
diff -r c6351a96bf80 -r a7e98deebb5c thys2/ClosedFormsBounds.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/ClosedFormsBounds.thy	Wed Mar 09 17:33:08 2022 +0000
@@ -0,0 +1,585 @@
+
+theory ClosedFormsBounds
+  imports "GeneralRegexBound" "ClosedForms"
+begin
+
+
+
+lemma alts_closed_form_bounded: shows
+"\<forall>r \<in> set rs. \<forall>s. rsize(rders_simp r s ) \<le> N \<Longrightarrow> 
+rsize (rders_simp (RALTS rs ) s) \<le> max (Suc ( N * (card (sizeNregex N)))) (rsize (RALTS rs) )"
+  apply(induct s)
+  apply simp
+  apply(insert alts_closed_form_variant)
+
+  
+  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 r0 [[c]]) ) ))) \<le>
+        Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
+         (star_updates s r0 [[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 (sizeNregex N))* N"
+  sorry
+
+
+lemma star_control_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 r0 [[c]]) ) {})  ) ) \<le> 
+(card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
+"
+  sorry
+
+lemma star_control_variant:
+  assumes "\<forall>s. rsize (rders_simp r0 s) \<le> N"
+  shows"Suc 
+      (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) 
+          (star_updates list r0 [[a]])) {}))) 
+\<le>  (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) "
+  apply(subgoal_tac    "(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) 
+          (star_updates list r0 [[a]])) {}))) 
+\<le>  ( (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
+  prefer 2
+  using assms star_control_bounded apply presburger
+  by simp
+
+
+
+lemma star_closed_form_bounded:
+  shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
+              rsize (rders_simp (RSTAR r0) s) \<le> 
+max (   (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))))   (rsize (RSTAR r0))"
+  apply(case_tac s)
+  apply simp
+  apply(subgoal_tac " rsize (rders_simp (RSTAR r0) (a # list)) = 
+rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))") 
+   prefer 2
+  using star_closed_form apply presburger
+  apply(subgoal_tac "rsize (rsimp (
+ RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list    r0 [[a]]) ) ))) 
+\<le>         Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
+         (star_updates list r0 [[a]]) ) {})  ) )")
+  prefer 2
+  using star_closed_form_bounded_by_rdistinct_list_estimate apply presburger
+  apply(subgoal_tac "Suc (sum_list 
+                 (map rsize
+                   (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates list r0 [[a]])) {}))) 
+\<le>  (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0)))  ")
+  apply auto[1]
+  using star_control_variant by blast
+
+
+
+
+
+
+lemma seq_list_estimate_control: shows 
+" rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
+           \<le> Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))"
+  
+  sorry
+
+lemma seq_estimate_bounded: 
+  assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1" and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
+  shows
+"Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
+ Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
+
+  sorry
+
+lemma seq_closed_form_bounded: shows
+"\<lbrakk>\<forall>s. rsize (rders_simp r1 s) \<le> N1 ; \<forall>s. rsize (rders_simp r2 s) \<le> N2\<rbrakk> \<Longrightarrow>
+rsize (rders_simp (RSEQ r1 r2) s) \<le> 
+max (Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))) (rsize (RSEQ r1 r2)) "
+  apply(case_tac s)
+  apply simp
+  apply(subgoal_tac " (rders_simp (RSEQ r1 r2) s) = 
+rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))")
+  prefer 2
+  using seq_closed_form_variant apply blast
+  apply(subgoal_tac "rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
+                    \<le>
+Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))")
+  apply(subgoal_tac "Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))
+\<le> Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
+  prefer 2
+  using seq_estimate_bounded apply blast
+   apply(subgoal_tac "rsize (rders_simp (RSEQ r1 r2) s) \<le> Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))")
+  using le_max_iff_disj apply blast
+   apply auto[1]
+  using seq_list_estimate_control by presburger
+
+
+lemma rders_simp_bounded: shows
+"\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
+  apply(induct r)
+       apply(rule_tac x = "Suc 0 " in exI)
+  using three_easy_cases0 apply force
+  using three_easy_cases1 apply blast
+  using three_easy_casesC apply blast
+  using seq_closed_form_bounded apply blast
+  apply (metis alts_closed_form_bounded size_list_estimation')
+  using star_closed_form_bounded by blast
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(*Obsolete materials*)
+
+
+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
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+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




