YES

Problem:
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 s(log(0())) -> s(0())
 log(s(x)) -> s(log(half(s(x))))

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   half#(s(s(x))) -> s#(half(x))
   s#(log(0())) -> s#(0())
   log#(s(x)) -> half#(s(x))
   log#(s(x)) -> log#(half(s(x)))
   log#(s(x)) -> s#(log(half(s(x))))
  TRS:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   s(log(0())) -> s(0())
   log(s(x)) -> s(log(half(s(x))))
  TDG Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    half#(s(s(x))) -> s#(half(x))
    s#(log(0())) -> s#(0())
    log#(s(x)) -> half#(s(x))
    log#(s(x)) -> log#(half(s(x)))
    log#(s(x)) -> s#(log(half(s(x))))
   TRS:
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
    s(log(0())) -> s(0())
    log(s(x)) -> s(log(half(s(x))))
   graph:
    log#(s(x)) -> log#(half(s(x))) ->
    log#(s(x)) -> s#(log(half(s(x))))
    log#(s(x)) -> log#(half(s(x))) -> log#(s(x)) -> log#(half(s(x)))
    log#(s(x)) -> log#(half(s(x))) -> log#(s(x)) -> half#(s(x))
    log#(s(x)) -> s#(log(half(s(x)))) -> s#(log(0())) -> s#(0())
    log#(s(x)) -> half#(s(x)) -> half#(s(s(x))) -> s#(half(x))
    log#(s(x)) -> half#(s(x)) -> half#(s(s(x))) -> half#(x)
    s#(log(0())) -> s#(0()) -> s#(log(0())) -> s#(0())
    half#(s(s(x))) -> s#(half(x)) -> s#(log(0())) -> s#(0())
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> s#(half(x))
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
   SCC Processor:
    #sccs: 3
    #rules: 3
    #arcs: 10/36
    DPs:
     log#(s(x)) -> log#(half(s(x)))
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     s(log(0())) -> s(0())
     log(s(x)) -> s(log(half(s(x))))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [log#](x0) = 1x0 + 0,
      
      [log](x0) = x0 + -16,
      
      [s](x0) = 3x0 + 4,
      
      [half](x0) = -3x0 + 1,
      
      [0] = 0
     orientation:
      log#(s(x)) = 4x + 5 >= 1x + 2 = log#(half(s(x)))
      
      half(0()) = 1 >= 0 = 0()
      
      half(s(0())) = 1 >= 0 = 0()
      
      half(s(s(x))) = 3x + 4 >= x + 4 = s(half(x))
      
      s(log(0())) = 4 >= 4 = s(0())
      
      log(s(x)) = 3x + 4 >= 3x + 4 = s(log(half(s(x))))
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       s(log(0())) -> s(0())
       log(s(x)) -> s(log(half(s(x))))
     Qed
    
    DPs:
     half#(s(s(x))) -> half#(x)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     s(log(0())) -> s(0())
     log(s(x)) -> s(log(half(s(x))))
    Subterm Criterion Processor:
     simple projection:
      pi(half#) = 0
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       s(log(0())) -> s(0())
       log(s(x)) -> s(log(half(s(x))))
     Qed
    
    DPs:
     s#(log(0())) -> s#(0())
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     s(log(0())) -> s(0())
     log(s(x)) -> s(log(half(s(x))))
    Subterm Criterion Processor:
     simple projection:
      pi(s#) = 0
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       s(log(0())) -> s(0())
       log(s(x)) -> s(log(half(s(x))))
     Qed