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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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447
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lemma flts_has_no_zero:
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shows "rdistinct (rflts rs) rset = rdistinct (rflts rs) (insert RZERO rset)"
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449
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447
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sorry
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449
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lemma not_mentioned_elem_distinct:
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shows "r \<noteq> a \<Longrightarrow> (r \<in> set (rdistinct rs {})) = (r \<in> set (rdistinct rs {a}))"
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sorry
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447
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lemma flts_vs_nflts:
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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> 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 (noalts_set \<union> alts_set) )))"
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apply(induct rs arbitrary: noalts_set)
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apply simp
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sorry
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449
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lemma distinct_simp_ineq_general:
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shows "rsimp ` no_simp = has_simp \<Longrightarrow>Suc (sum_list (map rsize (rdistinct (map rsimp rs) has_simp)))
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\<le> Suc (sum_list (map rsize (rdistinct rs no_simp)))"
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sorry
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447
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lemma without_flts_ineq:
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shows " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le>
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Suc (sum_list (map rsize (rdistinct ( rs ) {} )))"
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by (metis empty_iff flts_vs_nflts sup_bot_left)
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448
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447
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lemma distinct_simp_ineq:
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shows "Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))
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\<le> Suc (sum_list (map rsize (rdistinct rs {})))"
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448
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using distinct_simp_ineq_general by blast
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447
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lemma alts_simp_control:
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shows "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
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proof -
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have "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))"
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using alts_simp_ineq_unfold by auto
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then have "\<dots> \<le> Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))"
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using without_flts_ineq by blast
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show "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
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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)
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qed
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444
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446
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lemma rdistinct_equality1:
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shows "a \<notin> ss \<Longrightarrow> rdistinct (a # rs) ss = a # rdistinct rs (insert a ss) "
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by auto
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lemma larger_acc_smaller_distinct_res0:
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shows " ss \<subseteq> SS \<Longrightarrow> sum_list (map rsize (rdistinct rs SS)) \<le> sum_list (map rsize (rdistinct rs ss))"
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apply(induct rs arbitrary: ss SS)
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apply simp
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apply(case_tac "a \<in> ss")
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apply(subgoal_tac "a \<in> SS")
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apply simp
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apply blast
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apply(case_tac "a \<in> SS")
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apply simp
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apply(subgoal_tac "insert a ss \<subseteq> SS")
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apply simp
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apply (simp add: trans_le_add2)
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apply blast
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apply(simp)
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apply(subgoal_tac "insert a ss \<subseteq> insert a SS")
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apply blast
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by blast
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lemma larger_acc_smaller_distinct_res:
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shows " (sum_list (map rsize (rdistinct rs ss))) \<ge> (sum_list (map rsize (rdistinct rs (insert a ss))))"
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447
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apply(subgoal_tac "ss \<subseteq> (insert a ss)")
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apply(rule larger_acc_smaller_distinct_res0)
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apply simp
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by (simp add: subset_insertI)
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446
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lemma size_list_triangle1:
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shows "sum_list (map rsize (a # (rdistinct as ss))) \<ge> rsize a + sum_list (map rsize (rdistinct as (insert a ss)))"
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by (simp add: larger_acc_smaller_distinct_res)
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445
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lemma triangle_inequality_distinct:
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shows "sum_list (map rsize (rdistinct (a # rs) ss)) \<le> rsize a + (sum_list (map rsize (rdistinct rs ss)))"
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apply(case_tac "a \<in> ss")
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apply simp
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446
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apply(subst rdistinct_equality1)
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apply simp
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using size_list_triangle1 by auto
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445
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lemma same_regex_property_after_map:
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shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (map (f r2) Ss). P r"
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by auto
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lemma same_property_after_distinct:
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shows " \<forall>r \<in> set (map (f r2) Ss). P r \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
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apply(induct Ss arbitrary: xset)
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apply simp
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by auto
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lemma same_regex_property_after_distinct:
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shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
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apply(rule same_property_after_distinct)
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apply(rule same_regex_property_after_map)
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by simp
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449
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lemma Sum_cons:
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shows "distinct (a # as) \<Longrightarrow> \<Sum> (set ((a::nat) # as)) = a + \<Sum> (set as)"
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by simp
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lemma distinct_list_sizeNregex_bounded:
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shows "distinct rs \<and> (\<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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lemma distinct_list_size_len_bounded:
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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 "
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by (metis distinct_list_sizeNregex_bounded dual_order.trans mult.commute mult_le_mono1)
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lemma rdistinct_same_set:
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shows "(r \<in> set rs) = (r \<in> set (rdistinct rs {}))"
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apply(induct rs)
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apply simp
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apply(case_tac "a \<in> set rs")
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apply(case_tac "r = a")
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apply (simp)
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apply (simp add: not_mentioned_elem_distinct)
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using not_mentioned_elem_distinct by fastforce
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lemma distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size:
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shows "\<forall>r\<in> set rs. (rsize r ) \<le> N \<Longrightarrow> sum_list (map rsize (rdistinct rs {})) \<le>
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(card (sizeNregex N))* N"
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apply(subgoal_tac "distinct (rdistinct rs {})")
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prefer 2
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using rdistinct_does_the_job apply blast
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apply(subgoal_tac "length (rdistinct rs {}) \<le> card (sizeNregex N)")
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apply(rule distinct_list_size_len_bounded)
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apply(rule conjI)+
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apply simp
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apply(rule conjI)
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apply (meson rdistinct_same_set)
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apply blast
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apply(subgoal_tac "\<forall>r \<in> set (rdistinct rs {}). rsize r \<le> N")
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prefer 2
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apply (meson rdistinct_same_set)
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apply(subgoal_tac "length (rdistinct rs {}) = card (set (rdistinct rs {}))")
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prefer 2
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using set_related_list apply blast
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apply(simp only:)
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by (metis card_mono finite_size_n mem_Collect_eq sizeNregex_def subset_code(1))
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lemma star_closed_form_bounded_by_rdistinct_list_estimate:
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shows "rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
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(star_updates s r0 [[c]]) ) ))) \<le>
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Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
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(star_updates s r0 [[c]]) ) {}) ) )"
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by (metis alts_simp_control )
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lemma star_lambda_form:
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shows "\<forall> r \<in> set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) ls).
