--- a/progs/scala/re-bit2.scala Mon Feb 21 23:38:26 2022 +0000
+++ b/progs/scala/re-bit2.scala Wed Mar 02 11:43:41 2022 +0000
@@ -1,3 +1,7 @@
+
+
+
+
import scala.language.implicitConversions
import scala.language.reflectiveCalls
import scala.annotation.tailrec
@@ -225,71 +229,11 @@
case r1 :: rs2 => r1 :: flats(rs2)
}
-def projectFirstChild(rs: List[ARexp]) : List[ARexp] = rs match {
- case (ASEQ(bs, r1, r2p)::rs1) => r1::projectFirstChild(rs1)
- case Nil => Nil
- }
-
-
-
-def distinctBy2(xs: List[ARexp], acc: List[Rexp] = Nil): List[ARexp] = xs match {
- case Nil => Nil
- case (x::xs) => {
- val res = erase(x)
- if(acc.contains(res))
- distinctBy2(xs, acc)
- else
- x match {
- case ASEQ(bs0, AALTS(bs1, rs), r2) =>
- val newTerms = distinctBy2(rs.map(r1 => ASEQ(Nil, r1, r2)), acc)
- val rsPrime = projectFirstChild(newTerms)
- newTerms match {
- case Nil => distinctBy2(xs, acc)
- case t::Nil => ASEQ(bs0, fuse(bs1, rsPrime.head), r2)::distinctBy2(xs, erase(t)::acc)
- case ts => ASEQ(bs0, AALTS(bs1, rsPrime), r2)::distinctBy2(xs, newTerms.map(erase(_)):::acc)
- }
- case x => x::distinctBy2(xs, res::acc)
- }
- }
- }
-
-/*
-def strongBsimp(r: ARexp): ARexp =
- {
- r match {
- case ASEQ(bs1, r1, r2) => (strongBsimp(r1), strongBsimp(r2)) match {
- case (AZERO, _) => AZERO
- case (_, AZERO) => AZERO
- case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
- case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
- }
- case AALTS(bs1, rs) => {
- val rs_simp = rs.map(strongBsimp(_))
- val flat_res = flats(rs_simp)
- val dist_res = distinctBy2(flat_res)//distinctBy(flat_res, erase)
- dist_res match {
- case Nil => AZERO
- case s :: Nil => fuse(bs1, s)
- case rs => AALTS(bs1, rs)
- }
-
- }
- case r => r
- }
- }
-*/
-
def bsimp(r: ARexp): ARexp = r match {
case ASEQ(bs1, r1, r2) => (bsimp(r1), bsimp(r2)) match {
case (AZERO, _) => AZERO
case (_, AZERO) => AZERO
case (AONE(bs2), r2s) => fuse(bs1 ++ bs2, r2s)
- //case (AALTS(bs, rs), r) =>
- // AALTS(Nil, rs.map(ASEQ(bs1 ++ bs, _, r)))
- //case (ASEQ(bs2, r1, ASTAR(bs3, r2)), ASTAR(bs4, r3)) if erase(r2) == erase(r3) =>
- // ASEQ(bs1 ++ bs2, r1, ASTAR(bs3, r2))
- //case (ASEQ(bs2, r1, r2), r3) =>
- // ASEQ(bs1 ++ bs2, r1, ASEQ(Nil, r2, r3))
case (r1s, r2s) => ASEQ(bs1, r1s, r2s)
}
case AALTS(bs1, rs) => (flts(rs.map(bsimp))).distinctBy(erase) match {
@@ -297,7 +241,6 @@
case r::Nil => fuse(bs1, r)
case rs => AALTS(bs1, rs)
}
- //case (ASTAR(bs1, ASTAR(bs2, r))) => bsimp(ASTAR(bs1 ++ bs2, r))
case r => r
}
@@ -310,12 +253,101 @@
def blex_simp(r: ARexp, s: List[Char]) : Bits = s match {
case Nil => if (bnullable(r)) bmkeps(r)
else throw new Exception("Not matched")
- case c::cs => blex(bsimp(bder(c, r)), cs)
+ case c::cs => blex_simp(bsimp(bder(c, r)), cs)
}
def blexing_simp(r: Rexp, s: String) : Val =
decode(r, blex_simp(internalise(r), s.toList))
+//////////////////////
+// new simplification
+
+// collects first components of sequences.
