac a)
-       apply simp_all
-  done
-
-
-lemma rflts_mono:
-  shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize rs)"
-  apply(induct rs)
-  apply simp
-  apply(case_tac "a = RZERO")
-   apply simp
-  apply(case_tac "\<exists>rs1. a = RALTS rs1")
-  apply(erule exE)
-   apply simp
-  apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
-  prefer 2
-  using rflts_def_idiot apply blast
-  apply simp
-  done
-
-lemma rdistinct_smaller: shows "sum_list (map rsize (rdistinct rs ss)) \<le>
-sum_list (map rsize rs )"
-  apply (induct rs arbitrary: ss)
-   apply simp
-  by (simp add: trans_le_add2)
-
-lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
-  by (simp add: rdistinct_smaller)
-
-
-lemma rsimp_alts_mono :
-  shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa)  \<Longrightarrow>
-rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (sum_list (map rsize x))"
-  apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} )) 
-                    \<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
-  prefer 2
-  using rsimp_aalts_smaller apply auto[1]
-  apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")
-  prefer 2
-  using ralts_cap_mono apply blast
-  apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
-                     sum_list (map rsize ( (rflts (map rsimp x))))")
-  prefer 2
-  using rdistinct_smaller apply presburger
-  apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le> 
-                     sum_list (map rsize (map rsimp x))")
-  prefer 2
-  using rflts_mono apply blast
-  apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize x)")
-  prefer 2
-  
-  apply (simp add: sum_list_mono)
-  by linarith
-
-
-
-
-
-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
-  using rsimp_alts_mono by auto
-
-lemma idiot:
-  shows "rsimp_SEQ RONE r = r"
-  apply(case_tac r)
-       apply simp_all
-  done
-
-lemma no_alt_short_list_after_simp:
-  shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
-  sorry
-
-lemma no_further_dB_after_simp:
-  shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = 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 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 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 simp_helps_der_pierce:
-  shows " rsimp
-            (rder x
-              (rsimp_ALTs rs)) = 
-          rsimp 
-            (rsimp_ALTs 
-              (map (rder x )
-                rs
-              )
-            )"
-  sorry
-
-
-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
-
-corollary rsimp_inner_idem1:
-  shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
-  
-  sorry
-
-corollary rsimp_inner_idem2:
-  shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
-  sorry
-
-corollary rsimp_inner_idem3:
-  shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
-  by (meson map_idI rsimp_inner_idem2)
-
-corollary rsimp_inner_idem4:
-  shows "rsimp r = RALTS rs \<Longrightarrow> flts rs = rs"
-  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
-
-
-
-
-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 seq_update_seq_ders:
-  shows "rsimp (rder c ( rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # 
-(map (rders_simp r2) Ss))))) = 
-         rsimp (RALTS ((RSEQ (rders_simp r1 (s @ [c])) r2) # 
-(map (rders_simp r2) (seq_update c (rders_simp r1 s) Ss))))  "
-  sorry
-
-lemma seq_ders_closed_form1:
-  shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # 
-(map ( rders_simp r2 ) Ss)))"
-  apply(case_tac "rnullable r1")
-   apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] = 
-rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
-    prefer 2
-    apply (simp add: rsimp_idem)
-   apply(rule_tac x = "[[c]]" in exI)
-   apply simp
-  apply(subgoal_tac  " rders_simp (RSEQ r1 r2) [c] = 
-rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
-   apply blast
-  apply(simp add: rsimp_idem)
-  sorry
-
-
-
-
-
-
-
-
-lemma simp_flatten2:
-  shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
-  sorry
-
-
-lemma simp_flatten:
-  shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
-
-  sorry
-
-
-
-(*^^^^^^^^^nullable_seq_with_list1 related ^^^^^^^^^^^^^^^^*)
-
-
-
-
-
-
-
-
-
-
-
-lemma non_zero_size:
-  shows "rsize r \<ge> Suc 0"
-  apply(induct r)
-  apply auto done
-
-corollary size_geq1:
-  shows "rsize r \<ge> 1"
-  by (simp add: non_zero_size)
-
-
-lemma rexp_size_induct:
-  shows "\<And>N r x5 a list.
