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)
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [if#](x0, x1, x2) = 1/2x0 + x1 + 2x2,
    
    [log2#](x0, x1) = 2x0 + 2x1 + 1/2,
    
    [log#](x0) = 2x0 + 1,
    
    [inc#](x0) = 1/2x0,
    
    [le#](x0, x1) = 2x0,
    
    [half#](x0) = 1/2x0 + 1/2,
    
    [if](x0, x1, x2) = x1 + x2,
    
    [log2](x0, x1) = 2x0 + x1,
    
    [log](x0) = 3x0,
    
    [inc](x0) = x0,
    
    [false] = 2,
    
    [true] = 0,
    
    [le](x0, x1) = 2x0,
    
    [s](x0) = x0 + 1,
    
    [half](x0) = 1/2x0,
    
    [0] = 0
   orientation:
    half#(s(s(x))) = 1/2x + 3/2 >= 1/2x + 1/2 = half#(x)
    
    le#(s(x),s(y)) = 2x + 2 >= 2x = le#(x,y)
    
    inc#(s(x)) = 1/2x + 1/2 >= 1/2x = inc#(x)
    
    log#(x) = 2x + 1 >= 2x + 1/2 = log2#(x,0())
    
    log2#(x,y) = 2x + 2y + 1/2 >= 1/2y = inc#(y)
    
    log2#(x,y) = 2x + 2y + 1/2 >= 2x = le#(x,s(0()))
    
    log2#(x,y) = 2x + 2y + 1/2 >= 2x + 2y = if#(le(x,s(0())),x,inc(y))
    
    if#(false(),x,y) = x + 2y + 1 >= 1/2x + 1/2 = half#(x)
    
    if#(false(),x,y) = x + 2y + 1 >= x + 2y + 1/2 = log2#(half(x),y)
    
    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))
    
    le(0(),y) = 0 >= 0 = true()
    
    le(s(x),0()) = 2x + 2 >= 2 = false()
    
    le(s(x),s(y)) = 2x + 2 >= 2x = le(x,y)
    
    inc(0()) = 0 >= 0 = 0()
    
    inc(s(x)) = x + 1 >= x + 1 = s(inc(x))
    
    log(x) = 3x >= 2x = log2(x,0())
    
    log2(x,y) = 2x + y >= x + y = if(le(x,s(0())),x,inc(y))
    
    if(true(),x,s(y)) = x + y + 1 >= y = y
    
    if(false(),x,y) = x + y >= x + y = log2(half(x),y)
   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