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451
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(**)
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444
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theory ClosedFormsBounds
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imports "GeneralRegexBound" "ClosedForms"
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begin
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449
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lemma alts_ders_lambda_shape_ders:
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shows "\<forall>r \<in> set (map (\<lambda>r. rders_simp r ( s)) rs ). \<exists>r1 \<in> set rs. r = rders_simp r1 s"
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by (simp add: image_iff)
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lemma rlist_bound:
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shows "\<forall>r \<in> set rs. rsize r \<le> N \<Longrightarrow> sum_list (map rsize rs) \<le> N * (length rs)"
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apply(induct rs)
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apply simp
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by simp
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444
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lemma alts_closed_form_bounded: shows
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"\<forall>r \<in> set rs. \<forall>s. rsize(rders_simp r s ) \<le> N \<Longrightarrow>
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449
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rsize (rders_simp (RALTS rs ) s) \<le> max (Suc ( N * (length rs))) (rsize (RALTS rs) )"
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444
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apply(induct s)
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apply simp
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449
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apply(subst alts_closed_form_variant)
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apply force
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apply(subgoal_tac "rsize (rsimp (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs))) \<le> rsize ( (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs)))")
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447
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prefer 2
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449
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using rsimp_mono apply presburger
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apply(subgoal_tac "rsize (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs)) =
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Suc (sum_list (map rsize (map (\<lambda>r. rders_simp r (a # s)) rs)))")
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444
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prefer 2
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449
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using rsize.simps(4) apply blast
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apply(subgoal_tac "sum_list (map rsize (map (\<lambda>r. rders_simp r (a # s)) rs )) \<le> N * (length rs) ")
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apply linarith
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apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (a # s)) rs ). rsize r \<le> N")
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prefer 2
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apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (a # s)) rs ). \<exists>r1 \<in> set rs. r = rders_simp r1 (a # s)")
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prefer 2
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using alts_ders_lambda_shape_ders apply presburger
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apply metis
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apply(frule rlist_bound)
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by fastforce
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444
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447
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lemma alts_simp_ineq_unfold:
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shows "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))"
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using rsimp_aalts_smaller by auto
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444
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451
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lemma no_elem_distinct :
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shows "a \<notin> set rs \<Longrightarrow> rdistinct rs rset = rdistinct rs (insert a rset)"
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apply(induct rs arbitrary: rset)
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apply simp
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by force
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inductive good :: "rrexp \<Rightarrow> bool"
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where
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"RZERO \<notin> set rs \<Longrightarrow> good (RALTS rs)"
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|"good RZERO"
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|"good RONE"
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|"good (RCHAR c)"
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|"good (RSEQ r1 r2)"
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|"good (RSTAR r0)"
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lemma after_flts_no0:
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shows "\<forall>r \<in> set rs. good r \<Longrightarrow> RZERO \<notin> set (rflts rs)"
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apply(induct rs)
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apply simp
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apply(case_tac a)
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apply simp
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apply simp
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apply simp
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apply simp
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apply simp
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using good.simps apply blast
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apply simp
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done
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447
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lemma flts_has_no_zero:
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451
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shows "\<forall>r \<in> set rs. good r \<Longrightarrow> rdistinct (rflts rs) rset = rdistinct (rflts rs) (insert RZERO rset)"
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apply(subgoal_tac "RZERO \<notin> set (rflts rs)")
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apply (meson no_elem_distinct)
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apply(insert after_flts_no0)
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by blast
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lemma shape_of_alts:
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shows "\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs) \<Longrightarrow> RZERO \<notin> alts_set \<and> RONE \<notin> alts_set"
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by fastforce
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lemma quantified2_implies1:
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shows "\<forall>r. P \<and> Q \<Longrightarrow> \<forall>r. P"
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by auto
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lemma quantified_quantifiedAB_A:
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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"
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by fastforce
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lemma list_distinct_removal:
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shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rs1) rset = rdistinct rs1 rset"
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apply(induct rs arbitrary: rset)
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apply simp
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by simp
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lemma rdistinct_mono_list:
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shows " sum_list (map rsize (rdistinct (x5 @ rs) rset )) \<le>
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( sum_list (map rsize x5)) + (sum_list (map rsize (rdistinct rs ((set x5 ) \<union> rset))))"
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apply(induct x5 arbitrary: rs rset)
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apply simp
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apply(case_tac "a \<in> rset")
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apply simp
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apply (simp add: add.assoc insert_absorb trans_le_add2)
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apply simp
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by (metis Un_insert_right)
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lemma flts_size_reduction_alts:
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shows " \<And>a rs noalts_set alts_set corr_set x5.
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\<lbrakk>\<And>noalts_set alts_set corr_set.
