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))))
  TDG 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))))
   graph:
    log#(s(s(x))) -> log#(s(half(x))) ->
    log#(s(s(x))) -> log#(s(half(x)))
    log#(s(s(x))) -> log#(s(half(x))) -> log#(s(s(x))) -> half#(x)
    log#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
   SCC Processor:
    #sccs: 2
    #rules: 2
    #arcs: 4/9
    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))))
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [log#](x0) = 3x0,
      
      [log](x0) = x0 + -16,
      
      [s](x0) = 2x0 + -8,
      
      [half](x0) = x0 + -10,
      
      [0] = 2
     orientation:
      log#(s(s(x))) = 7x + -3 >= 5x + -5 = log#(s(half(x)))
      
      half(0()) = 2 >= 2 = 0()
      
      half(s(s(x))) = 4x + -6 >= 2x + -8 = s(half(x))
      
      log(s(0())) = 4 >= 2 = 0()
      
      log(s(s(x))) = 4x + -6 >= 4x + -6 = 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
    
    DPs:
     half#(s(s(x))) -> 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))))
    Subterm Criterion Processor:
     simple projection:
      pi(half#) = 0
     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