theory SizeBound6CT
imports "Lexer" "PDerivs"
begin
section \<open>Bit-Encodings\<close>
fun orderedSufAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
where
"orderedSufAux (Suc i) ss = (drop i ss) # (orderedSufAux i ss)"
|"orderedSufAux 0 ss = Nil"
fun
orderedSuf :: "char list \<Rightarrow> char list list"
where
"orderedSuf s = orderedSufAux (length s) s"
fun orderedPrefAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
where
"orderedPrefAux (Suc i) ss = (take i ss) # (orderedPrefAux i ss)"
|"orderedPrefAux 0 ss = Nil"
fun orderedPref :: "char list \<Rightarrow> char list list"
where
"orderedPref s = orderedPrefAux (length s) s"
lemma shape_of_pref_1list:
shows "orderedPref [c] = [[]]"
apply auto
done
lemma shape_of_suf_1list:
shows "orderedSuf [c] = [[c]]"
by auto
lemma shape_of_suf_2list:
shows "orderedSuf [c2, c3] = [[c3], [c2,c3]]"
by auto
lemma shape_of_prf_2list:
shows "orderedPref [c1, c2] = [[c1], []]"
by auto
lemma shape_of_suf_3list:
shows "orderedSuf [c1, c2, c3] = [[c3], [c2, c3], [c1, c2, c3]]"
by auto
lemma throwing_elem_around:
shows "orderedSuf (s1 @ [a] @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf ( s1 @ [a]) ))"
and "orderedSuf (s1 @ [a] @ s) = (orderedSuf ([a] @ s) @ (map (\<lambda>s11. s11 @ ([a] @ s))) (orderedSuf s1) )"
sorry
lemma suf_cons:
shows "orderedSuf (s1 @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf s1))"
apply(induct s arbitrary: s1)
apply simp
apply(subgoal_tac "s1 @ a # s = (s1 @ [a]) @ s")
prefer 2
apply simp
apply(subgoal_tac "orderedSuf (s1 @ a # s) = orderedSuf ((s1 @ [a]) @ s)")
prefer 2
apply presburger
apply(drule_tac x="s1 @ [a]" in meta_spec)
sorry
lemma shape_of_prf_3list:
shows "orderedPref [c1, c2, c3] = [[c1, c2], [c1], []]"
by auto
fun zip_concat :: "char list list \<Rightarrow> char list list \<Rightarrow> char list list"
where
"zip_concat (s1#ss1) (s2#ss2) = (s1@s2) # (zip_concat ss1 ss2)"
| "zip_concat [] [] = []"
| "zip_concat [] (s2#ss2) = s2 # (zip_concat [] ss2)"
| "zip_concat (s1#ss1) [] = s1 # (zip_concat ss1 [])"
lemma compliment_pref_suf:
shows "zip_concat (orderedPref s) (orderedSuf s) = replicate (length s) s "
apply(induct s)
apply auto[1]
sorry
datatype rrexp =
RZERO
| RONE
| RCHAR char
| RSEQ rrexp rrexp
| RALTS "rrexp list"
| RSTAR rrexp
abbreviation
"RALT r1 r2 \<equiv> RALTS [r1, r2]"
fun
rnullable :: "rrexp \<Rightarrow> bool"
where
"rnullable (RZERO) = False"
| "rnullable (RONE ) = True"
| "rnullable (RCHAR c) = False"
| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
| "rnullable (RSTAR r) = True"
fun
rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
where
"rder c (RZERO) = RZERO"
| "rder c (RONE) = RZERO"
| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
| "rder c (RSEQ r1 r2) =
(if rnullable r1
then RALT (RSEQ (rder c r1) r2) (rder c r2)
else RSEQ (rder c r1) r2)"
| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
fun
rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
where
"rders r [] = r"
| "rders r (c#s) = rders (rder c r) s"
fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
where
"rdistinct [] acc = []"
| "rdistinct (x#xs) acc =
(if x \<in> acc then rdistinct xs acc
else x # (rdistinct xs ({x} \<union> acc)))"
lemma rdistinct_idem:
shows "rdistinct (x # (rdistinct rs {x})) {} = x # (rdistinct rs {x})"
sorry
fun rflts :: "rrexp list \<Rightarrow> rrexp list"
where
"rflts [] = []"
| "rflts (RZERO # rs) = rflts rs"
| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
| "rflts (r1 # rs) = r1 # rflts rs"
fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
where