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(\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
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(\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set) \<Longrightarrow>
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Suc (sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> corr_set))))
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\<le> Suc (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))));
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(\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
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(\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set);
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a = RALTS x5\<rbrakk>
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\<Longrightarrow> Suc (sum_list (map rsize (rdistinct (rflts (a # rs)) (noalts_set \<union> corr_set))))
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\<le> Suc (sum_list
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(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))))"
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apply(case_tac "a \<in> alts_set")
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apply simp
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apply(subgoal_tac "set x5 \<subseteq> corr_set")
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apply(subst list_distinct_removal)
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apply auto[1]
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apply presburger
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apply fastforce
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apply (subgoal_tac "a \<notin> noalts_set")
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prefer 2
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apply blast
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apply simp
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apply(subgoal_tac "sum_list (map rsize (rdistinct (x5 @ rflts rs) (noalts_set \<union> corr_set)))
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\<le> sum_list (map rsize x5 ) + sum_list (map rsize (rdistinct (rflts rs) ((set x5) \<union> (noalts_set \<union> corr_set))))")
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prefer 2
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using rdistinct_mono_list apply presburger
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apply(subgoal_tac "insert (RALTS x5) (noalts_set \<union> alts_set) = noalts_set \<union> (insert (RALTS x5) alts_set)")
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apply(simp only:)
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apply(subgoal_tac "sum_list (map rsize x5) +
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sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> (set x5))))) \<le>
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Suc (sum_list (map rsize x5) +
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sum_list
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(map rsize
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(rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set)))))")
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apply (simp add: Un_left_commute inf_sup_aci(5))
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apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> set x5))))
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\<le> sum_list
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(map rsize
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(rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set))))")
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apply linarith
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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)")
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apply presburger
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apply (meson insert_iff sup.cobounded2 sup.coboundedI1)
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by blast
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449
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451
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lemma flts_vs_nflts1:
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shows "(\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs)
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\<and> (\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set))
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\<Longrightarrow> (sum_list (map rsize (rdistinct ( rflts rs) (noalts_set \<union> corr_set) )))
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\<le> (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set) ))))"
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apply(induct rs arbitrary: noalts_set alts_set corr_set)
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apply simp
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apply(case_tac a)
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apply(case_tac "RZERO \<in> noalts_set")
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apply simp
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apply(subgoal_tac "RZERO \<notin> alts_set")
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apply simp
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apply fastforce
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apply(case_tac "RONE \<in> noalts_set")
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apply simp
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apply(subgoal_tac "RONE \<notin> alts_set")
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prefer 2
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apply fastforce
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apply(case_tac "RONE \<in> corr_set")
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apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
