--- a/ChengsongPhdThesis/ChengsongPhDThesis.tex Tue Mar 08 00:50:40 2022 +0000
+++ b/ChengsongPhdThesis/ChengsongPhDThesis.tex Wed Mar 09 17:33:08 2022 +0000
@@ -38,6 +38,7 @@
\def\lexer{\mathit{lexer}}
\def\mkeps{\mathit{mkeps}}
+\def\DFA{\textit{DFA}}
\def\bmkeps{\textit{bmkeps}}
\def\retrieve{\textit{retrieve}}
\def\blexer{\textit{blexer}}
@@ -55,7 +56,11 @@
\def\S{\mathit{S}}
\def\rup{r^\uparrow}
-
+\newcommand{\PDER}{\textit{PDER}}
+\newcommand{\flts}{\textit{flts}}
+\newcommand{\distinctBy}{\textit{distinctBy}}
+\newcommand{\map}{\textit{map}}
+\newcommand{\size}{\textit{size}}
\def\awidth{\mathit{awidth}}
\def\pder{\mathit{pder}}
\def\maxterms{\mathit{maxterms}}
@@ -101,6 +106,7 @@
\section{Introduction}
\subsection{Basic Regex Introduction}
+
Suppose (basic) regular expressions are given by the following grammar:
\[ r ::= \ZERO \mid \ONE
\mid c
@@ -117,12 +123,8 @@
& & $\textit{else} \; \textit{output} \; \textit{NO}$
\end{tabular}
\end{center}
-
-
-\subsection{The practical problem}
-Introduce the problem of matching a pattern with a string.
-Why important?
-Need to be fast, real-time, on large inputs.
+Omnipresent use of regexes in modern
+software.
Examples: Snort, Bro, etc?
\subsubsection{The rules for network intrusion analysis tools }
TODO: read rules libraries such as Snort and the explanation for some of the rules
@@ -130,6 +132,28 @@
TODO: any other libraries?
+There has been many widely used libraries such as
+Henry Spencer's regexp(3), RE2, etc.
+They are fast and successful, but ugly corner cases
+allowing the $\textit{ReDoS}$ attack exist, and
+is a non-negligible protion.
+\subsection{The practical problem}
+These corner cases either
+\begin{itemize}
+\item
+go unnoticed until they
+cause considerable grief in real life
+\item
+or force the regex library writers to pose
+restrictions on the input, limiting the
+choice a programmer has when using regexes.
+\end{itemize}
+
+Motivation:
+We want some library that supports as many constructs as possible,
+but still gives formal guarantees on the correctness and running
+time.
+
\subsection{Regexes that brought down CloudFlare}
@@ -217,6 +241,29 @@
suffix up to length $n$ of input string is.
"Countdown States activation problem":
$.*a.{100}$ requires $2^100$ + DFA states.
+Example:
+Converting $((a|b )*b.{10}){3}$ to a $\DFA$
+gives the error:
+\begin{verbatim}
+147972 states before minimization, 79107 states in minimized DFA
+Old file "search.java" saved as "search.java~"
+Writing code to "search.java"
+
+Unexpected exception encountered. This indicates a bug in JFlex.
+Please consider filing an issue at http://github.com/jflex-de/jflex/issues/new
+
+
+character value expected
+java.lang.IllegalArgumentException: character value expected
+ at jflex.generator.PackEmitter.emitUC(PackEmitter.java:105)
+ at jflex.generator.CountEmitter.emit(CountEmitter.java:116)
+ at jflex.generator.Emitter.emitDynamicInit(Emitter.java:530)
+ at jflex.generator.Emitter.emit(Emitter.java:1369)
+ at jflex.generator.LexGenerator.generate(LexGenerator.java:115)
+ at jflex.Main.generate(Main.java:320)
+ at jflex.Main.main(Main.java:336)
+\end{verbatim}
+
\subsubsection{variant of DFA's}
counting set automata
\\
@@ -471,22 +518,134 @@
And we have a similar argument for the sequence case.
