(**)
theory ClosedFormsBounds
imports "GeneralRegexBound" "ClosedForms"
begin
lemma alts_ders_lambda_shape_ders:
shows "\<forall>r \<in> set (map (\<lambda>r. rders_simp r ( s)) rs ). \<exists>r1 \<in> set rs. r = rders_simp r1 s"
by (simp add: image_iff)
lemma rlist_bound:
shows "\<forall>r \<in> set rs. rsize r \<le> N \<Longrightarrow> sum_list (map rsize rs) \<le> N * (length rs)"
apply(induct rs)
apply simp
by simp
lemma alts_closed_form_bounded: shows
"\<forall>r \<in> set rs. \<forall>s. rsize(rders_simp r s ) \<le> N \<Longrightarrow>
rsize (rders_simp (RALTS rs ) s) \<le> max (Suc ( N * (length rs))) (rsize (RALTS rs) )"
apply(induct s)
apply simp
apply(subst alts_closed_form_variant)
apply force
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)))")
prefer 2
using rsimp_mono apply presburger
apply(subgoal_tac "rsize (RALTS (map (\<lambda>r. rders_simp r (a # s)) rs)) =
Suc (sum_list (map rsize (map (\<lambda>r. rders_simp r (a # s)) rs)))")
prefer 2
using rsize.simps(4) apply blast
apply(subgoal_tac "sum_list (map rsize (map (\<lambda>r. rders_simp r (a # s)) rs )) \<le> N * (length rs) ")
apply linarith
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>r. rders_simp r (a # s)) rs ). rsize r \<le> N")
prefer 2
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)")
prefer 2
using alts_ders_lambda_shape_ders apply presburger
apply metis
apply(frule rlist_bound)
by fastforce
lemma alts_simp_ineq_unfold:
shows "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))"
using rsimp_aalts_smaller by auto
lemma no_elem_distinct :
shows "a \<notin> set rs \<Longrightarrow> rdistinct rs rset = rdistinct rs (insert a rset)"
apply(induct rs arbitrary: rset)
apply simp
by force
inductive good :: "rrexp \<Rightarrow> bool"
where
"RZERO \<notin> set rs \<Longrightarrow> good (RALTS rs)"
|"good RZERO"
|"good RONE"
|"good (RCHAR c)"
|"good (RSEQ r1 r2)"
|"good (RSTAR r0)"
lemma after_flts_no0:
shows "\<forall>r \<in> set rs. good r \<Longrightarrow> RZERO \<notin> set (rflts rs)"
apply(induct rs)
apply simp
apply(case_tac a)
apply simp
apply simp
apply simp
apply simp
apply simp
using good.simps apply blast
apply simp
done
lemma flts_has_no_zero:
shows "\<forall>r \<in> set rs. good r \<Longrightarrow> rdistinct (rflts rs) rset = rdistinct (rflts rs) (insert RZERO rset)"
apply(subgoal_tac "RZERO \<notin> set (rflts rs)")
apply (meson no_elem_distinct)
apply(insert after_flts_no0)
by blast
lemma shape_of_alts:
shows "\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs) \<Longrightarrow> RZERO \<notin> alts_set \<and> RONE \<notin> alts_set"
by fastforce
lemma quantified2_implies1:
shows "\<forall>r. P \<and> Q \<Longrightarrow> \<forall>r. P"
by auto
lemma quantified_quantifiedAB_A:
shows " (\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set) \<Longrightarrow> \<forall>a \<in> alts_set. \<exists>xs. a = RALTS xs"
by fastforce
lemma list_distinct_removal:
shows "set rs \<subseteq> rset \<Longrightarrow> rdistinct (rs @ rs1) rset = rdistinct rs1 rset"
apply(induct rs arbitrary: rset)
apply simp
by simp
lemma rdistinct_mono_list:
shows " sum_list (map rsize (rdistinct (x5 @ rs) rset )) \<le>
( sum_list (map rsize x5)) + (sum_list (map rsize (rdistinct rs ((set x5 ) \<union> rset))))"
apply(induct x5 arbitrary: rs rset)
apply simp
apply(case_tac "a \<in> rset")
apply simp
apply (simp add: add.assoc insert_absorb trans_le_add2)
apply simp
by (metis Un_insert_right)
lemma flts_size_reduction_alts:
shows " \<And>a rs noalts_set alts_set corr_set x5.
