theory ClosedForms
imports "Lexer" "PDerivs"
begin
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"
fun vsuf :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list" where
"vsuf [] _ = []"
|"vsuf (c#cs) r1 = (if (rnullable r1) then (vsuf cs (rder c r1)) @ [c # cs]
else (vsuf cs (rder c r1))
) "
lemma seq_closed_form: shows
"rsimp (rders_simp (RSEQ r1 r2) s) =
rsimp ( RALTS ( (RSEQ (rders_simp r1 s) r2) #
(map (rders r2) (vsuf s r1))
)
)"
apply(induct s)
apply simp
sorry
fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> 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) )"
fun star_updates :: "char list \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
where
"star_updates [] r Ss = Ss"
| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
lemma star_closed_form:
shows "rders_simp (RSTAR r0) (c#s) =
rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates s r [[c]]) ) ))"
apply(induct s)
apply simp
sorry
lemma star_closed_form_bounded_by_rdistinct_list_estimate:
shows "rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
(star_updates s r [[c]]) ) ))) \<le>
Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
(star_updates s r [[c]]) ) {}) ) )"
sorry
lemma distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size:
shows "\<forall>r\<in> set rs. (rsize r ) \<le> N \<Longrightarrow> sum_list (map rsize (rdistinct rs {})) \<le>
(card (rexp_enum N))* N"
sorry
lemma ind_hypo_on_ders_leads_to_stars_bounded:
shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
(star_updates s r [[c]]) ) {}) ) ) \<le>
(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
"
sorry
lemma r0_bounded_star_bounded:
shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
\<forall>s. rsize (rders_simp (RSTAR r0) s) \<le>
(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))"
sorry
(*some basic facts about rsimp*)
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 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 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 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 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 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 simp_helps_der_pierce:
shows " rsimp
(rder x
(rsimp_ALTs rs)) =
rsimp
(rsimp_ALTs
(map (rder x )
rs
)
)"
sorry
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
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 seq_update_seq_ders:
shows "rsimp (rder c ( rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
(map (rders_simp r2) Ss))))) =
rsimp (RALTS ((RSEQ (rders_simp r1 (s @ [c])) r2) #
(map (rders_simp r2) (seq_update c (rders_simp r1 s) Ss)))) "
sorry
lemma seq_ders_closed_form1:
shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
(map ( rders_simp r2 ) Ss)))"
apply(case_tac "rnullable r1")
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
prefer 2
apply (simp add: rsimp_idem)
apply(rule_tac x = "[[c]]" in exI)
apply simp
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
apply blast
apply(simp add: rsimp_idem)
sorry
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
(*^^^^^^^^^nullable_seq_with_list1 related ^^^^^^^^^^^^^^^^*)
lemma non_zero_size:
shows "rsize r \<ge> Suc 0"
apply(induct r)
apply auto done
corollary size_geq1:
shows "rsize r \<ge> 1"
by (simp add: non_zero_size)
lemma rexp_size_induct:
shows "\<And>N r x5 a list.