+def coll(r: Rexp, rs: List[Rexp]) : List[Rexp] = rs match {
+ case Nil => Nil
+ case SEQ(r1, r2) :: rs =>
+ if (r == r2) r1 :: coll(r, rs) else coll(r, rs)
+ case r1 :: rs => coll(r, rs)
+}
+
+def bsimp_ASEQ1(bs: Bits, r1: ARexp, r2: ARexp) : ARexp = r1 match {
+ case AZERO => AZERO
+ case AONE(bs1) => fuse(bs ::: bs1, r2)
+ case _ => ASEQ(bs, r1, r2)
+}
+
+def bsimp_AALTs(bs: Bits, rs: List[ARexp]) : ARexp = rs match {
+ case Nil => AZERO
+ case r::Nil => fuse(bs, r)
+ case _ => AALTS(bs, rs)
+}
+
+def prune(r: ARexp, all: List[Rexp]) : ARexp = r match {
+ case ASEQ(bs, r1, r2) => {
+ val termsTruncated = coll(erase(r2), all)
+ val pruned = prune(r1, termsTruncated)
+ bsimp_ASEQ1(bs, pruned, r2)
+ }
+ case AALTS(bs, rs) => {
+ val rsp = rs.map(prune(_, all)).filter(_ != AZERO)
+ bsimp_AALTs(bs, rsp)
+ }
+ case r =>
+ if (all.contains(erase(r))) r else AZERO
+}
+
+def oneSimp(r: Rexp) : Rexp = r match {
+ case SEQ(ONE, r) => r
+ case SEQ(r1, r2) => SEQ(oneSimp(r1), r2)
+ case r => r
+}
+
+def breakup(r: Rexp) : List[Rexp] = r match {
+ case SEQ(r1, r2) => breakup(r1).map(SEQ(_, r2))
+ case ALT(r1, r2) => breakup(r1) ::: breakup(r2)
+ case _ => r::Nil
+}
+
+def addToAcc(r: ARexp, acc: List[Rexp]) : List[Rexp] =
+ breakup(erase(r)).filterNot(r => acc.contains(oneSimp(r)))
+
+def dBStrong(rs: List[ARexp], acc: List[Rexp]) : List[ARexp] = rs match {
+ case Nil => Nil
+ case r::rs => if (acc.contains(erase(r))) dBStrong(rs, acc)
+ else prune(r, addToAcc(r, acc)) match {
+ case AZERO => dBStrong(rs, addToAcc(r, acc) ::: acc)
+ case r1 => r1 :: dBStrong(rs, addToAcc(r, acc) ::: acc)
+ }
+}
+
+def bsimp_ASEQ(bs: Bits, r1: ARexp, r2: ARexp) : ARexp = (r1, r2) match {
+ case (AZERO, _) => AZERO
+ case (_, AZERO) => AZERO
+ case (AONE(bs1), r2) => fuse(bs ::: bs1, r2)
+ case _ => ASEQ(bs, r1, r2)
+}
+
+def bsimp2(r: ARexp): ARexp = r match {
+ case ASEQ(bs1, r1, r2) => bsimp_ASEQ(bs1, bsimp2(r1), bsimp2(r2))
+ case AALTS(bs1, rs) => bsimp_AALTs(bs1, dBStrong(flts(rs.map(bsimp2(_))), Nil))
+ case r => r
+}
+
+def bders_simp2(r: ARexp, s: List[Char]) : ARexp = s match {
+ case Nil => r
+ case c::cs => bders_simp2(bsimp2(bder(c, r)), cs)
+}
+
+
+def blex_simp2(r: ARexp, s: List[Char]) : Bits = s match {
+ case Nil => if (bnullable(r)) bmkeps(r)
+ else throw new Exception("Not matched")
+ case c::cs => blex_simp2(bsimp2(bder(c, r)), cs)
+}
+
+def blexing_simp2(r: Rexp, s: String) : Val =
+ decode(r, blex_simp2(internalise(r), s.toList))
+
//println(blexing_simp(reg, "aab"))
@@ -377,6 +409,9 @@
case STAR(r1) => 1 + size(r1)
}
+def asize(r: ARexp) : Int = size(erase(r))
+def psize(rs: Set[Rexp]) : Int = rs.map(size(_)).sum
+
def pp(r: ARexp): String = r match {
case ASEQ(_, ACHAR(_, a1),ASEQ(_, r1, r2)) => s"${a1}${pp(r1)}${pp(r2)}"
case ASEQ(_, ACHAR(_, a1),ACHAR(_, a2)) => s"${a1}${a2}"
@@ -389,6 +424,27 @@
}
+val TEST = STAR("a" | "aa")
+println(asize(bders(("a" * 0).toList, internalise(TEST))))
+println(asize(bders(("a" * 1).toList, internalise(TEST))))
