# HG changeset patch
# User Chengsong
# Date 1646133257 0
# Node ID 65e786a583652298dfeb4101da09bc1fd8a5e998
# Parent  0cce1aee0fb26ffc6f49d6cc4fefda9e2807125e
hi

diff -r 0cce1aee0fb2 -r 65e786a58365 thys2/SizeBound6CT.thy
--- a/thys2/SizeBound6CT.thy	Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/SizeBound6CT.thy	Tue Mar 01 11:14:17 2022 +0000
@@ -46,7 +46,7 @@
 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) )"
@@ -65,7 +65,7 @@
    apply presburger
   apply(drule_tac x="s1 @ [a]" in meta_spec)
   sorry
-
+*)
 
 
 lemma shape_of_prf_3list:
@@ -141,11 +141,6 @@
      (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
-
 
 
 
@@ -199,7 +194,14 @@
 | "rlist_size [] = 0"
 
 thm neq_Nil_conv
-
+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 rsimp_aalts_smaller:
   shows "rsize (rsimp_ALTs  rs) \<le> rsize (RALTS rs)"
@@ -249,7 +251,7 @@
   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
@@ -267,16 +269,83 @@
   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 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
+   apply presburger
+  using rsimp_alts_mono by auto
 
 lemma idiot:
   shows "rsimp_SEQ RONE r = r"
@@ -284,21 +353,15 @@
        apply simp_all
   done
 
-lemma no_dup_after_simp:
-  shows "RALTS rs = rsimp r \<Longrightarrow> distinct rs"
+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 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>
@@ -552,6 +615,10 @@
         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_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))"
@@ -831,14 +898,7 @@
         )"
   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:
@@ -1017,7 +1077,7 @@
 "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) )"
+                               else   (s@[c]) # (star_update c r Ss) )"
 
 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)"
@@ -1025,16 +1085,68 @@
   by force
 
 lemma star_update_case2:
-  shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = s # (star_update c r Ss)"
+  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))] ))"
-  sorry
+  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))] ))"
-  sorry
+  using head_one_more_simp rsimp_alts_idem by auto
+
 
 lemma evolution_step1:
   shows "rsimp
@@ -1056,6 +1168,225 @@
           (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)
@@ -1075,11 +1406,8 @@
   apply(subst List.list.map(2))
   apply(subst evolution_step2)
    apply simp
-  apply(case_tac "rnullable (rders_simp r a)")
-   apply(subst star_update_case1)
-    apply simp
-   apply(subst List.list.map)+
-  sledgehammer
+
+
   sorry
 
 
@@ -1094,7 +1422,8 @@
 
 lemma ralts_vs_rsimpalts:
   shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
-  sorry
+  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
@@ -1156,14 +1485,119 @@
   using star_list_push_der apply presburger
 
 
+  by (metis ralts_vs_rsimpalts starseq_list_evolution)
+
+
+lemma starder_is_a_list:
+  shows " \<exists>Ss. rders_simp (RSTAR r) s = rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) \<or> rders_simp (RSTAR r) s = RSTAR r"
+  apply(case_tac s)
+  prefer 2
+  apply (metis neq_Nil_conv starder_is_a_list_of_stars_or_starseqs)
+  apply simp
+  done
+
+
+(** start about bounds here**)
+
+
+lemma list_simp_size:
+  shows "rlist_size (map rsimp rs) \<le> rlist_size rs"
+  apply(induct rs)
+   apply simp
+  apply simp
+  apply (subgoal_tac "rsize (rsimp a) \<le> rsize a")
+  prefer 2
+  using rsimp_mono apply fastforce
+  using add_le_mono by presburger
+
+lemma inside_list_simp_inside_list:
+  shows "r \<in> set rs \<Longrightarrow> rsimp r \<in> set (map rsimp rs)"
+  apply (induct rs)
+   apply simp
+  apply auto
+  done
+
+
+lemma rsize_star_seq_list:
+  shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow>  \<exists>N3.\<forall>Ss.  
+rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3"
   sorry
 