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apply(simp only:)
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apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) =
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rdistinct (rflts rs) (noalts_set \<union> corr_set)")
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apply(simp only:)
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apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
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RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
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apply(simp only:)
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apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) =
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rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
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apply (simp only:)
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apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
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apply(simp only:)
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apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) =
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insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
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apply(simp only:)
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apply(subgoal_tac " (sum_list
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(map rsize ( rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))
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\<le> (sum_list
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(map rsize (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))")
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apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
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apply (metis (no_types, opaque_lifting) add_Suc_right le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
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apply fastforce
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apply fastforce
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apply (metis Un_iff insert_absorb)
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apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
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apply (meson UnCI rdistinct.simps(2))
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using rflts.simps(4) apply presburger
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apply simp
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apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
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apply(simp only:)
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apply (metis Un_insert_left insertE rrexp.distinct(15))
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apply fastforce
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apply(case_tac "a \<in> noalts_set")
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apply simp
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apply(subgoal_tac "a \<notin> alts_set")
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prefer 2
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apply blast
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apply(case_tac "a \<in> corr_set")
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apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
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prefer 2
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apply fastforce
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apply(simp only:)
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apply(subgoal_tac " sum_list
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(map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
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\<le>
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sum_list
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(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
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apply(subgoal_tac
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"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
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\<le>
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sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
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apply fastforce
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apply simp
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apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
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apply(simp only:)
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apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
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apply(simp only:)
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apply (metis insertE rrexp.distinct(21))
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apply blast
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251 |
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apply fastforce
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apply force
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apply simp
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apply (metis Un_insert_left insert_iff rrexp.distinct(21))
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apply(case_tac "a \<in> noalts_set")
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apply simp
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apply(subgoal_tac "a \<notin> alts_set")
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prefer 2
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apply blast
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apply(case_tac "a \<in> corr_set")
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apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