+println(asize(bders(("a" * 2).toList, internalise(TEST))))
+println(asize(bders(("a" * 3).toList, internalise(TEST))))
+println(asize(bders(("a" * 4).toList, internalise(TEST))))
+
+println(asize(bders_simp(internalise(TEST), ("a" * 0).toList)))
+println(asize(bders_simp(internalise(TEST), ("a" * 1).toList)))
+println(asize(bders_simp(internalise(TEST), ("a" * 2).toList)))
+println(asize(bders_simp(internalise(TEST), ("a" * 3).toList)))
+println(asize(bders_simp(internalise(TEST), ("a" * 4).toList)))
+
+
+println(asize(bders_simp2(internalise(TEST), ("a" * 0).toList)))
+println(asize(bders_simp2(internalise(TEST), ("a" * 1).toList)))
+println(asize(bders_simp2(internalise(TEST), ("a" * 2).toList)))
+println(asize(bders_simp2(internalise(TEST), ("a" * 3).toList)))
+println(asize(bders_simp2(internalise(TEST), ("a" * 4).toList)))
+println(asize(bders_simp2(internalise(TEST), ("a" * 5).toList)))
+
// Some Tests
//============
@@ -419,6 +475,7 @@
println(blexing(STARREG, "a" * 3))
println(blexing_simp(STARREG, "a" * 3))
+println(pders(List(STARREG), "a" * 3))
size(STARREG)
@@ -436,6 +493,9 @@
size(erase(bders_simp(internalise(STARREG), ("a" * 102).toList)))
size(erase(bders_simp(internalise(STARREG), ("a" * 103).toList)))
+size(erase(bders_simp2(internalise(STARREG), ("a" * 103).toList)))
+psize(pders(("a" * 103).toList, Set(STARREG)))
+
println(bders_simp(internalise(STARREG), ("a" * 1).toList))
println(bders_simp(internalise(STARREG), ("a" * 2).toList))
println(bders_simp(internalise(STARREG), ("a" * 3).toList))
@@ -709,3 +769,25 @@
encode(inj(dr, 'a', decode(dr_der, res1)))
*/
+
+
+
+
+
+
+
+/*
+def star(n: Long) = if ((n & 1L) == 1L) "*" else " "
+def stars(n: Long): String = if (n == 0L) "" else star(n) + " " + stars(n >> 1)
+def spaces(n: Int) = " " * n
+
+
+def sierpinski(n: Int) {
+ ((1 << n) - 1 to 0 by -1).foldLeft(1L) {
+ case (bitmap, remainingLines) =>
+ println(spaces(remainingLines) + stars(bitmap))
+ (bitmap << 1) ^ bitmap
+ }
+}
+
+*/
\ No newline at end of file
--- a/thys2/PDerivs.thy Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/PDerivs.thy Wed Mar 02 11:43:41 2022 +0000
@@ -400,8 +400,9 @@
apply(rule subset_trans)
thm pders_STAR
apply(rule pders_STAR)
- apply(simp)
- apply(auto simp add: pders_Set_def)[1]
+ apply(simp)
+ apply(auto simp add: pders_Set_def)[1]
+(* rest of SEQ case *)
apply(simp)
apply(rule conjI)
apply blast
--- a/thys2/Paper/Paper.thy Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/Paper/Paper.thy Wed Mar 02 11:43:41 2022 +0000
@@ -675,7 +675,7 @@
transforms a bitcoded regular expression into a (standard) regular
expression by just erasing the annotated bitsequences. We omit the
straightforward definition. For defining the algorithm, we also need
- the functions \textit{bnullable} and \textit{bmkeps}, which are the
+ the functions \textit{bnullable} and \textit{bmkeps}(\textit{s}), which are the
``lifted'' versions of \textit{nullable} and \textit{mkeps} acting on
bitcoded regular expressions.