-\subsection{stronger simplification}
+\subsection{stronger simplification needed}
+
+\subsubsection{Bounded List of Terms}
+We have seen that without simplification the size of $(a+aa)^*$
+grows exponentially and unbounded(where we omit certain nested
+parentheses among the four terms in the last explicitly written out regex):
+
+\def\ll{\stackrel{\_\backslash{} a}{\longrightarrow}}
+\begin{center}
+\begin{tabular}{rll}
+$(a + aa)^*$ & $\ll$ & $(\ONE + \ONE{}a) \cdot (a + aa)^*$\\
+& $\ll$ & $(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* \;+\; (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
+& $\ll$ & $(\ZERO + \ZERO{}a + \ZERO) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^* \;+\; $\\
+& & $\qquad(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
+& $\ll$ & \ldots \hspace{15mm}(regular expressions of sizes 98, 169, 283, 468, 767, \ldots)
+\end{tabular}
+\end{center}
+
+But if we look closely at the regex
+\begin{center}
+\begin{tabular}{rll}
+& & $\qquad(\ZERO + \ZERO{}a + \ONE) \cdot (a + aa)^* + (\ONE + \ONE{}a) \cdot (a + aa)^*$\\
+\end{tabular}
+\end{center}
+we realize that:
+\begin{itemize}
+\item
+The regex is equivalent to an alternative taking a long-flattened list,
+where each list is a sequence, and the second child of that sequence
+is always $(a+aa)^*$. In other words, the regex is a "linear combination"
+of terms of the form $(a+aa)\backslash s \cdot (a+aa)^*$ ($s$ is any string).
+\item
+The number of different terms of the shape $(a+aa) \backslash s \cdot (a+aa)^*$ is
+bounded because the first child $(a+aa) \backslash s$ can only be one of
+$(\ZERO + \ZERO{}a + \ZERO)$, $(\ZERO + \ZERO{}a + \ONE)$,
+$(\ONE + \ONE{}a)$ and $(\ZERO + \ZERO{}a)$.
+\item
+With simplification we have that the regex is additionally reduced to,
+
+where each term $\bsimp((a+aa)\backslash s ) $
+is further reduced to only
+$\ONE$, $\ONE + a$ and $\ZERO$.
+
+
+\end{itemize}
+Generalizing this to any regular expression of the form
+$\sum_{s\in L(r)} \bsimp(r\backslash s ) \cdot r^*$,
+we have the closed-form for star regex's string derivative as below:
+$\forall r \;s.\; \exists sl. \; s.t.\;\bsimp(r^* \backslash s) = \bsimp(\sum_{s'\in sl}(r\backslash s') \cdot r^* )$.
+
+The regex $\bsimp(\sum_{s' \in sl}(r\backslash s') \cdot r^*)$ is bounded by
+$\distinctBy(\flts(\sum_{s'\in sl}(\bsimp(r \backslash s')) \cdot r^*))$,
+which again is bounded by $\distinctBy(\sum_{s'\in sl}(\bsimp(r\backslash s')) \cdot r^*)$.
+This might give us a polynomial bound on the length of the list
+$\distinctBy[(\bsimp(r\backslash s')) \cdot r^* | {s'\in sl} ]$, if the terms in
+$\distinctBy[(\bsimp (r\backslash s')) | {s' \in sl}]$ has a polynomial bound.
+This is unfortunately not true under our current $\distinctBy$ function:
+If we let $r_n = ( (aa)^* + (aaa)^* + \ldots + \underline{(a\ldots a)^*}{n \,a's}) $,
+then we have that $\textit{maxterms} r_n = \textit{sup} (\textit{length} [\bsimp(r_n\backslash s') | s' \in sl]) =
+L.C.M(1,\ldots, n)$. According to \href{http://oeis.org/A003418}{OEISA003418}
+this grows exponentially quickly. So we have found a regex $r_n$ where
+$\textit{maxterms} (r_n ^* \backslash s) \geq 2^{(n-1)}$.
+
+
+\subsubsection{stronger version of \distinctBy}
+\href{https://www.researchgate.net/publication/340874991_Partial_derivatives_of_regular_expressions_and_finite_automaton_constructions}{Antimirove}
+has proven a linear bound on the number of terms in the "partial derivatives" of a regular expression:
+\begin{center}
+$\size (\PDER(r)) \leq \awidth (r)$.
