--- a/ChengsongPhdThesis/ChengsongPhDThesis.tex Mon Mar 07 12:27:27 2022 +0000
+++ b/ChengsongPhdThesis/ChengsongPhDThesis.tex Tue Mar 08 00:50:40 2022 +0000
@@ -378,10 +378,99 @@
\subsection{an improved version of bit-coded algorithm: with simp!}
-\subsection{a correctness proof for bitcoded}
+\subsection{a correctness proof for bitcoded algorithm}
\subsection{finiteness proof }
+\subsubsection{closed form}
+We can give the derivative of regular expressions
+with respect to string a closed form with respect to simplification:
+\begin{itemize}
+\item
+closed form for sequences:
+\begin{verbatim}
+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))
+ )
+ )"
+\end{verbatim}
+where the recursive function $\textit{vsuf}$ is defined as
+\begin{verbatim}
+fun vsuf :: "char list -> rrexp -> 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))
+ ) "
+\end{verbatim}
+\item
+closed form for stars:
+\begin{verbatim}
+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]]) ) ))"
+\end{verbatim}
+where the recursive function $\textit{star}\_\textit{updates}$ is defined as
+\begin{verbatim}
+fun star_update :: "char -> rrexp -> 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) )"
+
+fun star_updates :: "char list -> rrexp -> char list list -> char list list"
+ where
+"star_updates [] r Ss = Ss"
+| "star_updates (c # cs) r Ss = star_updates cs r (star_update c r Ss)"
+
+\end{verbatim}
+
+
+\end{itemize}
+These closed form is a formalization of the intuition
+ that we can push in the derivatives
+of compound regular expressions to its sub-expressions, and the resulting
+expression is a linear combination of those sub-expressions' derivatives.
+\subsubsection{Estimation of closed forms' size}
+And thanks to $\textit{distinctBy}$ helping with deduplication,
+the linear combination can be bounded by the set enumerating all
+regular expressions up to a certain size :
+\begin{verbatim}
+
+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]]) ) ))) <=
+ Suc (sum_list (map rsize (rdistinct (map (\lambda s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
+ (star_updates s r [[c]]) ) {}) ) )"
+
+ lemma distinct_list_rexp_up_to_certain_size_bouded_by_set_enumerating_up_to_that_size:
+ shows "\forallr\in set rs. (rsize r ) <= N ==> sum_list (map rsize (rdistinct rs {})) <=
+ (card (rexp_enum N))* N"
+
+ lemma ind_hypo_on_ders_leads_to_stars_bounded:
+ shows "\foralls. rsize (rders_simp r0 s) <= N ==>
+ (sum_list (map rsize (rdistinct (map (\lambda s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
+ (star_updates s r [[c]]) ) {}) ) ) <=
+(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
+"
+\end{verbatim}
+
+With the above 3 lemmas, we can argue that the inductive hypothesis
+$r_0$'s derivatives is bounded above leads to $r_0^*$'s
+derivatives being bounded above.
+\begin{verbatim}
+
+lemma r0_bounded_star_bounded:
+ shows "\foralls. rsize (rders_simp r0 s) <= N ==>
+ \foralls. rsize (rders_simp (RSTAR r0) s) <=
+(card (rexp_enum (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))"
+\end{verbatim}
+
+
+And we have a similar argument for the sequence case.
\subsection{stronger simplification}
@@ -401,32 +490,7 @@
Deciding whether a string is in the language of the regex
can be intuitively done by constructing an NFA\cite{Thompson_1968}:
-and simulate the running of it:
-\begin{figure}
-\centering
-\includegraphics[scale=0.5]{pics/regex_nfa_base.png}
-\end{figure}
-
-\begin{figure}
-\centering
-\includegraphics[scale=0.5]{pics/regex_nfa_seq1.png}
-\end{figure}
-
-\begin{figure}
-\centering
-\includegraphics[scale=0.5]{pics/regex_nfa_seq2.png}
-\end{figure}
-
-\begin{figure}
-\centering
-\includegraphics[scale=0.5]{pics/regex_nfa_alt.png}
-\end{figure}
-
-
-\begin{figure}
-\centering
-\includegraphics[scale=0.5]{pics/regex_nfa_star.png}
-\end{figure}
+and simulate the running of it.
Which should be simple enough that modern programmers
have no problems with it at all?
@@ -1898,6 +1962,9 @@
analysis approach by implementing them in the same language and then compare
their performance.
+
+\section{discarded}
+haha
\bibliographystyle{plain}
\bibliography{root,regex_time_complexity}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/ClosedForms.thy Tue Mar 08 00:50:40 2022 +0000
@@ -0,0 +1,1006 @@
+
+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