"rsimp_ALTs [] = RZERO"
| "rsimp_ALTs [r] = r"
| "rsimp_ALTs rs = RALTS rs"
fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
where
"rsimp_SEQ RZERO _ = RZERO"
| "rsimp_SEQ _ RZERO = RZERO"
| "rsimp_SEQ RONE r2 = r2"
| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
fun rsimp :: "rrexp \<Rightarrow> rrexp"
where
"rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
| "rsimp r = r"
fun
rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
where
"rders_simp r [] = r"
| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
fun rsize :: "rrexp \<Rightarrow> nat" where
"rsize RZERO = 1"
| "rsize (RONE) = 1"
| "rsize (RCHAR c) = 1"
| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
| "rsize (RSTAR r) = Suc (rsize r)"
fun rlist_size :: "rrexp list \<Rightarrow> nat" where
"rlist_size (r # rs) = rsize r + rlist_size rs"
| "rlist_size [] = 0"
thm neq_Nil_conv
lemma rsimp_aalts_smaller:
shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
apply(induct rs)
apply simp
apply simp
apply(case_tac "rs = []")
apply simp
apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
apply(erule exE)+
apply simp
apply simp
by(meson neq_Nil_conv)
lemma finite_list_of_ders:
shows"\<exists>dersset. ( (finite dersset) \<and> (\<forall>s. (rders_simp r s) \<in> dersset) )"
sorry
(*
lemma rders_shape:
shows "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
rsimp (RALTS ((RSEQ (rders r1 s) r2) #
(map (rders r2) (orderedSuf s))) )"
apply(induct s arbitrary: r1 r2 rule: rev_induct)
apply simp
apply simp
sorry
*)
fun rders_cond_list :: "rrexp \<Rightarrow> bool list \<Rightarrow> char list list \<Rightarrow> rrexp list"
where
"rders_cond_list r2 (True # bs) (s # strs) = (rders_simp r2 s) # (rders_cond_list r2 bs strs)"
| "rders_cond_list r2 (False # bs) (s # strs) = rders_cond_list r2 bs strs"
| "rders_cond_list r2 [] s = []"
| "rders_cond_list r2 bs [] = []"
fun nullable_bools :: "rrexp \<Rightarrow> char list list \<Rightarrow> bool list"
where
"nullable_bools r (s#strs) = (rnullable (rders r s)) # (nullable_bools r strs) "
|"nullable_bools r [] = []"
fun cond_list :: "rrexp \<Rightarrow> rrexp \<Rightarrow> char list \<Rightarrow> rrexp list"
where
"cond_list r1 r2 s = rders_cond_list r2 (nullable_bools r1 (orderedPref s) ) (orderedSuf s)"
thm rsimp_SEQ.simps
lemma rSEQ_mono:
shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"
apply auto
apply(induct r1)
apply auto
apply(case_tac "r2")
apply simp_all
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
done
lemma rsimp_mono:
shows "rsize (rsimp r) \<le> rsize r"
apply(induct r)
apply simp_all
apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
apply force
using rSEQ_mono
apply presburger
sorry
lemma idiot:
shows "rsimp_SEQ RONE r = r"
apply(case_tac r)
apply simp_all
done
lemma no_dup_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> distinct rs"
sorry
lemma no_further_dB_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
sorry
lemma longlist_withstands_rsimp_alts:
shows "length rs \<ge> 2 \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
sorry
lemma no_alt_short_list_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
sorry
lemma idiot2:
shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
\<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
apply(case_tac r1)
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
apply(case_tac r2)
apply simp_all
done
lemma rsimp_aalts_another:
shows "\<forall>r \<in> (set (map rsimp ((RSEQ (rders r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) )) ). (rsize r) < N "
sorry