\<lbrakk>\<And>noalts_set alts_set corr_set.
(\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
(\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set) \<Longrightarrow>
Suc (sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> corr_set))))
\<le> Suc (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set)))));
(\<forall>r\<in>noalts_set. \<forall>xs. r \<noteq> RALTS xs) \<and>
(\<forall>a\<in>alts_set. \<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set);
a = RALTS x5\<rbrakk>
\<Longrightarrow> Suc (sum_list (map rsize (rdistinct (rflts (a # rs)) (noalts_set \<union> corr_set))))
\<le> Suc (sum_list
(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set)))))"
apply(case_tac "a \<in> alts_set")
apply simp
apply(subgoal_tac "set x5 \<subseteq> corr_set")
apply(subst list_distinct_removal)
apply auto[1]
apply presburger
apply fastforce
apply (subgoal_tac "a \<notin> noalts_set")
prefer 2
apply blast
apply simp
apply(subgoal_tac "sum_list (map rsize (rdistinct (x5 @ rflts rs) (noalts_set \<union> corr_set)))
\<le> sum_list (map rsize x5 ) + sum_list (map rsize (rdistinct (rflts rs) ((set x5) \<union> (noalts_set \<union> corr_set))))")
prefer 2
using rdistinct_mono_list apply presburger
apply(subgoal_tac "insert (RALTS x5) (noalts_set \<union> alts_set) = noalts_set \<union> (insert (RALTS x5) alts_set)")
apply(simp only:)
apply(subgoal_tac "sum_list (map rsize x5) +
sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> (set x5))))) \<le>
Suc (sum_list (map rsize x5) +
sum_list
(map rsize
(rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set)))))")
apply (simp add: Un_left_commute inf_sup_aci(5))
apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts rs) (noalts_set \<union> (corr_set \<union> set x5))))
\<le> sum_list
(map rsize
(rdistinct rs (insert RZERO (noalts_set \<union> insert (RALTS x5) alts_set))))")
apply linarith
apply(subgoal_tac "\<forall>r \<in> insert (RALTS x5) alts_set. \<exists>xs1.( r = RALTS xs1 \<and> set xs1 \<subseteq> corr_set \<union> set x5)")
apply presburger
apply (meson insert_iff sup.cobounded2 sup.coboundedI1)
by blast
lemma flts_vs_nflts1:
shows "(\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs)
\<and> (\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set))
\<Longrightarrow> (sum_list (map rsize (rdistinct ( rflts rs) (noalts_set \<union> corr_set) )))
\<le> (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set) ))))"
apply(induct rs arbitrary: noalts_set alts_set corr_set)
apply simp
apply(case_tac a)
apply(case_tac "RZERO \<in> noalts_set")
apply simp
apply(subgoal_tac "RZERO \<notin> alts_set")
apply simp
apply fastforce
apply(case_tac "RONE \<in> noalts_set")
apply simp
apply(subgoal_tac "RONE \<notin> alts_set")
prefer 2
apply fastforce
apply(case_tac "RONE \<in> corr_set")
apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
apply(simp only:)
apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) =
rdistinct (rflts rs) (noalts_set \<union> corr_set)")
apply(simp only:)
apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
apply(simp only:)
apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) =
rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
apply (simp only:)
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
apply(simp only:)
apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) =
insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
apply(simp only:)
apply(subgoal_tac " (sum_list
(map rsize ( rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))
\<le> (sum_list
(map rsize (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))")
apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
apply (metis (no_types, opaque_lifting) le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
apply fastforce
apply fastforce
apply (metis Un_iff insert_absorb)
apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
apply (meson UnCI rdistinct.simps(2))
using rflts.simps(4) apply presburger
apply simp