 
+lemma rdistinct_bound_by_no_simp:
+  shows "
+
+   rlist_size (rdistinct  (map rsimp rs) (set (map rsimp ss)))
+ \<le> (rlist_size (rdistinct rs (set ss)))
+"
+  apply(induct rs arbitrary: ss)
+   apply simp
+  apply(case_tac "a \<in> set ss")
+   apply(subgoal_tac "rsimp a \<in> set (map rsimp ss)")
+  prefer 2
+  using inside_list_simp_inside_list apply blast
+
+   apply simp
+  apply simp
+  by (metis List.set_insert add_le_mono image_insert insert_absorb rsimp_mono trans_le_add2)
+
+
+lemma starder_closed_form_bound_aux1:
+  shows 
+"\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
+  Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {}))) "
+  
+  sorry
+
+lemma starder_closed_form_bound:
+  shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.  
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3"
+  apply(subgoal_tac " \<exists>N3.\<forall>Ss.  
+rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3")
+  prefer 2
+  
+  using rsize_star_seq_list apply auto[1]
+  apply(erule exE)
+  apply(rule_tac x = "Suc N3" in exI)
+  apply(subgoal_tac "\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
+ Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {})))")
+  prefer 2
+  using starder_closed_form_bound_aux1 apply blast
+  by (meson less_trans_Suc linorder_not_le not_less_eq)
+
+  
+thm starder_closed_form_bound_aux1
+
+(*
+ "ralts_vs_rsimpalts", , and "starder_closed_form_bound_aux1", which could be due to a bug in Sledgehammer or to inconsistent axioms (including "sorry"s) 
+*)
+
+lemma starder_size_bound:
+  shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.  
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
+rsize (RSTAR r0) < N3"
+  apply(subgoal_tac " \<exists>N3.\<forall>Ss.  
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3")
+  prefer 2
+  using starder_closed_form_bound apply blast
+  apply(erule exE)
+  apply(rule_tac x = "max N3 (Suc (rsize (RSTAR r0)))" in exI)
+  using less_max_iff_disj 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
+  apply(subgoal_tac  " \<exists>N3. \<forall>Ss. 
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
+rsize (RSTAR r0) < N3")
+  prefer 2
+  using starder_size_bound apply blast
+  apply(erule exE)
+  apply(rule_tac x = N3 in exI)
+  by (metis starder_is_a_list)
 
 
 lemma rderssimp_zero:
diff -r 0cce1aee0fb2 -r 65e786a58365 thys2/blexer2.sc
--- a/thys2/blexer2.sc	Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/blexer2.sc	Tue Mar 01 11:14:17 2022 +0000
@@ -622,7 +622,9 @@
 
   def bders_simp(s: List[Char], r: ARexp) : ARexp = s match {
     case Nil => r
-    case c::s => bders_simp(s, bsimp(bder(c, r)))
+    case c::s => 
+      println(erase(r))
+      bders_simp(s, bsimp(bder(c, r)))
   }
   
   def bdersSimp(s: String, r: Rexp) : ARexp = bders_simp(s.toList, internalise(r))
@@ -818,59 +820,60 @@
   val pderSTAR = pderUNIV(STARREG)
 
   val refSize = pderSTAR.map(size(_)).sum
-  println("different partial derivative terms:")
-  pderSTAR.foreach(r => r match {
+  // println("different partial derivative terms:")
+  // pderSTAR.foreach(r => r match {
       
-        case SEQ(head, rstar) =>
-          println(shortRexpOutput(head) ++ "~STARREG")
-        case STAR(rstar) =>
-          println("STARREG")
+  //       case SEQ(head, rstar) =>
+  //         println(shortRexpOutput(head) ++ "~STARREG")
+  //       case STAR(rstar) =>
+  //         println("STARREG")
       