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\<exists>s2. r = RSEQ (rders_simp r0 s2) (RSTAR r0) "
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by (meson ex_map_conv)
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lemma star_lambda_ders:
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shows " \<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
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\<forall>r\<in>set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates s r0 [[c]])).
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rsize r \<le> Suc (N + rsize (RSTAR r0))"
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apply(insert star_lambda_form)
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apply(simp)
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done
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lemma star_control_bounded:
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shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
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(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
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(star_updates s r0 [[c]]) ) {}) ) ) \<le>
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(card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
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"
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apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
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(star_updates s r0 [[c]]) ). (rsize r ) \<le> Suc (N + rsize (RSTAR r0))")
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prefer 2
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using star_lambda_ders apply blast
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using distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size by blast
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lemma star_control_variant:
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assumes "\<forall>s. rsize (rders_simp r0 s) \<le> N"
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shows"Suc
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(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
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(star_updates list r0 [[a]])) {})))
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\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) "
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apply(subgoal_tac "(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
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(star_updates list r0 [[a]])) {})))
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\<le> ( (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
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prefer 2
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using assms star_control_bounded apply presburger
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by simp
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lemma star_closed_form_bounded:
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shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
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rsize (rders_simp (RSTAR r0) s) \<le>
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max ( (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0))))) (rsize (RSTAR r0))"
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apply(case_tac s)
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apply simp
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apply(subgoal_tac " rsize (rders_simp (RSTAR r0) (a # list)) =
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rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))")
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291 |
prefer 2
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using star_closed_form apply presburger
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apply(subgoal_tac "rsize (rsimp (
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RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))
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\<le> Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
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(star_updates list r0 [[a]]) ) {}) ) )")
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prefer 2
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using star_closed_form_bounded_by_rdistinct_list_estimate apply presburger
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apply(subgoal_tac "Suc (sum_list
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(map rsize
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(rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates list r0 [[a]])) {})))
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\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
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apply auto[1]
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using star_control_variant by blast
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lemma seq_list_estimate_control: shows
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" rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
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\<le> Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))"
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by(metis alts_simp_control)
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445
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lemma map_ders_is_list_of_ders:
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shows "\<forall>s. rsize (rders_simp r2 s) \<le> N2 \<Longrightarrow>
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\<forall>r \<in> set (rdistinct (map (rders_simp r2) Ss) {}). rsize r \<le> N2"
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apply(rule same_regex_property_after_distinct)
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by simp
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444
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lemma seq_estimate_bounded:
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assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1" and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
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shows
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"Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
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Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
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445
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apply(subgoal_tac " (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
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(Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
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apply force
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apply(subgoal_tac " (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
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(rsize (RSEQ (rders_simp r1 s) r2)) + (sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) )")
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prefer 2
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using triangle_inequality_distinct apply blast
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apply(subgoal_tac " sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) \<le> N2 * card (sizeNregex N2) ")
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apply(subgoal_tac "rsize (RSEQ (rders_simp r1 s) r2) \<le> Suc (N1 + rsize r2)")
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apply linarith
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apply (simp add: assms(1))
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apply(subgoal_tac "\<forall>r \<in> set (rdistinct (map (rders_simp r2) (vsuf s r1)) {}). rsize r \<le> N2")
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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)
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using assms(2) map_ders_is_list_of_ders by blast
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444
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lemma seq_closed_form_bounded: shows
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"\<lbrakk>\<forall>s. rsize (rders_simp r1 s) \<le> N1 ; \<forall>s. rsize (rders_simp r2 s) \<le> N2\<rbrakk> \<Longrightarrow>
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rsize (rders_simp (RSEQ r1 r2) s) \<le>
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max (Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))) (rsize (RSEQ r1 r2)) "
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apply(case_tac s)
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apply simp
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apply(subgoal_tac " (rders_simp (RSEQ r1 r2) s) =
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rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))")
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347 |
prefer 2
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348 |
using seq_closed_form_variant apply blast
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349 |
apply(subgoal_tac "rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
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350 |
\<le>
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351 |
Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))")
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352 |
apply(subgoal_tac "Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))
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353 |
\<le> Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
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354 |
prefer 2
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355 |
using seq_estimate_bounded apply blast
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356 |
apply(subgoal_tac "rsize (rders_simp (RSEQ r1 r2) s) \<le> Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))")
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357 |
using le_max_iff_disj apply blast
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358 |
apply auto[1]
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359 |
using seq_list_estimate_control by presburger
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360 |
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361 |
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362 |
lemma rders_simp_bounded: shows
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363 |
"\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
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364 |
apply(induct r)
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365 |
apply(rule_tac x = "Suc 0 " in exI)
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366 |
using three_easy_cases0 apply force
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367 |
using three_easy_cases1 apply blast
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368 |
using three_easy_casesC apply blast
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369 |
using seq_closed_form_bounded apply blast
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370 |
apply (metis alts_closed_form_bounded size_list_estimation')
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371 |
using star_closed_form_bounded by blast
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372 |
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373 |
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374 |
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375 |
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376 |
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380 |
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381 |
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382 |
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383 |
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384 |
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385 |
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386 |
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387 |
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388 |
(*Obsolete materials*)
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389 |
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390 |
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391 |
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392 |
end
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