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
apply(simp only:)
apply (metis Un_insert_left insertE rrexp.distinct(15))
apply fastforce
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
prefer 2
apply blast
apply(case_tac "a \<in> corr_set")
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
prefer 2
apply fastforce
apply(simp only:)
apply(subgoal_tac " sum_list
(map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
\<le>
sum_list
(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
apply(subgoal_tac
"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
\<le>
sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
apply fastforce
apply simp
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
apply(simp only:)
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
apply(simp only:)
apply (metis insertE rrexp.distinct(21))
apply blast
apply fastforce
apply force
apply simp
apply (metis Un_insert_left insert_iff rrexp.distinct(21))
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
prefer 2
apply blast
apply(case_tac "a \<in> corr_set")
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
prefer 2
apply fastforce
apply(simp only:)
apply(subgoal_tac " sum_list
(map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
\<le>
sum_list
(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
apply(subgoal_tac
"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
\<le>
sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
apply fastforce
apply simp
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
apply(simp only:)
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
apply(simp only:)
apply (metis insertE rrexp.distinct(25))
apply blast
apply fastforce
apply force
apply simp
apply (metis Un_insert_left insertE rrexp.distinct(25))
using Suc_le_mono flts_size_reduction_alts apply presburger
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
prefer 2
apply blast
apply(case_tac "a \<in> corr_set")
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
prefer 2
apply fastforce
apply(simp only:)
apply(subgoal_tac " sum_list
(map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))
\<le>
sum_list
(map rsize (rdistinct (a # rs) (insert RZERO (noalts_set \<union> alts_set))))")
apply(subgoal_tac
"sum_list (map rsize (rdistinct (rflts (a # rs)) ((insert a noalts_set) \<union> corr_set)))
\<le>
sum_list (map rsize (rdistinct (a # rs) (insert RZERO ((insert a noalts_set) \<union> alts_set))))")
apply fastforce
apply simp
apply(subgoal_tac "(insert a (noalts_set \<union> alts_set)) = (insert a noalts_set) \<union> alts_set")
apply(simp only:)
apply(subgoal_tac "noalts_set \<union> corr_set = (insert a noalts_set) \<union> corr_set")
apply(simp only:)
apply (metis insertE rrexp.distinct(29))
apply blast
apply fastforce
apply force
apply simp
apply (metis Un_insert_left insert_iff rrexp.distinct(29))
done
lemma flts_vs_nflts:
shows "(\<forall>r \<in> noalts_set. \<forall>xs. r \<noteq> RALTS xs)
\<and> (\<forall>a \<in> alts_set. (\<exists>xs. a = RALTS xs \<and> set xs \<subseteq> corr_set))
\<Longrightarrow> Suc (sum_list (map rsize (rdistinct ( rflts rs) (noalts_set \<union> corr_set) )))
\<le> Suc (sum_list (map rsize (rdistinct rs (insert RZERO (noalts_set \<union> alts_set) ))))"
apply(induct rs arbitrary: noalts_set alts_set corr_set)
apply simp
apply(case_tac a)
apply(case_tac "RZERO \<in> noalts_set")
apply simp
apply(subgoal_tac "RZERO \<notin> alts_set")
apply simp
apply fastforce
apply(case_tac "RONE \<in> noalts_set")
apply simp
apply(subgoal_tac "RONE \<notin> alts_set")
prefer 2
apply fastforce
apply(case_tac "RONE \<in> corr_set")
apply(subgoal_tac "rflts (a # rs) = RONE # rflts rs")
apply(simp only:)
apply(subgoal_tac "rdistinct (RONE # rflts rs) (noalts_set \<union> corr_set) =
rdistinct (rflts rs) (noalts_set \<union> corr_set)")
apply(simp only:)
apply(subgoal_tac "rdistinct (RONE # rs) (insert RZERO (noalts_set \<union> alts_set)) =
RONE # (rdistinct rs (insert RONE (insert RZERO (noalts_set \<union> alts_set)))) ")
