# HG changeset patch
# User Chengsong
# Date 1646911121 0
# Node ID e072cfc2f2ee8a360b399fa93e62d67374c40c04
# Parent  a7e98deebb5c0d3361d060e3bbdcd0a89050ff5e
closedforms

diff -r a7e98deebb5c -r e072cfc2f2ee thys2/ClosedForms.thy
--- a/thys2/ClosedForms.thy	Wed Mar 09 17:33:08 2022 +0000
+++ b/thys2/ClosedForms.thy	Thu Mar 10 11:18:41 2022 +0000
@@ -35,7 +35,7 @@
 lemma seq_closed_form: shows 
 "rsimp (rders_simp (RSEQ r1 r2) s) = 
 rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) # 
-                (map (rders r2) (vsuf s r1)) 
+                (map (rders_simp r2) (vsuf s r1)) 
               )  
       )"
   apply(induct s)
@@ -46,6 +46,17 @@
 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
+  apply(induct s rule: rev_induct)
+   apply simp
+  apply(subst rders_simp_append)
+  apply(subst rders_simp_one_char)
+  apply(subst rsimp_idem[symmetric])
+  apply(subst rders_simp_one_char[symmetric])
+  apply(subst rders_simp_append[symmetric])
+  apply(insert seq_closed_form)
+  apply(subgoal_tac "rsimp (rders_simp (RSEQ r1 r2) (xs @ [x]))
+ = rsimp (RALTS (RSEQ (rders_simp r1 (xs @ [x])) r2 # map (rders_simp r2) (vsuf (xs @ [x]) r1)))")
+   apply force
+  by presburger
 
 end
\ No newline at end of file
diff -r a7e98deebb5c -r e072cfc2f2ee thys2/ClosedFormsBounds.thy
--- a/thys2/ClosedFormsBounds.thy	Wed Mar 09 17:33:08 2022 +0000
+++ b/thys2/ClosedFormsBounds.thy	Thu Mar 10 11:18:41 2022 +0000
@@ -28,6 +28,7 @@
 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
 
 
@@ -88,13 +89,57 @@
   
   sorry
 
+lemma triangle_inequality_distinct:
+  shows "sum_list (map rsize (rdistinct (a # rs) ss)) \<le> rsize a + (sum_list (map rsize (rdistinct rs ss)))"
+  apply(arbitrary: ss)
+   apply simp
+  apply(case_tac "a \<in> ss")
+   apply simp
+
+  sorry
+
+lemma same_regex_property_after_map:
+  shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set  (map (f r2) Ss). P r"
+  by auto
+
+lemma same_property_after_distinct:
+  shows " \<forall>r \<in> set  (map (f r2) Ss). P r \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
+  apply(induct Ss arbitrary: xset)
+   apply simp
+  by auto
+
+lemma same_regex_property_after_distinct:
+  shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
+  apply(rule same_property_after_distinct)
+  apply(rule same_regex_property_after_map)
+  by simp
+
+lemma map_ders_is_list_of_ders:
+  shows  "\<forall>s. rsize (rders_simp r2 s) \<le> N2 \<Longrightarrow>
+\<forall>r \<in> set (rdistinct (map (rders_simp r2) Ss) {}). rsize r \<le> N2"
+  apply(rule same_regex_property_after_distinct)
+  by simp
+
 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)))"
+  apply(subgoal_tac " (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
+  (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
+   apply force
+  apply(subgoal_tac " (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
+                      (rsize (RSEQ (rders_simp r1 s) r2)) + (sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) )")
+  prefer 2
+  using triangle_inequality_distinct apply blast
+  apply(subgoal_tac " sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) \<le> N2 * card (sizeNregex N2) ")
+   apply(subgoal_tac "rsize (RSEQ (rders_simp r1 s) r2) \<le> Suc (N1 + rsize r2)")
+    apply linarith
+   apply (simp add: assms(1))
+  apply(subgoal_tac "\<forall>r \<in> set (rdistinct (map (rders_simp r2) (vsuf s r1)) {}). rsize r \<le> N2")
+  apply (metis (no_types, opaque_lifting) assms(2) distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size ex_map_conv mult.commute)
+  using assms(2) map_ders_is_list_of_ders by blast
 
-  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>
@@ -145,441 +190,8 @@
 
 
 
-
-
-
-
-
-
-
-
-
-
 (*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