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theory ClosedForms
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imports "Lexer" "PDerivs"
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begin
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datatype rrexp =
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RZERO
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| RONE
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| RCHAR char
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| RSEQ rrexp rrexp
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| RALTS "rrexp list"
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| RSTAR rrexp
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abbreviation
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"RALT r1 r2 \<equiv> RALTS [r1, r2]"
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fun
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rnullable :: "rrexp \<Rightarrow> bool"
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where
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"rnullable (RZERO) = False"
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| "rnullable (RONE ) = True"
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| "rnullable (RCHAR c) = False"
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| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
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| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
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| "rnullable (RSTAR r) = True"
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fun
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rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
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where
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"rder c (RZERO) = RZERO"
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| "rder c (RONE) = RZERO"
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| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
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| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
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| "rder c (RSEQ r1 r2) =
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(if rnullable r1
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then RALT (RSEQ (rder c r1) r2) (rder c r2)
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else RSEQ (rder c r1) r2)"
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| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
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fun
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rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
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where
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"rders r [] = r"
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| "rders r (c#s) = rders (rder c r) s"
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fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
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where
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"rdistinct [] acc = []"
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| "rdistinct (x#xs) acc =
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(if x \<in> acc then rdistinct xs acc
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else x # (rdistinct xs ({x} \<union> acc)))"
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fun rflts :: "rrexp list \<Rightarrow> rrexp list"
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where
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"rflts [] = []"
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| "rflts (RZERO # rs) = rflts rs"
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| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
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| "rflts (r1 # rs) = r1 # rflts rs"
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fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
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where
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"rsimp_ALTs [] = RZERO"
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| "rsimp_ALTs [r] = r"
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| "rsimp_ALTs rs = RALTS rs"
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fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
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where
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"rsimp_SEQ RZERO _ = RZERO"
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| "rsimp_SEQ _ RZERO = RZERO"
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| "rsimp_SEQ RONE r2 = r2"
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| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
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fun rsimp :: "rrexp \<Rightarrow> rrexp"
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where
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"rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
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| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
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| "rsimp r = r"
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fun
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rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
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where
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"rders_simp r [] = r"
