diff -r dabd25e8e4e9 -r 7a016eeb118d thys2/ClosedFormsBounds.thy
--- a/thys2/ClosedFormsBounds.thy	Sat Mar 12 14:33:54 2022 +0000
+++ b/thys2/ClosedFormsBounds.thy	Tue Mar 15 16:37:41 2022 +0000
@@ -1,14 +1,11 @@
-
+(**)
 theory ClosedFormsBounds
   imports "GeneralRegexBound" "ClosedForms"
 begin
-
 lemma alts_ders_lambda_shape_ders:
   shows "\<forall>r \<in> set (map (\<lambda>r. rders_simp r ( s)) rs ). \<exists>r1 \<in> set rs. r = rders_simp r1 s"
   by (simp add: image_iff)
 
-
-
 lemma rlist_bound:
   shows "\<forall>r \<in> set rs. rsize r \<le> N \<Longrightarrow> sum_list (map rsize rs) \<le> N * (length rs)"
   apply(induct rs)
@@ -46,20 +43,391 @@
   shows "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))"
   using rsimp_aalts_smaller by auto
 
+lemma no_elem_distinct :
+  shows "a \<notin> set rs \<Longrightarrow> rdistinct rs rset = rdistinct rs (insert a rset)"
+  apply(induct rs arbitrary: rset)
+   apply simp
+  by force
+  
+
+
+
+inductive good :: "rrexp \<Rightarrow> bool"
+  where
+"RZERO \<notin> set rs \<Longrightarrow> good (RALTS rs)"
+|"good RZERO"
+|"good RONE"
+|"good (RCHAR c)"
+|"good (RSEQ r1 r2)"
+|"good (RSTAR r0)"
+
+
+lemma after_flts_no0:
+  shows "\<forall>r \<in> set rs. good r \<Longrightarrow> RZERO \<notin> set (rflts rs)"
+  apply(induct rs)
+   apply simp
+  apply(case_tac a)
+       apply simp
+  apply simp
+  apply simp
+    apply simp
+  apply simp
+  using good.simps apply blast
+  apply simp
+  done
+
 lemma flts_has_no_zero:
-  shows "rdistinct (rflts rs) rset = rdistinct (rflts rs) (insert RZERO rset)"
+  shows "\<forall>r \<in> set rs. good r \<Longrightarrow> rdistinct (rflts rs) rset = rdistinct (rflts rs) (insert RZERO rset)"
+  apply(subgoal_tac "RZERO \<notin> set (rflts rs)")  
+   apply (meson no_elem_distinct)
+  apply(insert after_flts_no0)
+  by blast
+
+lemma shape_of_alts:
+  shows "\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs) \<Longrightarrow> RZERO \<notin> alts_set \<and> RONE \<notin> alts_set"
+  by fastforce
+
+lemma quantified2_implies1:
+  shows "\<forall>r. P \<and> Q \<Longrightarrow> \<forall>r. P"
+  by auto
+
+lemma quantified_quantifiedAB_A:
+  shows " (\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set) \<Longrightarrow> \<forall>a \<in> alts_set. \<exists>xs. a = RALTS xs"
+  by fastforce
+
+
+
+
+lemma list_distinct_removal:
+  shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rs1) rset = rdistinct rs1 rset"
+  apply(induct rs arbitrary: rset)
+   apply simp
+  by simp
+
+
+lemma rdistinct_mono_list:
+  shows " sum_list (map rsize (rdistinct (x5 @ rs) rset )) \<le>
+        ( sum_list (map rsize x5)) + (sum_list (map rsize (rdistinct  rs ((set x5 ) \<union> rset))))"
+  apply(induct x5 arbitrary: rs rset)
+   apply simp
+  apply(case_tac "a \<in> rset")
+   apply simp
+   apply (simp add: add.assoc insert_absorb trans_le_add2)
+  apply simp
+  by (metis Un_insert_right)
+
+
+lemma flts_size_reduction_alts:
+  shows " \<And>a rs noalts_set alts_set corr_set x5.
+       \<lbrakk>\<And>noalts_set alts_set corr_set.
+           (\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
+           (\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set) \<Longrightarrow>
+           Suc (sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> corr_set))))
+           \<le> Suc (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))));
+        (\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
+        (\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set);
+        a = RALTS x5\<rbrakk>
+       \<Longrightarrow> Suc (sum_list (map rsize (rdistinct (rflts (a # rs)) (noalts_set \<union> corr_set))))
+           \<le> Suc (sum_list
+                    (map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))))"
+  apply(case_tac "a \<in> alts_set")
+   apply simp
+   apply(subgoal_tac "set x5 \<subseteq> corr_set")
+  apply(subst list_distinct_removal)
+  apply auto[1]
+    apply presburger
+   apply fastforce
+  apply (subgoal_tac "a \<notin> noalts_set")
+  prefer 2
+  apply blast
+  apply simp
+  apply(subgoal_tac "sum_list (map rsize (rdistinct (x5 @ rflts rs) (noalts_set \<union> corr_set))) 
+                   \<le> sum_list (map rsize x5 ) + sum_list (map rsize (rdistinct (rflts rs) ((set x5) \<union> (noalts_set \<union> corr_set))))")
+  prefer 2
+  using rdistinct_mono_list apply presburger
+  apply(subgoal_tac "insert (RALTS x5) (noalts_set \<union> alts_set) = noalts_set \<union> (insert (RALTS x5) alts_set)")
+   apply(simp only:)
+  apply(subgoal_tac "sum_list (map rsize x5) + 
+sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> (set x5))))) \<le>
+Suc (sum_list (map rsize x5) +
+                   sum_list
+                    (map rsize
+                      (rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set)))))")
+  
+  apply (simp add: Un_left_commute inf_sup_aci(5))
+   apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> set x5))))
+\<le> sum_list
+                    (map rsize
+                      (rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set))))")
+    apply linarith
+   apply(subgoal_tac "\<forall>r \<in>  insert (RALTS x5) alts_set. \<exists>xs1.( r = RALTS xs1 \<and> set xs1 \<subseteq> corr_set \<union> set x5)")
+    apply presburger
+   apply (meson insert_iff sup.cobounded2 sup.coboundedI1)
+  by blast
+
+
+
 