-       \<lbrakk> rsize r = Suc N; r = RALTS x5;
-        x5 = a # list\<rbrakk>  \<Longrightarrow>\<exists>i j. rsize a = i \<and> rsize (RALTS list) = j \<and> i + j =  Suc N \<and> i \<le> N \<and> j \<le> N"
-  apply(rule_tac x = "rsize a" in exI)
-  apply(rule_tac x = "rsize (RALTS list)" in exI)
-  apply(subgoal_tac "rsize a \<ge> 1")
-   prefer 2
-  using One_nat_def non_zero_size apply presburger
-  apply(subgoal_tac "rsize (RALTS list) \<ge> 1 ")
-  prefer 2
-  using size_geq1 apply blast
-  apply simp
-  done
-
-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 SEQ_set_cartesian where
-"SEQ_set_cartesian A n  = {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
-  "sizeNregex N \<equiv> {r. rsize r \<le> N}"
-
-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))"
+lemma seq_closed_form_variant: shows
+"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) = 
+rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
   sorry
 
-
-lemma chars_finite:
-  shows "finite (RCHAR ` (UNIV::(char set)))"
-  apply(simp)
-  done
-
-thm full_SetCompr_eq 
-
-lemma size1finite:
-  shows "finite (sizeNregex (Suc 0))"
-  apply(subst sizenregex_induct)
-  apply(subst finite_Un)+
-  apply(subgoal_tac "sizeNregex 0 = {}")
-  apply(rule conjI)+
-  apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
-      apply simp
-      apply (simp add: full_SetCompr_eq)
-  apply (simp add: SEQ_set_def)
-    apply (simp add: ALT_set_def)  
-   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 star_update_case1:
-  shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
-  
-  by force
-
-lemma star_update_case2:
-  shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
-  by simp
-
-lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
-  apply(case_tac r)
-       apply simp+
-  done
-
-lemma rsimp_alts_idem_aux1:
-  shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
-  by force
-
-
-
-lemma rsimp_alts_idem_aux2:
-  shows "rsimp a = rsimp (RALTS [a])"
-  apply(simp)
-  apply(case_tac "rsimp a")
-       apply simp+
-  apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
-  by simp
-
-lemma rsimp_alts_idem:
-  shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
-  apply(induct as)
-   apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
-  prefer 2
-    apply simp
-  using bubble_break rsimp_alts_idem_aux2 apply auto[1]
-  apply(case_tac as)
-   apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
-  prefer 2
-    apply simp
-  using head_one_more_simp apply fastforce
-  apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
-  prefer 2
-  
-  using rsimp_ALTs.simps(3) apply presburger
-  
-  apply(simp only:)
-  apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
-  prefer 2
-  using rsimp_ALTs.simps(3) apply presburger
-  apply(simp only:)
-  apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
-  prefer 2
-  
-  using rsimp_ALTs.simps(3) apply presburger
-  apply(simp only:)
-  using simp_flatten2
-  apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list))))  =  rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
-  prefer 2
-
-  apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
-  apply (simp only:)
-  done
-
-
-lemma rsimp_alts_idem2:
-  shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
-  using head_one_more_simp rsimp_alts_idem by auto
-
-
-lemma evolution_step1:
-  shows "rsimp
-        (rsimp_ALTs
-          (rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
-         rsimp 
-        (rsimp_ALTs
-          (rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [(rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)))]))   "
-  using rsimp_alts_idem by auto
-
-lemma evolution_step2:
-  assumes " 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)))"
-  shows "rsimp 
-        (rsimp_ALTs 
-          (rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) = 
-                 rsimp 
-        (rsimp_ALTs
-          (rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [ rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]))  "
-  by (simp add: assms rsimp_alts_idem)
-
-lemma rsimp_seq_aux1:
-  shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
-  apply simp
-  done
-
-lemma multiple_alts_simp_flatten:
-  shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
-  by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
-
-
-lemma evo3_main_aux1:
-  shows "rsimp
-            (RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
-              (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
-           rsimp
-            (RALTS
-              (RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
-               RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
-  apply(subgoal_tac "rsimp
-            (RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
-              (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
-rsimp
-            (RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
-              (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
-  prefer 2
-   apply (simp add: rsimp_idem)
-  apply (simp only:)
-  apply(subst multiple_alts_simp_flatten)
-  by simp
-
-
-lemma evo3_main_nullable:
-  shows "
-\<And>a Ss.