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| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
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fun rsize :: "rrexp \<Rightarrow> nat" where
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"rsize RZERO = 1"
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| "rsize (RONE) = 1"
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| "rsize (RCHAR c) = 1"
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| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
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| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
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| "rsize (RSTAR r) = Suc (rsize r)"
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fun rlist_size :: "rrexp list \<Rightarrow> nat" where
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"rlist_size (r # rs) = rsize r + rlist_size rs"
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| "rlist_size [] = 0"
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fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
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"vsuf [] _ = []"
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|"vsuf (c#cs) r1 = (if (rnullable r1) then (vsuf cs (rder c r1)) @ [c # cs]
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else (vsuf cs (rder c r1))
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) "
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lemma seq_closed_form: shows
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"rsimp (rders_simp (RSEQ r1 r2) s) =
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rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) #
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(map (rders r2) (vsuf s r1))
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)
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)"
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apply(induct s)
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apply simp
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sorry
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fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list" where
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"star_update c r [] = []"
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|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s))
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then (s@[c]) # [c] # (star_update c r Ss)
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else (s@[c]) # (star_update c r Ss) )"
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fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
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where
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"star_updates [] r Ss = Ss"
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| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
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lemma star_closed_form:
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shows "rders_simp (RSTAR r0) (c#s) =
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rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r [[c]]) ) ))"
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apply(induct s)
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apply simp
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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 r [[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 r [[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 (rexp_enum N))* N"
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sorry
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lemma ind_hypo_on_ders_leads_to_stars_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 r [[c]]) ) {}) ) ) \<le>
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(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
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"
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sorry
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lemma r0_bounded_star_bounded:
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shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
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\<forall>s. rsize (rders_simp (RSTAR r0) s) \<le>
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(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))"
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sorry
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(*some basic facts about rsimp*)
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lemma hand_made_def_rlist_size:
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shows "rlist_size rs = sum_list (map rsize rs)"
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proof (induct rs)
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case Nil show ?case by simp
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next
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case (Cons a rs) thus ?case
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by simp
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qed
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lemma rder_rsimp_ALTs_commute:
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shows " (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
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apply(induct rs)
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apply simp
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apply(case_tac rs)