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prefer 2
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264 |
apply fastforce
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apply(simp only:)
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apply(subgoal_tac " sum_list
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(map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
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\<le>
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sum_list
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(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
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271 |
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apply(subgoal_tac
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"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
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\<le>
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sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
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apply fastforce
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277 |
apply simp
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278 |
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
|
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|
279 |
apply(simp only:)
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|
280 |
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
|
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|
281 |
apply(simp only:)
|
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|
282 |
|
|
|
283 |
|
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|
284 |
apply (metis insertE rrexp.distinct(25))
|
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|
285 |
apply blast
|
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|
286 |
apply fastforce
|
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|
287 |
apply force
|
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|
288 |
apply simp
|
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|
289 |
|
|
|
290 |
apply (metis Un_insert_left insertE rrexp.distinct(25))
|
|
|
291 |
|
|
|
292 |
using Suc_le_mono flts_size_reduction_alts apply presburger
|
|
|
293 |
apply(case_tac "a \<in> noalts_set")
|
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|
294 |
apply simp
|
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|
295 |
apply(subgoal_tac "a \<notin> alts_set")
|
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|
296 |
prefer 2
|
|
|
297 |
apply blast
|
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|
298 |
apply(case_tac "a \<in> corr_set")
|
|
|
299 |
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
|
|
|
300 |
prefer 2
|
|
|
301 |
apply fastforce
|
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|
302 |
apply(simp only:)
|
|
|
303 |
apply(subgoal_tac " sum_list
|
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|
304 |
(map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
|
|
|
305 |
\<le>
|
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|
306 |
sum_list
|
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|
307 |
(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
|
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|
308 |
|
|
|
309 |
apply(subgoal_tac
|
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|
310 |
"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
|
|
|
311 |
\<le>
|
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|
312 |
sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
|
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|
313 |
apply fastforce
|
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|
314 |
apply simp
|
|
|
315 |
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
|
|
|
316 |
apply(simp only:)
|
|
|
317 |
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
|
|
|
318 |
apply(simp only:)
|
|
|
319 |
apply (metis insertE rrexp.distinct(29))
|
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|
320 |
|
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|
321 |
apply blast
|
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|
322 |
|
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|
323 |
apply fastforce
|
|
|
324 |
apply force
|
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|
325 |
apply simp
|
|
|
326 |
apply (metis Un_insert_left insert_iff rrexp.distinct(29))
|
|
|
327 |
done
|
|
|
328 |
|
|
|
329 |
|
|
|
330 |
|
|
|
331 |
|
|
447
|
332 |
|
|
|
333 |
lemma flts_vs_nflts:
|
|
451
|
334 |
shows "(\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs)
|
|
|
335 |
\<and> (\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set))
|
|
447
|
336 |
\<Longrightarrow> Suc (sum_list (map rsize (rdistinct ( rflts rs) (noalts_set \<union> corr_set) )))
|
|
451
|
337 |
\<le> Suc (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set) ))))"
|
|
|
338 |
apply(induct rs arbitrary: noalts_set alts_set corr_set)
|
|
447
|
339 |
apply simp
|
|
451
|
340 |
apply(case_tac a)
|
|
|
341 |
apply(case_tac "RZERO \<in> noalts_set")
|
|
|
342 |
apply simp
|
|
|
343 |
apply(subgoal_tac "RZERO \<notin> alts_set")
|
|
|
344 |
apply simp
|
|
|
345 |
apply fastforce
|
|
|
346 |
apply(case_tac "RONE \<in> noalts_set")
|
|
|
347 |
apply simp
|
|
|
348 |
apply(subgoal_tac "RONE \<notin> alts_set")
|
|
|
349 |
prefer 2
|
|
|
350 |
apply fastforce
|
|
|
351 |
apply(case_tac "RONE \<in> corr_set")
|
|
|
352 |
apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
|
|
|
353 |
apply(simp only:)
|
|
|
354 |
apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) =
|
|
|
355 |
rdistinct (rflts rs) (noalts_set \<union> corr_set)")
|
|
|
356 |
apply(simp only:)
|
|
|
357 |
apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
|
|
|
358 |
RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
|
|
|
359 |
apply(simp only:)
|
|
|
360 |
apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) =
|
|
|
361 |
rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
