--- a/Brexp.scala	Wed Mar 13 13:33:54 2019 +0000
+++ b/Brexp.scala	Wed Mar 13 15:27:09 2019 +0000
@@ -52,8 +52,6 @@
     case BSTAR(r) => BSEQ(brder(c, r), BSTAR(r))
   }
   def bflat(rs: List[BRexp]): List[BRexp] = {
-    //println("bs: " + bs + "rs: "+ rs + "length of bs, rs" + bs.length + ", " + rs.length)
-    //assert(bs.length == rs.length - 1 || bs.length == rs.length)//bs == Nil, rs == Nil  May add early termination later.
     rs match {
       case Nil => Nil//TODO: Look at this! bs should have been exhausted one bit earlier.
       case BZERO :: rs1 => bflat(rs1)
@@ -130,31 +128,52 @@
     //case BSTAR(r) => BSTAR(r)
     case r => r
   }
+  //we want to bound the size by a function bspill s.t.
+  //bspill(ders_simp(r,s)) ⊂  PD(r) 
+  //and bspill is size-preserving (up to a constant factor)
+  //so we bound |ders_simp(r,s)| by |PD(r)|
+  //where we already have a nice bound for |PD(r)|: t^2*n^2 in Antimirov's paper
 
-  def break_chain(bs: Bit, rs: List[BRexp]): List[Rexp] = {
-    //do it in a functional style
-    if(bs == S)
-      rs.flatMap(bspill)
-    else
-      List(ALTS(rs.map(berase)))
-  }
+  //the function bspill converts a brexp into a list of rexps
+  //the conversion mainly about this: (r1+r2)*r3*r4*.....rn -> {r1*r3*r4*....rn, r2*r3*r4*.....rn} 
+  //but this is not always allowed
+  //sometimes, we just leave the expression as it is: 
+  //eg1: (a+b)*c -> {(a+b)*c}
+  //eg2: r1+r2 -> {r1+r2} instead of {r1, r2}
+  //why?
+  //if we always return {a, b} when we encounter a+b, the property
+  //bspill(ders_simp(r,s)) ⊂  PD(r) 
+  //does not always hold
+  //for instance 
+  //if we call the bspill that always returns {r1, r2} when encountering r1+r2 "bspilli"
+  //then bspilli( ders_simp( (c+cb)+c(a+b), c ) ) == bspilli(1+b+a) = {1, b, a}
+  //However the letter a does not belong to PD( (c+cb)+c(a+b) )
+  //then we can no longer say ders_simp(r,s)'s size is bounded by PD(r) because the former contains something the latter does not have
+  //In order to make sure the inclusion holds, we have to find out why new terms appeared in the bspilli set that don't exist in the PD(r) set
+  //Why a does not belong to PD((c+cb)+c(a+b))?
+  //PD(r1+r2) = PD(r1) union PD(r2) => PD((c+cb)+c(a+b)) = PD(c+cb) union PD(c(a+b))
+  //So the only possibility for PD to include a must be in the latter part of the regex, namely, c(a+b)
+  //we have lemma that PD(r) = union of pders(s, r) where s is taken over all strings whose length does not exceed depth(r)
+  //so PD(r) ⊂ pder(empty_string, r) union pder(c, r) union pder(ca, r) union pder(cb, r) where r = c(a+b)
+  //RHS = {1} union pder(c, c(a+b))
+  //Observe that pder(c, c(a+b)) == {a+b}
+  //a and b are together, from the original regular expression (c+cb)+c(a+b).
+  //but by our simplification, we first flattened this a+b into the same level with 1+b, then
+  //removed duplicates of b, thereby destroying the structure in a+b and making this term a, instead of a+b
+  //But in PD(r) we only have a+b, no a
+  //one ad hoc solution might be to try this bspill(ders_simp(r,s)) ⊂  PD(r) union {all letters}
+  //But this does not hold either according to experiment.
+  //So we need to make sure the structure r1+r2 is not simplified away if it is from the original expression
   
-  //this function converts a brexp into a list of rexps
-  //the conversion mainly about this: (a+b)*c -> {a*c, b*c} if a+b is generated from scratch (e.g. when deriving a seq)
-  //or is from the original regex but have been "touched" (i.e. have been derived)
-  //Why make this distinction? Look at the following example:
-  //r = (c+cb)+c(a+b)
-  // r\c = (1+b)+(a+b)
-  //after simplification
-  // (r\c)simp= 1+b+a
-  //we lost the structure that tells us 1+b should be grouped together and a grouped as itself
-  //however in our brexp simplification, 
-  //(r\c)br_simp = 1+b+(a+b)
-  //we do not allow the bracket to be opened as it is from the original expression and have not been touched
+
+  
   def bspill(r: BRexp): List[Rexp] = {
       r match {
-          case BALTS(bs, rs) => {
-            break_chain(bs, rs)
+          case BALTS(b, rs) => {
+            if(b == S)
+              rs.flatMap(bspill)
+            else
+              List(ALTS(rs.map(berase)))
           }
           case BSEQ(r2, r3) => bspill(r2).map( a => if(a == ONE) berase(r3) else SEQ(a, berase(r3)) )
           case BZERO => List()