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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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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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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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sorry
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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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lemma rexp_size_induct:
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shows "\<And>N r x5 a list.
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\<lbrakk> rsize r = Suc N; r = RALTS x5;
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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"
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apply(rule_tac x = "rsize a" in exI)
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apply(rule_tac x = "rsize (RALTS list)" in exI)
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apply(subgoal_tac "rsize a \<ge> 1")
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prefer 2
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using One_nat_def non_zero_size apply presburger
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apply(subgoal_tac "rsize (RALTS list) \<ge> 1 ")
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prefer 2
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using size_geq1 apply blast
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apply simp
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done
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lemma star_update_case1:
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shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
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by force
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lemma star_update_case2:
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shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
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by simp
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lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
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apply(case_tac r)
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apply simp+
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done
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lemma rsimp_alts_idem_aux1:
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shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
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by force
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lemma rsimp_alts_idem_aux2:
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shows "rsimp a = rsimp (RALTS [a])"
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apply(simp)
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apply(case_tac "rsimp a")
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apply simp+
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apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
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by simp
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lemma rsimp_alts_idem:
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shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
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apply(induct as)
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apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
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prefer 2
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apply simp
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using bubble_break rsimp_alts_idem_aux2 apply auto[1]
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apply(case_tac as)
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apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
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prefer 2
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apply simp
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using head_one_more_simp apply fastforce
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apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
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prefer 2
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using rsimp_ALTs.simps(3) apply presburger
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apply(simp only:)
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apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
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prefer 2
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using rsimp_ALTs.simps(3) apply presburger
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apply(simp only:)
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apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
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prefer 2
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using rsimp_ALTs.simps(3) apply presburger
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apply(simp only:)
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using simp_flatten2
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apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list)))) = rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
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prefer 2
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apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
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apply (simp only:)
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done
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lemma rsimp_alts_idem2:
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shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
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using head_one_more_simp rsimp_alts_idem by auto
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lemma evolution_step1:
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shows "rsimp
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(rsimp_ALTs
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(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
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rsimp
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(rsimp_ALTs
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(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)))])) "
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using rsimp_alts_idem by auto
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lemma evolution_step2:
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assumes " rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
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rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
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shows "rsimp
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(rsimp_ALTs
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(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
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rsimp
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(rsimp_ALTs
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(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)))])) "
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by (simp add: assms rsimp_alts_idem)
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lemma rsimp_seq_aux1:
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shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
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apply simp
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done
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lemma multiple_alts_simp_flatten:
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shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
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by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
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lemma evo3_main_aux1:
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shows "rsimp
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(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
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(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
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rsimp
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301 |
(RALTS
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302 |
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
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303 |
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
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304 |
apply(subgoal_tac "rsimp
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305 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
