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:
assumes "\<forall>r \<in> set rs. rsize r \<le> N"
shows "rsizes rs \<le> N * (length rs)"
using assms
apply(induct rs)
apply simp
by simp
lemma alts_closed_form_bounded:
assumes "\<forall>r \<in> set rs. \<forall>s. rsize (rders_simp r s) \<le> N"
shows "rsize (rders_simp (RALTS rs) s) \<le> max (Suc (N * (length rs))) (rsize (RALTS rs))"
proof (cases s)
case Nil
then show "rsize (rders_simp (RALTS rs) s) \<le> max (Suc (N * length rs)) (rsize (RALTS rs))"
by simp
next
case (Cons a s)
from assms have "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (a # s)) rs ). rsize r \<le> N"
by (metis alts_ders_lambda_shape_ders)
then have a: "rsizes (map (\<lambda>r. rders_simp r (a # s)) rs ) \<le> N * (length rs)"
by (metis length_map rlist_bound)
have "rsize (rders_simp (RALTS rs) (a # s))
= rsize (rsimp (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs)))"
by (metis alts_closed_form_variant list.distinct(1))
also have "... \<le> rsize (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs))"
using rsimp_mono by blast
also have "... = Suc (rsizes (map (\<lambda>r. rders_simp r (a # s)) rs))"
by simp
also have "... \<le> Suc (N * (length rs))"
using a by blast
finally have "rsize (rders_simp (RALTS rs) (a # s)) \<le> max (Suc (N * length rs)) (rsize (RALTS rs))"
by auto
then show ?thesis using local.Cons by simp
qed
lemma alts_simp_ineq_unfold:
shows "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct (rflts (map rsimp rs)) {}))"
using rsimp_aalts_smaller by auto
lemma rdistinct_mono_list:
shows "rsizes (rdistinct (x5 @ rs) rset) \<le> rsizes x5 + rsizes (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:
assumes a: "\<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 (rsizes (rdistinct (rflts rs) (noalts_set \<union> corr_set)))
\<le> Suc (rsizes (rdistinct rs (insert RZERO (noalts_set \<union> alts_set))))"
and b: "\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs"
and c: "\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set"
and d: "a = RALTS x5"
shows "rsizes (rdistinct (rflts (a # rs)) (noalts_set \<union> corr_set))
\<le> rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))"
apply(case_tac "a \<in> alts_set")
using a b c d
apply simp
apply(subgoal_tac "set x5 \<subseteq> corr_set")
apply(subst rdistinct_concat)
apply auto[1]
apply presburger
apply fastforce
using a b c d
apply (subgoal_tac "a \<notin> noalts_set")
prefer 2
apply blast
apply simp
apply(subgoal_tac "rsizes (rdistinct (x5 @ rflts rs) (noalts_set \<union> corr_set))
\<le> rsizes x5 + rsizes (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 "rsizes x5 + rsizes (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> (set x5)))) \<le>
rsizes x5 + rsizes (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 "rsizes (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> set x5))) \<le>
rsizes (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
lemma flts_vs_nflts1:
assumes "\<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 "rsizes (rdistinct (rflts rs) (noalts_set \<union> corr_set))
\<le> rsizes (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))"
using assms
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 "rsizes (rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))
\<le> rsizes (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))")
apply (smt (verit, ccfv_threshold) dual_order.trans insertE rrexp.distinct(17))
apply (metis (no_types, opaque_lifting) 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(17))
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 "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
rsizes (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 nonalt.simps(1) nonalt.simps(4))
apply blast
apply fastforce
apply force
apply simp
apply (metis Un_insert_left insertE nonalt.simps(1) nonalt.simps(4))