lemma shape_derssimpseq_onechar:
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
and "rders_simp (RSEQ r1 r2) [c] =
rsimp (RALTS ((RSEQ (rders r1 [c]) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
apply simp
apply(simp add: rders.simps)
apply(case_tac "rsimp (rder c r1) = RZERO")
apply auto
apply(case_tac "rsimp (rder c r1) = RONE")
apply auto
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp r2")
prefer 2
using idiot
apply simp
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp r2]) {})")
prefer 2
apply auto
apply(case_tac "rsimp r2")
apply auto
apply(subgoal_tac "rdistinct x5 {} = x5")
prefer 2
using no_further_dB_after_simp
apply metis
apply(subgoal_tac "rsimp_ALTs (rdistinct x5 {}) = rsimp_ALTs x5")
prefer 2
apply fastforce
apply auto
apply (metis no_alt_short_list_after_simp)
apply (case_tac "rsimp r2 = RZERO")
apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RZERO")
prefer 2
apply(case_tac "rsimp ( rder c r1)")
apply auto
apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RSEQ (rsimp (rder c r1)) (rsimp r2)")
prefer 2
apply auto
sorry
lemma shape_derssimpseq_onechar2:
shows "rders_simp (RSEQ r1 r2) [c] =
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
sorry
lemma rders__onechar:
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
by simp
lemma rders_append:
"rders c (s1 @ s2) = rders (rders c s1) s2"
apply(induct s1 arbitrary: c s2)
apply(simp_all)
done
lemma rders_simp_append:
"rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
apply(induct s1 arbitrary: c s2)
apply(simp_all)
done
lemma inside_simp_removal:
shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
sorry
lemma set_related_list:
shows "distinct rs \<Longrightarrow> length rs = card (set rs)"
by (simp add: distinct_card)
(*this section deals with the property of distinctBy: creates a list without duplicates*)
lemma rdistinct_never_added_twice:
shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
by force
lemma rdistinct_does_the_job:
shows "distinct (rdistinct rs s)"
apply(induct rs arbitrary: s)
apply simp
apply simp
sorry
lemma simp_helps_der_pierce:
shows " rsimp
(rder x
(rsimp_ALTs rs)) =
rsimp
(rsimp_ALTs
(map (rder x )
rs
)
)"
sorry
lemma simp_helps_der_pierce_dB:
shows " rsimp
(rsimp_ALTs
(map (rder x)
(rdistinct rs {}))) =
rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
sorry
lemma simp_helps_der_pierce_flts:
shows " rsimp
(rsimp_ALTs
(rdistinct
(map (rder x)
(rflts rs )
) {}
)
) =
rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}) )"
sorry
lemma unfold_ders_simp_inside_only:
shows " (rders_simp (RSEQ r1 r2) xs =
rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))
\<Longrightarrow> rsimp (rder x (rders_simp (RSEQ r1 r2) xs)) = rsimp (rder x (rsimp (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))))"
by presburger
lemma unfold_ders_simp_inside_only_nosimp:
shows " (rders_simp (RSEQ r1 r2) xs =
rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))
\<Longrightarrow> rsimp (rder x (rders_simp (RSEQ r1 r2) xs)) = rsimp (rder x (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))"
using inside_simp_removal by presburger
lemma rders_simp_one_char:
shows "rders_simp r [c] = rsimp (rder c r)"
apply auto
done
lemma rsimp_idem:
shows "rsimp (rsimp r) = rsimp r"
sorry
lemma head_one_more_simp:
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
by (simp add: rsimp_idem)
lemma head_one_more_dersimp:
shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
thm cond_list.simps
lemma suffix_plus1char:
shows "\<not> (rnullable (rders r1 s)) \<Longrightarrow> cond_list r1 r2 (s@[c]) = map (rder c) (cond_list r1 r2 s)"
