YES

Problem:
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 pred(s(x)) -> x
 minus(x,0()) -> x
 minus(x,s(y)) -> pred(minus(x,y))
 mod(0(),y) -> 0()
 mod(s(x),0()) -> 0()
 mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
 if_mod(true(),s(x),s(y)) -> mod(minus(x,y),s(y))
 if_mod(false(),s(x),s(y)) -> s(x)

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