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 "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
rsizes (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(31))
apply blast
apply fastforce
apply force
apply simp
apply (metis Un_insert_left insertE rrexp.distinct(31))
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 "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
rsizes (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(37))
apply blast
apply fastforce
apply force
apply simp
apply (metis Un_insert_left insert_iff rrexp.distinct(37))
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 "rsizes (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))) \<le>
rsizes (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))")
apply(subgoal_tac "rsizes (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)) \<le>
rsizes (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 nonalt.simps(1) nonalt.simps(7))
apply blast
apply blast
apply force
apply(auto)
by (metis Un_insert_left insert_iff rrexp.distinct(39))
lemma flts_vs_nflts:
assumes "\<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 "rsizes (rdistinct (rflts rs) (noalts_set \<union> corr_set))
\<le> rsizes (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))"
by (simp add: assms flts_vs_nflts1)
lemma distinct_simp_ineq_general:
assumes "rsimp ` no_simp = has_simp" "finite no_simp"
shows "rsizes (rdistinct (map rsimp rs) has_simp) \<le> rsizes (rdistinct rs no_simp)"
using assms
apply(induct rs no_simp arbitrary: has_simp rule: rdistinct.induct)
apply simp
apply(auto)
using add_le_mono rsimp_mono by presburger
lemma larger_acc_smaller_distinct_res0:
assumes "ss \<subseteq> SS"
shows "rsizes (rdistinct rs SS) \<le> rsizes (rdistinct rs ss)"
using assms
apply(induct rs arbitrary: ss SS)
apply simp
by (metis distinct_early_app1 rdistinct_smaller)
lemma without_flts_ineq:
shows "rsizes (rdistinct (rflts rs) {}) \<le> rsizes (rdistinct rs {})"
proof -
have "rsizes (rdistinct (rflts rs) {}) \<le> rsizes (rdistinct rs (insert RZERO {}))"
by (metis empty_iff flts_vs_nflts sup_bot_left)
also have "... \<le> rsizes (rdistinct rs {})"
by (simp add: larger_acc_smaller_distinct_res0)
finally show ?thesis
by blast
qed
lemma distinct_simp_ineq:
shows "rsizes (rdistinct (map rsimp rs) {}) \<le> rsizes (rdistinct rs {})"
using distinct_simp_ineq_general by blast
lemma alts_simp_control:
shows "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct rs {}))"
proof -
have "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct (rflts (map rsimp rs)) {}))"
using alts_simp_ineq_unfold by auto
moreover have "\<dots> \<le> Suc (rsizes (rdistinct (map rsimp rs) {}))"
using without_flts_ineq by blast
ultimately show "rsize (rsimp (RALTS rs)) \<le> Suc (rsizes (rdistinct rs {}))"
by (meson Suc_le_mono distinct_simp_ineq le_trans)
qed
lemma larger_acc_smaller_distinct_res:
shows "rsizes (rdistinct rs (insert a ss)) \<le> rsizes (rdistinct rs ss)"
by (simp add: larger_acc_smaller_distinct_res0 subset_insertI)
lemma triangle_inequality_distinct:
shows "rsizes (rdistinct (a # rs) ss) \<le> rsize a + rsizes (rdistinct rs ss)"
apply(case_tac "a \<in> ss")
apply simp
by (simp add: larger_acc_smaller_distinct_res)
lemma distinct_list_size_len_bounded:
assumes "\<forall>r \<in> set rs. rsize r \<le> N" "length rs \<le> lrs"
shows "rsizes rs \<le> lrs * N "
using assms
by (metis rlist_bound dual_order.trans mult.commute mult_le_mono1)
lemma rdistinct_same_set:
shows "r \<in> set rs \<longleftrightarrow> r \<in> set (rdistinct rs {})"
apply(induct rs)
apply simp
by (metis rdistinct_set_equality)
(* distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size *)
lemma distinct_list_rexp_upto:
assumes "\<forall>r\<in> set rs. (rsize r) \<le> N"
shows "rsizes (rdistinct rs {}) \<le> (card (sizeNregex N)) * N"
apply(subgoal_tac "distinct (rdistinct rs {})")
prefer 2
