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)))"
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 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
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 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)"
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 ralts_cap_mono:
shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "
by simp
lemma rflts_def_idiot:
shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
\<Longrightarrow> rflts (a # rs) = a # rflts rs"
apply(case_tac a)
apply simp_all
done
lemma rflts_mono:
shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize rs)"
apply(induct rs)
apply simp
apply(case_tac "a = RZERO")
apply simp
apply(case_tac "\<exists>rs1. a = RALTS rs1")
apply(erule exE)
apply simp
apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
prefer 2
using rflts_def_idiot apply blast
apply simp
done
lemma rdistinct_smaller: shows "sum_list (map rsize (rdistinct rs ss)) \<le>
sum_list (map rsize rs )"
apply (induct rs arbitrary: ss)
apply simp
by (simp add: trans_le_add2)
lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
by (simp add: rdistinct_smaller)
lemma rsimp_alts_mono :
shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa) \<Longrightarrow>
rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (sum_list (map rsize x))"
apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} ))
\<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
prefer 2
using rsimp_aalts_smaller apply auto[1]
apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")
prefer 2
using ralts_cap_mono apply blast
apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
sum_list (map rsize ( (rflts (map rsimp x))))")
prefer 2
using rdistinct_smaller apply presburger
apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le>
sum_list (map rsize (map rsimp x))")
prefer 2
using rflts_mono apply blast
apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize x)")
prefer 2
apply (simp add: sum_list_mono)
by linarith
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
using rsimp_alts_mono by auto
lemma idiot:
shows "rsimp_SEQ RONE r = r"
apply(case_tac r)
apply simp_all
done
lemma no_alt_short_list_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
sorry
lemma no_further_dB_after_simp:
shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = 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
corollary rsimp_inner_idem1:
shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
sorry
corollary rsimp_inner_idem2:
shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
sorry
corollary rsimp_inner_idem3:
shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
by (meson map_idI rsimp_inner_idem2)
corollary rsimp_inner_idem4:
shows "rsimp r = RALTS rs \<Longrightarrow> flts rs = rs"
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_flatten2:
shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
sorry
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])) ) )"
using RHS1 LHS1_def_der_seq cond_list_head_last simp_flatten
by (metis append.left_neutral append_Cons)
(*^^^^^^^^^nullable_seq_with_list1 related ^^^^^^^^^^^^^^^^*)
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)
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
(*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)
by (metis add.assoc less_Suc_eq less_max_iff_disj 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 (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))"
apply simp
apply(rule_tac x = "[[c]]" in exI)
apply auto
apply(case_tac "rsimp (rder c r)")
apply simp
apply auto
apply(subgoal_tac "rsimp (RSEQ x41 x42) = RSEQ x41 x42")
prefer 2
apply (metis rsimp_idem)
apply(subgoal_tac "rsimp x41 = x41")
prefer 2
using rsimp_inner_idem1 apply blast
apply(subgoal_tac "rsimp x42 = x42")
prefer 2
using rsimp_inner_idem1 apply blast
apply simp
apply(subgoal_tac "map rsimp x5 = x5")
prefer 2
using rsimp_inner_idem3 apply blast
apply simp
apply(subgoal_tac "rflts x5 = x5")
prefer 2
using rsimp_inner_idem4 apply blast
apply simp
using rsimp_inner_idem4 by auto
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
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@[c]) # (star_update c r Ss) )"
lemma star_update_case1:
shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
by force
lemma star_update_case2:
shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
by simp
lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
apply(case_tac r)
apply simp+
done
lemma rsimp_alts_idem_aux1:
shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
by force
lemma rsimp_alts_idem_aux2:
shows "rsimp a = rsimp (RALTS [a])"
apply(simp)
apply(case_tac "rsimp a")
apply simp+
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
by simp
lemma rsimp_alts_idem:
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
apply(induct as)
apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
prefer 2
apply simp
using bubble_break rsimp_alts_idem_aux2 apply auto[1]
apply(case_tac as)
apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
prefer 2
apply simp
using head_one_more_simp apply fastforce
apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
using simp_flatten2
apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list)))) = rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
prefer 2
apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
apply (simp only:)
done
lemma rsimp_alts_idem2:
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
using head_one_more_simp rsimp_alts_idem by auto
lemma evolution_step1:
shows "rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [(rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)))])) "
using rsimp_alts_idem by auto
lemma evolution_step2:
assumes " rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
shows "rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [ rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])) "
by (simp add: assms rsimp_alts_idem)
lemma rsimp_seq_aux1:
shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
apply simp
done
lemma multiple_alts_simp_flatten:
shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
lemma evo3_main_aux1:
shows "rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
rsimp
(RALTS
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
apply(subgoal_tac "rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
rsimp
(RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
prefer 2
apply (simp add: rsimp_idem)
apply (simp only:)
apply(subst multiple_alts_simp_flatten)
by simp
lemma evo3_main_nullable:
shows "
\<And>a Ss.