|
|
|
362 |
apply (simp only:)
|
|
|
363 |
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
|
|
|
364 |
apply(simp only:)
|
|
|
365 |
apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) =
|
|
|
366 |
insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
|
|
|
367 |
apply(simp only:)
|
|
|
368 |
apply(subgoal_tac " Suc (sum_list
|
|
|
369 |
(map rsize ( rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))
|
|
|
370 |
\<le> Suc (sum_list
|
|
|
371 |
(map rsize (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))")
|
|
|
372 |
apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
|
|
|
373 |
apply (metis (no_types, opaque_lifting) add_Suc_right le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
|
|
|
374 |
apply fastforce
|
|
|
375 |
apply fastforce
|
|
|
376 |
apply (metis Un_iff insert_absorb)
|
|
|
377 |
apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
|
|
|
378 |
apply (meson UnCI rdistinct.simps(2))
|
|
|
379 |
using rflts.simps(4) apply presburger
|
|
|
380 |
apply simp
|
|
|
381 |
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
|
|
|
382 |
apply(simp only:)
|
|
|
383 |
apply (metis Un_insert_left insertE rrexp.distinct(15))
|
|
|
384 |
apply fastforce
|
|
|
385 |
apply(case_tac "a \<in> noalts_set")
|
|
|
386 |
apply simp
|
|
|
387 |
apply(subgoal_tac "a \<notin> alts_set")
|
|
|
388 |
prefer 2
|
|
|
389 |
apply blast
|
|
|
390 |
apply(case_tac "a \<in> corr_set")
|
|
|
391 |
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
|
|
|
392 |
prefer 2
|
|
|
393 |
apply fastforce
|
|
|
394 |
apply(simp only:)
|
|
|
395 |
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
|
|
|
396 |
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
|
|
|
397 |
apply(case_tac "a \<in> noalts_set")
|
|
|
398 |
apply simp
|
|
|
399 |
apply(subgoal_tac "a \<notin> alts_set")
|
|
|
400 |
prefer 2
|
|
|
401 |
apply blast
|
|
|
402 |
apply(case_tac "a \<in> corr_set")
|
|
|
403 |
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
|
|
|
404 |
prefer 2
|
|
|
405 |
apply fastforce
|
|
|
406 |
apply(simp only:)
|
|
|
407 |
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
|
|
|
408 |
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
|
|
|
409 |
prefer 2
|
|
|
410 |
apply(case_tac "a \<in> noalts_set")
|
|
|
411 |
apply simp
|
|
|
412 |
apply(subgoal_tac "a \<notin> alts_set")
|
|
|
413 |
prefer 2
|
|
|
414 |
apply blast
|
|
|
415 |
apply(case_tac "a \<in> corr_set")
|
|
|
416 |
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
|
|
|
417 |
prefer 2
|
|
|
418 |
apply fastforce
|
|
|
419 |
apply(simp only:)
|
|
|
420 |
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
|
|
|
421 |
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
|
|
|
422 |
using flts_size_reduction_alts apply presburger
|
|
|
423 |
done
|
|
447
|
424 |
|
|
451
|
425 |
|
|
|
426 |
|
|
|
427 |
|
|
|
428 |
|
|
|
429 |
(* apply(rutac x = "\<lambda>a rs noalts_set alts_set corr_set. set xs ")*)
|
|
|
430 |
|
|
447
|
431 |
|
|
449
|
432 |
lemma distinct_simp_ineq_general:
|
|
450
|
433 |
shows "rsimp ` no_simp = has_simp \<and> finite no_simp \<Longrightarrow>Suc (sum_list (map rsize (rdistinct (map rsimp rs) has_simp)))
|
|
449
|
434 |
\<le> Suc (sum_list (map rsize (rdistinct rs no_simp)))"
|
|
450
|
435 |
apply(induct rs arbitrary: no_simp has_simp)
|
|
|
436 |
apply simp
|
|
|
437 |
apply(case_tac "a \<in> no_simp")
|
|
|
438 |
apply(subgoal_tac "rsimp a \<in> has_simp")
|
|
|
439 |
apply auto[1]
|
|
|
440 |
apply blast
|
|
|
441 |
apply(case_tac "rsimp a \<in> no_simp")
|
|
|
442 |
apply(subgoal_tac "rsimp a \<in> has_simp")
|
|
|
443 |
prefer 2
|
|
451
|
444 |
apply (simp add: rev_image_eqI rsimp_idem)
|
|
|
445 |
apply simp
|
|
|
446 |
apply (metis finite_insert image_insert insert_absorb trans_le_add2)
|
|
|
447 |
apply(case_tac "rsimp a \<notin> has_simp")
|
|
|
448 |
apply simp
|
|
|
449 |
apply (metis add_mono_thms_linordered_semiring(1) finite.insertI image_insert rsimp_mono)
|
|
|
450 |
apply simp
|
|
|
451 |
by (metis finite.insertI image_insert insert_absorb trans_le_add2)
|
|
449
|
452 |
|
|
447
|
453 |
|
|
450
|
454 |
lemma not_mentioned_elem_distinct_strong:
|
|
|
455 |
shows "r \<noteq> a \<Longrightarrow> (r \<in> set (rdistinct rs rset)) = (r \<in> set (rdistinct rs (insert a rset)))"
|
|
|
456 |
apply(induct rs arbitrary: rset)
|
|
|
457 |
apply simp
|
|
|
458 |
by force
|
|
|
459 |
|
|
|
460 |
lemma not_mentioned_elem_distinct:
|
|
|
461 |
shows "r \<noteq> a \<Longrightarrow> (r \<in> set (rdistinct rs {})) = (r \<in> set (rdistinct rs (insert a {})))"
|
|
|
462 |
by (metis not_mentioned_elem_distinct_strong)
|
|
|
463 |
|
|
|
464 |
|
|
|
465 |
|
|
|
466 |
|
|
451
|
467 |
lemma larger_acc_smaller_distinct_res0:
|
|
|
468 |
shows " ss \<subseteq> SS \<Longrightarrow> sum_list (map rsize (rdistinct rs SS)) \<le> sum_list (map rsize (rdistinct rs ss))"
|
|
|
469 |
apply(induct rs arbitrary: ss SS)
|
|
|
470 |
apply simp
|
|
|
471 |
apply(case_tac "a \<in> ss")
|
|
|
472 |
apply(subgoal_tac "a \<in> SS")
|
|
|
473 |
apply simp
|
|
|
474 |
apply blast
|
|
|
475 |
apply(case_tac "a \<in> SS")
|
|
|
476 |
apply simp
|
|
|
477 |
apply(subgoal_tac "insert a ss \<subseteq> SS")
|
|
|
478 |
apply simp
|
|
|
479 |
apply (simp add: trans_le_add2)
|
|
|
480 |
apply blast
|
|
|
481 |
apply(simp)
|
|
|
482 |
apply(subgoal_tac "insert a ss \<subseteq> insert a SS")
|
|
|
483 |
apply blast
|
|
|
484 |
by blast
|
|
450
|
485 |
|
|
|
486 |
|
|
447
|
487 |
lemma without_flts_ineq:
|
|
|
488 |
shows " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le>
|
|
|
489 |
Suc (sum_list (map rsize (rdistinct ( rs ) {} )))"
|
|
451
|
490 |
proof -
|
|
|
491 |
have " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le>
|
|
|
492 |
Suc (sum_list (map rsize (rdistinct rs (insert RZERO {}))))"
|
|
|
493 |
by (metis empty_iff flts_vs_nflts sup_bot_left)
|
|
|
494 |
also have "... \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
|
|
|
495 |
by (simp add: larger_acc_smaller_distinct_res0)
|
|
|
496 |
finally show ?thesis sledgehammer
|
|
|
497 |
by blast
|
|
|
498 |
qed
|
|
|
