YES

Problem:
 pred(s(x)) -> x
 minus(x,0()) -> x
 minus(x,s(y)) -> pred(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:
   minus#(x,s(y)) -> minus#(x,y)
   minus#(x,s(y)) -> pred#(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:
   pred(s(x)) -> x
   minus(x,0()) -> x
   minus(x,s(y)) -> pred(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:
    minus#(x,s(y)) -> minus#(x,y)
    minus#(x,s(y)) -> pred#(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:
    minus(x,0()) -> x
    minus(x,s(y)) -> pred(minus(x,y))
    pred(s(x)) -> x
    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(x,0()) -> x
     minus(x,s(y)) -> pred(minus(x,y))
     pred(s(x)) -> x
     quot(0(),s(y)) -> 0()
     quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    interpretation:
     [log#](x0) = 3x0,
     
     [quot#](x0, x1) = x0 + 2x1,
     
     [minus#](x0, x1) = x0 + 2x1,
     
     [pred#](x0) = 0,
     
     [quot](x0, x1) = x0,
     
     [minus](x0, x1) = x0,
     
     [0] = 0,
     
     [pred](x0) = x0,
     
     [s](x0) = 2x0 + 1
    orientation:
     minus#(x,s(y)) = x + 4y + 2 >= x + 2y = minus#(x,y)
     
     minus#(x,s(y)) = x + 4y + 2 >= 0 = pred#(minus(x,y))
     
     quot#(s(x),s(y)) = 2x + 4y + 3 >= x + 2y = minus#(x,y)
     
     quot#(s(x),s(y)) = 2x + 4y + 3 >= x + 4y + 2 = quot#(minus(x,y),s(y))
     
     log#(s(s(x))) = 12x + 9 >= x + 6 = quot#(x,s(s(0())))
     
     log#(s(s(x))) = 12x + 9 >= 6x + 3 = log#(s(quot(x,s(s(0())))))
     
     minus(x,0()) = x >= x = x
     
     minus(x,s(y)) = x >= x = pred(minus(x,y))
     
     pred(s(x)) = 2x + 1 >= x = x
     
     quot(0(),s(y)) = 0 >= 0 = 0()
     
     quot(s(x),s(y)) = 2x + 1 >= 2x + 1 = s(quot(minus(x,y),s(y)))
    problem:
     DPs:
      
     TRS:
      minus(x,0()) -> x
      minus(x,s(y)) -> pred(minus(x,y))
      pred(s(x)) -> x
      quot(0(),s(y)) -> 0()
      quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
    Qed