+\end{center}
+
+The proof is by structural induction on the regular expression r.
+The hard cases are the sequence case $r_1\cdot r_2$ and star case $r^*$.
+The central idea that allows the induction to go through for this bound is on the inclusion:
+\begin{center}
+$\pder_{s@[c]} (a\cdot b) \subseteq (\pder_{s@[c]} a ) \cdot b \cup (\bigcup_{s' \in Suf(s@[c])} (\pder_{s'} \; b))$
+\end{center}
+
+This way,
+
+\begin{center}
+\begin{tabular}{lcl}
+$| \pder_{s@[c]} (a\cdot b) |$ & $ \leq$ & $ | (\pder_{s@[c]} a ) \cdot b \cup (\bigcup_{s' \in Suf(s@[c])} (\pder_{s'} \; b))|$\\
+& $\leq$ & $| (\pder_{s@[c]} a ) \cdot b| + | (\bigcup_{s' \in Suf(s@[c])} (\pder_{s'} \; b))|$\\
+& $=$ & $\awidth(a) + \awidth(b)$ \\
+& $=$ & $\awidth(a+b)$
+\end{tabular}
+\end{center}
+
+we have that a compound regular expression $a\cdot b$'s subterms
+ is bounded by its sub-expression's derivatives terms.
+
+This argument can be modified to bound the terms in
+our version of regex with strong simplification:
+\begin{center}
+\begin{tabular}{lcl}
+$| \maxterms (\bsimp (a\cdot b) \backslash s)|$ & $=$ & $ |maxterms(\bsimp( (a\backslash s \cdot b) + \sum_{s'\in sl}(b\backslash s') ))|$\\
+& $\leq$ & $| (\pder_{s@[c]} a ) \cdot b| + | (\bigcup_{s' \in Suf(s@[c])} (\pder_{s'} \; b))|$\\
+& $=$ & $\awidth(a) + \awidth(b)$ \\
+& $=$ & $\awidth(a+b)$
+\end{tabular}
+\end{center}
+
+
\subsection{cubic bound}
+Bounding the regex's subterms by
+its alphabetic width.
+
+The entire expression's size can be bounded by
+number of subterms times each subterms' size.
-
-
-
+\section{Support for bounded repetitions and other constructs}
+Example:
+$.*a.\{100\}$ after translation to $\DFA$ and minimization will
+always take over $2^{100}$ states.
-
+\section{Towards a library with fast running time practically}
-\subsection{Too Detailed too basic? regex to NFA translation}
+registers and cache-related optimizations?
+JVM related optimizations?
+
+\section{Past Report materials}
Deciding whether a string is in the language of the regex
can be intuitively done by constructing an NFA\cite{Thompson_1968}:
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/BasicIdentities.thy Wed Mar 09 17:33:08 2022 +0000
@@ -0,0 +1,441 @@
+theory BasicIdentities 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)"
+
+
+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_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
+
+
+
+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))
+ ) "
+
+
+
+
+
+
+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)"
+
+
+
+
+
+
+(*some basic facts about rsimp*)
+
+
+
+
+end
\ No newline at end of file
--- a/thys2/ClosedForms.thy Tue Mar 08 00:50:40 2022 +0000
+++ b/thys2/ClosedForms.thy Wed Mar 09 17:33:08 2022 +0000
@@ -1,117 +1,36 @@
-
-theory ClosedForms
- imports "Lexer" "PDerivs"
+theory ClosedForms imports
+"BasicIdentities"
begin
+lemma alts_closed_form: shows
+"rsimp (rders_simp (RALTS rs) s) =
+rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
+ apply(induct s rule: rev_induct)
+ apply simp
+ apply simp
+ apply(subst rders_simp_append)
+ apply(subgoal_tac " rsimp (rders_simp (rders_simp (RALTS rs) xs) [x]) =
+ rsimp(rders_simp (rsimp_ALTs (rdistinct (rflts (map (rsimp \<circ> (\<lambda>r. rders_simp r xs)) rs)) {})) [x])")
+ prefer 2
+ apply (metis inside_simp_removal rders_simp_one_char)
+ apply(simp only: )
+ sorry
-datatype rrexp =