apply simp
sorry
lemma suffix_plus1charn:
shows "rnullable (rders r1 s) \<Longrightarrow> cond_list r1 r2 (s@[c]) = (rder c r2) # (map (rder c) (cond_list r1 r2 s))"
sorry
lemma ders_simp_nullability:
shows "rnullable (rders r s) = rnullable (rders_simp r s)"
sorry
lemma first_elem_seqder:
shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
# rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
by auto
lemma first_elem_seqder1:
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
by (simp add: rsimp_idem)
lemma first_elem_seqdersimps:
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
using first_elem_seqder1 rders_simp_append by auto
lemma first_elem_seqder_nullable:
shows "rnullable (rders_simp r1 xs) \<Longrightarrow> cond_list r1 r2 (xs @ [x]) = rder x r2 # map (rder x) (cond_list r1 r2 xs)"
sorry
(*nullable_seq_with_list1 related *)
lemma LHS0_def_der_alt:
shows "rsimp (rder x (RALTS ( (RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) =
rsimp (RALTS ((rder x (RSEQ (rders_simp r1 xs) r2)) # (map (rder x) (cond_list r1 r2 xs))))"
by fastforce
lemma LHS1_def_der_seq:
shows "rnullable (rders_simp r1 xs) \<Longrightarrow>
rsimp (rder x (RALTS ( (RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) =
rsimp(RALTS ((RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) # [rder x r2]) ) # (map (rder x ) (cond_list r1 r2 xs))))"
by (simp add: rders_simp_append rsimp_idem)
lemma cond_list_head_last:
shows "rnullable (rders r1 s) \<Longrightarrow>
RALTS (r # (cond_list r1 r2 (s @ [c]))) = RALTS (r # ((rder c r2) # (map (rder c) (cond_list r1 r2 s))))"
using suffix_plus1charn by blast
lemma simp_flatten:
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
sorry
lemma RHS1:
shows "rnullable (rders_simp r1 xs) \<Longrightarrow>
rsimp (RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) #
(cond_list r1 r2 (xs @[x])))) =
rsimp (RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) #
( ((rder x r2) # (map (rder x) (cond_list r1 r2 xs)))))) "
using first_elem_seqder_nullable by presburger
lemma nullable_seq_with_list1:
shows " rnullable (rders_simp r1 s) \<Longrightarrow>
rsimp (rder c (RALTS ( (RSEQ (rders_simp r1 s) r2) # (cond_list r1 r2 s)) )) =
rsimp (RALTS ( (RSEQ (rders_simp r1 (s @ [c])) r2) # (cond_list r1 r2 (s @ [c])) ) )"
by (metis LHS1_def_der_seq append.left_neutral append_Cons first_elem_seqder_nullable simp_flatten)
lemma nullable_seq_with_list:
shows "rnullable (rders_simp r1 xs) \<Longrightarrow> rsimp (rder x (RALTS ((RSEQ (rders_simp r1 xs) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)) ))) =
rsimp (RALTS ((RSEQ (rders_simp r1 (xs@[x])) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref (xs@[x]))) (orderedSuf (xs@[x]))) ) )
"
apply(subgoal_tac "rsimp (rder x (RALTS ((RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) =
rsimp (RALTS ((RSEQ (rders_simp r1 (xs@[x])) r2) # (cond_list r1 r2 (xs@[x]))))")
apply auto[1]
using nullable_seq_with_list1 by auto
lemma r1r2ders_whole:
"rsimp
(RALTS
(rder x (RSEQ (rders_simp r1 xs) r2) #
map (rder x) (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))) =
rsimp( RALTS( ( (RSEQ (rders_simp r1 (xs@[x])) r2)
# (rders_cond_list r2 (nullable_bools r1 (orderedPref (xs @ [x]))) (orderedSuf (xs @ [x])))))) "
using ders_simp_nullability first_elem_seqdersimps nullable_seq_with_list1 suffix_plus1char by auto
lemma rders_simp_same_simpders:
shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
apply(induct s rule: rev_induct)
apply simp
apply(case_tac "xs = []")
apply simp