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apply simp
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apply auto
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done
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lemma rsimp_aalts_smaller:
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shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
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apply(induct rs)
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apply simp
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apply simp
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apply(case_tac "rs = []")
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apply simp
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apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
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apply(erule exE)+
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apply simp
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apply simp
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by(meson neq_Nil_conv)
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lemma rSEQ_mono:
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shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"
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apply auto
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apply(induct r1)
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apply auto
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apply(case_tac "r2")
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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done
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lemma ralts_cap_mono:
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shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "
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by simp
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lemma rflts_def_idiot:
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shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
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\<Longrightarrow> rflts (a # rs) = a # rflts rs"
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apply(case_tac a)
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apply simp_all
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done
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lemma rflts_mono:
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shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize rs)"
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apply(induct rs)
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apply simp
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apply(case_tac "a = RZERO")
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apply simp
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apply(case_tac "\<exists>rs1. a = RALTS rs1")
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apply(erule exE)
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apply simp
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apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
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prefer 2
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using rflts_def_idiot apply blast
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apply simp
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done
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lemma rdistinct_smaller: shows "sum_list (map rsize (rdistinct rs ss)) \<le>
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sum_list (map rsize rs )"
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apply (induct rs arbitrary: ss)
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apply simp
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by (simp add: trans_le_add2)
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lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
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by (simp add: rdistinct_smaller)
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lemma rsimp_alts_mono :
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shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa) \<Longrightarrow>
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rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (sum_list (map rsize x))"
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apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} ))
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\<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
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prefer 2
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using rsimp_aalts_smaller apply auto[1]
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apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")
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prefer 2
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using ralts_cap_mono apply blast
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apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
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sum_list (map rsize ( (rflts (map rsimp x))))")
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prefer 2
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using rdistinct_smaller apply presburger
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apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le>
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sum_list (map rsize (map rsimp x))")
