YES

Problem:
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 inc(0()) -> 0()
 inc(s(x)) -> s(inc(x))
 zero(0()) -> true()
 zero(s(x)) -> false()
 p(0()) -> 0()
 p(s(x)) -> x
 bits(x) -> bitIter(x,0())
 bitIter(x,y) -> if(zero(x),x,inc(y))
 if(true(),x,y) -> p(y)
 if(false(),x,y) -> bitIter(half(x),y)

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   inc#(s(x)) -> inc#(x)
   bits#(x) -> bitIter#(x,0())
   bitIter#(x,y) -> inc#(y)
   bitIter#(x,y) -> zero#(x)
   bitIter#(x,y) -> if#(zero(x),x,inc(y))
   if#(true(),x,y) -> p#(y)
   if#(false(),x,y) -> half#(x)
   if#(false(),x,y) -> bitIter#(half(x),y)
  TRS:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   inc(0()) -> 0()
   inc(s(x)) -> s(inc(x))
   zero(0()) -> true()
   zero(s(x)) -> false()
   p(0()) -> 0()
   p(s(x)) -> x
   bits(x) -> bitIter(x,0())
   bitIter(x,y) -> if(zero(x),x,inc(y))
   if(true(),x,y) -> p(y)
   if(false(),x,y) -> bitIter(half(x),y)
  Usable Rule Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    inc#(s(x)) -> inc#(x)
    bits#(x) -> bitIter#(x,0())
    bitIter#(x,y) -> inc#(y)
    bitIter#(x,y) -> zero#(x)
    bitIter#(x,y) -> if#(zero(x),x,inc(y))
    if#(true(),x,y) -> p#(y)
    if#(false(),x,y) -> half#(x)
    if#(false(),x,y) -> bitIter#(half(x),y)
   TRS:
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    zero(0()) -> true()
    zero(s(x)) -> false()
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
   Matrix Interpretation Processor: dim=1
    
    usable rules:
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     zero(0()) -> true()
     zero(s(x)) -> false()
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
    interpretation:
     [if#](x0, x1, x2) = 1/2x0 + x1 + 1/2x2 + 1/2,
     
     [bitIter#](x0, x1) = 2x0 + 1/2x1 + 1,
     
     [bits#](x0) = 3x0 + 3/2,
     
     [p#](x0) = 1/2x0,
     
     [zero#](x0) = 0,
     
     [inc#](x0) = 1/2x0,
     
     [half#](x0) = x0,
     
     [false] = 2,
     
     [true] = 0,
     
     [zero](x0) = 2x0,
     
     [inc](x0) = x0,
     
     [s](x0) = x0 + 2,
     
     [half](x0) = 1/2x0,
     
     [0] = 0
    orientation:
     half#(s(s(x))) = x + 4 >= x = half#(x)
     
     inc#(s(x)) = 1/2x + 1 >= 1/2x = inc#(x)
     
     bits#(x) = 3x + 3/2 >= 2x + 1 = bitIter#(x,0())
     
     bitIter#(x,y) = 2x + 1/2y + 1 >= 1/2y = inc#(y)
     
     bitIter#(x,y) = 2x + 1/2y + 1 >= 0 = zero#(x)
     
     bitIter#(x,y) = 2x + 1/2y + 1 >= 2x + 1/2y + 1/2 = if#(zero(x),x,inc(y))
     
     if#(true(),x,y) = x + 1/2y + 1/2 >= 1/2y = p#(y)
     
     if#(false(),x,y) = x + 1/2y + 3/2 >= x = half#(x)
     
     if#(false(),x,y) = x + 1/2y + 3/2 >= x + 1/2y + 1 = bitIter#(half(x),y)
     
     inc(0()) = 0 >= 0 = 0()
     
     inc(s(x)) = x + 2 >= x + 2 = s(inc(x))
     
     zero(0()) = 0 >= 0 = true()
     
     zero(s(x)) = 2x + 4 >= 2 = false()
     
     half(0()) = 0 >= 0 = 0()
     
     half(s(0())) = 1 >= 0 = 0()
     
     half(s(s(x))) = 1/2x + 2 >= 1/2x + 2 = s(half(x))
    problem:
     DPs:
      
     TRS:
      inc(0()) -> 0()
      inc(s(x)) -> s(inc(x))
      zero(0()) -> true()
      zero(s(x)) -> false()
      half(0()) -> 0()
      half(s(0())) -> 0()
      half(s(s(x))) -> s(half(x))
    Qed