using rdistinct_does_the_job apply blast
apply(subgoal_tac "length (rdistinct rs {}) \<le> card (sizeNregex N)")
apply(rule distinct_list_size_len_bounded)
using assms
apply (meson rdistinct_same_set)
apply blast
apply(subgoal_tac "\<forall>r \<in> set (rdistinct rs {}). rsize r \<le> N")
prefer 2
using assms
apply (meson rdistinct_same_set)
apply(subgoal_tac "length (rdistinct rs {}) = card (set (rdistinct rs {}))")
prefer 2
apply (simp add: distinct_card)
apply(simp)
by (metis card_mono finite_size_n mem_Collect_eq sizeNregex_def subsetI)
lemma star_control_bounded:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows "rsizes (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates s r [[c]])) {})
\<le> (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r)))"
by (smt (verit) add_Suc_shift add_mono_thms_linordered_semiring(3) assms distinct_list_rexp_upto image_iff list.set_map plus_nat.simps(2) rsize.simps(5))
lemma star_closed_form_bounded:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows "rsize (rders_simp (RSTAR r) s) \<le>
max ((Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r))))) (rsize (RSTAR r))"
proof(cases s)
case Nil
then show "rsize (rders_simp (RSTAR r) s)
\<le> max (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * Suc (N + rsize (RSTAR r))) (rsize (RSTAR r))"
by simp
next
case (Cons a list)
then have "rsize (rders_simp (RSTAR r) s) =
rsize (rsimp (RALTS ((map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])))))"
using star_closed_form by fastforce
also have "... \<le> Suc (rsizes (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])) {}))"
using alts_simp_control by blast
also have "... \<le> Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r)))"
using star_control_bounded[OF assms] by (metis add_mono le_add1 mult_Suc plus_1_eq_Suc)
also have "... \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * Suc (N + rsize (RSTAR r))) (rsize (RSTAR r))"
by simp
finally show ?thesis by simp
qed
thm ntimes_closed_form
thm rsize.simps
lemma nupdates_snoc:
shows " (nupdates (xs @ [x]) r optlist) = nupdate x r (nupdates xs r optlist)"
by (simp add: nupdates_append)
lemma nupdate_elems:
shows "\<forall>opt \<in> set (nupdate c r optlist). opt = None \<or> (\<exists>s n. opt = Some (s, n))"
using nonempty_string.cases by auto
lemma nupdates_elems:
shows "\<forall>opt \<in> set (nupdates s r optlist). opt = None \<or> (\<exists>s n. opt = Some (s, n))"
by (meson nonempty_string.cases)
lemma opterm_optlist_result_shape:
shows "\<forall>r' \<in> set (map (optermsimp r) optlist). r' = RZERO \<or> (\<exists>s m. r' = RSEQ (rders_simp r s) (RNTIMES r m))"
apply(induct optlist)
apply simp
apply(case_tac a)
apply simp+
by fastforce
lemma opterm_optlist_result_shape2:
shows "\<And>optlist. \<forall>r' \<in> set (map (optermsimp r) optlist). r' = RZERO \<or> (\<exists>s m. r' = RSEQ (rders_simp r s) (RNTIMES r m))"
using opterm_optlist_result_shape by presburger
lemma nupdate_n_leq_n:
shows "\<forall>r \<in> set (nupdate c' r [Some ([c], n)]). r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
apply(case_tac n)
apply simp
apply simp
done
(*
lemma nupdate_induct_leqn:
shows "\<lbrakk>\<forall>opt \<in> set optlist. opt = None \<or> (\<exists>s' m. opt = Some(s', m) \<and> m \<le> n) \<rbrakk> \<Longrightarrow>
\<forall>opt \<in> set (nupdate c' r optlist). opt = None \<or> (\<exists>s' m. opt = Some (s', m) \<and> m \<le> n)"
apply (case_tac optlist)
apply simp
apply(case_tac a)
apply simp
sledgehammer
*)
lemma nupdates_n_leq_n:
shows "\<forall>r \<in> set (nupdates s r [Some ([c], n)]). r = None \<or>( \<exists>s' m. r = Some (s', m) \<and> m \<le> n)"
apply(induct s rule: rev_induct)
apply simp
apply(subst nupdates_append)
by (metis nupdates_elems_leqn nupdates_snoc)
lemma ntimes_closed_form_list_elem_shape:
shows "\<forall>r' \<in> set (map (optermsimp r) (nupdates s r [Some ([c], n)])).