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prefer 2
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using rflts_mono apply blast
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apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize x)")
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prefer 2
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apply (simp add: sum_list_mono)
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by linarith
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lemma rsimp_mono:
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shows "rsize (rsimp r) \<le> rsize r"
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apply(induct r)
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apply simp_all
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apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
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apply force
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using rSEQ_mono
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apply presburger
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using rsimp_alts_mono by auto
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lemma idiot:
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shows "rsimp_SEQ RONE r = r"
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apply(case_tac r)
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apply simp_all
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done
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lemma no_alt_short_list_after_simp:
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shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
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sorry
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lemma no_further_dB_after_simp:
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shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
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sorry
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lemma idiot2:
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shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
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\<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
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apply(case_tac r1)
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apply(case_tac r2)
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apply simp_all
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327 |
apply(case_tac r2)
|
|
|
328 |
apply simp_all
|
|
|
329 |
apply(case_tac r2)
|
|
|
330 |
apply simp_all
|
|
|
331 |
apply(case_tac r2)
|
|
|
332 |
apply simp_all
|
|
|
333 |
apply(case_tac r2)
|
|
|
334 |
apply simp_all
|
|
|
335 |
done
|
|
|
336 |
|
|
|
337 |
lemma rders__onechar:
|
|
|
338 |
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
|
|
|
339 |
by simp
|
|
|
340 |
|
|
|
341 |
lemma rders_append:
|
|
|
342 |
"rders c (s1 @ s2) = rders (rders c s1) s2"
|
|
|
343 |
apply(induct s1 arbitrary: c s2)
|
|
|
344 |
apply(simp_all)
|
|
|
345 |
done
|
|
|
346 |
|
|
|
347 |
lemma rders_simp_append:
|
|
|
348 |
"rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
|
|
|
349 |
apply(induct s1 arbitrary: c s2)
|
|
|
350 |
apply(simp_all)
|
|
|
351 |
done
|
|
|
352 |
|
|
|
353 |
lemma inside_simp_removal:
|
|
|
354 |
shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
|
|
|
355 |
sorry
|
|
|
356 |
|
|
|
357 |
lemma set_related_list:
|
|
|
358 |
shows "distinct rs \<Longrightarrow> length rs = card (set rs)"
|
|
|
359 |
by (simp add: distinct_card)
|
|
|
360 |
(*this section deals with the property of distinctBy: creates a list without duplicates*)
|
|
|
361 |
lemma rdistinct_never_added_twice:
|
|
|
362 |
shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
|
|
|
363 |
by force
|
|
|
364 |
|
|
|
365 |
|
|
|
366 |
lemma rdistinct_does_the_job:
|
|
|
367 |
shows "distinct (rdistinct rs s)"
|
|
|
368 |
apply(induct rs arbitrary: s)
|
|
|
369 |
apply simp
|
|
|
370 |
apply simp
|
|
|
371 |
sorry
|
|
|
372 |
|
|
|
373 |
lemma rders_simp_same_simpders:
|
|
|
374 |
shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
|
|
|
375 |
apply(induct s rule: rev_induct)
|
|
|
376 |
apply simp
|
|
|
377 |
apply(case_tac "xs = []")
|
|
|
378 |
apply simp
|
|
|
379 |
apply(simp add: rders_append rders_simp_append)
|
|
|
380 |
using inside_simp_removal by blast
|
|
|
381 |
|
|
|
382 |
lemma simp_helps_der_pierce:
|
|
|
383 |
shows " rsimp
|
|
|
384 |
(rder x
|
|
|
385 |
(rsimp_ALTs rs)) =
|
|
|
386 |
rsimp
|
|
|
387 |
(rsimp_ALTs
|
|
|
388 |
(map (rder x )
|
|
|
389 |
rs
|
|
|
390 |
)
|
|
|
391 |
)"
|
|
|
392 |
sorry
|
|
|
393 |
|
|
|
394 |
|
|
|
395 |
lemma rders_simp_one_char:
|
|
|
396 |
shows "rders_simp r [c] = rsimp (rder c r)"
|
|
|
397 |
apply auto
|
|
|
398 |
done
|
|
|
399 |
|
|
|
400 |
lemma rsimp_idem:
|
|
|
401 |
shows "rsimp (rsimp r) = rsimp r"
|
|
|
402 |
sorry
|
|
|
403 |
|
|
|
404 |
corollary rsimp_inner_idem1:
|
|
|
405 |
shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
|
|
|
406 |
|
|
|
407 |
sorry
|
|
|
408 |
|
|
|
409 |
corollary rsimp_inner_idem2:
|
|
|
410 |
shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
|
|
|
411 |
sorry
|
|
|
412 |
|
|
|
413 |
corollary rsimp_inner_idem3:
|
|
|
414 |
shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
|
|
|
415 |
by (meson map_idI rsimp_inner_idem2)
|
|
|
416 |
|
|
|
417 |
corollary rsimp_inner_idem4:
|
|
|
418 |
shows "rsimp r = RALTS rs \<Longrightarrow> flts rs = rs"
|
|
|
419 |
sorry
|
|
|
420 |
|
|
|
421 |
|
|
|
422 |
lemma head_one_more_simp:
|
|
|
423 |
