theory ClosedFormsBounds
imports "GeneralRegexBound" "ClosedForms"
begin
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 * (card (sizeNregex N)))) (rsize (RALTS rs) )"
apply(induct s)
apply simp
apply(insert alts_closed_form_variant)
sorry
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]]) ) {}) ) )"
sorry
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"
sorry
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)))
"
sorry
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_list_estimate_control: shows
" rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
\<le> Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))"
sorry
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)))"
sorry
lemma seq_closed_form_bounded: shows
"\<lbrakk>\<forall>s. rsize (rders_simp r1 s) \<le> N1 ; \<forall>s. rsize (rders_simp r2 s) \<le> N2\<rbrakk> \<Longrightarrow>
rsize (rders_simp (RSEQ r1 r2) s) \<le>
max (Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))) (rsize (RSEQ r1 r2)) "
apply(case_tac s)
apply simp
apply(subgoal_tac " (rders_simp (RSEQ r1 r2) s) =
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))")
prefer 2
using seq_closed_form_variant apply blast
apply(subgoal_tac "rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
\<le>
Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))")
apply(subgoal_tac "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)))")
prefer 2
using seq_estimate_bounded apply blast
apply(subgoal_tac "rsize (rders_simp (RSEQ r1 r2) s) \<le> Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))")
using le_max_iff_disj apply blast
apply auto[1]
using seq_list_estimate_control by presburger
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
using seq_closed_form_bounded apply blast
apply (metis alts_closed_form_bounded size_list_estimation')
using star_closed_form_bounded by blast
(*Obsolete materials*)
lemma rexp_size_induct:
shows "\<And>N r x5 a list.
\<lbrakk> rsize r = Suc N; r = RALTS x5;
x5 = a # list\<rbrakk> \<Longrightarrow>\<exists>i j. rsize a = i \<and> rsize (RALTS list) = j \<and> i + j = Suc N \<and> i \<le> N \<and> j \<le> N"
apply(rule_tac x = "rsize a" in exI)
apply(rule_tac x = "rsize (RALTS list)" in exI)
apply(subgoal_tac "rsize a \<ge> 1")
prefer 2
using One_nat_def non_zero_size apply presburger
apply(subgoal_tac "rsize (RALTS list) \<ge> 1 ")
prefer 2
using size_geq1 apply blast
apply simp
done
lemma star_update_case1:
shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
by force
lemma star_update_case2:
shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
by simp
lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
apply(case_tac r)
apply simp+
done
lemma rsimp_alts_idem_aux1:
shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
by force
lemma rsimp_alts_idem_aux2:
shows "rsimp a = rsimp (RALTS [a])"
apply(simp)
apply(case_tac "rsimp a")
apply simp+
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
by simp
lemma rsimp_alts_idem:
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
apply(induct as)
apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
prefer 2
apply simp
using bubble_break rsimp_alts_idem_aux2 apply auto[1]
apply(case_tac as)
apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
prefer 2
apply simp
using head_one_more_simp apply fastforce
apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
using simp_flatten2
apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list)))) = rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
prefer 2
apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
apply (simp only:)
done
lemma rsimp_alts_idem2:
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
using head_one_more_simp rsimp_alts_idem by auto
lemma evolution_step1:
shows "rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [(rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)))])) "
using rsimp_alts_idem by auto
lemma evolution_step2:
assumes " rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
shows "rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [ rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])) "
by (simp add: assms rsimp_alts_idem)
lemma rsimp_seq_aux1:
shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
apply simp
done
lemma multiple_alts_simp_flatten:
shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
lemma evo3_main_aux1:
shows "rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
rsimp
(RALTS
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
apply(subgoal_tac "rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
rsimp
(RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
prefer 2
apply (simp add: rsimp_idem)
apply (simp only:)
apply(subst multiple_alts_simp_flatten)
by simp
lemma evo3_main_nullable:
shows "
\<And>a Ss.