- RZERO
-| RONE
-| RCHAR char
-| RSEQ rrexp rrexp
-| RALTS "rrexp list"
-| RSTAR rrexp
-
-abbreviation
- "RALT r1 r2 \<equiv> RALTS [r1, r2]"
+lemma alts_closed_form_variant: shows
+"s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s =
+rsimp (RALTS (map (\<lambda>r. rders_simp r s) rs))"
+ sorry
-fun
- rnullable :: "rrexp \<Rightarrow> bool"
-where
- "rnullable (RZERO) = False"
-| "rnullable (RONE ) = True"
-| "rnullable (RCHAR c) = False"
-| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
-| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
-| "rnullable (RSTAR r) = True"
-
-
-fun
- rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
-where
- "rder c (RZERO) = RZERO"
-| "rder c (RONE) = RZERO"
-| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
-| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
-| "rder c (RSEQ r1 r2) =
- (if rnullable r1
- then RALT (RSEQ (rder c r1) r2) (rder c r2)
- else RSEQ (rder c r1) r2)"
-| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
-
-
-fun
- rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
-where
- "rders r [] = r"
-| "rders r (c#s) = rders (rder c r) s"
-
-fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
- where
- "rdistinct [] acc = []"
-| "rdistinct (x#xs) acc =
- (if x \<in> acc then rdistinct xs acc
- else x # (rdistinct xs ({x} \<union> acc)))"
-
-
-
+lemma 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 r0 [[c]]) ) ))"
+ apply(induct s)
+ apply simp
+ sorry
-fun rflts :: "rrexp list \<Rightarrow> rrexp list"
- where
- "rflts [] = []"
-| "rflts (RZERO # rs) = rflts rs"
-| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
-| "rflts (r1 # rs) = r1 # rflts rs"
-
-
-fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
- where
- "rsimp_ALTs [] = RZERO"
-| "rsimp_ALTs [r] = r"
-| "rsimp_ALTs rs = RALTS rs"
-
-fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
- where
- "rsimp_SEQ RZERO _ = RZERO"
-| "rsimp_SEQ _ RZERO = RZERO"
-| "rsimp_SEQ RONE r2 = r2"
-| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
-
-
-fun rsimp :: "rrexp \<Rightarrow> rrexp"
- where
- "rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
-| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
-| "rsimp r = r"
-
-
-fun
- rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
-where
- "rders_simp r [] = r"
-| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
-
-fun rsize :: "rrexp \<Rightarrow> nat" where
- "rsize RZERO = 1"
-| "rsize (RONE) = 1"
-| "rsize (RCHAR c) = 1"
-| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
-| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
-| "rsize (RSTAR r) = Suc (rsize r)"
-
-
-fun rlist_size :: "rrexp list \<Rightarrow> nat" where
-"rlist_size (r # rs) = rsize r + rlist_size rs"
-| "rlist_size [] = 0"
-
-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) =
@@ -124,883 +43,9 @@
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))"
+lemma seq_closed_form_variant: shows
+"s \<noteq> [] \<Longrightarrow> (rders_simp (RSEQ r1 r2) s) =
+rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))"
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
+end
\ No newline at end of file
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/ClosedFormsBounds.thy Wed Mar 09 17:33:08 2022 +0000
@@ -0,0 +1,585 @@
+
+theory ClosedFormsBounds
+ imports "GeneralRegexBound" "ClosedForms"
+begin
+
+
+
+lemma alts_closed_form_bounded: shows