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
|
|
|
424 |
by (simp add: rsimp_idem)
|
|
|
425 |
|
|
|
426 |
lemma head_one_more_dersimp:
|
|
|
427 |
shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
|
|
|
428 |
using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
|
|
|
429 |
|
|
|
430 |
|
|
|
431 |
|
|
|
432 |
|
|
|
433 |
lemma ders_simp_nullability:
|
|
|
434 |
shows "rnullable (rders r s) = rnullable (rders_simp r s)"
|
|
|
435 |
sorry
|
|
|
436 |
|
|
|
437 |
lemma first_elem_seqder:
|
|
|
438 |
shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
|
|
|
439 |
# rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
|
|
|
440 |
by auto
|
|
|
441 |
|
|
|
442 |
lemma first_elem_seqder1:
|
|
|
443 |
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
|
|
|
444 |
map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
|
|
|
445 |
by (simp add: rsimp_idem)
|
|
|
446 |
|
|
|
447 |
lemma first_elem_seqdersimps:
|
|
|
448 |
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
|
|
|
449 |
map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
|
|
|
450 |
using first_elem_seqder1 rders_simp_append by auto
|
|
|
451 |
|
|
|
452 |
|
|
|
453 |
|
|
|
454 |
|
|
|
455 |
|
|
|
456 |
lemma seq_update_seq_ders:
|
|
|
457 |
shows "rsimp (rder c ( rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
|
|
|
458 |
(map (rders_simp r2) Ss))))) =
|
|
|
459 |
rsimp (RALTS ((RSEQ (rders_simp r1 (s @ [c])) r2) #
|
|
|
460 |
(map (rders_simp r2) (seq_update c (rders_simp r1 s) Ss)))) "
|
|
|
461 |
sorry
|
|
|
462 |
|
|
|
463 |
lemma seq_ders_closed_form1:
|
|
|
464 |
shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
|
|
|
465 |
(map ( rders_simp r2 ) Ss)))"
|
|
|
466 |
apply(case_tac "rnullable r1")
|
|
|
467 |
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
|
|
|
468 |
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
|
|
|
469 |
prefer 2
|
|
|
470 |
apply (simp add: rsimp_idem)
|
|
|
471 |
apply(rule_tac x = "[[c]]" in exI)
|
|
|
472 |
apply simp
|
|
|
473 |
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
|
|
|
474 |
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
|
|
|
475 |
apply blast
|
|
|
476 |
apply(simp add: rsimp_idem)
|
|
|
477 |
sorry
|
|
|
478 |
|
|
|
479 |
|
|
|
480 |
|
|
|
481 |
|
|
|
482 |
|
|
|
483 |
|
|
|
484 |
|
|
|
485 |
|
|
|
486 |
lemma simp_flatten2:
|
|
|
487 |
shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
|
|
|
488 |
sorry
|
|
|
489 |
|
|
|
490 |
|
|
|
491 |
lemma simp_flatten:
|
|
|
492 |
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
|
|
|
493 |
|
|
|
494 |
sorry
|
|
|
495 |
|
|
|
496 |
|
|
|
497 |
|
|
|
498 |
(*^^^^^^^^^nullable_seq_with_list1 related ^^^^^^^^^^^^^^^^*)
|
|
|
499 |
|
|
|
500 |
|
|
|
501 |
|
|
|
502 |
|
|
|
503 |
|
|
|
504 |
|
|
|
505 |
|
|
|
506 |
|
|
|
507 |
|
|
|
508 |
|
|
|
509 |
|
|
|
510 |
lemma non_zero_size:
|
|
|
511 |
shows "rsize r \<ge> Suc 0"
|
|
|
512 |
apply(induct r)
|
|
|
513 |
apply auto done
|
|
|
514 |
|
|
|
515 |
corollary size_geq1:
|
|
|
516 |
shows "rsize r \<ge> 1"
|
|
|
517 |
by (simp add: non_zero_size)
|
|
|
518 |
|
|
|
519 |
|
|
|
520 |
lemma rexp_size_induct:
|
|
|
521 |
shows "\<And>N r x5 a list.
|
|
|
522 |
\<lbrakk> rsize r = Suc N; r = RALTS x5;
|
|
|
523 |
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"
|
|
|
524 |
apply(rule_tac x = "rsize a" in exI)
|
|
|
525 |
apply(rule_tac x = "rsize (RALTS list)" in exI)
|
|
|
526 |
apply(subgoal_tac "rsize a \<ge> 1")
|
|
|
527 |
prefer 2
|
|
|
528 |
using One_nat_def non_zero_size apply presburger
|
|
|
529 |
apply(subgoal_tac "rsize (RALTS list) \<ge> 1 ")
|
|
|
530 |
prefer 2
|
|
|
531 |
using size_geq1 apply blast
|
|
|
532 |
apply simp
|
|
|
533 |
done
|
|
|
534 |
|
|
|
535 |
definition SEQ_set where
|
|
|
536 |
"SEQ_set A n \<equiv> {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A \<and> rsize r1 + rsize r2 \<le> n}"
|
|
|
537 |
|
|
|
538 |
definition SEQ_set_cartesian where
|
|
|
539 |
"SEQ_set_cartesian A n = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"
|
|
|
540 |
|
|
|
541 |
definition ALT_set where
|
|
|
542 |
"ALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> sum_list (map rsize rs) \<le> n}"
|
|
|
543 |
|
|
|
544 |
|
|
|
545 |
definition
|
|
|
546 |
"sizeNregex N \<equiv> {r. rsize r \<le> N}"
|
|
|
547 |
|
|
|
548 |
lemma sizenregex_induct:
|
|
|
549 |
shows "sizeNregex (Suc n) = sizeNregex n \<union> {RZERO, RONE, RALTS []} \<union> {RCHAR c| c. True} \<union>
|
|
|
550 |
SEQ_set ( sizeNregex n) n \<union> ALT_set (sizeNregex n) n \<union> (RSTAR ` (sizeNregex n))"
|
|
|
551 |
sorry
|
|
|
552 |
|
|
|
553 |
|
|
|
554 |
lemma chars_finite:
|
|
|
555 |
shows "finite (RCHAR ` (UNIV::(char set)))"
|
|
|
556 |
apply(simp)
|
|
|
557 |
done
|
|
|
558 |
|
|
|
559 |
thm full_SetCompr_eq
|
|
|
560 |
|
|
|
561 |
lemma size1finite:
|
|
|
562 |
shows "finite (sizeNregex (Suc 0))"
|
|
|
563 |
apply(subst sizenregex_induct)
|
|
|
564 |
apply(subst finite_Un)+
|
|
|
565 |
apply(subgoal_tac "sizeNregex 0 = {}")
|
|
|
566 |
apply(rule conjI)+
|
|
|
567 |
apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
|
|
|
568 |
apply simp
|
|
|
569 |
apply (simp add: full_SetCompr_eq)
|
|
|
570 |
apply (simp add: SEQ_set_def)
|
|
|
571 |
apply (simp add: ALT_set_def)
|
|
|
572 |
apply(simp add: full_SetCompr_eq)
|
|
|
573 |
using non_zero_size not_less_eq_eq sizeNregex_def by fastforce
|
|
|
574 |
|
|
|
575 |
lemma seq_included_in_cart:
|
|
|
576 |
shows "SEQ_set A n \<subseteq> SEQ_set_cartesian A n"
|
|
|
577 |
using SEQ_set_cartesian_def SEQ_set_def by fastforce
|
|
|
578 |
|
|
|
579 |
lemma finite_seq:
|
|
|
580 |
shows " finite (sizeNregex n) \<Longrightarrow> finite (SEQ_set (sizeNregex n) n)"
|
|
|