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
\<Longrightarrow> rsimp
(rsimp_ALTs
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r))
= RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
prefer 2
apply simp
apply(simp only:)
apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
prefer 2
using star_update_case1 apply presburger
apply(simp only:)
apply(subst List.list.map(2))+
apply(subgoal_tac "rsimp
(rsimp_ALTs
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp
(RALTS
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac " rsimp
(rsimp_ALTs
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))
=
rsimp
(RALTS
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply (simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
prefer 2
apply (simp add: rsimp_idem)
apply(simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
prefer 2
using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
apply(simp only:)
apply(subgoal_tac " rsimp
(RALTS
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) =
rsimp
(RALTS
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
prefer 2
apply (smt (z3) idiot2 list.simps(9) rrexp.distinct(9) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_idem)
apply(simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) ")
prefer 2
using rsimp_idem apply force
apply(simp only:)
using evo3_main_aux1 by blast
lemma evo3_main_not1:
shows " \<not>rnullable (rders_simp r a) \<Longrightarrow> rder x (RSEQ (rders_simp r a) (RSTAR r)) = RSEQ (rder x (rders_simp r a)) (RSTAR r)"
by fastforce
lemma evo3_main_not2:
shows "\<not>rnullable (rders_simp r a) \<Longrightarrow> rsimp
(rsimp_ALTs
(rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
(rsimp_ALTs
((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
lemma evo3_main_not3:
shows "rsimp
(rsimp_ALTs
(rsimp_SEQ r1 (RSTAR r) # rs)) =
rsimp (rsimp_ALTs
(RSEQ r1 (RSTAR r) # rs))"
by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
lemma evo3_main_notnullable:
shows "\<And>a Ss.
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
\<Longrightarrow> rsimp
(rsimp_ALTs
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(subst star_update_case2)
apply simp
apply(subst List.list.map(2))
apply(subst evo3_main_not2)
apply simp
apply(subst evo3_main_not3)
using rsimp_alts_idem by presburger
lemma evo3_aux2:
shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
by simp
lemma evo3_aux3:
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
by (metis list.simps(8) list.simps(9) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem)
lemma evo3_aux4:
shows " rsimp
(rsimp_ALTs
[RSEQ (rder x r) (RSTAR r),
rsimp (rsimp_ALTs rs)]) =
rsimp
(rsimp_ALTs
(rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
lemma evo3_aux5:
shows "rders_simp r a \<noteq> RONE \<and> rders_simp r a \<noteq> RZERO \<Longrightarrow> rsimp_SEQ (rders_simp r a) (RSTAR r) = RSEQ (rders_simp r a) (RSTAR r)"
using idiot2 by blast
lemma evolution_step3:
shows" \<And>a Ss.
rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) \<Longrightarrow>
rsimp
(rsimp_ALTs
[rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(case_tac "rders_simp r a = RONE")
apply(subst rsimp_seq_aux1)
apply simp
apply(subst rder.simps(6))
apply(subgoal_tac "rnullable (rders_simp r a)")
prefer 2
using rnullable.simps(2) apply presburger
apply(subst star_update_case1)
apply simp
apply(subst List.list.map)+
apply(subst rders_simp_append)
apply(subst evo3_aux2)
apply simp
apply(subst evo3_aux3)
apply(subst evo3_aux4)
apply simp
apply(case_tac "rders_simp r a = RZERO")
apply (simp add: rsimp_alts_idem2)
apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
prefer 2
using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
apply(subst evo3_aux5)
apply simp
apply(case_tac "rnullable (rders_simp r a) ")
using evo3_main_nullable apply blast
using evo3_main_notnullable apply blast
done
(*
proof (prove)
goal (1 subgoal):
1. map f (a # s) = f a # map f s
Auto solve_direct: the current goal can be solved directly with
HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
*)
lemma starseq_list_evolution:
fixes r :: rrexp and Ss :: "char list list" and x :: char
shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)) )"
apply(induct Ss)
apply simp
apply(subst List.list.map(2))
apply(subst evolution_step2)
apply simp
sorry
lemma star_seqs_produce_star_seqs:
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
by (meson comp_apply)
lemma map_der_lambda_composition:
shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
by force
lemma ralts_vs_rsimpalts:
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
lemma linearity_of_list_of_star_or_starseqs:
fixes r::rrexp and Ss::"char list list" and x::char
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