-  sorry
+lemma flts_vs_nflts1:
+  shows "(\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs)
+ \<and> (\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set))  
+\<Longrightarrow>  (sum_list (map rsize (rdistinct ( rflts rs) (noalts_set \<union> corr_set)  )))
+         \<le>  (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set) ))))"
+    apply(induct rs arbitrary: noalts_set alts_set corr_set)
+   apply simp
+  apply(case_tac a)
+       apply(case_tac "RZERO \<in> noalts_set")
+        apply simp
+       apply(subgoal_tac "RZERO \<notin> alts_set")
+        apply simp
+       apply fastforce
+      apply(case_tac "RONE \<in> noalts_set")
+       apply simp
+      apply(subgoal_tac "RONE \<notin> alts_set")
+  prefer 2
+  apply fastforce
+      apply(case_tac "RONE \<in> corr_set")
+       apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
+        apply(simp only:)
+        apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) = 
+                           rdistinct (rflts rs) (noalts_set \<union> corr_set)")
+         apply(simp only:)
+  apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
+                     RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
+          apply(simp only:)
+  apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) = 
+                     rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
+  apply (simp only:)
+  apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
+            apply(simp only:)
+  apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) = 
+                     insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
+             apply(simp only:)
+  apply(subgoal_tac "  (sum_list
+                    (map rsize ( rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))
+                   \<le>  (sum_list
+                    (map rsize (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))")
+  apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
+  apply (metis (no_types, opaque_lifting) add_Suc_right le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
+            apply fastforce
+           apply fastforce
+  apply (metis Un_iff insert_absorb)
+         apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
+        apply (meson UnCI rdistinct.simps(2))
+  using rflts.simps(4) apply presburger
+      apply simp
+      apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
+  apply(simp only:)
+  apply (metis Un_insert_left insertE rrexp.distinct(15))
+      apply fastforce
+     apply(case_tac "a \<in> noalts_set")
+      apply simp
+  apply(subgoal_tac "a \<notin> alts_set")
+      prefer 2
+      apply blast
+  apply(case_tac "a \<in> corr_set")
+      apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set  \<union> corr_set)")
+  prefer 2
+  apply fastforce
+      apply(simp only:)
+      apply(subgoal_tac "  sum_list
+               (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
+\<le>
+                sum_list
+               (map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
+
+       apply(subgoal_tac 
+"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
+\<le>
+ sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
+  apply fastforce
+       apply simp
+  apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
+        apply(simp only:)
+        apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
+  apply(simp only:)
+  apply (metis insertE rrexp.distinct(21))
+        apply blast
+  
+  apply fastforce
+  apply force
+     apply simp
+     apply (metis Un_insert_left insert_iff rrexp.distinct(21))
+    apply(case_tac "a \<in> noalts_set")
+     apply simp
+  apply(subgoal_tac "a \<notin> alts_set")
+      prefer 2
+      apply blast
+  apply(case_tac "a \<in> corr_set")
+      apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set  \<union> corr_set)")
+  prefer 2
+  apply fastforce
+      apply(simp only:)
+      apply(subgoal_tac "  sum_list
+               (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
+\<le>
+                sum_list
+               (map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
+
+       apply(subgoal_tac 
+"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
+\<le>
+ sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
+  apply fastforce
+       apply simp
+  apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
+        apply(simp only:)
+        apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
+  apply(simp only:)
+
+
+  apply (metis insertE rrexp.distinct(25))
+  apply blast
+  apply fastforce
+  apply force
+     apply simp
+  
+    apply (metis Un_insert_left insertE rrexp.distinct(25))
+
+  using Suc_le_mono flts_size_reduction_alts apply presburger
+     apply(case_tac "a \<in> noalts_set")
+      apply simp
+  apply(subgoal_tac "a \<notin> alts_set")
+      prefer 2
+      apply blast
+  apply(case_tac "a \<in> corr_set")
+      apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set  \<union> corr_set)")
+  prefer 2
+  apply fastforce
+      apply(simp only:)
+      apply(subgoal_tac "  sum_list
+               (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
+\<le>
+                sum_list
+               (map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
+
+       apply(subgoal_tac 
+"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
+\<le>
+ sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
+  apply fastforce
+       apply simp
+  apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
+        apply(simp only:)
+        apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
+  apply(simp only:)
+  apply (metis insertE rrexp.distinct(29))
+
+        apply blast
+  
+  apply fastforce
+  apply force
+     apply simp
+  apply (metis Un_insert_left insert_iff rrexp.distinct(29))
+  done
+
+
+
+
 