581 |
apply(rule finite_subset)
|
|
|
582 |
sorry
|
|
|
583 |
|
|
|
584 |
|
|
|
585 |
lemma finite_size_n:
|
|
|
586 |
shows "finite (sizeNregex n)"
|
|
|
587 |
apply(induct n)
|
|
|
588 |
apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
|
|
|
589 |
apply(subst sizenregex_induct)
|
|
|
590 |
apply(subst finite_Un)+
|
|
|
591 |
apply(rule conjI)+
|
|
|
592 |
apply simp
|
|
|
593 |
apply simp
|
|
|
594 |
apply (simp add: full_SetCompr_eq)
|
|
|
595 |
|
|
|
596 |
sorry
|
|
|
597 |
|
|
|
598 |
|
|
|
599 |
|
|
|
600 |
|
|
|
601 |
|
|
|
602 |
|
|
|
603 |
|
|
|
604 |
|
|
|
605 |
|
|
|
606 |
|
|
|
607 |
|
|
|
608 |
|
|
|
609 |
|
|
|
610 |
|
|
|
611 |
|
|
|
612 |
|
|
|
613 |
|
|
|
614 |
|
|
|
615 |
|
|
|
616 |
|
|
|
617 |
|
|
|
618 |
lemma star_update_case1:
|
|
|
619 |
shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
|
|
|
620 |
|
|
|
621 |
by force
|
|
|
622 |
|
|
|
623 |
lemma star_update_case2:
|
|
|
624 |
shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
|
|
|
625 |
by simp
|
|
|
626 |
|
|
|
627 |
lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
|
|
|
628 |
apply(case_tac r)
|
|
|
629 |
apply simp+
|
|
|
630 |
done
|
|
|
631 |
|
|
|
632 |
lemma rsimp_alts_idem_aux1:
|
|
|
633 |
shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
|
|
|
634 |
by force
|
|
|
635 |
|
|
|
636 |
|
|
|
637 |
|
|
|
638 |
lemma rsimp_alts_idem_aux2:
|
|
|
639 |
shows "rsimp a = rsimp (RALTS [a])"
|
|
|
640 |
apply(simp)
|
|
|
641 |
apply(case_tac "rsimp a")
|
|
|
642 |
apply simp+
|
|
|
643 |
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
|
|
|
644 |
by simp
|
|
|
645 |
|
|
|
646 |
lemma rsimp_alts_idem:
|
|
|
647 |
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
|
|
|
648 |
apply(induct as)
|
|
|
649 |
apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
|
|
|
650 |
prefer 2
|
|
|
651 |
apply simp
|
|
|
652 |
using bubble_break rsimp_alts_idem_aux2 apply auto[1]
|
|
|
653 |
apply(case_tac as)
|
|
|
654 |
apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
|
|
|
655 |
prefer 2
|
|
|
656 |
apply simp
|
|
|
657 |
using head_one_more_simp apply fastforce
|
|
|
658 |
apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
|
|
|
659 |
prefer 2
|
|
|
660 |
|
|
|
661 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
662 |
|
|
|
663 |
apply(simp only:)
|
|
|
664 |
apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
|
|
|
665 |
prefer 2
|
|
|
666 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
667 |
apply(simp only:)
|
|
|
668 |
apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
|
|
|
669 |
prefer 2
|
|
|
670 |
|
|
|
671 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
672 |
apply(simp only:)
|
|
|
673 |
using simp_flatten2
|
|
|
674 |
apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list)))) = rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
|
|
|
675 |
prefer 2
|
|
|
676 |
|
|
|
677 |
apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
|
|
|
678 |
apply (simp only:)
|
|
|
679 |
done
|
|
|
680 |
|
|
|
681 |
|
|
|
682 |
lemma rsimp_alts_idem2:
|
|
|
683 |
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
|
|
|
684 |
using head_one_more_simp rsimp_alts_idem by auto
|
|
|
685 |
|
|
|
686 |
|
|
|
687 |
lemma evolution_step1:
|
|
|
688 |
shows "rsimp
|
|
|
689 |
(rsimp_ALTs
|
|
|
690 |
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
691 |
rsimp
|
|
|
692 |
(rsimp_ALTs
|
|
|
693 |
(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)))])) "
|
|
|
694 |
using rsimp_alts_idem by auto
|
|
|
695 |
|
|
|
696 |
lemma evolution_step2:
|
|
|
697 |
assumes " rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
698 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
|
|
|
699 |
shows "rsimp
|
|
|
700 |
(rsimp_ALTs
|
|
|
701 |
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
702 |
rsimp
|
|
|
703 |
(rsimp_ALTs
|
|
|
704 |
(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)))])) "
|
|
|
705 |
by (simp add: assms rsimp_alts_idem)
|
|
|
706 |
|
|
|
707 |
lemma rsimp_seq_aux1:
|
|
|
708 |
shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
|
|
|
709 |
apply simp
|
|
|
710 |
done
|
|
|
711 |
|
|
|
712 |
lemma multiple_alts_simp_flatten:
|
|
|
713 |
shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
|
|
|
714 |
by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
|
|
|
715 |
|
|
|
716 |
|
|
|
717 |
lemma evo3_main_aux1:
|
|
|
718 |
shows "rsimp
|
|
|
719 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
720 |
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
|
|
|
721 |
rsimp
|
|
|
722 |
(RALTS
|
|
|
723 |
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
724 |
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
|
|
|
725 |
apply(subgoal_tac "rsimp
|
|
|
726 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
727 |
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
|
|
|
728 |
rsimp
|
|
|
729 |
(RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
730 |
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
|
|
|
731 |
prefer 2
|
|
|
732 |
apply (simp add: rsimp_idem)
|
|
|
733 |
apply (simp only:)
|
|
|
734 |
apply(subst multiple_alts_simp_flatten)
|
|
|
735 |
by simp
|
|
|
736 |
|
|
|
737 |
|
|
|
738 |
lemma evo3_main_nullable:
|
|
|
739 |
shows "
|
|
|
740 |
\<And>a Ss.
|
|
|
741 |
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
742 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
|
|
|
743 |
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
|
|
|
744 |
\<Longrightarrow> rsimp
|
|
|
745 |
(rsimp_ALTs
|
|
|
746 |
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
|
|
|
747 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
|
|
|
748 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