\<lbrakk> rsize r = Suc N; r = RALTS x5;
x5 = a # list\<rbrakk> \<Longrightarrow>\<exists>i j. rsize a = i \<and> rsize (RALTS list) = j \<and> i + j = Suc N \<and> i \<le> N \<and> j \<le> N"
apply(rule_tac x = "rsize a" in exI)
apply(rule_tac x = "rsize (RALTS list)" in exI)
apply(subgoal_tac "rsize a \<ge> 1")
prefer 2
using One_nat_def non_zero_size apply presburger
apply(subgoal_tac "rsize (RALTS list) \<ge> 1 ")
prefer 2
using size_geq1 apply blast
apply simp
done
definition SEQ_set where
"SEQ_set A n \<equiv> {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A \<and> rsize r1 + rsize r2 \<le> n}"
definition SEQ_set_cartesian where
"SEQ_set_cartesian A n = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"
definition ALT_set where
"ALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> sum_list (map rsize rs) \<le> n}"
definition
"sizeNregex N \<equiv> {r. rsize r \<le> N}"
lemma sizenregex_induct:
shows "sizeNregex (Suc n) = sizeNregex n \<union> {RZERO, RONE, RALTS []} \<union> {RCHAR c| c. True} \<union>
SEQ_set ( sizeNregex n) n \<union> ALT_set (sizeNregex n) n \<union> (RSTAR ` (sizeNregex n))"
sorry
lemma chars_finite:
shows "finite (RCHAR ` (UNIV::(char set)))"
apply(simp)
done
thm full_SetCompr_eq
lemma size1finite:
shows "finite (sizeNregex (Suc 0))"
apply(subst sizenregex_induct)
apply(subst finite_Un)+
apply(subgoal_tac "sizeNregex 0 = {}")
apply(rule conjI)+
apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
apply simp
apply (simp add: full_SetCompr_eq)
apply (simp add: SEQ_set_def)
apply (simp add: ALT_set_def)
apply(simp add: full_SetCompr_eq)
using non_zero_size not_less_eq_eq sizeNregex_def by fastforce
lemma seq_included_in_cart:
shows "SEQ_set A n \<subseteq> SEQ_set_cartesian A n"
using SEQ_set_cartesian_def SEQ_set_def by fastforce
lemma finite_seq:
shows " finite (sizeNregex n) \<Longrightarrow> finite (SEQ_set (sizeNregex n) n)"
apply(rule finite_subset)
sorry
lemma finite_size_n:
shows "finite (sizeNregex n)"
apply(induct n)
apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
apply(subst sizenregex_induct)
apply(subst finite_Un)+
apply(rule conjI)+
apply simp
apply simp
apply (simp add: full_SetCompr_eq)
sorry
lemma star_update_case1:
shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
by force
lemma star_update_case2:
shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
by simp
lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
apply(case_tac r)
apply simp+
done
lemma rsimp_alts_idem_aux1:
shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
by force
lemma rsimp_alts_idem_aux2:
shows "rsimp a = rsimp (RALTS [a])"
apply(simp)
apply(case_tac "rsimp a")
apply simp+
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
by simp
lemma rsimp_alts_idem:
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
apply(induct as)
apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
prefer 2
apply simp
using bubble_break rsimp_alts_idem_aux2 apply auto[1]
apply(case_tac as)
apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
prefer 2
apply simp
using head_one_more_simp apply fastforce
apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
using simp_flatten2
apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list)))) = rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
prefer 2
apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
apply (simp only:)
done
lemma rsimp_alts_idem2:
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
using head_one_more_simp rsimp_alts_idem by auto
lemma evolution_step1:
shows "rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [(rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)))])) "
using rsimp_alts_idem by auto
lemma evolution_step2:
assumes " rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
shows "rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp
(rsimp_ALTs
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [ rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])) "
by (simp add: assms rsimp_alts_idem)
lemma rsimp_seq_aux1:
shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
apply simp
done
lemma multiple_alts_simp_flatten:
shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
lemma evo3_main_aux1:
shows "rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
rsimp
(RALTS
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
apply(subgoal_tac "rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
rsimp
(RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
prefer 2
apply (simp add: rsimp_idem)
apply (simp only:)
apply(subst multiple_alts_simp_flatten)
by simp
lemma evo3_main_nullable:
shows "
\<And>a Ss.
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
\<Longrightarrow> rsimp
(rsimp_ALTs
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r))
= RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
prefer 2
apply simp
apply(simp only:)
apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
prefer 2
using star_update_case1 apply presburger
apply(simp only:)
apply(subst List.list.map(2))+
apply(subgoal_tac "rsimp
(rsimp_ALTs
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp
(RALTS
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply(simp only:)
apply(subgoal_tac " rsimp
(rsimp_ALTs
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))
=
rsimp
(RALTS
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
prefer 2
using rsimp_ALTs.simps(3) apply presburger
apply (simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
prefer 2
apply (simp add: rsimp_idem)
apply(simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
prefer 2
using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
apply(simp only:)
apply(subgoal_tac " rsimp
(RALTS
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) =
rsimp
(RALTS
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
prefer 2
apply (smt (z3) idiot2 list.simps(9) rrexp.distinct(9) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_idem)
apply(simp only:)
apply(subgoal_tac " rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
rsimp
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) ")
prefer 2
using rsimp_idem apply force
apply(simp only:)
using evo3_main_aux1 by blast
lemma evo3_main_not1:
shows " \<not>rnullable (rders_simp r a) \<Longrightarrow> rder x (RSEQ (rders_simp r a) (RSTAR r)) = RSEQ (rder x (rders_simp r a)) (RSTAR r)"
by fastforce
lemma evo3_main_not2:
shows "\<not>rnullable (rders_simp r a) \<Longrightarrow> rsimp
(rsimp_ALTs
(rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
(rsimp_ALTs
((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
lemma evo3_main_not3:
shows "rsimp
(rsimp_ALTs
(rsimp_SEQ r1 (RSTAR r) # rs)) =
rsimp (rsimp_ALTs
(RSEQ r1 (RSTAR r) # rs))"
by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
lemma evo3_main_notnullable:
shows "\<And>a Ss.