%
@@ -695,14 +695,16 @@
&
\begin{tabular}{@ {}l@ {\hspace{1mm}}c@ {\hspace{1mm}}l@ {}}
$\textit{bmkeps}\,(\textit{ONE}\,bs)$ & $\dn$ & $bs$\\
- $\textit{bmkeps}\,(\textit{ALTs}\,bs\,r\!::\!\rs)$ & $\dn$ &
- $\textit{if}\;\textit{bnullable}\,r$\\
- & &$\textit{then}\;bs\,@\,\textit{bmkeps}\,r$\\
- & &$\textit{else}\;bs\,@\,\textit{bmkeps}\,\rs$\\
+ $\textit{bmkeps}\,(\textit{ALTs}\,bs\,\rs)$ & $\dn$ &
+ $bs\,@\,\textit{bmkepss}\,\rs$\\
$\textit{bmkeps}\,(\textit{SEQ}\,bs\,r_1\,r_2)$ & $\dn$ &\\
\multicolumn{3}{r}{$bs \,@\,\textit{bmkeps}\,r_1\,@\, \textit{bmkeps}\,r_2$}\\
$\textit{bmkeps}\,(\textit{STAR}\,bs\,r)$ & $\dn$ &
- $bs \,@\, [\S]$
+ $bs \,@\, [\S]$\\
+ $\textit{bmkepss}\,(r\!::\!\rs)$ & $\dn$ &
+ $\textit{if}\;\textit{bnullable}\,r$\\
+ & &$\textit{then}\;\textit{bmkeps}\,r$\\
+ & &$\textit{else}\;\textit{bmkepss}\,\rs$
\end{tabular}
\end{tabular}
\end{center}
@@ -1040,7 +1042,7 @@
\noindent where we scan the list from left to right (because we
have to remove later copies). In @{text distinctBy}, @{text f} is a
function and @{text acc} is an accumulator for regular
- expressions---essentially a set of regular expression that we have already seen
+ expressions---essentially a set of regular expressions that we have already seen
while scanning the list. Therefore we delete an element, say @{text x},
from the list provided @{text "f x"} is already in the accumulator;
otherwise we keep @{text x} and scan the rest of the list but
@@ -1182,12 +1184,12 @@
simplified regular expressions in small steps (unlike the @{text bsimp}-function which
does the same in a big step), and show that each of
the small steps preserves the bitcodes that lead to the final POSIX value.
- The rewrite system is organised such that $\leadsto$ is for bitcode regular
+ The rewrite system is organised such that $\leadsto$ is for bitcoded regular
expressions and $\stackrel{s}{\leadsto}$ for lists of bitcoded regular
expressions. The former essentially implements the simplifications of
@{text "bsimpSEQ"} and @{text flts}; while the latter implements the
simplifications in @{text "bsimpALTs"}. We can show that any bitcoded
- regular expression reduces in one or more steps to the simplified
+ regular expression reduces in zero or more steps to the simplified
regular expression generated by @{text bsimp}:
\begin{lemma}\label{lemone}
@@ -1317,9 +1319,9 @@
text {*
-In this section let us sketch our argument on why the size of the simplified
+In this section let us sketch our argument for why the size of the simplified
derivatives with the aggressive simplification function is finite. Suppose
-we have a size functions for bitcoded regular expressions, written
+we have a size function for bitcoded regular expressions, written
@{text "|r|"}, which counts the number of nodes if we regard $r$ as a tree
(we omit the precise definition). For this we show that for every $r$
there exists a bound $N$
@@ -1418,7 +1420,7 @@
\cite[Page 14]{Sulzmann2014}.
Given the growth of the
derivatives in some cases even after aggressive simplification, this
- is a hard to believe fact. A similar claim of about a theoretical runtime
+ is a hard to believe fact. A similar claim about a theoretical runtime
of @{text "O(n\<^sup>2)"} is made for the Verbatim lexer, which calculates POSIX matches and is based on
derivatives \cite{verbatim}. In this case derivatives are not simplified.