|
|
|
749 |
apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r))
|
|
|
750 |
= RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
|
|
|
751 |
prefer 2
|
|
|
752 |
apply simp
|
|
|
753 |
apply(simp only:)
|
|
|
754 |
apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
|
|
|
755 |
prefer 2
|
|
|
756 |
using star_update_case1 apply presburger
|
|
|
757 |
apply(simp only:)
|
|
|
758 |
apply(subst List.list.map(2))+
|
|
|
759 |
apply(subgoal_tac "rsimp
|
|
|
760 |
(rsimp_ALTs
|
|
|
761 |
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
|
|
|
762 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
|
|
|
763 |
rsimp
|
|
|
764 |
(RALTS
|
|
|
765 |
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
|
|
|
766 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
|
|
|
767 |
prefer 2
|
|
|
768 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
769 |
apply(simp only:)
|
|
|
770 |
apply(subgoal_tac " rsimp
|
|
|
771 |
(rsimp_ALTs
|
|
|
772 |
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
773 |
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))
|
|
|
774 |
=
|
|
|
775 |
rsimp
|
|
|
776 |
(RALTS
|
|
|
777 |
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
778 |
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
|
|
|
779 |
|
|
|
780 |
prefer 2
|
|
|
781 |
using rsimp_ALTs.simps(3) apply presburger
|
|
|
782 |
apply (simp only:)
|
|
|
783 |
apply(subgoal_tac " rsimp
|
|
|
784 |
(RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
|
|
|
785 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
|
|
|
786 |
rsimp
|
|
|
787 |
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
|
|
|
788 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
|
|
|
789 |
prefer 2
|
|
|
790 |
apply (simp add: rsimp_idem)
|
|
|
791 |
apply(simp only:)
|
|
|
792 |
apply(subgoal_tac " rsimp
|
|
|
793 |
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
|
|
|
794 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
|
|
|
795 |
rsimp
|
|
|
796 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
797 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
|
|
|
798 |
prefer 2
|
|
|
799 |
using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
|
|
|
800 |
apply(simp only:)
|
|
|
801 |
apply(subgoal_tac " rsimp
|
|
|
802 |
(RALTS
|
|
|
803 |
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
804 |
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) =
|
|
|
805 |
rsimp
|
|
|
806 |
(RALTS
|
|
|
807 |
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
|
|
|
808 |
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
|
|
|
809 |
prefer 2
|
|
|
810 |
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)
|
|
|
811 |
apply(simp only:)
|
|
|
812 |
apply(subgoal_tac " rsimp
|
|
|
813 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
814 |
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
|
|
|
815 |
rsimp
|
|
|
816 |
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
|
|
|
817 |
( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) ")
|
|
|
818 |
prefer 2
|
|
|
819 |
using rsimp_idem apply force
|
|
|
820 |
apply(simp only:)
|
|
|
821 |
using evo3_main_aux1 by blast
|
|
|
822 |
|
|
|
823 |
|
|
|
824 |
lemma evo3_main_not1:
|
|
|
825 |
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)"
|
|
|
826 |
by fastforce
|
|
|
827 |
|
|
|
828 |
|
|
|
829 |
lemma evo3_main_not2:
|
|
|
830 |
shows "\<not>rnullable (rders_simp r a) \<Longrightarrow> rsimp
|
|
|
831 |
(rsimp_ALTs
|
|
|
832 |
(rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
|
|
|
833 |
(rsimp_ALTs
|
|
|
834 |
((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
|
|
|
835 |
by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
|
|
|
836 |
|
|
|
837 |
lemma evo3_main_not3:
|
|
|
838 |
shows "rsimp
|
|
|
839 |
(rsimp_ALTs
|
|
|
840 |
(rsimp_SEQ r1 (RSTAR r) # rs)) =
|
|
|
841 |
rsimp (rsimp_ALTs
|
|
|
842 |
(RSEQ r1 (RSTAR r) # rs))"
|
|
|
843 |
by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
|
|
|
844 |
|
|
|
845 |
|
|
|
846 |
lemma evo3_main_notnullable:
|
|
|
847 |
shows "\<And>a Ss.
|
|
|
848 |
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
849 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
|
|
|
850 |
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
|
|
|
851 |
\<Longrightarrow> rsimp
|
|
|
852 |
(rsimp_ALTs
|
|
|
853 |
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
|
|
|
854 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
|
|
|
855 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
|
|
|
856 |
apply(subst star_update_case2)
|
|
|
857 |
apply simp
|
|
|
858 |
apply(subst List.list.map(2))
|
|
|
859 |
apply(subst evo3_main_not2)
|
|
|
860 |
apply simp
|
|
|
861 |
apply(subst evo3_main_not3)
|
|
|
862 |
using rsimp_alts_idem by presburger
|
|
|
863 |
|
|
|
864 |
|
|
|
865 |
lemma evo3_aux2:
|
|
|
866 |
shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
|
|
|
867 |
by simp
|
|
|
868 |
lemma evo3_aux3:
|
|
|
869 |
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
|
|
|
870 |
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)
|
|
|
871 |
|
|
|
872 |
lemma evo3_aux4:
|
|
|
873 |
shows " rsimp
|
|
|
874 |
(rsimp_ALTs
|
|
|
875 |
[RSEQ (rder x r) (RSTAR r),
|
|
|
876 |
rsimp (rsimp_ALTs rs)]) =
|
|
|
877 |
rsimp
|
|
|
878 |
(rsimp_ALTs
|
|
|
879 |
(rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
|
|
|
880 |
by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
|
|
|
881 |
|
|
|
882 |
lemma evo3_aux5:
|
|
|
883 |
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)"
|
|
|
884 |
using idiot2 by blast
|
|
|
885 |
|
|
|
886 |
|
|
|
887 |
lemma evolution_step3:
|
|
|
888 |
shows" \<And>a Ss.