 lemma flts_vs_nflts:
-  shows "\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs
- \<and> (\<forall>a \<in> alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set)  
+  shows "(\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs)
+ \<and> (\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set))  
 \<Longrightarrow> Suc (sum_list (map rsize (rdistinct ( rflts rs) (noalts_set \<union> corr_set)  )))
-         \<le> Suc (sum_list (map rsize (rdistinct rs (noalts_set \<union> alts_set) )))"
-  apply(induct rs arbitrary: noalts_set)
+         \<le> Suc (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set) ))))"
+  apply(induct rs arbitrary: noalts_set alts_set corr_set)
    apply simp
+  apply(case_tac a)
+       apply(case_tac "RZERO \<in> noalts_set")
+        apply simp
+       apply(subgoal_tac "RZERO \<notin> alts_set")
+        apply simp
+       apply fastforce
+      apply(case_tac "RONE \<in> noalts_set")
+       apply simp
+      apply(subgoal_tac "RONE \<notin> alts_set")
+  prefer 2
+  apply fastforce
+      apply(case_tac "RONE \<in> corr_set")
+       apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
+        apply(simp only:)
+        apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) = 
+                           rdistinct (rflts rs) (noalts_set \<union> corr_set)")
+         apply(simp only:)
+  apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
+                     RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
+          apply(simp only:)
+  apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) = 
+                     rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
+  apply (simp only:)
+  apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
+            apply(simp only:)
+  apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) = 
+                     insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
+             apply(simp only:)
+  apply(subgoal_tac " Suc (sum_list
+                    (map rsize ( rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))
+                   \<le> Suc (sum_list
+                    (map rsize (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))")
+  apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
+  apply (metis (no_types, opaque_lifting) add_Suc_right le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
+            apply fastforce
+           apply fastforce
+  apply (metis Un_iff insert_absorb)
+         apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
+        apply (meson UnCI rdistinct.simps(2))
+  using rflts.simps(4) apply presburger
+      apply simp
+      apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
+  apply(simp only:)
+  apply (metis Un_insert_left insertE rrexp.distinct(15))
+      apply fastforce
+     apply(case_tac "a \<in> noalts_set")
+      apply simp
+  apply(subgoal_tac "a \<notin> alts_set")
+      prefer 2
+      apply blast
+  apply(case_tac "a \<in> corr_set")
+      apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set  \<union> corr_set)")
+  prefer 2
+  apply fastforce
+      apply(simp only:)
+  apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
+  apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
+     apply(case_tac "a \<in> noalts_set")
+      apply simp
+  apply(subgoal_tac "a \<notin> alts_set")
+      prefer 2
+      apply blast
+  apply(case_tac "a \<in> corr_set")
+      apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set  \<union> corr_set)")
+  prefer 2
+  apply fastforce
+      apply(simp only:)
+  apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
+  apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
+   prefer 2
+     apply(case_tac "a \<in> noalts_set")
+      apply simp
+  apply(subgoal_tac "a \<notin> alts_set")
+      prefer 2
+      apply blast
+  apply(case_tac "a \<in> corr_set")
+      apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set  \<union> corr_set)")
+  prefer 2
+  apply fastforce
+      apply(simp only:)
+  apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
+  apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
+  using flts_size_reduction_alts apply presburger
+  done
 