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306 |
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
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307 |
rsimp
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308 |
(RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
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309 |
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
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310 |
prefer 2
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311 |
apply (simp add: rsimp_idem)
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312 |
apply (simp only:)
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313 |
apply(subst multiple_alts_simp_flatten)
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314 |
by simp
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315 |
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316 |
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317 |
lemma evo3_main_nullable:
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|
318 |
shows "
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|
319 |
\<And>a Ss.
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|
320 |
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
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|
321 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
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322 |
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
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323 |
\<Longrightarrow> rsimp
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324 |
(rsimp_ALTs
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325 |
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
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326 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
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|
327 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
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328 |
apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r))
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329 |
= RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
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330 |
prefer 2
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331 |
apply simp
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332 |
apply(simp only:)
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333 |
apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
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334 |
prefer 2
|
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|
335 |
using star_update_case1 apply presburger
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336 |
apply(simp only:)
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|
337 |
apply(subst List.list.map(2))+
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338 |
apply(subgoal_tac "rsimp
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339 |
(rsimp_ALTs
|
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340 |
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
|
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|
341 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
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|
342 |
rsimp
|
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|
343 |
(RALTS
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|
344 |
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
|
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|
345 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
|
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346 |
prefer 2
|
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|
347 |
using rsimp_ALTs.simps(3) apply presburger
|
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|
348 |
apply(simp only:)
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|
349 |
apply(subgoal_tac " rsimp
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|
350 |
(rsimp_ALTs
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|
351 |
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
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|
352 |
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))
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|
353 |
=
|
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|
354 |
rsimp
|
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|
355 |
(RALTS
|
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|
356 |
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
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|
357 |
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
|
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|
358 |
|
|
|
359 |
prefer 2
|
|
|
360 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
361 |
apply (simp only:)
|
|
|
362 |
apply(subgoal_tac " rsimp
|
|
|
363 |
(RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
|
|
|
364 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
|
|
|
365 |
rsimp
|
|
|
366 |
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
|
|
|
367 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
|
|
|
368 |
prefer 2
|
|
|
369 |
apply (simp add: rsimp_idem)
|
|
|
370 |
apply(simp only:)
|
|
|
371 |
apply(subgoal_tac " rsimp
|
|
|
372 |
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
|
|
|
373 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
|
|
|
374 |
rsimp
|
|
|
375 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
376 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
|
|
|
377 |
prefer 2
|
|
|
378 |
using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
|
|
|
379 |
apply(simp only:)
|
|
|
380 |
apply(subgoal_tac " rsimp
|
|
|
381 |
(RALTS
|
|
|
382 |
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
383 |
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) =
|
|
|
384 |
rsimp
|
|
|
385 |
(RALTS
|
|
|
386 |
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
387 |
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
|
|
|
388 |
prefer 2
|
|
|
389 |
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)
|
|
|
390 |
apply(simp only:)
|
|
|
391 |
apply(subgoal_tac " rsimp
|
|
|
392 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
393 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
|
|
|
394 |
rsimp
|
|
|
395 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
396 |
( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) ")
|
|
|
397 |
prefer 2
|
|
|
398 |
using rsimp_idem apply force
|
|
|
399 |
apply(simp only:)
|
|
|
400 |
using evo3_main_aux1 by blast
|
|
|
401 |
|
|
|
402 |
|
|
|
403 |
lemma evo3_main_not1:
|
|
|
404 |
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)"
|
|
|
405 |
by fastforce
|
|
|
406 |
|
|
|
407 |
|
|
|
408 |
lemma evo3_main_not2:
|
|
|
409 |
shows "\<not>rnullable (rders_simp r a) \<Longrightarrow> rsimp
|
|
|
410 |
(rsimp_ALTs
|
|
|
411 |
(rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
|
|
|
412 |
(rsimp_ALTs
|
|
|
413 |
((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
|
|
|
414 |
by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
|
|
|
415 |
|
|
|
416 |
lemma evo3_main_not3:
|
|
|
417 |
shows "rsimp
|
|
|
418 |
(rsimp_ALTs
|
|
|
419 |
(rsimp_SEQ r1 (RSTAR r) # rs)) =
|
|
|
420 |
rsimp (rsimp_ALTs
|
|
|
421 |
(RSEQ r1 (RSTAR r) # rs))"
|
|
|
422 |
by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
|
|
|
423 |
|
|
|
424 |
|
|
|
425 |
lemma evo3_main_notnullable:
|
|
|
426 |
shows "\<And>a Ss.
|
|
|
427 |
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
428 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
|
|
|
429 |
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
|
|
|
430 |
\<Longrightarrow> rsimp
|
|
|
431 |
(rsimp_ALTs
|
|
|
432 |
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
|
|
|
433 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
|
|
|
434 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
|
|
|
435 |
apply(subst star_update_case2)
|
|
|
436 |
apply simp
|
|
|
437 |
apply(subst List.list.map(2))
|
|
|
438 |
apply(subst evo3_main_not2)
|
|
|
439 |
apply simp
|
|
|
440 |
apply(subst evo3_main_not3)
|
|
|
441 |
using rsimp_alts_idem by presburger
|
|
|
442 |
|
|
|
443 |
|
|
|
444 |
lemma evo3_aux2:
|
|
|
445 |
shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
|
|
|
446 |
by simp
|
|
|
447 |
lemma evo3_aux3:
|
|
|
448 |
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
|
|
|
449 |
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)
|
|
|
450 |
|
|
|
451 |
lemma evo3_aux4:
|
|
|
452 |
shows " rsimp
|
|
|
453 |
(rsimp_ALTs
|
|
|
454 |
[RSEQ (rder x r) (RSTAR r),
|
|
|
455 |
rsimp (rsimp_ALTs rs)]) =
|
|
|
456 |
rsimp
|
|
|
457 |
(rsimp_ALTs
|
|
|
458 |
(rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
|
|
|
459 |
by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
|
|
|
460 |
|
|
|
461 |
lemma evo3_aux5:
|
|
|
462 |
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)"
|
|
|
463 |
using idiot2 by blast
|
|
|
464 |
|
|
|
465 |
|
|
|
466 |
lemma evolution_step3:
|
|
|
467 |
shows" \<And>a Ss.