apply(subst rder_rsimp_ALTs_commute)
apply(subst map_der_lambda_composition)
using starseq_list_evolution
apply(rule_tac x = "star_update x r Ss" in exI)
apply(subst ralts_vs_rsimpalts)
by simp
(*certified correctness---does not depend on any previous sorry*)
lemma star_list_push_der: shows " \<lbrakk>xs \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss));
xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
\<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) =
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
prefer 2
apply blast
apply(erule exE)
apply(subgoal_tac "rders_simp (RSTAR r) (xs @ [x]) = rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using rders_simp_append
using rders_simp_one_char apply presburger
apply(rule_tac x= "Ss" in exI)
apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using inside_simp_removal rsimp_idem apply presburger
apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using rder.simps(4) apply presburger
apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
apply (metis rsimp_idem)
by (metis map_der_lambda_composition)
lemma simp_in_lambdas :
shows "
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) ) =
rsimp (RALTS (map (\<lambda>s1. (rsimp (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))))) Ss))"
by (metis (no_types, lifting) comp_apply list.map_comp map_eq_conv rsimp.simps(2) rsimp_idem)
lemma starder_is_a_list_of_stars_or_starseqs:
shows "s \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) s = rsimp (RALTS( (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(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) (xs @ [x]) =
rsimp (RALTS (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))")
prefer 2
using star_list_push_der apply presburger
by (metis ralts_vs_rsimpalts starseq_list_evolution)
lemma starder_is_a_list:
shows " \<exists>Ss. rders_simp (RSTAR r) s = rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) \<or> rders_simp (RSTAR r) s = RSTAR r"
apply(case_tac s)
prefer 2
apply (metis neq_Nil_conv starder_is_a_list_of_stars_or_starseqs)
apply simp
done
(** start about bounds here**)
lemma list_simp_size:
shows "rlist_size (map rsimp rs) \<le> rlist_size rs"
apply(induct rs)
apply simp
apply simp
apply (subgoal_tac "rsize (rsimp a) \<le> rsize a")
prefer 2
using rsimp_mono apply fastforce
using add_le_mono by presburger
lemma inside_list_simp_inside_list:
shows "r \<in> set rs \<Longrightarrow> rsimp r \<in> set (map rsimp rs)"
apply (induct rs)
apply simp
apply auto
done
lemma rsize_star_seq_list:
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3"
sorry
lemma rdistinct_bound_by_no_simp:
shows "
rlist_size (rdistinct (map rsimp rs) (set (map rsimp ss)))
\<le> (rlist_size (rdistinct rs (set ss)))
"
apply(induct rs arbitrary: ss)
apply simp
apply(case_tac "a \<in> set ss")
apply(subgoal_tac "rsimp a \<in> set (map rsimp ss)")
prefer 2
using inside_list_simp_inside_list apply blast
apply simp
apply simp
by (metis List.set_insert add_le_mono image_insert insert_absorb rsimp_mono trans_le_add2)
lemma starder_closed_form_bound_aux1:
shows
"\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {}))) "
sorry
lemma starder_closed_form_bound:
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3"
apply(subgoal_tac " \<exists>N3.\<forall>Ss.
rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3")
prefer 2
using rsize_star_seq_list apply auto[1]
apply(erule exE)
apply(rule_tac x = "Suc N3" in exI)
apply(subgoal_tac "\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {})))")
prefer 2
using starder_closed_form_bound_aux1 apply blast
by (meson less_trans_Suc linorder_not_le not_less_eq)
thm starder_closed_form_bound_aux1
(*
"ralts_vs_rsimpalts", , and "starder_closed_form_bound_aux1", which could be due to a bug in Sledgehammer or to inconsistent axioms (including "sorry"s)
*)
lemma starder_size_bound:
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
rsize (RSTAR r0) < N3"
apply(subgoal_tac " \<exists>N3.\<forall>Ss.
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3")
prefer 2
using starder_closed_form_bound apply blast
apply(erule exE)
apply(rule_tac x = "max N3 (Suc (rsize (RSTAR r0)))" in exI)
using less_max_iff_disj 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"
apply(subgoal_tac " \<exists>N3. \<forall>Ss.
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
rsize (RSTAR r0) < N3")
prefer 2
using starder_size_bound apply blast
apply(erule exE)
apply(rule_tac x = N3 in exI)
by (metis starder_is_a_list)
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
lemma finite_list_of_ders:
fixes r
shows"\<exists>dersset. ( (finite dersset) \<and> (\<forall>s. (rders_simp r s) \<in> dersset) )"
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