apply(simp add: rders_append rders_simp_append)
using inside_simp_removal by blast
lemma shape_derssimp_seq:
shows "\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )"
apply(induct s arbitrary: r1 r2 rule: rev_induct)
apply simp
apply(case_tac "xs = []")
using shape_derssimpseq_onechar2 apply force
apply(simp only: rders_simp_append)
apply(simp only: rders_simp_one_char)
apply(subgoal_tac "rsimp (rder x (rders_simp (RSEQ r1 r2) xs))
= rsimp (rder x (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))")
prefer 2
using unfold_ders_simp_inside_only_nosimp apply presburger
apply(subgoal_tac "rsimp (rder x (RALTS (RSEQ (rders_simp r1 xs) r2
# rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))) =
rsimp ( (RALTS (rder x (RSEQ (rders_simp r1 xs) r2)
# (map (rder x) (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))))
")
prefer 2
apply simp
using r1r2ders_whole rders_simp_append rders_simp_one_char by presburger
(*
apply(subgoal_tac " rsimp
(rder x
(rsimp_ALTs
(rdistinct
(rflts
(rsimp_SEQ (rsimp (rders_simp r1 xs)) (rsimp r2) #
map rsimp
(rders_cond_list r2 (nullable_bools r1 (orderedPrefAux (length xs) xs)) (orderedSufAux (length xs) xs))))
{}))) =
rsimp
(
(rsimp_ALTs
(map (rder x)
(rdistinct
(rflts
(rsimp_SEQ (rsimp (rders_simp r1 xs)) (rsimp r2) #
map rsimp
(rders_cond_list r2 (nullable_bools r1 (orderedPrefAux (length xs) xs)) (orderedSufAux (length xs) xs))))
{})
)
)
) ")
prefer 2
*)
lemma shape_derssimp_alts:
shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders r s) rs))"
apply(case_tac "s")
apply simp
apply simp
sorry
(*
fun rexp_encode :: "rrexp \<Rightarrow> nat"
where
"rexp_encode RZERO = 0"
|"rexp_encode RONE = 1"
|"rexp_encode (RCHAR c) = 2"
|"rexp_encode (RSEQ r1 r2) = ( 2 ^ (rexp_encode r1)) "
*)
lemma finite_chars:
shows " \<exists>N. ( (\<forall>r \<in> (set rs). \<exists>c. r = RCHAR c) \<and> (distinct rs) \<longrightarrow> length rs < N)"
apply(rule_tac x = "Suc 256" in exI)
sorry
definition all_chars :: "int \<Rightarrow> char list"
where "all_chars n = map char_of [0..n]"
(*
fun rexp_enum :: "nat \<Rightarrow> rrexp list"
where
"rexp_enum 0 = []"
|"rexp_enum (Suc 0) = RALTS [] # RZERO # (RONE # (map RCHAR (all_chars 255)))"
|"rexp_enum (Suc n) = [(RSEQ r1 r2). r1 \<in> set (rexp_enum i) \<and> r2 \<in> set (rexp_enum j) \<and> i + j = n]"
*)
fun rexp_enum :: "nat \<Rightarrow> rrexp set"
where
"rexp_enum 0 = {}"
|"rexp_enum (Suc 0) = {RALTS []} \<union> {RZERO} \<union> {RONE} \<union> { (RCHAR c) |c. c \<in> set (all_chars 255)}"
|"rexp_enum (Suc n) = {(RSEQ r1 r2)|r1 r2 i j. r1 \<in> (rexp_enum i) \<and> r2 \<in> (rexp_enum j) \<and> i + j = n}"
lemma finite_sized_rexp_forms_finite_set:
shows " \<exists>SN. ( \<forall>r.( rsize r < N \<longrightarrow> r \<in> SN)) \<and> (finite SN)"
apply(induct N)
apply simp
apply auto
(*\<lbrakk>\<forall>r. rsize r < N \<longrightarrow> r \<in> SN; finite SN\<rbrakk> \<Longrightarrow> \<exists>SN. (\<forall>r. rsize r < Suc N \<longrightarrow> r \<in> SN) \<and> finite SN*)
(* \<And>N. \<exists>SN. (\<forall>r. rsize r < N \<longrightarrow> r \<in> SN) \<and> finite SN \<Longrightarrow> \<exists>SN. (\<forall>r. rsize r < Suc N \<longrightarrow> r \<in> SN) \<and> finite SN*)
sorry
lemma finite_size_finite_regx:
shows " \<exists>l. \<forall>rs. ((\<forall>r \<in> (set rs). rsize r < N) \<and> (distinct rs) \<longrightarrow> (length rs) < l) "
sorry
(*below probably needs proved concurrently*)
lemma finite_r1r2_ders_list:
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
\<Longrightarrow> \<exists>l. \<forall>s.