|
|
|
468 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
469 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) \<Longrightarrow>
|
|
|
470 |
rsimp
|
|
|
471 |
(rsimp_ALTs
|
|
|
472 |
[rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
|
|
|
473 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
|
|
|
474 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
|
|
|
475 |
apply(case_tac "rders_simp r a = RONE")
|
|
|
476 |
apply(subst rsimp_seq_aux1)
|
|
|
477 |
apply simp
|
|
|
478 |
apply(subst rder.simps(6))
|
|
|
479 |
apply(subgoal_tac "rnullable (rders_simp r a)")
|
|
|
480 |
prefer 2
|
|
|
481 |
using rnullable.simps(2) apply presburger
|
|
|
482 |
apply(subst star_update_case1)
|
|
|
483 |
apply simp
|
|
|
484 |
|
|
|
485 |
apply(subst List.list.map)+
|
|
|
486 |
apply(subst rders_simp_append)
|
|
|
487 |
apply(subst evo3_aux2)
|
|
|
488 |
apply simp
|
|
|
489 |
apply(subst evo3_aux3)
|
|
|
490 |
apply(subst evo3_aux4)
|
|
|
491 |
apply simp
|
|
|
492 |
apply(case_tac "rders_simp r a = RZERO")
|
|
|
493 |
|
|
|
494 |
apply (simp add: rsimp_alts_idem2)
|
|
|
495 |
apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
|
|
|
496 |
prefer 2
|
|
|
497 |
using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
|
|
|
498 |
using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
|
|
|
499 |
apply(subst evo3_aux5)
|
|
|
500 |
apply simp
|
|
|
501 |
apply(case_tac "rnullable (rders_simp r a) ")
|
|
|
502 |
using evo3_main_nullable apply blast
|
|
|
503 |
using evo3_main_notnullable apply blast
|
|
|
504 |
done
|
|
|
505 |
|
|
|
506 |
(*
|
|
|
507 |
proof (prove)
|
|
|
508 |
goal (1 subgoal):
|
|
|
509 |
1. map f (a # s) = f a # map f s
|
|
|
510 |
Auto solve_direct: the current goal can be solved directly with
|
|
|
511 |
HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
|
|
|
512 |
List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
|
|
|
513 |
List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
|
|
|
514 |
*)
|
|
|
515 |
lemma starseq_list_evolution:
|
|
|
516 |
fixes r :: rrexp and Ss :: "char list list" and x :: char
|
|
|
517 |
shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
|
|
|
518 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)) )"
|
|
|
519 |
apply(induct Ss)
|
|
|
520 |
apply simp
|
|
|
521 |
apply(subst List.list.map(2))
|
|
|
522 |
apply(subst evolution_step2)
|
|
|
523 |
apply simp
|
|
|
524 |
|
|
|
525 |
|
|
|
526 |
sorry
|
|
|
527 |
|
|
|
528 |
|
|
|
529 |
lemma star_seqs_produce_star_seqs:
|
|
|
530 |
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
|
|
|
531 |
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
|
|
|
532 |
by (meson comp_apply)
|
|
|
533 |
|
|
|
534 |
lemma map_der_lambda_composition:
|
|
|
535 |
shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
|
|
|
536 |
by force
|
|
|
537 |
|
|
|
538 |
lemma ralts_vs_rsimpalts:
|
|
|
539 |
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
|
|
|
540 |
by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
|
|
|
541 |
|
|
|
542 |
|
|
|
543 |
lemma linearity_of_list_of_star_or_starseqs:
|
|
|
544 |
fixes r::rrexp and Ss::"char list list" and x::char
|
|
|
545 |
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
|
|
|
546 |
rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
|
|
|
547 |
apply(subst rder_rsimp_ALTs_commute)
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548 |
apply(subst map_der_lambda_composition)
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549 |
using starseq_list_evolution
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550 |
apply(rule_tac x = "star_update x r Ss" in exI)
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551 |
apply(subst ralts_vs_rsimpalts)
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552 |
by simp
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553 |
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554 |
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555 |
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556 |
(*certified correctness---does not depend on any previous sorry*)
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557 |
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));
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558 |
xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
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559 |
\<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) =
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560 |
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
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561 |
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
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562 |
prefer 2
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563 |
apply blast
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564 |
apply(erule exE)
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|
565 |
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))))")
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|
566 |
prefer 2
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|
567 |
using rders_simp_append
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568 |
using rders_simp_one_char apply presburger
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|
569 |
apply(rule_tac x= "Ss" in exI)
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|
570 |
apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
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571 |
rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
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572 |
prefer 2
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|
573 |
using inside_simp_removal rsimp_idem apply presburger
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|
574 |
apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
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|
575 |
rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
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|
576 |
prefer 2
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|
577 |
using rder.simps(4) apply presburger
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|
578 |
apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
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|
579 |
rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
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|
580 |
apply (metis rsimp_idem)
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|
581 |
by (metis map_der_lambda_composition)
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|
582 |
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583 |
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|
584 |
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|
585 |
end
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