|
|
|
889 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
|
|
|
890 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) \<Longrightarrow>
|
|
|
891 |
rsimp
|
|
|
892 |
(rsimp_ALTs
|
|
|
893 |
[rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
|
|
|
894 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
|
|
|
895 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
|
|
|
896 |
apply(case_tac "rders_simp r a = RONE")
|
|
|
897 |
apply(subst rsimp_seq_aux1)
|
|
|
898 |
apply simp
|
|
|
899 |
apply(subst rder.simps(6))
|
|
|
900 |
apply(subgoal_tac "rnullable (rders_simp r a)")
|
|
|
901 |
prefer 2
|
|
|
902 |
using rnullable.simps(2) apply presburger
|
|
|
903 |
apply(subst star_update_case1)
|
|
|
904 |
apply simp
|
|
|
905 |
|
|
|
906 |
apply(subst List.list.map)+
|
|
|
907 |
apply(subst rders_simp_append)
|
|
|
908 |
apply(subst evo3_aux2)
|
|
|
909 |
apply simp
|
|
|
910 |
apply(subst evo3_aux3)
|
|
|
911 |
apply(subst evo3_aux4)
|
|
|
912 |
apply simp
|
|
|
913 |
apply(case_tac "rders_simp r a = RZERO")
|
|
|
914 |
|
|
|
915 |
apply (simp add: rsimp_alts_idem2)
|
|
|
916 |
apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
|
|
|
917 |
prefer 2
|
|
|
918 |
using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
|
|
|
919 |
using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
|
|
|
920 |
apply(subst evo3_aux5)
|
|
|
921 |
apply simp
|
|
|
922 |
apply(case_tac "rnullable (rders_simp r a) ")
|
|
|
923 |
using evo3_main_nullable apply blast
|
|
|
924 |
using evo3_main_notnullable apply blast
|
|
|
925 |
done
|
|
|
926 |
|
|
|
927 |
(*
|
|
|
928 |
proof (prove)
|
|
|
929 |
goal (1 subgoal):
|
|
|
930 |
1. map f (a # s) = f a # map f s
|
|
|
931 |
Auto solve_direct: the current goal can be solved directly with
|
|
|
932 |
HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
|
|
|
933 |
List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
|
|
|
934 |
List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
|
|
|
935 |
*)
|
|
|
936 |
lemma starseq_list_evolution:
|
|
|
937 |
fixes r :: rrexp and Ss :: "char list list" and x :: char
|
|
|
938 |
shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
|
|
|
939 |
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)) )"
|
|
|
940 |
apply(induct Ss)
|
|
|
941 |
apply simp
|
|
|
942 |
apply(subst List.list.map(2))
|
|
|
943 |
apply(subst evolution_step2)
|
|
|
944 |
apply simp
|
|
|
945 |
|
|
|
946 |
|
|
|
947 |
sorry
|
|
|
948 |
|
|
|
949 |
|
|
|
950 |
lemma star_seqs_produce_star_seqs:
|
|
|
951 |
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
|
|
|
952 |
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
|
|
|
953 |
by (meson comp_apply)
|
|
|
954 |
|
|
|
955 |
lemma map_der_lambda_composition:
|
|
|
956 |
shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
|
|
|
957 |
by force
|
|
|
958 |
|
|
|
959 |
lemma ralts_vs_rsimpalts:
|
|
|
960 |
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
|
|
|
961 |
by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
|
|
|
962 |
|
|
|
963 |
|
|
|
964 |
lemma linearity_of_list_of_star_or_starseqs:
|
|
|
965 |
fixes r::rrexp and Ss::"char list list" and x::char
|
|
|
966 |
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
|
|
|
967 |
rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
|
|
|
968 |
apply(subst rder_rsimp_ALTs_commute)
|
|
|
969 |
apply(subst map_der_lambda_composition)
|
|
|
970 |
using starseq_list_evolution
|
|
|
971 |
apply(rule_tac x = "star_update x r Ss" in exI)
|
|
|
972 |
apply(subst ralts_vs_rsimpalts)
|
|
|
973 |
by simp
|
|
|
974 |
|
|
|
975 |
|
|
|
976 |
|
|
|
977 |
(*certified correctness---does not depend on any previous sorry*)
|
|
|
978 |
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));
|
|
|
979 |
xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
|
|
|
980 |
\<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) =
|
|
|
981 |
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
|
|
|
982 |
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
|
|
|
983 |
prefer 2
|
|
|
984 |
apply blast
|
|
|
985 |
apply(erule exE)
|
|
|
986 |
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))))")
|
|
|
987 |
prefer 2
|
|
|
988 |
using rders_simp_append
|
|
|
989 |
using rders_simp_one_char apply presburger
|
|
|
990 |
apply(rule_tac x= "Ss" in exI)
|
|
|
991 |
apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
|
|
|
992 |
rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
|
|
|
993 |
prefer 2
|
|
|
994 |
using inside_simp_removal rsimp_idem apply presburger
|
|
|
995 |
apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
|
|
|
996 |
rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
|
|
|
997 |
prefer 2
|
|
|
998 |
using rder.simps(4) apply presburger
|
|
|
999 |
apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
|
|
|
1000 |
rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
|
|
|
1001 |
apply (metis rsimp_idem)
|
|
|
1002 |
by (metis map_der_lambda_composition)
|
|
|
1003 |
|
|
|
1004 |
|
|
|
1005 |
|
|
|
1006 |
end
|