499 |
|
|
|
500 |
|
|
447
|
501 |
|
|
|
502 |
|
|
448
|
503 |
|
|
|
504 |
|
|
447
|
505 |
lemma distinct_simp_ineq:
|
|
|
506 |
shows "Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))
|
|
|
507 |
\<le> Suc (sum_list (map rsize (rdistinct rs {})))"
|
|
448
|
508 |
|
|
|
509 |
using distinct_simp_ineq_general by blast
|
|
|
510 |
|
|
|
511 |
|
|
447
|
512 |
|
|
|
513 |
|
|
|
514 |
lemma alts_simp_control:
|
|
|
515 |
shows "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
|
|
|
516 |
proof -
|
|
|
517 |
have "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))"
|
|
|
518 |
|
|
|
519 |
using alts_simp_ineq_unfold by auto
|
|
|
520 |
then have "\<dots> \<le> Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))"
|
|
|
521 |
using without_flts_ineq by blast
|
|
|
522 |
|
|
|
523 |
show "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
|
|
|
524 |
by (meson \<open>Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {}))) \<le> Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))\<close> \<open>rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))\<close> distinct_simp_ineq order_trans)
|
|
|
525 |
qed
|
|
444
|
526 |
|
|
|
527 |
|
|
|
528 |
|
|
446
|
529 |
lemma rdistinct_equality1:
|
|
|
530 |
shows "a \<notin> ss \<Longrightarrow> rdistinct (a # rs) ss = a # rdistinct rs (insert a ss) "
|
|
|
531 |
by auto
|
|
|
532 |
|
|
451
|
533 |
|
|
446
|
534 |
|
|
|
535 |
|
|
|
536 |
lemma larger_acc_smaller_distinct_res:
|
|
|
537 |
shows " (sum_list (map rsize (rdistinct rs ss))) \<ge> (sum_list (map rsize (rdistinct rs (insert a ss))))"
|
|
447
|
538 |
apply(subgoal_tac "ss \<subseteq> (insert a ss)")
|
|
|
539 |
apply(rule larger_acc_smaller_distinct_res0)
|
|
|
540 |
apply simp
|
|
|
541 |
by (simp add: subset_insertI)
|
|
446
|
542 |
|
|
|
543 |
lemma size_list_triangle1:
|
|
|
544 |
shows "sum_list (map rsize (a # (rdistinct as ss))) \<ge> rsize a + sum_list (map rsize (rdistinct as (insert a ss)))"
|
|
|
545 |
by (simp add: larger_acc_smaller_distinct_res)
|
|
|
546 |
|
|
|
547 |
|
|
445
|
548 |
lemma triangle_inequality_distinct:
|
|
|
549 |
shows "sum_list (map rsize (rdistinct (a # rs) ss)) \<le> rsize a + (sum_list (map rsize (rdistinct rs ss)))"
|
|
|
550 |
apply(case_tac "a \<in> ss")
|
|
|
551 |
apply simp
|
|
446
|
552 |
apply(subst rdistinct_equality1)
|
|
|
553 |
apply simp
|
|
|
554 |
using size_list_triangle1 by auto
|
|
445
|
555 |
|
|
|
556 |
lemma same_regex_property_after_map:
|
|
|
557 |
shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (map (f r2) Ss). P r"
|
|
|
558 |
by auto
|
|
|
559 |
|
|
|
560 |
lemma same_property_after_distinct:
|
|
|
561 |
shows " \<forall>r \<in> set (map (f r2) Ss). P r \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
|
|
|
562 |
apply(induct Ss arbitrary: xset)
|
|
|
563 |
apply simp
|
|
|
564 |
by auto
|
|
|
565 |
|
|
|
566 |
lemma same_regex_property_after_distinct:
|
|
|
567 |
shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
|
|
|
568 |
apply(rule same_property_after_distinct)
|
|
|
569 |
apply(rule same_regex_property_after_map)
|
|
|
570 |
by simp
|
|
|
571 |
|
|
449
|
572 |
|
|
|
573 |
|
|
|
574 |
lemma Sum_cons:
|
|
|
575 |
shows "distinct (a # as) \<Longrightarrow> \<Sum> (set ((a::nat) # as)) = a + \<Sum> (set as)"
|
|
|
576 |
by simp
|
|
|
577 |
|
|
|
578 |
|
|
|
579 |
lemma distinct_list_sizeNregex_bounded:
|
|
|
580 |
shows "distinct rs \<and> (\<forall> r \<in> (set rs). rsize r \<le> N) \<Longrightarrow> sum_list (map rsize rs) \<le> N * length rs"
|
|
|
581 |
apply(induct rs)
|
|
|
582 |
apply simp
|
|
|
583 |
by simp
|
|
|
584 |
|
|
|
585 |
|
|
|
586 |
lemma distinct_list_size_len_bounded:
|
|
|
587 |
shows "distinct rs \<and> (\<forall>r \<in> set rs. rsize r \<le> N) \<and> length rs \<le> lrs \<Longrightarrow> sum_list (map rsize rs) \<le> lrs * N "
|
|
|
588 |
by (metis distinct_list_sizeNregex_bounded dual_order.trans mult.commute mult_le_mono1)
|
|
|
589 |
|
|
|
590 |
|
|
|
591 |
|
|
|
592 |
lemma rdistinct_same_set:
|
|
|
593 |
shows "(r \<in> set rs) = (r \<in> set (rdistinct rs {}))"
|
|
|
594 |
apply(induct rs)
|
|
|
595 |
apply simp
|
|
|
596 |
apply(case_tac "a \<in> set rs")
|
|
|
597 |
apply(case_tac "r = a")
|
|
|
598 |
apply (simp)
|
|
|
599 |
apply (simp add: not_mentioned_elem_distinct)
|
|
|
600 |
using not_mentioned_elem_distinct by fastforce
|
|
|
601 |
|
|
|
602 |
|
|
|
603 |
|
|
|
604 |
lemma distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size:
|
|
|
605 |
shows "\<forall>r\<in> set rs. (rsize r ) \<le> N \<Longrightarrow> sum_list (map rsize (rdistinct rs {})) \<le>
|
|
|
606 |
(card (sizeNregex N))* N"
|
|
|
607 |
apply(subgoal_tac "distinct (rdistinct rs {})")
|
|
|
608 |
prefer 2
|
|
|
609 |
using rdistinct_does_the_job apply blast
|
|
|
610 |
apply(subgoal_tac "length (rdistinct rs {}) \<le> card (sizeNregex N)")
|
|
|
611 |
apply(rule distinct_list_size_len_bounded)
|
|
|
612 |
apply(rule conjI)+
|
|
|
613 |
apply simp
|
|
|
614 |
apply(rule conjI)
|
|
|
615 |
apply (meson rdistinct_same_set)
|
|
|
616 |
apply blast
|
|
|
617 |
apply(subgoal_tac "\<forall>r \<in> set (rdistinct rs {}). rsize r \<le> N")
|
|
|
618 |
prefer 2
|
|
|
619 |
apply (meson rdistinct_same_set)
|
|
|
620 |
apply(subgoal_tac "length (rdistinct rs {}) = card (set (rdistinct rs {}))")
|
|
|
621 |
prefer 2
|
|
|
622 |
using set_related_list apply blast
|
|
|
623 |
apply(simp only:)
|
|
|
624 |
by (metis card_mono finite_size_n mem_Collect_eq sizeNregex_def subset_code(1))
|
|
|
625 |
|
|
|
626 |
|
|
|
627 |
|
|
|
628 |
|
|
|
629 |
|
|
|
630 |
|
|
|
631 |
lemma star_closed_form_bounded_by_rdistinct_list_estimate:
|
|
|
632 |
shows "rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
|
|
|
633 |
(star_updates s r0 [[c]]) ) ))) \<le>
|
|
|
634 |
Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
|
|
|
635 |
(star_updates s r0 [[c]]) ) {}) ) )"
|
|
|
636 |
by (metis alts_simp_control )
|
|
|
637 |
|
|
|
638 |
|
|
|
639 |
|
|
|
640 |
|
|
|
641 |
lemma star_lambda_form:
|
|
|
642 |
shows "\<forall> r \<in> set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) ls).