apply(simp only:)
apply(subgoal_tac "rdistinct (rflts rs) (noalts_set \<union> corr_set) =
rdistinct (rflts rs) (insert RONE (noalts_set \<union> corr_set))")
apply (simp only:)
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
apply(simp only:)
apply(subgoal_tac "insert RONE (insert RZERO (noalts_set \<union> alts_set)) =
insert RZERO ((insert RONE noalts_set) \<union> alts_set)")
apply(simp only:)
apply(subgoal_tac " Suc (sum_list
(map rsize ( rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))
\<le> Suc (sum_list
(map rsize (RONE # rdistinct rs (insert RZERO (insert RONE noalts_set \<union> alts_set)))))")
apply (smt (verit, best) dual_order.trans insert_iff rrexp.distinct(15))
apply (metis (no_types, opaque_lifting) add_Suc_right le_add_same_cancel2 list.simps(9) sum_list.Cons zero_le)
apply fastforce
apply fastforce
apply (metis Un_iff insert_absorb)
apply (metis UnE insertE insert_is_Un rdistinct.simps(2) rrexp.distinct(1))
apply (meson UnCI rdistinct.simps(2))
using rflts.simps(4) apply presburger
apply simp
apply(subgoal_tac "insert RONE (noalts_set \<union> corr_set) = (insert RONE noalts_set) \<union> corr_set")
apply(simp only:)
apply (metis Un_insert_left insertE rrexp.distinct(15))
apply fastforce
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
prefer 2
apply blast
apply(case_tac "a \<in> corr_set")
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
prefer 2
apply fastforce
apply(simp only:)
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
prefer 2
apply blast
apply(case_tac "a \<in> corr_set")
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
prefer 2
apply fastforce
apply(simp only:)
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
prefer 2
apply(case_tac "a \<in> noalts_set")
apply simp
apply(subgoal_tac "a \<notin> alts_set")
prefer 2
apply blast
apply(case_tac "a \<in> corr_set")
apply(subgoal_tac "noalts_set \<union> corr_set = insert a ( noalts_set \<union> corr_set)")
prefer 2
apply fastforce
apply(simp only:)
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
apply (metis add_mono_thms_linordered_semiring(1) flts_vs_nflts1 le_numeral_extra(4) plus_1_eq_Suc)
using flts_size_reduction_alts apply presburger
done
(* apply(rutac x = "\<lambda>a rs noalts_set alts_set corr_set. set xs ")*)
lemma distinct_simp_ineq_general:
shows "rsimp ` no_simp = has_simp \<and> finite no_simp \<Longrightarrow>Suc (sum_list (map rsize (rdistinct (map rsimp rs) has_simp)))
\<le> Suc (sum_list (map rsize (rdistinct rs no_simp)))"
apply(induct rs arbitrary: no_simp has_simp)
apply simp
apply(case_tac "a \<in> no_simp")
apply(subgoal_tac "rsimp a \<in> has_simp")
apply auto[1]
apply blast
apply(case_tac "rsimp a \<in> no_simp")
apply(subgoal_tac "rsimp a \<in> has_simp")
prefer 2
apply (simp add: rev_image_eqI rsimp_idem)
apply simp
apply (metis finite_insert image_insert insert_absorb trans_le_add2)
apply(case_tac "rsimp a \<notin> has_simp")
apply simp
apply (metis add_mono_thms_linordered_semiring(1) finite.insertI image_insert rsimp_mono)
apply simp
by (metis finite.insertI image_insert insert_absorb trans_le_add2)
lemma not_mentioned_elem_distinct_strong:
shows "r \<noteq> a \<Longrightarrow> (r \<in> set (rdistinct rs rset)) = (r \<in> set (rdistinct rs (insert a rset)))"
apply(induct rs arbitrary: rset)
apply simp
by force
lemma not_mentioned_elem_distinct:
shows "r \<noteq> a \<Longrightarrow> (r \<in> set (rdistinct rs {})) = (r \<in> set (rdistinct rs (insert a {})))"
by (metis not_mentioned_elem_distinct_strong)
lemma larger_acc_smaller_distinct_res0:
shows " ss \<subseteq> SS \<Longrightarrow> sum_list (map rsize (rdistinct rs SS)) \<le> sum_list (map rsize (rdistinct rs ss))"
apply(induct rs arbitrary: ss SS)
apply simp
apply(case_tac "a \<in> ss")
apply(subgoal_tac "a \<in> SS")
apply simp
apply blast
apply(case_tac "a \<in> SS")
apply simp
apply(subgoal_tac "insert a ss \<subseteq> SS")
apply simp
apply (simp add: trans_le_add2)
apply blast
apply(simp)