-    }
-    )
-  println("the total number of terms is")
-  //println(refSize)
-  println(pderSTAR.size)
+  //   }
+  //   )
+  // println("the total number of terms is")
+  // //println(refSize)
+  // println(pderSTAR.size)
 
   val A : Rexp= ("c" | (ONE | "b") ~ "d") ~((ONE).%)
   val B : Rexp = ((ONE).%)
   val C : Rexp = ("d") ~ ((ONE).%)
   val PRUNE_REG : Rexp = (C | B | A)
   val APRUNE_REG = internalise(PRUNE_REG)
-  // val program_solution = pruneRexp(APRUNE_REG, breakIntoTerms(PRUNE_REG))
-  // println("program executes and gives: as disired!")
-  // println(shortRexpOutput(erase(program_solution)))
-  val simpedPruneReg = strongBsimp(APRUNE_REG)
-  println(shortRexpOutput(erase(simpedPruneReg)))
-  for(i <- List(100, 900 ) ){// 100, 400, 800, 840, 841, 900
-    val prog0 = "a" * i
-    //println(s"test: $prog0")
-    println(s"testing with $i a's" )
-    //val bd = bdersSimp(prog0, STARREG)//DB
-    val sbd = bdersSimpS(prog0, STARREG)//strongDB
-    starPrint(erase(sbd))
-    val subTerms = breakIntoTerms(erase(sbd))
-    //val subTermsLarge = breakIntoTerms(erase(bd))
+  // // val program_solution = pruneRexp(APRUNE_REG, breakIntoTerms(PRUNE_REG))
+  // // println("program executes and gives: as disired!")
+  // // println(shortRexpOutput(erase(program_solution)))
+  // val simpedPruneReg = strongBsimp(APRUNE_REG)
+
+  // println(shortRexpOutput(erase(simpedPruneReg)))
+  // for(i <- List(1,2 ) ){// 100, 400, 800, 840, 841, 900
+  //   val prog0 = "a" * i
+  //   //println(s"test: $prog0")
+  //   println(s"testing with $i a's" )
+  //   //val bd = bdersSimp(prog0, STARREG)//DB
+  //   val sbd = bdersSimpS(prog0, STARREG)//strongDB
+  //   starPrint(erase(sbd))
+  //   val subTerms = breakIntoTerms(erase(sbd))
+  //   //val subTermsLarge = breakIntoTerms(erase(bd))
     
-    println(s"subterms of regex with strongDB: ${subTerms.length}")//, standard DB: ${subTermsLarge.length}")
+  //   println(s"subterms of regex with strongDB: ${subTerms.length}")//, standard DB: ${subTermsLarge.length}")
 
 
 
-    println("the number of distinct subterms for bsimp with strongDB")
-    println(subTerms.distinct.size)
-    //println(subTermsLarge.distinct.size)
-    println("which coincides with the number of PDER terms")
+  //   println("the number of distinct subterms for bsimp with strongDB")
+  //   println(subTerms.distinct.size)
+  //   //println(subTermsLarge.distinct.size)
+  //   println("which coincides with the number of PDER terms")
 
 
-    // println(shortRexpOutput(erase(sbd)))
-    // println(shortRexpOutput(erase(bd)))
+  //   // println(shortRexpOutput(erase(sbd)))
+  //   // println(shortRexpOutput(erase(bd)))
     
-    println("pdersize, original, strongSimp")
-    println(refSize, size(STARREG),  asize(sbd))
+  //   println("pdersize, original, strongSimp")
+  //   println(refSize, size(STARREG),  asize(sbd))
 
-    val vres = strong_blexing_simp( STARREG, prog0)
-    println(vres)
-  }
+  //   val vres = strong_blexing_simp( STARREG, prog0)
+  //   println(vres)
+  // }
 //   println(vs.length)
 //   println(vs)
    
@@ -878,6 +881,9 @@
   // val prog1 = """read  n; write n"""  
   // println(s"test: $prog1")
   // println(lexing_simp(WHILE_REGS, prog1))
+  val display = ("a"| "ab").%
+  val adisplay = internalise(display)
+  bders_simp( "aaaaa".toList, adisplay)
 }
 
 small()