Clearly our result of having finite
--- a/thys2/SizeBound4.thy Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/SizeBound4.thy Wed Mar 02 11:43:41 2022 +0000
@@ -1019,7 +1019,7 @@
apply(simp only: map_append)
apply(simp only: map_single)
apply(rule rs_in_rstar)
- thm rrewrite_srewrite.intros
+ thm rrewrite_srewrite.intros
apply(rule rrewrite_srewrite.ss6)
using as
apply(auto simp add: der_correctness Der_def)
--- a/thys2/SizeBound6CT.thy Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/SizeBound6CT.thy Wed Mar 02 11:43:41 2022 +0000
@@ -47,24 +47,69 @@
shows "orderedSuf [c1, c2, c3] = [[c3], [c2, c3], [c1, c2, c3]]"
by auto
+
+fun ordsuf :: "char list \<Rightarrow> char list list"
+ where
+ "ordsuf [] = []"
+| "ordsuf (x # xs) = (ordsuf xs) @ [(x # xs)]"
+
+lemma
+ shows "ordsuf [c] = [[c]]"
+ and "ordsuf [c2, c3] = [[c3], [c2,c3]]"
+ and "ordsuf [c1, c2, c3] = [[c3], [c2, c3], [c1, c2, c3]]"
+ by auto
+
+lemma ordsuf_last:
+ shows "ordsuf (xs @ [x]) = [x] # (map (\<lambda>s. s @ [x]) (ordsuf xs))"
+ apply(induct xs)
+ apply(auto)
+ done
+
+lemma ordsuf_append:
+ shows "ordsuf (s1 @ s) = (ordsuf s) @ (map (\<lambda>s11. s11 @ s) (ordsuf s1))"
+apply(induct s1 arbitrary: s rule: rev_induct)
+ apply(simp)
+ apply(drule_tac x="[x] @ s" in meta_spec)
+ apply(simp)
+ apply(subst ordsuf_last)
+ apply(simp)
+ done
+
+
+lemma
+ "orderedSuf xs = ordsuf xs"
+ apply(induct xs rule: rev_induct)
+ apply(simp)
+ apply(simp)
+ apply(subst ordsuf_last)
+ apply(simp)
+ oops
+
+(*
+(*
lemma throwing_elem_around:
shows "orderedSuf (s1 @ [a] @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf ( s1 @ [a]) ))"
and "orderedSuf (s1 @ [a] @ s) = (orderedSuf ([a] @ s) @ (map (\<lambda>s11. s11 @ ([a] @ s))) (orderedSuf s1) )"
+ apply(auto)
+ prefer 2
+
sorry
-
+*)
lemma suf_cons:
shows "orderedSuf (s1 @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf s1))"
+ apply(induct s1 arbitrary: s rule: rev_induct)
+ apply(simp)
+ apply(drule_tac x="[x] @ s" in meta_spec)
+ apply(simp)
+
+
apply(induct s arbitrary: s1)
apply simp
- apply(subgoal_tac "s1 @ a # s = (s1 @ [a]) @ s")
- prefer 2
- apply simp
- apply(subgoal_tac "orderedSuf (s1 @ a # s) = orderedSuf ((s1 @ [a]) @ s)")
- prefer 2
- apply presburger
apply(drule_tac x="s1 @ [a]" in meta_spec)
- sorry
+ apply(simp only: append_assoc append.simps)
+
+ using throwing_elem_around(2) by force
@@ -141,11 +186,6 @@
(if x \<in> acc then rdistinct xs acc
else x # (rdistinct xs ({x} \<union> acc)))"
-lemma rdistinct_idem:
- shows "rdistinct (x # (rdistinct rs {x})) {} = x # (rdistinct rs {x})"
-
- sorry
-
@@ -199,7 +239,14 @@
| "rlist_size [] = 0"
thm neq_Nil_conv
-
+lemma hand_made_def_rlist_size:
+ shows "rlist_size rs = sum_list (map rsize rs)"
+proof (induct rs)
+ case Nil show ?case by simp
+next
+ case (Cons a rs) thus ?case
+ by simp
+qed
lemma rsimp_aalts_smaller:
shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
@@ -249,7 +296,7 @@
where
"cond_list r1 r2 s = rders_cond_list r2 (nullable_bools r1 (orderedPref s) ) (orderedSuf s)"
-thm rsimp_SEQ.simps
+
lemma rSEQ_mono:
shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"
apply auto
@@ -267,16 +314,83 @@
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 simp_all
+
apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
apply force
using rSEQ_mono
- apply presburger
- sorry
+ apply presburger
+ using rsimp_alts_mono by auto
lemma idiot:
shows "rsimp_SEQ RONE r = r"
@@ -284,21 +398,15 @@
apply simp_all
done
-lemma no_dup_after_simp:
- shows "RALTS rs = rsimp r \<Longrightarrow> distinct rs"
+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 longlist_withstands_rsimp_alts:
- shows "length rs \<ge> 2 \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
- sorry
-
-lemma no_alt_short_list_after_simp:
- shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
- sorry
lemma idiot2:
shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
@@ -552,6 +660,10 @@
RALTS (r # (cond_list r1 r2 (s @ [c]))) = RALTS (r # ((rder c r2) # (map (rder c) (cond_list r1 r2 s))))"
using suffix_plus1charn by blast
+lemma simp_flatten2:
+ shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
+ sorry
+
lemma simp_flatten:
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
@@ -831,14 +943,7 @@
)"
sorry
-lemma hand_made_def_rlist_size:
- shows "rlist_size rs = sum_list (map rsize rs)"
-proof (induct rs)
- case Nil show ?case by simp
-next
- case (Cons a rs) thus ?case
- by simp
-qed
+
(*this section deals with the property of distinctBy: creates a list without duplicates*)
lemma distinct_mono:
@@ -1017,7 +1122,7 @@
"star_update c r [] = []"
|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s))
then (s@[c]) # [c] # (star_update c r Ss)
- else s # (star_update c r Ss) )"
+ else (s@[c]) # (star_update c r Ss) )"
lemma star_update_case1:
shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
@@ -1025,16 +1130,68 @@
by force
lemma star_update_case2:
- shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = s # (star_update c r Ss)"
+ 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))] ))"
- sorry
+ 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))] ))"
- sorry
+ using head_one_more_simp rsimp_alts_idem by auto
+
lemma evolution_step1:
shows "rsimp
@@ -1056,6 +1213,225 @@
(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)
@@ -1075,11 +1451,8 @@
apply(subst List.list.map(2))
apply(subst evolution_step2)
apply simp
- apply(case_tac "rnullable (rders_simp r a)")
- apply(subst star_update_case1)
- apply simp
- apply(subst List.list.map)+
- sledgehammer
+
+
sorry
@@ -1094,7 +1467,8 @@
lemma ralts_vs_rsimpalts:
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
- sorry
+ 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
@@ -1156,14 +1530,119 @@
using star_list_push_der apply presburger
+ by (metis ralts_vs_rsimpalts starseq_list_evolution)
+
+
+lemma starder_is_a_list:
+ shows " \<exists>Ss. rders_simp (RSTAR r) s = rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) \<or> rders_simp (RSTAR r) s = RSTAR r"
+ apply(case_tac s)
+ prefer 2
+ apply (metis neq_Nil_conv starder_is_a_list_of_stars_or_starseqs)
+ apply simp
+ done
+
+
+(** start about bounds here**)
+
+
+lemma list_simp_size:
+ shows "rlist_size (map rsimp rs) \<le> rlist_size rs"
+ apply(induct rs)
+ apply simp
+ apply simp
+ apply (subgoal_tac "rsize (rsimp a) \<le> rsize a")
+ prefer 2
+ using rsimp_mono apply fastforce
+ using add_le_mono by presburger
+
+lemma inside_list_simp_inside_list:
+ shows "r \<in> set rs \<Longrightarrow> rsimp r \<in> set (map rsimp rs)"
+ apply (induct rs)
+ apply simp
+ apply auto
+ done
+
+
+lemma rsize_star_seq_list:
+ shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
+rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3"
sorry
+lemma rdistinct_bound_by_no_simp:
+ shows "
+
+ rlist_size (rdistinct (map rsimp rs) (set (map rsimp ss)))
+ \<le> (rlist_size (rdistinct rs (set ss)))
+"
+ apply(induct rs arbitrary: ss)
+ apply simp
+ apply(case_tac "a \<in> set ss")
+ apply(subgoal_tac "rsimp a \<in> set (map rsimp ss)")
+ prefer 2
+ using inside_list_simp_inside_list apply blast
+
+ apply simp
+ apply simp
+ by (metis List.set_insert add_le_mono image_insert insert_absorb rsimp_mono trans_le_add2)
+
+
+lemma starder_closed_form_bound_aux1:
+ shows
+"\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
+ Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {}))) "
+
+ sorry
+
+lemma starder_closed_form_bound:
+ shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3"
+ apply(subgoal_tac " \<exists>N3.\<forall>Ss.
+rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3")
+ prefer 2
+
+ using rsize_star_seq_list apply auto[1]
+ apply(erule exE)
+ apply(rule_tac x = "Suc N3" in exI)
+ apply(subgoal_tac "\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
+ Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {})))")
+ prefer 2
+ using starder_closed_form_bound_aux1 apply blast
+ by (meson less_trans_Suc linorder_not_le not_less_eq)
+
+
+thm starder_closed_form_bound_aux1
+
+(*
+ "ralts_vs_rsimpalts", , and "starder_closed_form_bound_aux1", which could be due to a bug in Sledgehammer or to inconsistent axioms (including "sorry"s)
+*)
+
+lemma starder_size_bound:
+ shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
+rsize (RSTAR r0) < N3"
+ apply(subgoal_tac " \<exists>N3.\<forall>Ss.
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3")
+ prefer 2
+ using starder_closed_form_bound apply blast
+ apply(erule exE)
+ apply(rule_tac x = "max N3 (Suc (rsize (RSTAR r0)))" in exI)
+ using less_max_iff_disj by blast
+
+
+
+
lemma finite_star:
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 )
\<Longrightarrow> \<exists>N3. \<forall>s.(rsize (rders_simp (RSTAR r0) s)) < N3"
-
- sorry
+ apply(subgoal_tac " \<exists>N3. \<forall>Ss.
+rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
+rsize (RSTAR r0) < N3")
+ prefer 2
+ using starder_size_bound apply blast
+ apply(erule exE)
+ apply(rule_tac x = N3 in exI)
+ by (metis starder_is_a_list)
lemma rderssimp_zero:
--- a/thys2/SizeBoundStrong.thy Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/SizeBoundStrong.thy Wed Mar 02 11:43:41 2022 +0000
@@ -585,17 +585,20 @@
| "bsimp_ASEQ1 bs r1 r2 = ASEQ bs r1 r2"
-fun collect where
+fun collect :: "rexp \<Rightarrow> rexp list \<Rightarrow> rexp list" where
\<open>collect _ [] = []\<close>
-| \<open>collect erasedR2 ((SEQ r1 r2) # rs) = (if r2 = erasedR2 then r1 # (collect erasedR2 rs)
- else collect erasedR2 rs)\<close>
+| \<open>collect erasedR2 ((SEQ r1 r2) # rs) =
+ (if r2 = erasedR2 then r1 # (collect erasedR2 rs) else collect erasedR2 rs)\<close>
| \<open>collect erasedR2 (r # rs) = collect erasedR2 rs\<close>
fun pruneRexp where
- \<open>pruneRexp (ASEQ bs r1 r2) allowableTerms =
-( let termsTruncated = (collect (erase r2) allowableTerms) in (let pruned = pruneRexp r1 termsTruncated in (bsimp_ASEQ1 bs pruned r2)) )\<close>
-| \<open>pruneRexp (AALTs bs rs) allowableTerms = (let rsp = (filter (\<lambda>r. r \<noteq> AZERO) (map (\<lambda>r. pruneRexp r allowableTerms) rs) ) in bsimp_AALTs bs rsp )
+ "pruneRexp (ASEQ bs r1 r2) allowableTerms =
+ (let termsTruncated = (collect (erase r2) allowableTerms) in
+ (let pruned = pruneRexp r1 termsTruncated in
+ (bsimp_ASEQ1 bs pruned r2)))"
+| \<open>pruneRexp (AALTs bs rs) allowableTerms =
+ (let rsp = (filter (\<lambda>r. r \<noteq> AZERO) (map (\<lambda>r. pruneRexp r allowableTerms) rs) ) in bsimp_AALTs bs rsp )
\<close>
| \<open>pruneRexp r allowableTerms = (if (erase r) \<in> (set allowableTerms) then r else AZERO)\<close>
--- a/thys2/blexer2.sc Mon Feb 21 23:38:26 2022 +0000
+++ b/thys2/blexer2.sc Wed Mar 02 11:43:41 2022 +0000
@@ -622,7 +622,9 @@
def bders_simp(s: List[Char], r: ARexp) : ARexp = s match {
case Nil => r
- case c::s => bders_simp(s, bsimp(bder(c, r)))
+ case c::s =>
+ println(erase(r))
+ bders_simp(s, bsimp(bder(c, r)))
}
def bdersSimp(s: String, r: Rexp) : ARexp = bders_simp(s.toList, internalise(r))
@@ -818,59 +820,60 @@
val pderSTAR = pderUNIV(STARREG)
val refSize = pderSTAR.map(size(_)).sum
- println("different partial derivative terms:")
- pderSTAR.foreach(r => r match {
+ // println("different partial derivative terms:")
+ // pderSTAR.foreach(r => r match {
- case SEQ(head, rstar) =>
- println(shortRexpOutput(head) ++ "~STARREG")
- case STAR(rstar) =>
- println("STARREG")
+ // case SEQ(head, rstar) =>
+ // println(shortRexpOutput(head) ++ "~STARREG")
+ // case STAR(rstar) =>
+ // println("STARREG")
- }
- )
- println("the total number of terms is")
- //println(refSize)
- println(pderSTAR.size)
+ // }
+ // )
+ // println("the total number of terms is")
+ // //println(refSize)
+ // println(pderSTAR.size)
val A : Rexp= ("c" | (ONE | "b") ~ "d") ~((ONE).%)
val B : Rexp = ((ONE).%)
val C : Rexp = ("d") ~ ((ONE).%)
val PRUNE_REG : Rexp = (C | B | A)
val APRUNE_REG = internalise(PRUNE_REG)
- // val program_solution = pruneRexp(APRUNE_REG, breakIntoTerms(PRUNE_REG))
- // println("program executes and gives: as disired!")