+"\<forall>r \<in> set rs. \<forall>s. rsize(rders_simp r s ) \<le> N \<Longrightarrow>
+rsize (rders_simp (RALTS rs ) s) \<le> max (Suc ( N * (card (sizeNregex N)))) (rsize (RALTS rs) )"
+ apply(induct s)
+ apply simp
+ apply(insert alts_closed_form_variant)
+
+
+ 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 r0 [[c]]) ) ))) \<le>
+ Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
+ (star_updates s r0 [[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 (sizeNregex N))* N"
+ sorry
+
+
+lemma star_control_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 r0 [[c]]) ) {}) ) ) \<le>
+(card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0)))
+"
+ sorry
+
+lemma star_control_variant:
+ assumes "\<forall>s. rsize (rders_simp r0 s) \<le> N"
+ shows"Suc
+ (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
+ (star_updates list r0 [[a]])) {})))
+\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) "
+ apply(subgoal_tac "(sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0))
+ (star_updates list r0 [[a]])) {})))
+\<le> ( (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
+ prefer 2
+ using assms star_control_bounded apply presburger
+ by simp
+
+
+
+lemma star_closed_form_bounded:
+ shows "\<forall>s. rsize (rders_simp r0 s) \<le> N \<Longrightarrow>
+ rsize (rders_simp (RSTAR r0) s) \<le>
+max ( (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * (Suc (N + rsize (RSTAR r0))))) (rsize (RSTAR r0))"
+ apply(case_tac s)
+ apply simp
+ apply(subgoal_tac " rsize (rders_simp (RSTAR r0) (a # list)) =
+rsize (rsimp ( RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))")
+ prefer 2
+ using star_closed_form apply presburger
+ apply(subgoal_tac "rsize (rsimp (
+ RALTS ( (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) ) (star_updates list r0 [[a]]) ) )))
+\<le> Suc (sum_list (map rsize (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0) )
+ (star_updates list r0 [[a]]) ) {}) ) )")
+ prefer 2
+ using star_closed_form_bounded_by_rdistinct_list_estimate apply presburger
+ apply(subgoal_tac "Suc (sum_list
+ (map rsize
+ (rdistinct (map (\<lambda>s1. RSEQ (rders_simp r0 s1) (RSTAR r0)) (star_updates list r0 [[a]])) {})))
+\<le> (Suc (card (sizeNregex (Suc (N + rsize (RSTAR r0))))) * Suc (N + rsize (RSTAR r0))) ")
+ apply auto[1]
+ using star_control_variant by blast
+
+
+
+
+
+
+lemma seq_list_estimate_control: shows
+" rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
+ \<le> Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))"
+
+ sorry
+
+lemma seq_estimate_bounded:
+ assumes "\<forall>s. rsize (rders_simp r1 s) \<le> N1" and "\<forall>s. rsize (rders_simp r2 s) \<le> N2"
+ shows
+"Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {}))) \<le>
+ Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))"
+
+ sorry
+
+lemma seq_closed_form_bounded: shows
+"\<lbrakk>\<forall>s. rsize (rders_simp r1 s) \<le> N1 ; \<forall>s. rsize (rders_simp r2 s) \<le> N2\<rbrakk> \<Longrightarrow>
+rsize (rders_simp (RSEQ r1 r2) s) \<le>
+max (Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))) (rsize (RSEQ r1 r2)) "
+ apply(case_tac s)
+ apply simp
+ apply(subgoal_tac " (rders_simp (RSEQ r1 r2) s) =
+rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) # (map (rders_simp r2) (vsuf s r1))))")
+ prefer 2
+ using seq_closed_form_variant apply blast
+ apply(subgoal_tac "rsize (rsimp (RALTS (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1))))
