YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 minus(0(),y) -> 0()
 minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
 if_minus(true(),s(x),y) -> 0()
 if_minus(false(),s(x),y) -> s(minus(x,y))
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 log(s(0())) -> 0()
 log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))

Proof:
 DP Processor:
  DPs:
   le#(s(x),s(y)) -> le#(x,y)
   minus#(s(x),y) -> le#(s(x),y)
   minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y)
   if_minus#(false(),s(x),y) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   log#(s(s(x))) -> quot#(x,s(s(0())))
   log#(s(s(x))) -> log#(s(quot(x,s(s(0())))))
  TRS:
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   minus(0(),y) -> 0()
   minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
   if_minus(true(),s(x),y) -> 0()
   if_minus(false(),s(x),y) -> s(minus(x,y))
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   log(s(0())) -> 0()
   log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [log#](x0) = x0,
    
    [quot#](x0, x1) = 2x0,
    
    [if_minus#](x0, x1, x2) = x0 + 2x1,
    
    [minus#](x0, x1) = 6x0 + 1,
    
    [le#](x0, x1) = 2x0 + 2,
    
    [log](x0) = x0,
    
    [quot](x0, x1) = x0,
    
    [if_minus](x0, x1, x2) = x1,
    
    [minus](x0, x1) = x0,
    
    [false] = 2,
    
    [s](x0) = 3x0 + 4,
    
    [true] = 0,
    
    [le](x0, x1) = 4x0,
    
    [0] = 0
   orientation:
    le#(s(x),s(y)) = 6x + 10 >= 2x + 2 = le#(x,y)
    
    minus#(s(x),y) = 18x + 25 >= 6x + 10 = le#(s(x),y)
    
    minus#(s(x),y) = 18x + 25 >= 18x + 24 = if_minus#(le(s(x),y),s(x),y)
    
    if_minus#(false(),s(x),y) = 6x + 10 >= 6x + 1 = minus#(x,y)
    
    quot#(s(x),s(y)) = 6x + 8 >= 6x + 1 = minus#(x,y)
    
    quot#(s(x),s(y)) = 6x + 8 >= 2x = quot#(minus(x,y),s(y))
    
    log#(s(s(x))) = 9x + 16 >= 2x = quot#(x,s(s(0())))
    
    log#(s(s(x))) = 9x + 16 >= 3x + 4 = log#(s(quot(x,s(s(0())))))
    
    le(0(),y) = 0 >= 0 = true()
    
    le(s(x),0()) = 12x + 16 >= 2 = false()
    
    le(s(x),s(y)) = 12x + 16 >= 4x = le(x,y)
    
    minus(0(),y) = 0 >= 0 = 0()
    
    minus(s(x),y) = 3x + 4 >= 3x + 4 = if_minus(le(s(x),y),s(x),y)
    
    if_minus(true(),s(x),y) = 3x + 4 >= 0 = 0()
    
    if_minus(false(),s(x),y) = 3x + 4 >= 3x + 4 = s(minus(x,y))
    
    quot(0(),s(y)) = 0 >= 0 = 0()
    
    quot(s(x),s(y)) = 3x + 4 >= 3x + 4 = s(quot(minus(x,y),s(y)))
    
    log(s(0())) = 4 >= 0 = 0()
    
    log(s(s(x))) = 9x + 16 >= 9x + 16 = s(log(s(quot(x,s(s(0()))))))
   problem:
    DPs:
     
    TRS:
     le(0(),y) -> true()
     le(s(x),0()) -> false()
     le(s(x),s(y)) -> le(x,y)
     minus(0(),y) -> 0()
     minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
     if_minus(true(),s(x),y) -> 0()
     if_minus(false(),s(x),y) -> s(minus(x,y))
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
     log(s(0())) -> 0()
     log(s(s(x))) -> s(log(s(quot(x,s(s(0()))))))
   Qed