r' = RZERO \<or> (\<exists>s' m. r' = RSEQ (rders_simp r s') (RNTIMES r m) \<and> m \<le> n)"
apply(insert opterm_optlist_result_shape2)
apply(case_tac s)
apply(auto)
apply (metis rders_simp_one_char)
by (metis case_prod_conv nupdates.simps(2) nupdates_n_leq_n option.simps(4) option.simps(5))
lemma ntimes_trivial1:
shows "rsize RZERO \<le> N + rsize (RNTIMES r n)"
by simp
lemma ntimes_trivial20:
shows "m \<le> n \<Longrightarrow> rsize (RNTIMES r m) \<le> rsize (RNTIMES r n)"
by simp
lemma ntimes_trivial2:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows " r' = RSEQ (rders_simp r s1) (RNTIMES r m) \<and> m \<le> n
\<Longrightarrow> rsize r' \<le> Suc (N + rsize (RNTIMES r n))"
apply simp
by (simp add: add_mono_thms_linordered_semiring(1) assms)
lemma ntimes_closed_form_list_elem_bounded:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows "\<forall>r' \<in> set (map (optermsimp r) (nupdates s r [Some ([c], n)])). rsize r' \<le> Suc (N + rsize (RNTIMES r n))"
apply(rule ballI)
apply(subgoal_tac "r' = RZERO \<or> (\<exists>s' m. r' = RSEQ (rders_simp r s') (RNTIMES r m) \<and> m \<le> n)")
prefer 2
using ntimes_closed_form_list_elem_shape apply blast
apply(case_tac "r' = RZERO")
using le_SucI ntimes_trivial1 apply presburger
apply(subgoal_tac "\<exists>s1 m. r' = RSEQ (rders_simp r s1) (RNTIMES r m) \<and> m \<le> n")
apply(erule exE)+
using assms ntimes_trivial2 apply presburger
by blast
lemma P_holds_after_distinct:
assumes "\<forall>r \<in> set rs. P r"
shows "\<forall>r \<in> set (rdistinct rs rset). P r"
by (simp add: assms rdistinct_set_equality1)
lemma ntimes_control_bounded:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows "rsizes (rdistinct (map (optermsimp r) (nupdates s r [Some ([c], n)])) {})
\<le> (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n)))"
apply(subgoal_tac "\<forall>r' \<in> set (rdistinct (map (optermsimp r) (nupdates s r [Some ([c], n)])) {}).
rsize r' \<le> Suc (N + rsize (RNTIMES r n))")
apply (meson distinct_list_rexp_upto rdistinct_same_set)
apply(subgoal_tac "\<forall>r' \<in> set (map (optermsimp r) (nupdates s r [Some ([c], n)])). rsize r' \<le> Suc (N + rsize (RNTIMES r n))")
apply (simp add: rdistinct_set_equality)
by (metis assms nat_le_linear not_less_eq_eq ntimes_closed_form_list_elem_bounded)
lemma ntimes_closed_form_bounded0:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows " (rders_simp (RNTIMES r 0) s) = RZERO \<or> (rders_simp (RNTIMES r 0) s) = RNTIMES r 0
"
apply(induct s)
apply simp
by (metis always0 list.simps(3) rder.simps(7) rders.simps(2) rders_simp_same_simpders rsimp.simps(3))
lemma ntimes_closed_form_bounded1:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows " rsize (rders_simp (RNTIMES r 0) s) \<le> max (rsize RZERO) (rsize (RNTIMES r 0))"
by (metis assms max.cobounded1 max.cobounded2 ntimes_closed_form_bounded0)
lemma self_smaller_than_bound:
shows "\<forall>s. rsize (rders_simp r s) \<le> N \<Longrightarrow> rsize r \<le> N"
apply(drule_tac x = "[]" in spec)
apply simp
done
lemma ntimes_closed_form_bounded_nil_aux:
shows "max (rsize RZERO) (rsize (RNTIMES r 0)) = 1 + rsize r"
by auto
lemma ntimes_closed_form_bounded_nil:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows " rsize (rders_simp (RNTIMES r 0) s) \<le> 1 + rsize r"
using assms ntimes_closed_form_bounded1 by auto
lemma ntimes_ineq1:
shows "(rsize (RNTIMES r n)) \<ge> 1 + rsize r"
by simp
lemma ntimes_ineq2:
shows "1 + rsize r \<le>
max ((Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n))))) (rsize (RNTIMES r n))"
by (meson le_max_iff_disj ntimes_ineq1)
lemma ntimes_closed_form_bounded:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows "rsize (rders_simp (RNTIMES r (Suc n)) s) \<le>
max ((Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n))))) (rsize (RNTIMES r n))"
proof(cases s)
case Nil
then show "rsize (rders_simp (RNTIMES r (Suc n)) s)
\<le> max (Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * Suc (N + rsize (RNTIMES r n))) (rsize (RNTIMES r n))"
by simp
next
case (Cons a list)
then have "rsize (rders_simp (RNTIMES r (Suc n)) s) =
rsize (rsimp (RALTS ((map (optermsimp r) (nupdates list r [Some ([a], n)])))))"
using ntimes_closed_form by fastforce
also have "... \<le> Suc (rsizes (rdistinct ((map (optermsimp r) (nupdates list r [Some ([a], n)]))) {}))"
