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)
   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))
     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))) -> half#(x)
     log#(s(s(x))) -> log#(s(half(x))) -> log#(s(s(x))) -> 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))))
     KBO Processor:
      argument filtering:
       pi(0) = []
       pi(half) = 0
       pi(s) = [0]
       pi(log) = 0
       pi(log#) = [0]
      weight function:
       w0 = 1
       w(log#) = w(log) = w(0) = 1
       w(s) = w(half) = 0
      precedence:
       log# ~ s > log ~ 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