YES

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

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   log#(s(s(x))) -> half#(x)
   log#(s(s(x))) -> log#(s(half(x)))
  TRS:
   half(0()) -> 0()
   half(s(s(x))) -> s(half(x))
   log(s(0())) -> 0()
   log(s(s(x))) -> s(log(s(half(x))))
  Usable Rule Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    log#(s(s(x))) -> half#(x)
    log#(s(s(x))) -> log#(s(half(x)))
   TRS:
    half(0()) -> 0()
    half(s(s(x))) -> s(half(x))
   CDG Processor:
    DPs:
     half#(s(s(x))) -> half#(x)
     log#(s(s(x))) -> half#(x)
     log#(s(s(x))) -> log#(s(half(x)))
    TRS:
     half(0()) -> 0()
     half(s(s(x))) -> s(half(x))
    graph:
     log#(s(s(x))) -> log#(s(half(x))) ->
     log#(s(s(x))) -> half#(x)
     log#(s(s(x))) -> log#(s(half(x))) -> log#(s(s(x))) -> log#(s(half(x)))
    Restore Modifier:
     DPs:
      half#(s(s(x))) -> half#(x)
      log#(s(s(x))) -> half#(x)
      log#(s(s(x))) -> log#(s(half(x)))
     TRS:
      half(0()) -> 0()
      half(s(s(x))) -> s(half(x))
      log(s(0())) -> 0()
      log(s(s(x))) -> s(log(s(half(x))))
     SCC Processor:
      #sccs: 1
      #rules: 1
      #arcs: 2/9
      DPs:
       log#(s(s(x))) -> log#(s(half(x)))
      TRS:
       half(0()) -> 0()
       half(s(s(x))) -> s(half(x))
       log(s(0())) -> 0()
       log(s(s(x))) -> s(log(s(half(x))))
      Matrix Interpretation Processor:
       dimension: 1
       interpretation:
        [log#](x0) = x0 + 1,
        
        [log](x0) = x0,
        
        [s](x0) = x0 + 1,
        
        [half](x0) = x0,
        
        [0] = 0
       orientation:
        log#(s(s(x))) = x + 3 >= x + 2 = log#(s(half(x)))
        
        half(0()) = 0 >= 0 = 0()
        
        half(s(s(x))) = x + 2 >= x + 1 = s(half(x))
        
        log(s(0())) = 1 >= 0 = 0()
        
        log(s(s(x))) = x + 2 >= x + 2 = s(log(s(half(x))))
       problem:
        DPs:
         
        TRS:
         half(0()) -> 0()
         half(s(s(x))) -> s(half(x))
         log(s(0())) -> 0()
         log(s(s(x))) -> s(log(s(half(x))))
       Qed