(length (rdistinct (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) {}) ) < l "
sorry
(*
\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
rsimp (RALTS ((RSEQ (rders r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )
*)
lemma sum_list_size2:
shows "\<forall>z \<in>set rs. (rsize z ) \<le> Nr \<Longrightarrow> rlist_size rs \<le> (length rs) * Nr"
apply(induct rs)
apply simp
by simp
lemma sum_list_size:
fixes rs
shows " \<forall>r \<in> (set rs). (rsize r) \<le> Nr \<and> (length rs) \<le> l \<Longrightarrow> rlist_size rs \<le> l * Nr"
by (metis dual_order.trans mult.commute mult_le_mono2 mult_zero_right sum_list_size2 zero_le)
lemma seq_second_term_chain1:
shows " \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) \<le>
rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))"
sorry
lemma seq_second_term_chain2:
shows "\<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) =
rlist_size (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))"
oops
lemma seq_second_term_bounded:
fixes r2 r1
assumes "\<forall>s. rsize (rders_simp r2 s) < N2"
shows "\<exists>N3. \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) < N3"
sorry
lemma seq_first_term_bounded:
fixes r1 r2
shows "\<exists>Nr. \<forall>s. rsize (rders_simp r1 s) \<le> Nr \<Longrightarrow> \<exists>Nr'. \<forall>s. rsize (RSEQ (rders_simp r1 s) r2) \<le> Nr'"
apply(erule exE)
apply(rule_tac x = "Nr + (rsize r2) + 1" in exI)
by simp
lemma alts_triangle_inequality:
shows "rsize (RALTS (r # rs)) \<le> rsize r + rlist_size rs + 1 "
apply(subgoal_tac "rsize (RALTS (r # rs) ) = rsize r + rlist_size rs + 1")
apply auto[1]
apply(induct rs)
apply simp
apply auto
done
lemma seq_equal_term_nosimp_entire_bounded:
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
\<Longrightarrow> \<exists>N3. \<forall>s. rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rdistinct ((rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) ) {}) ) ) \<le> N3"
apply(subgoal_tac "\<exists>N3. \<forall>s. rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) ) \<le>
rsize (RSEQ (rders_simp r1 s) r2) +
rlist_size (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) + 1")
prefer 2
using alts_triangle_inequality apply presburger
using seq_first_term_bounded
using seq_second_term_bounded
apply(subgoal_tac "\<exists>N3. \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) < N3")
prefer 2
apply meson
apply(subgoal_tac "\<exists>Nr'. \<forall>s. rsize (RSEQ (rders_simp r1 s) r2) \<le> Nr'")
prefer 2
apply (meson order_le_less)
apply(erule exE)
apply(erule exE)
apply(erule exE)
apply(rule_tac x = "N3a + Nr'" in exI)
sorry
lemma alts_simp_bounded_by_sloppy1_version:
shows "\<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le>
rsize (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
)
{}
)
) "
sorry
lemma alts_simp_bounded_by_sloppy1:
shows "rsize (rsimp (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
)
{}
)
)) \<le>
rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
)
)"
sorry
lemma hand_made_def_rlist_size:
shows "rlist_size rs = sum_list (map rsize rs)"
proof (induct rs)
case Nil show ?case by simp
next
case (Cons a rs) thus ?case
by simp
qed
(*this section deals with the property of distinctBy: creates a list without duplicates*)
lemma distinct_mono:
shows "rlist_size (rdistinct (a # s) {}) \<le> rlist_size (a # (rdistinct s {}) )"
sorry
lemma distinct_acc_mono:
shows "A \<subseteq> B \<Longrightarrow> rlist_size (rdistinct s A) \<ge> rlist_size (rdistinct s B)"
apply(induct s arbitrary: A B)
apply simp
apply(case_tac "a \<in> A")
apply(subgoal_tac "a \<in> B")
apply simp
apply blast
apply(subgoal_tac "rlist_size (rdistinct (a # s) A) = rlist_size (a # (rdistinct s (A \<union> {a})))")
apply(case_tac "a \<in> B")
apply(subgoal_tac "rlist_size (rdistinct (a # s) B) = rlist_size ( (rdistinct s B))")
apply (metis Un_insert_right dual_order.trans insert_subset le_add_same_cancel2 rlist_size.simps(1) sup_bot_right zero_order(1))
apply simp
apply auto
by (meson insert_mono)
lemma distinct_mono2:
shows " rlist_size (rdistinct s {a}) \<le> rlist_size (rdistinct s {})"
apply(induct s)
apply simp
apply simp
using distinct_acc_mono by auto
lemma distinct_mono_spares_first_elem:
shows "rsize (RALTS (rdistinct (a # s) {})) \<le> rsize (RALTS (a # (rdistinct s {})))"
apply simp
apply (subgoal_tac "rlist_size ( (rdistinct s {a})) \<le> rlist_size ( (rdistinct s {})) ")
using hand_made_def_rlist_size apply auto[1]
using distinct_mono2 by auto
lemma sloppy1_bounded_by_sloppiest:
shows "rsize (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
)
{}
)
) \<le> rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {})
)
)"
sorry
lemma alts_simp_bounded_by_sloppiest_version:
shows "\<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le>
rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}) ) ) "
by (meson alts_simp_bounded_by_sloppy1_version order_trans sloppy1_bounded_by_sloppiest)
lemma seq_equal_term_entire_bounded:
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
\<Longrightarrow> \<exists>N3. \<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le> N3"
using seq_equal_term_nosimp_entire_bounded
apply(subgoal_tac " \<exists>N3. \<forall>s. rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}) ) ) \<le> N3")
apply(erule exE)
prefer 2
apply blast
apply(subgoal_tac "\<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le>
rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}) ) ) ")
prefer 2
using alts_simp_bounded_by_sloppiest_version apply blast
apply(rule_tac x = "Suc N3 " in exI)
apply(rule allI)
apply(subgoal_tac " rsize
(rsimp
(RALTS
(RSEQ (rders_simp r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))))
\<le> rsize
(RALTS
(RSEQ (rders_simp r1 s) r2 #
rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}))")
prefer 2
apply presburger
apply(subgoal_tac " rsize
(RALTS
(RSEQ (rders_simp r1 s) r2 #
rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {})) \<le> N3")
apply linarith
apply simp
done
lemma M1seq:
fixes r1 r2
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
\<Longrightarrow> \<exists>N3.\<forall>s.(rsize (rders_simp (RSEQ r1 r2) s)) < N3"
apply(frule seq_equal_term_entire_bounded)
apply(erule exE)
apply(rule_tac x = "max (N3+2) (Suc (Suc (rsize r1) + (rsize r2)))" in exI)
apply(rule allI)
apply(case_tac "s = []")
prefer 2
apply (metis add_2_eq_Suc' le_imp_less_Suc less_SucI max.strict_coboundedI1 shape_derssimp_seq(2))
by (metis add.assoc less_Suc_eq max.strict_coboundedI2 plus_1_eq_Suc rders_simp.simps(1) rsize.simps(5))
(* apply (simp add: less_SucI shape_derssimp_seq(2))
apply (meson less_SucI less_max_iff_disj)
apply simp
