YES

Problem:
 half(0()) -> 0()
 half(s(0())) -> 0()
 half(s(s(x))) -> s(half(x))
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 inc(0()) -> 0()
 inc(s(x)) -> s(inc(x))
 log(x) -> log2(x,0())
 log2(x,y) -> if(le(x,s(0())),x,inc(y))
 if(true(),x,s(y)) -> y
 if(false(),x,y) -> log2(half(x),y)

Proof:
 DP Processor:
  DPs:
   half#(s(s(x))) -> half#(x)
   le#(s(x),s(y)) -> le#(x,y)
   inc#(s(x)) -> inc#(x)
   log#(x) -> log2#(x,0())
   log2#(x,y) -> inc#(y)
   log2#(x,y) -> le#(x,s(0()))
   log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
   if#(false(),x,y) -> half#(x)
   if#(false(),x,y) -> log2#(half(x),y)
  TRS:
   half(0()) -> 0()
   half(s(0())) -> 0()
   half(s(s(x))) -> s(half(x))
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   inc(0()) -> 0()
   inc(s(x)) -> s(inc(x))
   log(x) -> log2(x,0())
   log2(x,y) -> if(le(x,s(0())),x,inc(y))
   if(true(),x,s(y)) -> y
   if(false(),x,y) -> log2(half(x),y)
  TDG Processor:
   DPs:
    half#(s(s(x))) -> half#(x)
    le#(s(x),s(y)) -> le#(x,y)
    inc#(s(x)) -> inc#(x)
    log#(x) -> log2#(x,0())
    log2#(x,y) -> inc#(y)
    log2#(x,y) -> le#(x,s(0()))
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    if#(false(),x,y) -> half#(x)
    if#(false(),x,y) -> log2#(half(x),y)
   TRS:
    half(0()) -> 0()
    half(s(0())) -> 0()
    half(s(s(x))) -> s(half(x))
    le(0(),y) -> true()
    le(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    inc(0()) -> 0()
    inc(s(x)) -> s(inc(x))
    log(x) -> log2(x,0())
    log2(x,y) -> if(le(x,s(0())),x,inc(y))
    if(true(),x,s(y)) -> y
    if(false(),x,y) -> log2(half(x),y)
   graph:
    if#(false(),x,y) -> log2#(half(x),y) ->
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    if#(false(),x,y) -> log2#(half(x),y) ->
    log2#(x,y) -> le#(x,s(0()))
    if#(false(),x,y) -> log2#(half(x),y) -> log2#(x,y) -> inc#(y)
    if#(false(),x,y) -> half#(x) ->
    half#(s(s(x))) -> half#(x)
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y)) ->
    if#(false(),x,y) -> log2#(half(x),y)
    log2#(x,y) -> if#(le(x,s(0())),x,inc(y)) -> if#(false(),x,y) -> half#(x)
    log2#(x,y) -> inc#(y) -> inc#(s(x)) -> inc#(x)
    log2#(x,y) -> le#(x,s(0())) -> le#(s(x),s(y)) -> le#(x,y)
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> le#(x,s(0()))
    log#(x) -> log2#(x,0()) -> log2#(x,y) -> inc#(y)
    inc#(s(x)) -> inc#(x) -> inc#(s(x)) -> inc#(x)
    le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(x,y)
    half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
   SCC Processor:
    #sccs: 4
    #rules: 5
    #arcs: 14/81
    DPs:
     if#(false(),x,y) -> log2#(half(x),y)
     log2#(x,y) -> if#(le(x,s(0())),x,inc(y))
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(half(x),y)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [if#](x0, x1, x2) = x0 + -1x1 + -16,
      
      [log2#](x0, x1) = 1x0 + -1,
      
      [if](x0, x1, x2) = x0 + -1x1 + x2 + -1,
      
      [log2](x0, x1) = 1x0 + x1 + 1,
      
      [log](x0) = 1x0 + 15,
      
      [inc](x0) = x0,
      
      [false] = 1,
      
      [true] = 0,
      
      [le](x0, x1) = x0 + -4x1 + -16,
      
      [s](x0) = 2x0 + 2,
      
      [half](x0) = -2x0 + 0,
      
      [0] = 0
     orientation:
      if#(false(),x,y) = -1x + 1 >= -1x + 1 = log2#(half(x),y)
      
      log2#(x,y) = 1x + -1 >= x + -2 = if#(le(x,s(0())),x,inc(y))
      
      half(0()) = 0 >= 0 = 0()
      
      half(s(0())) = 0 >= 0 = 0()
      
      half(s(s(x))) = 2x + 2 >= x + 2 = s(half(x))
      
      le(0(),y) = -4y + 0 >= 0 = true()
      
      le(s(x),0()) = 2x + 2 >= 1 = false()
      
      le(s(x),s(y)) = 2x + -2y + 2 >= x + -4y + -16 = le(x,y)
      
      inc(0()) = 0 >= 0 = 0()
      
      inc(s(x)) = 2x + 2 >= 2x + 2 = s(inc(x))
      
      log(x) = 1x + 15 >= 1x + 1 = log2(x,0())
      
      log2(x,y) = 1x + y + 1 >= x + y + -1 = if(le(x,s(0())),x,inc(y))
      
      if(true(),x,s(y)) = -1x + 2y + 2 >= y = y
      
      if(false(),x,y) = -1x + y + 1 >= -1x + y + 1 = log2(half(x),y)
     problem:
      DPs:
       if#(false(),x,y) -> log2#(half(x),y)
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(half(x),y)
     SCC Processor:
      #sccs: 0
      #rules: 0
      #arcs: 2/1
      
    
    DPs:
     half#(s(s(x))) -> half#(x)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(half(x),y)
    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))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(half(x),y)
     Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(half(x),y)
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(half(x),y)
     Qed
    
    DPs:
     inc#(s(x)) -> inc#(x)
    TRS:
     half(0()) -> 0()
     half(s(0())) -> 0()
     half(s(s(x))) -> s(half(x))
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     inc(0()) -> 0()
     inc(s(x)) -> s(inc(x))
     log(x) -> log2(x,0())
     log2(x,y) -> if(le(x,s(0())),x,inc(y))
     if(true(),x,s(y)) -> y
     if(false(),x,y) -> log2(half(x),y)
    Subterm Criterion Processor:
     simple projection:
      pi(inc#) = 0
     problem:
      DPs:
       
      TRS:
       half(0()) -> 0()
       half(s(0())) -> 0()
       half(s(s(x))) -> s(half(x))
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       inc(0()) -> 0()
       inc(s(x)) -> s(inc(x))
       log(x) -> log2(x,0())
       log2(x,y) -> if(le(x,s(0())),x,inc(y))
       if(true(),x,s(y)) -> y
       if(false(),x,y) -> log2(half(x),y)
     Qed