-  sorry
+
+
+
+
+(*  apply(rutac x = "\<lambda>a rs noalts_set alts_set corr_set. set xs ")*)
+  
 
 lemma distinct_simp_ineq_general:
   shows "rsimp ` no_simp = has_simp \<and> finite no_simp \<Longrightarrow>Suc (sum_list (map rsize (rdistinct (map rsimp rs) has_simp)))
@@ -73,10 +441,14 @@
   apply(case_tac "rsimp a \<in> no_simp")
    apply(subgoal_tac "rsimp a \<in> has_simp")
   prefer 2
-  apply (simp add: rev_image_eqI rsimp_idem)
-  sledgehammer[timeout = 100]
-
-  sorry
+    apply (simp add: rev_image_eqI rsimp_idem)
+  apply simp
+   apply (metis finite_insert image_insert insert_absorb trans_le_add2)
+  apply(case_tac "rsimp a \<notin> has_simp")
+  apply simp
+   apply (metis add_mono_thms_linordered_semiring(1) finite.insertI image_insert rsimp_mono)
+  apply simp
+  by (metis finite.insertI image_insert insert_absorb trans_le_add2)
 
 
 lemma not_mentioned_elem_distinct_strong:
@@ -92,13 +464,40 @@
 
 
 
-
+lemma larger_acc_smaller_distinct_res0:
+  shows " ss \<subseteq> SS \<Longrightarrow> sum_list (map rsize (rdistinct rs SS)) \<le> sum_list (map rsize (rdistinct rs ss))"
+  apply(induct rs arbitrary: ss SS)
+  apply simp
+  apply(case_tac "a \<in> ss")
+   apply(subgoal_tac "a \<in> SS")
+    apply simp
+   apply blast
+  apply(case_tac "a \<in> SS")
+   apply simp
+   apply(subgoal_tac "insert a ss \<subseteq> SS")
+    apply simp
+  apply (simp add: trans_le_add2)
+  apply blast
+  apply(simp)
+  apply(subgoal_tac "insert a ss \<subseteq> insert a SS")
+   apply blast
+  by blast
 
 
 lemma without_flts_ineq:
   shows " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le> 
          Suc (sum_list (map rsize (rdistinct (    rs  ) {}  )))"
-  by (metis empty_iff flts_vs_nflts sup_bot_left)
+proof -
+  have " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le>  
+         Suc (sum_list (map rsize (rdistinct rs (insert RZERO {}))))"
+    by (metis empty_iff flts_vs_nflts sup_bot_left)
+  also have "... \<le>  Suc (sum_list (map rsize (rdistinct rs {})))" 
+    by (simp add: larger_acc_smaller_distinct_res0)
+  finally show ?thesis sledgehammer
+    by blast
+qed
+
+
 
 
 
@@ -131,24 +530,7 @@
   shows "a \<notin> ss \<Longrightarrow> rdistinct (a  # rs) ss = a # rdistinct rs (insert a ss) "
   by auto
 
-lemma larger_acc_smaller_distinct_res0:
-  shows " ss \<subseteq> SS \<Longrightarrow> sum_list (map rsize (rdistinct rs SS)) \<le> sum_list (map rsize (rdistinct rs ss))"
-  apply(induct rs arbitrary: ss SS)
-  apply simp
-  apply(case_tac "a \<in> ss")
-   apply(subgoal_tac "a \<in> SS")
-    apply simp
-   apply blast
-  apply(case_tac "a \<in> SS")
-   apply simp
-   apply(subgoal_tac "insert a ss \<subseteq> SS")
-    apply simp
-  apply (simp add: trans_le_add2)
-  apply blast
-  apply(simp)
-  apply(subgoal_tac "insert a ss \<subseteq> insert a SS")
-   apply blast
-  by blast
+
 
 
 lemma larger_acc_smaller_distinct_res:
@@ -393,7 +775,7 @@
 
 
 
-
+unused_thms