done*)
(*lemma empty_diff:
shows "s = [] \<Longrightarrow>
(rsize (rders_simp (RSEQ r1 r2) s)) \<le>
(max
(rsize (rsimp (RALTS (RSEQ (rders r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))))
(Suc (rsize r1 + rsize r2)) ) "
apply simp
done*)
(*For star related bound*)
lemma star_is_a_singleton_list_derc:
shows " \<exists>Ss. rders_simp (RSTAR r) [c] = rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)"
apply simp
apply(rule_tac x = "[[c]]" in exI)
apply auto
done
lemma rder_rsimp_ALTs_commute:
shows " (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
apply(induct rs)
apply simp
apply(case_tac rs)
apply simp
apply auto
done
lemma double_nested_ALTs_under_rsimp:
shows "rsimp (rsimp_ALTs ((RALTS rs1) # rs)) = rsimp (RALTS (rs1 @ rs))"
apply(case_tac rs1)
apply simp
apply (metis list.simps(8) list.simps(9) neq_Nil_conv rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3))
apply(case_tac rs)
apply simp
apply auto
sorry
fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list => char list list" where
"star_update c r [] = []"
|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s))
then (s@[c]) # [c] # (star_update c r Ss)
else s # (star_update c r Ss) )"
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
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 der_seqstar_res:
shows "rder x (RSEQ r1 r2) = RSEQ r3 r4"
oops
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_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)"
apply(simp add: rder_rsimp_ALTs_commute)
apply(induct Ss)
apply simp
apply (metis list.simps(8) rsimp_ALTs.simps(1))
sorry
lemma starder_is_a_list_of_stars_or_starseqs:
shows "s \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) s = rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)"
apply(induct s rule: rev_induct)
apply simp
apply(case_tac "xs = []")
using star_is_a_singleton_list_derc
apply(simp)
apply auto
apply(simp add: rders_simp_append)
using linearity_of_list_of_star_or_starseqs by blast
lemma finite_star:
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 )
\<Longrightarrow> \<exists>N3. \<forall>s.(rsize (rders_simp (RSTAR r0) s)) < N3"
sorry
lemma rderssimp_zero:
shows"rders_simp RZERO s = RZERO"
apply(induction s)
apply simp
by simp
lemma rderssimp_one:
shows"rders_simp RONE (a # s) = RZERO"
apply(induction s)
apply simp
by simp
lemma rderssimp_char:
shows "rders_simp (RCHAR c) s = RONE \<or> rders_simp (RCHAR c) s = RZERO \<or> rders_simp (RCHAR c) s = (RCHAR c)"
apply auto
by (metis rder.simps(2) rder.simps(3) rders_simp.elims rders_simp.simps(2) rderssimp_one rsimp.simps(4))
lemma finite_size_ders:
fixes r
shows " \<exists>Nr. \<forall>s. rsize (rders_simp r s) < Nr"
apply(induct r rule: rrexp.induct)
apply auto
apply(rule_tac x = "2" in exI)
using rderssimp_zero rsize.simps(1) apply presburger
apply(rule_tac x = "2" in exI)
apply (metis Suc_1 lessI rders_simp.elims rderssimp_one rsize.simps(1) rsize.simps(2))
apply(rule_tac x = "2" in meta_spec)
apply (metis lessI rderssimp_char rsize.simps(1) rsize.simps(2) rsize.simps(3))
using M1seq apply blast
prefer 2
apply (simp add: finite_star)
sorry
unused_thms
lemma seq_ders_shape:
shows "E"
oops
(*rsimp (rders (RSEQ r1 r2) s) =
rsimp RALT [RSEQ (rders r1 s) r2, rders r2 si, ...]
where si is the i-th shortest suffix of s such that si \<in> L r2"
*)
end