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
\<Longrightarrow> rsimp
(rsimp_ALTs
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(subst star_update_case2)
apply simp
apply(subst List.list.map(2))
apply(subst evo3_main_not2)
apply simp
apply(subst evo3_main_not3)
using rsimp_alts_idem by presburger
lemma evo3_aux2:
shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
by simp
lemma evo3_aux3:
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
by (metis list.simps(8) list.simps(9) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem)
lemma evo3_aux4:
shows " rsimp
(rsimp_ALTs
[RSEQ (rder x r) (RSTAR r),
rsimp (rsimp_ALTs rs)]) =
rsimp
(rsimp_ALTs
(rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
lemma evo3_aux5:
shows "rders_simp r a \<noteq> RONE \<and> rders_simp r a \<noteq> RZERO \<Longrightarrow> rsimp_SEQ (rders_simp r a) (RSTAR r) = RSEQ (rders_simp r a) (RSTAR r)"
using idiot2 by blast
lemma evolution_step3:
shows" \<And>a Ss.
rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) \<Longrightarrow>
rsimp
(rsimp_ALTs
[rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
apply(case_tac "rders_simp r a = RONE")
apply(subst rsimp_seq_aux1)
apply simp
apply(subst rder.simps(6))
apply(subgoal_tac "rnullable (rders_simp r a)")
prefer 2
using rnullable.simps(2) apply presburger
apply(subst star_update_case1)
apply simp
apply(subst List.list.map)+
apply(subst rders_simp_append)
apply(subst evo3_aux2)
apply simp
apply(subst evo3_aux3)
apply(subst evo3_aux4)
apply simp
apply(case_tac "rders_simp r a = RZERO")
apply (simp add: rsimp_alts_idem2)
apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
prefer 2
using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
apply(subst evo3_aux5)
apply simp
apply(case_tac "rnullable (rders_simp r a) ")
using evo3_main_nullable apply blast
using evo3_main_notnullable apply blast
done
(*
proof (prove)
goal (1 subgoal):
1. map f (a # s) = f a # map f s
Auto solve_direct: the current goal can be solved directly with
HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
*)
lemma starseq_list_evolution:
fixes r :: rrexp and Ss :: "char list list" and x :: char
shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)) )"
apply(induct Ss)
apply simp
apply(subst List.list.map(2))
apply(subst evolution_step2)
apply simp
sorry
lemma star_seqs_produce_star_seqs:
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
by (meson comp_apply)
lemma map_der_lambda_composition:
shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
by force
lemma ralts_vs_rsimpalts:
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
lemma linearity_of_list_of_star_or_starseqs:
fixes r::rrexp and Ss::"char list list" and x::char
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
apply(subst rder_rsimp_ALTs_commute)
apply(subst map_der_lambda_composition)
using starseq_list_evolution
apply(rule_tac x = "star_update x r Ss" in exI)
apply(subst ralts_vs_rsimpalts)
by simp
(*certified correctness---does not depend on any previous sorry*)
lemma star_list_push_der: shows " \<lbrakk>xs \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss));
xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
\<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) =
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
prefer 2
apply blast
apply(erule exE)
apply(subgoal_tac "rders_simp (RSTAR r) (xs @ [x]) = rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using rders_simp_append
using rders_simp_one_char apply presburger
apply(rule_tac x= "Ss" in exI)
apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using inside_simp_removal rsimp_idem apply presburger
apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
prefer 2
using rder.simps(4) apply presburger
apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
apply (metis rsimp_idem)
by (metis map_der_lambda_composition)
end