apply(subgoal_tac "insert a ss \<subseteq> insert a SS")
apply blast
by blast
lemma without_flts_ineq:
shows " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le>
Suc (sum_list (map rsize (rdistinct ( rs ) {} )))"
proof -
have " Suc (sum_list (map rsize (rdistinct (rflts rs) {}) )) \<le>
Suc (sum_list (map rsize (rdistinct rs (insert RZERO {}))))"
by (metis empty_iff flts_vs_nflts sup_bot_left)
also have "... \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
by (simp add: larger_acc_smaller_distinct_res0)
finally show ?thesis
by blast
qed
lemma distinct_simp_ineq:
shows "Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))
\<le> Suc (sum_list (map rsize (rdistinct rs {})))"
using distinct_simp_ineq_general by blast
lemma alts_simp_control:
shows "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
proof -
have "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct (rflts (map rsimp rs)) {})))"
using alts_simp_ineq_unfold by auto
moreover have "\<dots> \<le> Suc (sum_list (map rsize (rdistinct (map rsimp rs) {})))"
using without_flts_ineq by blast
ultimately show "rsize (rsimp (RALTS rs)) \<le> Suc (sum_list (map rsize (rdistinct rs {})))"
by (meson distinct_simp_ineq order.trans)
qed
lemma rdistinct_equality1:
shows "a \<notin> ss \<Longrightarrow> rdistinct (a # rs) ss = a # rdistinct rs (insert a ss) "
by auto
lemma larger_acc_smaller_distinct_res:
shows " (sum_list (map rsize (rdistinct rs ss))) \<ge> (sum_list (map rsize (rdistinct rs (insert a ss))))"
apply(subgoal_tac "ss \<subseteq> (insert a ss)")
apply(rule larger_acc_smaller_distinct_res0)
apply simp
by (simp add: subset_insertI)
lemma size_list_triangle1:
shows "sum_list (map rsize (a # (rdistinct as ss))) \<ge> rsize a + sum_list (map rsize (rdistinct as (insert a ss)))"
by (simp add: larger_acc_smaller_distinct_res)
lemma triangle_inequality_distinct:
shows "sum_list (map rsize (rdistinct (a # rs) ss)) \<le> rsize a + (sum_list (map rsize (rdistinct rs ss)))"
apply(case_tac "a \<in> ss")
apply simp
apply(subst rdistinct_equality1)
apply simp
using size_list_triangle1 by auto
lemma same_regex_property_after_map:
shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (map (f r2) Ss). P r"
by auto
lemma same_property_after_distinct:
shows " \<forall>r \<in> set (map (f r2) Ss). P r \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
apply(induct Ss arbitrary: xset)
apply simp
by auto
lemma same_regex_property_after_distinct:
shows "\<forall>s. P (f r2 s) \<Longrightarrow> \<forall>r \<in> set (rdistinct (map (f r2) Ss) xset). P r"
apply(rule same_property_after_distinct)
apply(rule same_regex_property_after_map)
by simp
lemma Sum_cons:
shows "distinct (a # as) \<Longrightarrow> \<Sum> (set ((a::nat) # as)) = a + \<Sum> (set as)"
by simp
lemma distinct_list_sizeNregex_bounded:
shows "distinct rs \<and> (\<forall> r \<in> (set rs). rsize r \<le> N) \<Longrightarrow> sum_list (map rsize rs) \<le> N * length rs"
apply(induct rs)
apply simp
by simp
lemma distinct_list_size_len_bounded:
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 "
by (metis distinct_list_sizeNregex_bounded dual_order.trans mult.commute mult_le_mono1)
lemma rdistinct_same_set:
shows "(r \<in> set rs) = (r \<in> set (rdistinct rs {}))"
apply(induct rs)
apply simp
apply(case_tac "a \<in> set rs")
apply(case_tac "r = a")
apply (simp)
apply (simp add: not_mentioned_elem_distinct)
using not_mentioned_elem_distinct by fastforce
lemma distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size:
shows "\<forall>r\<in> set rs. (rsize r ) \<le> N \<Longrightarrow> sum_list (map rsize (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)
apply(rule conjI)+
apply simp
apply(rule conjI)
apply (meson rdistinct_same_set)
apply blast
apply(subgoal_tac "\<forall>r \<in> set (rdistinct rs {}). rsize r \<le> N")
prefer 2
apply (meson rdistinct_same_set)
apply(subgoal_tac "length (rdistinct rs {}) = card (set (rdistinct rs {}))")
prefer 2
using set_related_list apply blast