|
|
|
643 |
\<exists>s2. r = RSEQ (rders_simp r0 s2) (RSTAR r0) "
|
|
|
644 |
by (meson ex_map_conv)
|
|
|
645 |
|
|
|
646 |
|
|
|
647 |
lemma star_lambda_ders:
|
|
|
648 |
shows " \<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
|
|
|
649 |
\<forall>r\<in>set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates s r0 [[c]])).
|
|
|
650 |
rsize r \<le> Suc (N + rsize (RSTAR r0))"
|
|
|
651 |
apply(insert star_lambda_form)
|
|
|
652 |
apply(simp)
|
|
|
653 |
done
|
|
|
654 |
|
|
|
655 |
|
|
|
656 |
|
|
|
657 |
|
|
|
658 |
lemma star_control_bounded:
|
|
|
659 |
shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
|
|
|
660 |
(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
|
|
|
661 |
(star_updates s r0 [[c]]) ) {}) ) ) \<le>
|
|
|
662 |
(card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
|
|
|
663 |
"
|
|
|
664 |
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
|
|
|
665 |
(star_updates s r0 [[c]]) ). (rsize r ) \<le> Suc (N + rsize (RSTAR r0))")
|
|
|
666 |
prefer 2
|
|
|
667 |
using star_lambda_ders apply blast
|
|
|
668 |
using distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size by blast
|
|
|
669 |
|
|
|
670 |
|
|
|
671 |
lemma star_control_variant:
|
|
|
672 |
assumes "\<forall>s. rsize (rders_simp r0 s) \<le> N"
|
|
|
673 |
shows"Suc
|
|
|
674 |
(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
|
|
|
675 |
(star_updates list r0 [[a]])) {})))
|
|
|
676 |
\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) "
|
|
|
677 |
apply(subgoal_tac "(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
|
|
|
678 |
(star_updates list r0 [[a]])) {})))
|
|
|
679 |
\<le> ( (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
|
|
|
680 |
prefer 2
|
|
|
681 |
using assms star_control_bounded apply presburger
|
|
|
682 |
by simp
|
|
|
683 |
|
|
|
684 |
|
|
|
685 |
|
|
|
686 |
lemma star_closed_form_bounded:
|
|
|
687 |
shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
|
|
|
688 |
rsize (rders_simp (RSTAR r0) s) \<le>
|
|
|
689 |
max ( (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0))))) (rsize (RSTAR r0))"
|
|
|
690 |
apply(case_tac s)
|
|
|
691 |
apply simp
|
|
|
692 |
apply(subgoal_tac " rsize (rders_simp (RSTAR r0) (a # list)) =
|
|
|
693 |
rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))")
|
|
|
694 |
prefer 2
|
|
|
695 |
using star_closed_form apply presburger
|
|
|
696 |
apply(subgoal_tac "rsize (rsimp (
|
|
|
697 |
RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))
|
|
|
698 |
\<le> Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
|
|
|
699 |
(star_updates list r0 [[a]]) ) {}) ) )")
|
|
|
700 |
prefer 2
|
|
|
701 |
using star_closed_form_bounded_by_rdistinct_list_estimate apply presburger
|
|
|
702 |
apply(subgoal_tac "Suc (sum_list
|
|
|
703 |
(map rsize
|
|
|
704 |
(rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates list r0 [[a]])) {})))
|
|
|
705 |
\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
|
|
|
706 |
apply auto[1]
|
|
|
707 |
using star_control_variant by blast
|
|
|
708 |
|
|
|
709 |
|
|
|
710 |
lemma seq_list_estimate_control: shows
|
|
|
711 |
" rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
|
|
|
712 |
\<le> Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))"
|
|
|
713 |
by(metis alts_simp_control)
|
|
|
714 |
|
|
445
|
715 |
lemma map_ders_is_list_of_ders:
|
|
|
716 |
shows "\<forall>s. rsize (rders_simp r2 s) \<le> N2 \<Longrightarrow>
|
|
|
717 |
\<forall>r \<in> set (rdistinct (map (rders_simp r2) Ss) {}). rsize r \<le> N2"
|
|
|
718 |
