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)
   EDG 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)) -> 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)) -> s#(log(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))) -> half#(x)
     log#(s(x)) -> half#(s(x)) -> half#(s(s(x))) -> s#(half(x))
     half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
     half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> s#(half(x))
    CDG 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())
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 4/36
      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))))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [log#](x0) = 2x0 + 1/2,
        
        [log](x0) = 2x0,
        
        [s](x0) = x0 + 1,
        
        [half](x0) = 1/2x0,
        
        [0] = 0
       orientation:
        log#(s(x)) = 2x + 5/2 >= x + 3/2 = log#(half(s(x)))
        
        half(0()) = 0 >= 0 = 0()
        
        half(s(0())) = 1/2 >= 0 = 0()
        
        half(s(s(x))) = 1/2x + 1 >= 1/2x + 1 = s(half(x))
        
        s(log(0())) = 1 >= 1 = s(0())
        
        log(s(x)) = 2x + 2 >= x + 2 = 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