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)
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [if#](x0, x1, x2) = x0 + x1 + 2x2 + 1/2,
    
    [bitIter#](x0, x1) = 2x0 + 2x1 + 3,
    
    [bits#](x0) = 3x0 + 7/2,
    
    [p#](x0) = 0,
    
    [zero#](x0) = 0,
    
    [inc#](x0) = x0 + 2,
    
    [half#](x0) = x0 + 1,
    
    [if](x0, x1, x2) = 3/2x1 + 1/2x2,
    
    [bitIter](x0, x1) = 2x0 + 1/2x1,
    
    [bits](x0) = 2x0,
    
    [p](x0) = 1/2x0,
    
    [false] = 3,
    
    [true] = 0,
    
    [zero](x0) = x0 + 2,
    
    [inc](x0) = x0,
    
    [s](x0) = 3x0 + 2,
    
    [half](x0) = 1/2x0,
    
    [0] = 0
   orientation:
    half#(s(s(x))) = 9x + 9 >= x + 1 = half#(x)
    
    inc#(s(x)) = 3x + 4 >= x + 2 = inc#(x)
    
    bits#(x) = 3x + 7/2 >= 2x + 3 = bitIter#(x,0())
    
    bitIter#(x,y) = 2x + 2y + 3 >= y + 2 = inc#(y)
    
    bitIter#(x,y) = 2x + 2y + 3 >= 0 = zero#(x)
    
    bitIter#(x,y) = 2x + 2y + 3 >= 2x + 2y + 5/2 = if#(zero(x),x,inc(y))
    
    if#(true(),x,y) = x + 2y + 1/2 >= 0 = p#(y)
    
    if#(false(),x,y) = x + 2y + 7/2 >= x + 1 = half#(x)
    
    if#(false(),x,y) = x + 2y + 7/2 >= x + 2y + 3 = bitIter#(half(x),y)
    
    half(0()) = 0 >= 0 = 0()
    
    half(s(0())) = 1 >= 0 = 0()
    
    half(s(s(x))) = 9/2x + 4 >= 3/2x + 2 = s(half(x))
    
    inc(0()) = 0 >= 0 = 0()
    
    inc(s(x)) = 3x + 2 >= 3x + 2 = s(inc(x))
    
    zero(0()) = 2 >= 0 = true()
    
    zero(s(x)) = 3x + 4 >= 3 = false()
    
    p(0()) = 0 >= 0 = 0()
    
    p(s(x)) = 3/2x + 1 >= x = x
    
    bits(x) = 2x >= 2x = bitIter(x,0())
    
    bitIter(x,y) = 2x + 1/2y >= 3/2x + 1/2y = if(zero(x),x,inc(y))
    
    if(true(),x,y) = 3/2x + 1/2y >= 1/2y = p(y)
    
    if(false(),x,y) = 3/2x + 1/2y >= x + 1/2y = bitIter(half(x),y)
   problem:
    DPs:
     
    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)
   Qed