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
\<Longrightarrow> rsimp
(rsimp_ALTs
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r))
= RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
prefer 2
apply simp
apply(simp only:)
apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
prefer 2
using star_update_case1 apply presburger
apply(simp only:)
apply(subst List.list.map(2))+
apply(subgoal_tac "rsimp
(rsimp_ALTs
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp
(RALTS
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac " rsimp
(rsimp_ALTs
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))
=
rsimp
(RALTS
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply (simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
prefer 2
apply (simp add: rsimp_idem)
apply(simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
prefer 2
using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
apply(simp only:)
apply(subgoal_tac " rsimp
(RALTS
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) =
rsimp
(RALTS
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
prefer 2
apply (smt (z3) idiot2 list.simps(9) rrexp.distinct(9) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_idem)
apply(simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) ")
prefer 2
using rsimp_idem apply force
apply(simp only:)
using evo3_main_aux1 by blast
lemma evo3_main_not1:
shows " \<not>rnullable (rders_simp r a) \<Longrightarrow> rder x (RSEQ (rders_simp r a) (RSTAR r)) = RSEQ (rder x (rders_simp r a)) (RSTAR r)"
by fastforce
lemma evo3_main_not2:
shows "\<not>rnullable (rders_simp r a) \<Longrightarrow> rsimp
(rsimp_ALTs
(rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
(rsimp_ALTs
((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
lemma evo3_main_not3:
shows "rsimp
(rsimp_ALTs
(rsimp_SEQ r1 (RSTAR r) # rs)) =
rsimp (rsimp_ALTs
(RSEQ r1 (RSTAR r) # rs))"
by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
lemma evo3_main_notnullable:
shows "\<And>a Ss.
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
\<Longrightarrow> rsimp
(rsimp_ALTs
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(subst star_update_case2)
apply simp
apply(subst List.list.map(2))
apply(subst evo3_main_not2)
apply simp
apply(subst evo3_main_not3)
using rsimp_alts_idem by presburger
lemma evo3_aux2:
shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
by simp
lemma evo3_aux3:
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
by (metis list.simps(8) list.simps(9) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem)
lemma evo3_aux4:
shows " rsimp
(rsimp_ALTs
[RSEQ (rder x r) (RSTAR r),
rsimp (rsimp_ALTs rs)]) =
rsimp
(rsimp_ALTs
(rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
lemma evo3_aux5:
shows "rders_simp r a \<noteq> RONE \<and> rders_simp r a \<noteq> RZERO \<Longrightarrow> rsimp_SEQ (rders_simp r a) (RSTAR r) = RSEQ (rders_simp r a) (RSTAR r)"
using idiot2 by blast
lemma evolution_step3:
shows" \<And>a Ss.
rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) \<Longrightarrow>
rsimp
(rsimp_ALTs
[rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(case_tac "rders_simp r a = RONE")
apply(subst rsimp_seq_aux1)
apply simp
apply(subst rder.simps(6))
apply(subgoal_tac "rnullable (rders_simp r a)")
prefer 2
using rnullable.simps(2) apply presburger
apply(subst star_update_case1)
apply simp
apply(subst List.list.map)+
apply(subst rders_simp_append)
apply(subst evo3_aux2)
apply simp
apply(subst evo3_aux3)
apply(subst evo3_aux4)
apply simp
apply(case_tac "rders_simp r a = RZERO")
apply (simp add: rsimp_alts_idem2)
apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
prefer 2
using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
apply(subst evo3_aux5)
apply simp
apply(case_tac "rnullable (rders_simp r a) ")
using evo3_main_nullable apply blast
using evo3_main_notnullable apply blast
done
(*
proof (prove)
goal (1 subgoal):
1. map f (a # s) = f a # map f s
Auto solve_direct: the current goal can be solved directly with
HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
*)
lemma starseq_list_evolution:
fixes r :: rrexp and Ss :: "char list list" and x :: char
shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)) )"
apply(induct Ss)
apply simp
apply(subst List.list.map(2))
apply(subst evolution_step2)
apply simp
sorry
lemma star_seqs_produce_star_seqs:
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
by (meson comp_apply)
lemma map_der_lambda_composition:
shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
by force
lemma ralts_vs_rsimpalts:
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
lemma linearity_of_list_of_star_or_starseqs:
fixes r::rrexp and Ss::"char list list" and x::char
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
apply(subst rder_rsimp_ALTs_commute)
apply(subst map_der_lambda_composition)
using starseq_list_evolution
apply(rule_tac x = "star_update x r Ss" in exI)
apply(subst ralts_vs_rsimpalts)
by simp
(*certified correctness---does not depend on any previous sorry*)
lemma star_list_push_der: shows " \<lbrakk>xs \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss));
xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
\<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) =
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
prefer 2
apply blast
apply(erule exE)
apply(subgoal_tac "rders_simp (RSTAR r) (xs @ [x]) = rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using rders_simp_append
using rders_simp_one_char apply presburger
apply(rule_tac x= "Ss" in exI)
apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using inside_simp_removal rsimp_idem apply presburger
apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using rder.simps(4) apply presburger
apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
apply (metis rsimp_idem)
by (metis map_der_lambda_composition)
end