+ \<le>
+Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))")
+ apply(subgoal_tac "Suc (sum_list (map rsize (rdistinct (RSEQ (rders_simp r1 s) r2 # map (rders_simp r2) (vsuf s r1)) {})))
+\<le> Suc (Suc (N1 + (rsize r2)) + (N2 * card (sizeNregex N2)))")
+ prefer 2
+ using seq_estimate_bounded apply blast
+ apply(subgoal_tac "rsize (rders_simp (RSEQ r1 r2) s) \<le> Suc (Suc (N1 + rsize r2) + N2 * card (sizeNregex N2))")
+ using le_max_iff_disj apply blast
+ apply auto[1]
+ using seq_list_estimate_control by presburger
+
+
+lemma rders_simp_bounded: shows
+"\<exists>N. \<forall>s. rsize (rders_simp r s) \<le> N"
+ apply(induct r)
+ apply(rule_tac x = "Suc 0 " in exI)
+ using three_easy_cases0 apply force
+ using three_easy_cases1 apply blast
+ using three_easy_casesC apply blast
+ using seq_closed_form_bounded apply blast
+ apply (metis alts_closed_form_bounded size_list_estimation')
+ using star_closed_form_bounded by blast
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(*Obsolete materials*)
+
+
+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
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/GeneralRegexBound.thy Wed Mar 09 17:33:08 2022 +0000
@@ -0,0 +1,97 @@
+theory GeneralRegexBound imports
+"BasicIdentities"
+begin
+
+
+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)
+
+
+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 three_easy_cases0: shows
+"rsize (rders_simp RZERO s) \<le> Suc 0"
+ sorry
+
+
+lemma three_easy_cases1: shows
+"rsize (rders_simp RONE s) \<le> Suc 0"
+ sorry
+
+lemma three_easy_casesC: shows
+"rsize (rders_simp (RCHAR c) s) \<le> Suc 0"
+
+ sorry
+
+
+
+
+end
+
--- a/thys2/SizeBoundStrong.thy Tue Mar 08 00:50:40 2022 +0000
+++ b/thys2/SizeBoundStrong.thy Wed Mar 09 17:33:08 2022 +0000
@@ -942,21 +942,14 @@
fun pAKC_aux :: "arexp list \<Rightarrow> arexp \<Rightarrow> rexp \<Rightarrow> arexp"
where
-"pAKC_aux rsa r ctx = (if (L (SEQ (erase r) ctx)) \<subseteq> (L (erase (AALTs [] rsa))) then AZERO else
- case r of (ASEQ bs r1 r2) \<Rightarrow> pAKC_aux rsa r1 (SEQ (erase r1) ctx) |
- (AALTs bs rs) \<Rightarrow> AALTs bs (map (\<lambda>r. pAKC_aux rsa r ctx) rs) |
+"pAKC_aux rsa r ctx = (if (L (SEQ (erase r) ( ctx) )) \<subseteq> (L (erase (AALTs [] rsa))) then AZERO else
+ case r of (ASEQ bs r1 r2) \<Rightarrow>
+ bsimp_ASEQ bs (pAKC_aux rsa r1 (SEQ (erase r1) ( ctx) )) r2 |
+ (AALTs bs rs) \<Rightarrow>
+ bsimp_AALTs bs (flts (map (\<lambda>r. pAKC_aux rsa r ( ctx) ) rs)) |
r \<Rightarrow> r
)"
-|"pAKC_aux rsa (ASEQ bs1 a1 a2) ctx = (if (L (SEQ (SEQ (erase a1) (erase a2)) ctx)) \<subseteq>
- L (erase (AALTs [] rsa))
- then AZERO
- else (pAKC_aux rsa a1 (ASEQ bs1 a2 ctx)))"
-| "pAKC_aux rsa (AALTs bs1 rs1) ctx = ctx "
-
-fun pruneAwayKnownComponents :: "arexp list \<Rightarrow> arexp \<Rightarrow> arexp" where
-"pruneAwayKnownComponents rsa (ASEQ bs r1 r2) = (pAKC_aux rsa r1 r2)"
-
inductive
@@ -974,7 +967,7 @@
| "AALTs bs [] \<leadsto> AZERO"
| "AALTs bs [r] \<leadsto> fuse bs r"
| "erase a1 = erase a2 \<Longrightarrow> AALTs bs (rsa@[a1]@rsb@[a2]@rsc) \<leadsto> AALTs bs (rsa@[a1]@rsb@rsc)"
-| "pruneAwayKnownComponents rsa a2 = a2' \<Longrightarrow> AALTs bs (rsa @ [a2] @ rsb) \<leadsto> AALTs bs (rsa @ [a2'] @ rsb)"
+| "pAKC_aux rsa a2 ONE = a2' \<Longrightarrow> AALTs bs (rsa @ [a2] @ rsb) \<leadsto> AALTs bs (rsa @ [a2'] @ rsb)"
inductive