using alts_simp_control by blast
also have "... \<le> Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * (Suc (N + rsize (RNTIMES r n)))"
using ntimes_control_bounded[OF assms]
by (metis add_mono le_add1 mult_Suc plus_1_eq_Suc)
also have "... \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RNTIMES r n))))) * Suc (N + rsize (RNTIMES r n))) (rsize (RNTIMES r n))"
by simp
finally show ?thesis by simp
qed
lemma ntimes_closed_form_boundedA:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N"
shows "\<exists>N'. \<forall>s. rsize (rders_simp (RNTIMES r n) s) \<le> N'"
apply(case_tac n)
using assms ntimes_closed_form_bounded_nil apply blast
using assms ntimes_closed_form_bounded by blast
lemma star_closed_form_nonempty_bounded:
assumes "\<forall>s. rsize (rders_simp r s) \<le> N" and "s \<noteq> []"
shows "rsize (rders_simp (RSTAR r) s) \<le>
((Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r))))) "
proof(cases s)
case Nil
then show ?thesis
using local.Nil by fastforce
next
case (Cons a list)
then have "rsize (rders_simp (RSTAR r) s) =
rsize (rsimp (RALTS ((map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])))))"
using star_closed_form by fastforce
also have "... \<le> Suc (rsizes (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r s1) (RSTAR r)) (star_updates list r [[a]])) {}))"
using alts_simp_control by blast
also have "... \<le> Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * (Suc (N + rsize (RSTAR r)))"
by (smt (z3) add_mono_thms_linordered_semiring(1) assms(1) le_add1 map_eq_conv mult_Suc plus_1_eq_Suc star_control_bounded)
also have "... \<le> max (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r))))) * Suc (N + rsize (RSTAR r))) (rsize (RSTAR r))"
by simp
finally show ?thesis by simp
qed
lemma seq_estimate_bounded:
assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1"
and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
shows
"rsizes (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})
\<le> (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
proof -
have a: "rsizes (rdistinct (map (rders_simp r2) (vsuf s r1)) {}) \<le> N2 * card (sizeNregex N2)"
by (metis assms(2) distinct_list_rexp_upto ex_map_conv mult.commute)
have "rsizes (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}) \<le>
rsize (RSEQ (rders_simp r1 s) r2) + rsizes (rdistinct (map (rders_simp r2) (vsuf s r1)) {})"
using triangle_inequality_distinct by blast
also have "... \<le> rsize (RSEQ (rders_simp r1 s) r2) + N2 * card (sizeNregex N2)"
by (simp add: a)
also have "... \<le> Suc (N1 + (rsize r2) + N2 * card (sizeNregex N2))"
by (simp add: assms(1))
finally show ?thesis
by force
qed
lemma seq_closed_form_bounded2:
assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1"
and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
shows "rsize (rders_simp (RSEQ r1 r2) s)
\<le> max (2 + N1 + (rsize r2) + (N2 * card (sizeNregex N2))) (rsize (RSEQ r1 r2))"
proof(cases s)
case Nil
then show "rsize (rders_simp (RSEQ r1 r2) s)
\<le> max (2 + N1 + (rsize r2) + (N2 * card (sizeNregex N2))) (rsize (RSEQ r1 r2))"
by simp
next
case (Cons a list)
then have "rsize (rders_simp (RSEQ r1 r2) s) =
rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1)))))"
using seq_closed_form_variant by (metis list.distinct(1))
also have "... \<le> Suc (rsizes (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))"
using alts_simp_control by blast
also have "... \<le> 2 + N1 + (rsize r2) + (N2 * card (sizeNregex N2))"
using seq_estimate_bounded[OF assms] by auto
ultimately show "rsize (rders_simp (RSEQ r1 r2) s)
\<le> max (2 + N1 + (rsize r2) + N2 * card (sizeNregex N2)) (rsize (RSEQ r1 r2))"
by auto
qed
lemma rders_simp_bounded:
shows "\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
apply(induct r)
apply(rule_tac x = "Suc 0 " in exI)
using three_easy_cases0 apply force
using three_easy_cases1 apply blast
using three_easy_casesC apply blast
apply(erule exE)+
apply(rule exI)
apply(rule allI)
apply(rule seq_closed_form_bounded2)
apply(assumption)
apply(assumption)
apply (metis alts_closed_form_bounded size_list_estimation')
using star_closed_form_bounded apply blast
using ntimes_closed_form_boundedA by blast
unused_thms
export_code rders_simp rsimp rder in Scala module_name Example
end