apply(simp only:)
by (metis card_mono finite_size_n mem_Collect_eq sizeNregex_def subset_code(1))
lemma star_closed_form_bounded_by_rdistinct_list_estimate:
shows "rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
(star_updates s r0 [[c]]) ) ))) \<le>
Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
(star_updates s r0 [[c]]) ) {}) ) )"
by (metis alts_simp_control )
lemma star_lambda_form:
shows "\<forall> r \<in> set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) ls).
\<exists>s2. r = RSEQ (rders_simp r0 s2) (RSTAR r0) "
by (meson ex_map_conv)
lemma star_lambda_ders:
shows " \<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
\<forall>r\<in>set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates s r0 [[c]])).
rsize r \<le> Suc (N + rsize (RSTAR r0))"
apply(insert star_lambda_form)
apply(simp)
done
lemma star_control_bounded:
shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
(star_updates s r0 [[c]]) ) {}) ) ) \<le>
(card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
"
apply(subgoal_tac "\<forall>r \<in> set (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
(star_updates s r0 [[c]]) ). (rsize r ) \<le> Suc (N + rsize (RSTAR r0))")
prefer 2
using star_lambda_ders apply blast
using distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size by blast
lemma star_control_variant:
assumes "\<forall>s. rsize (rders_simp r0 s) \<le> N"
shows"Suc
(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
(star_updates list r0 [[a]])) {})))
\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) "
apply(subgoal_tac "(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
(star_updates list r0 [[a]])) {})))
\<le> ( (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
prefer 2
using assms star_control_bounded apply presburger
by simp
lemma star_closed_form_bounded:
shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
rsize (rders_simp (RSTAR r0) s) \<le>
max ( (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0))))) (rsize (RSTAR r0))"
apply(case_tac s)
apply simp
apply(subgoal_tac " rsize (rders_simp (RSTAR r0) (a # list)) =
rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))")
prefer 2
using star_closed_form apply presburger
apply(subgoal_tac "rsize (rsimp (
RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))
\<le> Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
(star_updates list r0 [[a]]) ) {}) ) )")
prefer 2
using star_closed_form_bounded_by_rdistinct_list_estimate apply presburger
apply(subgoal_tac "Suc (sum_list
(map rsize
(rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates list r0 [[a]])) {})))
\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
apply auto[1]
using star_control_variant by blast
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
"Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
proof -
have a: "sum_list (map rsize (rdistinct (map (rders_simp r2) (vsuf s r1)) {})) \<le> N2 * card (sizeNregex N2)"
by (metis assms(2) distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size ex_map_conv mult.commute)
have "sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})) \<le>
rsize (RSEQ (rders_simp r1 s) r2) + sum_list (map rsize (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 (Suc (Suc (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 (Suc (Suc (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.discI)
also have "... \<le> Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))"
using alts_simp_control by blast
also have "... \<le> Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
using assms seq_estimate_bounded by blast
ultimately show "rsize (rders_simp (RSEQ r1 r2) s)
\<le> max (Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))) (rsize (RSEQ r1 r2))"
by auto
qed
lemma rders_simp_bounded:
fixes "r"
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 by blast
corollary rders_simp_finiteness:
shows "\<forall>r. \<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
using rders_simp_bounded by auto
end