-       \<lbrakk>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)));
-        rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
-       \<Longrightarrow> rsimp
-            (rsimp_ALTs
-              [rder x (RSEQ (rders_simp r a) (RSTAR r)),
-               rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
-           rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
-  apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r)) 
-                   = RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
-  prefer 2
-   apply simp
-  apply(simp only:)
-  apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
-   prefer 2
-  using star_update_case1 apply presburger
-  apply(simp only:)
-  apply(subst List.list.map(2))+
-  apply(subgoal_tac "rsimp
-            (rsimp_ALTs
-              [RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
-               rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) = 
-rsimp
-            (RALTS
-              [RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
-               rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
-  prefer 2
-  using rsimp_ALTs.simps(3) apply presburger
-  apply(simp only:)
-  apply(subgoal_tac " rsimp
-            (rsimp_ALTs
-              (rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
-               rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) 
-= 
- rsimp
-            (RALTS
-              (rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
-               rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
-
-  prefer 2
-  using rsimp_ALTs.simps(3) apply presburger
-  apply (simp only:)
-  apply(subgoal_tac " rsimp
-            (RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
-              (rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) = 
-             rsimp
-            (RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
-              (rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
-  prefer 2
-   apply (simp add: rsimp_idem)
-  apply(simp only:)
-  apply(subgoal_tac "             rsimp
-            (RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
-              (rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) = 
-             rsimp
-            (RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
-              (rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
-   prefer 2
-  using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
-  apply(simp only:)
-  apply(subgoal_tac " rsimp
-            (RALTS
-              (rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
-               rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) = 
- rsimp
-            (RALTS
-              (RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
-               RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
-  prefer 2
-  apply (smt (z3) idiot2 list.simps(9) rrexp.distinct(9) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_idem)
-  apply(simp only:)
-  apply(subgoal_tac "      rsimp
-            (RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
-              (rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
-     rsimp
-            (RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
-              ( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))  ")
-  prefer 2
-  using rsimp_idem apply force
-  apply(simp only:)
-  using evo3_main_aux1 by blast
-  
-
-lemma evo3_main_not1:
-  shows " \<not>rnullable (rders_simp r a) \<Longrightarrow> rder x (RSEQ (rders_simp r a) (RSTAR r)) = RSEQ (rder x (rders_simp r a)) (RSTAR r)"
-  by fastforce
-
-
-lemma evo3_main_not2:
-  shows "\<not>rnullable (rders_simp r a) \<Longrightarrow>  rsimp
-            (rsimp_ALTs
-              (rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
-            (rsimp_ALTs
-              ((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
-  by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
-
-lemma evo3_main_not3:
-  shows "rsimp
-            (rsimp_ALTs
-              (rsimp_SEQ r1 (RSTAR r) # rs)) = 
-         rsimp (rsimp_ALTs
-              (RSEQ r1 (RSTAR r) # rs))"
-  by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
-
-
-lemma evo3_main_notnullable:
-  shows "\<And>a Ss.
