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()))))))
  Usable Rule 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(s(x),0()) -> false()
    le(s(x),s(y)) -> le(x,y)
    le(0(),y) -> true()
    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)))
   Matrix Interpretation Processor: dim=1
    
    usable rules:
     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)))
    interpretation:
     [log#](x0) = 1/2x0 + 2,
     
     [quot#](x0, x1) = 2x0 + 1/2,
     
     [if_minus#](x0, x1, x2) = x1 + 1/2,
     
     [minus#](x0, x1) = 2x0 + 1,
     
     [le#](x0, x1) = 1/2x0,
     
     [quot](x0, x1) = 2x0 + 1/2,
     
     [if_minus](x0, x1, x2) = x1,
     
     [minus](x0, x1) = x0,
     
     [false] = 0,
     
     [s](x0) = 2x0 + 1,
     
     [true] = 0,
     
     [le](x0, x1) = 0,
     
     [0] = 1/2
    orientation:
     le#(s(x),s(y)) = x + 1/2 >= 1/2x = le#(x,y)
     
     minus#(s(x),y) = 4x + 3 >= x + 1/2 = le#(s(x),y)
     
     minus#(s(x),y) = 4x + 3 >= 2x + 3/2 = if_minus#(le(s(x),y),s(x),y)
     
     if_minus#(false(),s(x),y) = 2x + 3/2 >= 2x + 1 = minus#(x,y)
     
     quot#(s(x),s(y)) = 4x + 5/2 >= 2x + 1 = minus#(x,y)
     
     quot#(s(x),s(y)) = 4x + 5/2 >= 2x + 1/2 = quot#(minus(x,y),s(y))
     
     log#(s(s(x))) = 2x + 7/2 >= 2x + 1/2 = quot#(x,s(s(0())))
     
     log#(s(s(x))) = 2x + 7/2 >= 2x + 3 = log#(s(quot(x,s(s(0())))))
     
     le(s(x),0()) = 0 >= 0 = false()
     
     le(s(x),s(y)) = 0 >= 0 = le(x,y)
     
     le(0(),y) = 0 >= 0 = true()
     
     minus(0(),y) = 1/2 >= 1/2 = 0()
     
     minus(s(x),y) = 2x + 1 >= 2x + 1 = if_minus(le(s(x),y),s(x),y)
     
     if_minus(true(),s(x),y) = 2x + 1 >= 1/2 = 0()
     
     if_minus(false(),s(x),y) = 2x + 1 >= 2x + 1 = s(minus(x,y))
     
     quot(0(),s(y)) = 3/2 >= 1/2 = 0()
     
     quot(s(x),s(y)) = 4x + 5/2 >= 4x + 2 = s(quot(minus(x,y),s(y)))
    problem:
     DPs:
      
     TRS:
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      le(0(),y) -> true()
      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)))
    Qed