apply(rule same_regex_property_after_distinct)
|
|
|
719 |
by simp
|
|
|
720 |
|
|
444
|
721 |
lemma seq_estimate_bounded:
|
|
|
722 |
assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1" and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
|
|
|
723 |
shows
|
|
|
724 |
"Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
|
|
|
725 |
Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
|
|
445
|
726 |
apply(subgoal_tac " (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
|
|
|
727 |
(Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
|
|
|
728 |
apply force
|
|
|
729 |
apply(subgoal_tac " (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
|
|
|
730 |
(rsize (RSEQ (rders_simp r1 s) r2)) + (sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) )")
|
|
|
731 |
prefer 2
|
|
|
732 |
using triangle_inequality_distinct apply blast
|
|
|
733 |
apply(subgoal_tac " sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) \<le> N2 * card (sizeNregex N2) ")
|
|
|
734 |
apply(subgoal_tac "rsize (RSEQ (rders_simp r1 s) r2) \<le> Suc (N1 + rsize r2)")
|
|
|
735 |
apply linarith
|
|
|
736 |
apply (simp add: assms(1))
|
|
|
737 |
apply(subgoal_tac "\<forall>r \<in> set (rdistinct (map (rders_simp r2) (vsuf s r1)) {}). rsize r \<le> N2")
|
|
|
738 |
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)
|
|
|
739 |
using assms(2) map_ders_is_list_of_ders by blast
|
|
444
|
740 |
|
|
|
741 |
|
|
|
742 |
lemma seq_closed_form_bounded: shows
|
|
|
743 |
"\<lbrakk>\<forall>s. rsize (rders_simp r1 s) \<le> N1 ; \<forall>s. rsize (rders_simp r2 s) \<le> N2\<rbrakk> \<Longrightarrow>
|
|
|
744 |
rsize (rders_simp (RSEQ r1 r2) s) \<le>
|
|
|
745 |
max (Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))) (rsize (RSEQ r1 r2)) "
|
|
|
746 |
apply(case_tac s)
|
|
|
747 |
apply simp
|
|
|
748 |
apply(subgoal_tac " (rders_simp (RSEQ r1 r2) s) =
|
|
|
749 |
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))")
|
|
|
750 |
prefer 2
|
|
|
751 |
using seq_closed_form_variant apply blast
|
|
|
752 |
apply(subgoal_tac "rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
|
|
|
753 |
\<le>
|
|
|
754 |
Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))")
|
|
|
755 |
apply(subgoal_tac "Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))
|
|
|
756 |
\<le> Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
|
|
|
757 |
prefer 2
|
|
|
758 |
using seq_estimate_bounded apply blast
|
|
|
759 |
apply(subgoal_tac "rsize (rders_simp (RSEQ r1 r2) s) \<le> Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))")
|
|
|
760 |
using le_max_iff_disj apply blast
|
|
|
761 |
apply auto[1]
|
|
|
762 |
using seq_list_estimate_control by presburger
|
|
|
763 |
|
|
|
764 |
|
|
|
765 |
lemma rders_simp_bounded: shows
|
|
|
766 |
"\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
|
|
|
767 |
apply(induct r)
|
|
|
768 |
apply(rule_tac x = "Suc 0 " in exI)
|
|
|
769 |
using three_easy_cases0 apply force
|
|
|
770 |
using three_easy_cases1 apply blast
|
|
|
771 |
using three_easy_casesC apply blast
|
|
|
772 |
using seq_closed_form_bounded apply blast
|
|
|
773 |
apply (metis alts_closed_form_bounded size_list_estimation')
|
|
|
774 |
using star_closed_form_bounded by blast
|
|
|
775 |
|
|
|
776 |
|
|
|
777 |
|
|
451
|
778 |
unused_thms
|
|
444
|
779 |
|
|
|
780 |
|
|
|
781 |
|
|
|
782 |
|
|
|
783 |
|
|
|
784 |
|
|
|
785 |
|
|
|
786 |
|
|
|
787 |
|
|
|
788 |
|
|
|
789 |
|
|
|
790 |
|
|
|
791 |
(*Obsolete materials*)
|
|
|
792 |
|
|
|
793 |
|
|
|
794 |
|
|
|
795 |
end
|