-       \<lbrakk>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)));
-        rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
-       \<Longrightarrow> rsimp
-            (rsimp_ALTs
-              [rder x (RSEQ (rders_simp r a) (RSTAR r)),
-               rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
-           rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
-  apply(subst star_update_case2)
-   apply simp
-  apply(subst List.list.map(2))
-  apply(subst evo3_main_not2)
-   apply simp
-  apply(subst evo3_main_not3)
-  using rsimp_alts_idem by presburger
-
-
-lemma evo3_aux2:
-  shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
-  by simp
-lemma evo3_aux3:
-  shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
-  by (metis list.simps(8) list.simps(9) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem)
-
-lemma evo3_aux4:
-  shows " rsimp
-            (rsimp_ALTs
-              [RSEQ (rder x r) (RSTAR r),
-               rsimp (rsimp_ALTs rs)]) =
-           rsimp
-            (rsimp_ALTs
-              (rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
-  by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
-
-lemma evo3_aux5:
-  shows "rders_simp r a \<noteq> RONE \<and> rders_simp r a \<noteq> RZERO \<Longrightarrow> rsimp_SEQ (rders_simp r a) (RSTAR r) = RSEQ (rders_simp r a) (RSTAR r)"
-  using idiot2 by blast
-
-
-lemma evolution_step3:
-  shows" \<And>a Ss.
-       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))) \<Longrightarrow>
-       rsimp
-        (rsimp_ALTs
-          [rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
-           rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
-       rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
-  apply(case_tac "rders_simp r a = RONE")
-   apply(subst rsimp_seq_aux1)
-    apply simp
-  apply(subst rder.simps(6))
-   apply(subgoal_tac "rnullable (rders_simp r a)")
-    prefer 2
-  using rnullable.simps(2) apply presburger
-   apply(subst star_update_case1)
-    apply simp
-
-   apply(subst List.list.map)+
-  apply(subst rders_simp_append)
-   apply(subst evo3_aux2)
-    apply simp
-   apply(subst evo3_aux3)
-   apply(subst evo3_aux4)
-   apply simp
-  apply(case_tac "rders_simp r a = RZERO")
-
-   apply (simp add: rsimp_alts_idem2)
-   apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
-  prefer 2
-  using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
-  using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
-  apply(subst evo3_aux5)
-   apply simp
-  apply(case_tac "rnullable (rders_simp r a) ")
-  using evo3_main_nullable apply blast
-  using evo3_main_notnullable apply blast
-  done
-
-(*
-proof (prove)
-goal (1 subgoal):
- 1. map f (a # s) = f a # map f s 
-Auto solve_direct: the current goal can be solved directly with
-  HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
-  List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
-  List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
-*)
-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
-  apply(subst List.list.map(2))
-  apply(subst evolution_step2)
-   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 map_der_lambda_composition:
-  shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
-  by force
-
-lemma ralts_vs_rsimpalts:
-  shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
-  by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
-  
-
-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 (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
-  apply(subst rder_rsimp_ALTs_commute)
-  apply(subst map_der_lambda_composition)
-  using starseq_list_evolution
-  apply(rule_tac x = "star_update x r Ss" in exI)
-  apply(subst ralts_vs_rsimpalts)
-  by simp
-
-
-
-(*certified correctness---does not depend on any previous sorry*)
-lemma star_list_push_der: shows  " \<lbrakk>xs \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss));
-        xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
-     \<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) = 
-        rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
-  apply(subgoal_tac  "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
-  prefer 2
-  apply blast
-  apply(erule exE)
-  apply(subgoal_tac "rders_simp (RSTAR r) (xs @ [x]) = rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
-  prefer 2
-  using rders_simp_append
-  using rders_simp_one_char apply presburger
-  apply(rule_tac x= "Ss" in exI)
-  apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) = 
-                       rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
-  prefer 2
-  using inside_simp_removal rsimp_idem apply presburger
-  apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
-                     rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
-  prefer 2
-  using rder.simps(4) apply presburger
-  apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
-                     rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
-   apply (metis rsimp_idem)
-  by (metis map_der_lambda_composition)
-
-
-
-end
+end
\ No newline at end of file