- // println(shortRexpOutput(erase(program_solution)))
- val simpedPruneReg = strongBsimp(APRUNE_REG)
- println(shortRexpOutput(erase(simpedPruneReg)))
- for(i <- List(100, 900 ) ){// 100, 400, 800, 840, 841, 900
- val prog0 = "a" * i
- //println(s"test: $prog0")
- println(s"testing with $i a's" )
- //val bd = bdersSimp(prog0, STARREG)//DB
- val sbd = bdersSimpS(prog0, STARREG)//strongDB
- starPrint(erase(sbd))
- val subTerms = breakIntoTerms(erase(sbd))
- //val subTermsLarge = breakIntoTerms(erase(bd))
+ // // val program_solution = pruneRexp(APRUNE_REG, breakIntoTerms(PRUNE_REG))
+ // // println("program executes and gives: as disired!")
+ // // println(shortRexpOutput(erase(program_solution)))
+ // val simpedPruneReg = strongBsimp(APRUNE_REG)
+
+ // println(shortRexpOutput(erase(simpedPruneReg)))
+ // for(i <- List(1,2 ) ){// 100, 400, 800, 840, 841, 900
+ // val prog0 = "a" * i
+ // //println(s"test: $prog0")
+ // println(s"testing with $i a's" )
+ // //val bd = bdersSimp(prog0, STARREG)//DB
+ // val sbd = bdersSimpS(prog0, STARREG)//strongDB
+ // starPrint(erase(sbd))
+ // val subTerms = breakIntoTerms(erase(sbd))
+ // //val subTermsLarge = breakIntoTerms(erase(bd))
- println(s"subterms of regex with strongDB: ${subTerms.length}")//, standard DB: ${subTermsLarge.length}")
+ // println(s"subterms of regex with strongDB: ${subTerms.length}")//, standard DB: ${subTermsLarge.length}")
- println("the number of distinct subterms for bsimp with strongDB")
- println(subTerms.distinct.size)
- //println(subTermsLarge.distinct.size)
- println("which coincides with the number of PDER terms")
+ // println("the number of distinct subterms for bsimp with strongDB")
+ // println(subTerms.distinct.size)
+ // //println(subTermsLarge.distinct.size)
+ // println("which coincides with the number of PDER terms")
- // println(shortRexpOutput(erase(sbd)))
- // println(shortRexpOutput(erase(bd)))
+ // // println(shortRexpOutput(erase(sbd)))
+ // // println(shortRexpOutput(erase(bd)))
- println("pdersize, original, strongSimp")
- println(refSize, size(STARREG), asize(sbd))
+ // println("pdersize, original, strongSimp")
+ // println(refSize, size(STARREG), asize(sbd))
- val vres = strong_blexing_simp( STARREG, prog0)
- println(vres)
- }
+ // val vres = strong_blexing_simp( STARREG, prog0)
+ // println(vres)
+ // }
// println(vs.length)
// println(vs)
@@ -878,6 +881,9 @@
// val prog1 = """read n; write n"""
// println(s"test: $prog1")
// println(lexing_simp(WHILE_REGS, prog1))
+ val display = ("a"| "ab").%
+ val adisplay = internalise(display)
+ bders_simp( "aaaaa".toList, adisplay)
}
small()
Binary file thys2/paper.pdf has changed