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theory ClosedFormsBounds
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imports "GeneralRegexBound" "ClosedForms"
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begin
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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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rsize (rders_simp (RALTS rs ) s) \<le> max (Suc ( N * (card (sizeNregex N)))) (rsize (RALTS rs) )"
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apply(induct s)
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apply simp
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apply(insert alts_closed_form_variant)
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sorry
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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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sorry
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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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445
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444
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sorry
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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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sorry
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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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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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sorry
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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(arbitrary: ss)
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apply simp
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apply(case_tac "a \<in> ss")
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apply simp
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sorry
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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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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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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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prefer 2
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using seq_closed_form_variant apply blast
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apply(subgoal_tac "rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
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\<le>
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Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))")
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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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\<le> Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
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prefer 2
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using seq_estimate_bounded apply blast
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apply(subgoal_tac "rsize (rders_simp (RSEQ r1 r2) s) \<le> Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))")
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using le_max_iff_disj apply blast
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apply auto[1]
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using seq_list_estimate_control by presburger
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lemma rders_simp_bounded: shows
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"\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
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apply(induct r)
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apply(rule_tac x = "Suc 0 " in exI)
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using three_easy_cases0 apply force
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using three_easy_cases1 apply blast
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using three_easy_casesC apply blast
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using seq_closed_form_bounded apply blast
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apply (metis alts_closed_form_bounded size_list_estimation')
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using star_closed_form_bounded by blast
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(*Obsolete materials*)
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end
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