YES

Problem:
 lt(0(),s(x)) -> true()
 lt(x,0()) -> false()
 lt(s(x),s(y)) -> lt(x,y)
 logarithm(x) -> ifa(lt(0(),x),x)
 ifa(true(),x) -> help(x,1())
 ifa(false(),x) -> logZeroError()
 help(x,y) -> ifb(lt(y,x),x,y)
 ifb(true(),x,y) -> help(half(x),s(y))
 ifb(false(),x,y) -> y
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))

Proof:
 DP Processor:
  DPs:
   lt#(s(x),s(y)) -> lt#(x,y)
   logarithm#(x) -> lt#(0(),x)
   logarithm#(x) -> ifa#(lt(0(),x),x)
   ifa#(true(),x) -> help#(x,1())
   help#(x,y) -> lt#(y,x)
   help#(x,y) -> ifb#(lt(y,x),x,y)
   ifb#(true(),x,y) -> half#(x)
   ifb#(true(),x,y) -> help#(half(x),s(y))
   half#(s(s(x))) -> half#(x)
  TRS:
   lt(0(),s(x)) -> true()
   lt(x,0()) -> false()
   lt(s(x),s(y)) -> lt(x,y)
   logarithm(x) -> ifa(lt(0(),x),x)
   ifa(true(),x) -> help(x,1())
   ifa(false(),x) -> logZeroError()
   help(x,y) -> ifb(lt(y,x),x,y)
   ifb(true(),x,y) -> help(half(x),s(y))
   ifb(false(),x,y) -> y
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
  Usable Rule Processor:
   DPs:
    lt#(s(x),s(y)) -> lt#(x,y)
    logarithm#(x) -> lt#(0(),x)
    logarithm#(x) -> ifa#(lt(0(),x),x)
    ifa#(true(),x) -> help#(x,1())
    help#(x,y) -> lt#(y,x)
    help#(x,y) -> ifb#(lt(y,x),x,y)
    ifb#(true(),x,y) -> half#(x)
    ifb#(true(),x,y) -> help#(half(x),s(y))
    half#(s(s(x))) -> half#(x)
   TRS:
    lt(0(),s(x)) -> true()
    lt(x,0()) -> false()
    lt(s(x),s(y)) -> lt(x,y)
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     lt(0(),s(x)) -> true()
     lt(x,0()) -> false()
     lt(s(x),s(y)) -> lt(x,y)
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
    interpretation:
     [half#](x0) = -4x0 + 0,
     
     [ifb#](x0, x1, x2) = -2x0 + x1,
     
     [help#](x0, x1) = 1x0 + 1,
     
     [ifa#](x0, x1) = 1x0 + 6x1 + 0,
     
     [logarithm#](x0) = 12x0 + 11,
     
     [lt#](x0, x1) = x1 + 0,
     
     [half](x0) = -2x0 + 1,
     
     [1] = 8,
     
     [false] = 1,
     
     [true] = 5,
     
     [lt](x0, x1) = 2x1,
     
     [s](x0) = 3x0 + 4,
     
     [0] = 0
    orientation:
     lt#(s(x),s(y)) = 3y + 4 >= y + 0 = lt#(x,y)
     
     logarithm#(x) = 12x + 11 >= x + 0 = lt#(0(),x)
     
     logarithm#(x) = 12x + 11 >= 6x + 0 = ifa#(lt(0(),x),x)
     
     ifa#(true(),x) = 6x + 6 >= 1x + 1 = help#(x,1())
     
     help#(x,y) = 1x + 1 >= x + 0 = lt#(y,x)
     
     help#(x,y) = 1x + 1 >= x = ifb#(lt(y,x),x,y)
     
     ifb#(true(),x,y) = x + 3 >= -4x + 0 = half#(x)
     
     ifb#(true(),x,y) = x + 3 >= -1x + 2 = help#(half(x),s(y))
     
     half#(s(s(x))) = 2x + 3 >= -4x + 0 = half#(x)
     
     lt(0(),s(x)) = 5x + 6 >= 5 = true()
     
     lt(x,0()) = 2 >= 1 = false()
     
     lt(s(x),s(y)) = 5y + 6 >= 2y = lt(x,y)
     
     half(0()) = 1 >= 0 = 0()
     
     half(s(0())) = 2 >= 0 = 0()
     
     half(s(s(x))) = 4x + 5 >= 1x + 4 = s(half(x))
    problem:
     DPs:
      
     TRS:
      lt(0(),s(x)) -> true()
      lt(x,0()) -> false()
      lt(s(x),s(y)) -> lt(x,y)
      half(0()) -> 0()
      half(s(0())) -> 0()
      half(s(s(x))) -> s(half(x))
    Qed