YES

Problem:
 minus(n__0(),Y) -> 0()
 minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
 geq(X,n__0()) -> true()
 geq(n__0(),n__s(Y)) -> false()
 geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
 div(0(),n__s(Y)) -> 0()
 div(s(X),n__s(Y)) ->
 if(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),n__0())
 if(true(),X,Y) -> activate(X)
 if(false(),X,Y) -> activate(Y)
 0() -> n__0()
 s(X) -> n__s(X)
 activate(n__0()) -> 0()
 activate(n__s(X)) -> s(X)
 activate(X) -> X

Proof:
 DP Processor:
  DPs:
   minus#(n__0(),Y) -> 0#()
   minus#(n__s(X),n__s(Y)) -> activate#(Y)
   minus#(n__s(X),n__s(Y)) -> activate#(X)
   minus#(n__s(X),n__s(Y)) -> minus#(activate(X),activate(Y))
   geq#(n__s(X),n__s(Y)) -> activate#(Y)
   geq#(n__s(X),n__s(Y)) -> activate#(X)
   geq#(n__s(X),n__s(Y)) -> geq#(activate(X),activate(Y))
   div#(s(X),n__s(Y)) -> minus#(X,activate(Y))
   div#(s(X),n__s(Y)) -> div#(minus(X,activate(Y)),n__s(activate(Y)))
   div#(s(X),n__s(Y)) -> activate#(Y)
   div#(s(X),n__s(Y)) -> geq#(X,activate(Y))
   div#(s(X),n__s(Y)) ->
   if#(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),n__0())
   if#(true(),X,Y) -> activate#(X)
   if#(false(),X,Y) -> activate#(Y)
   activate#(n__0()) -> 0#()
   activate#(n__s(X)) -> s#(X)
  TRS:
   minus(n__0(),Y) -> 0()
   minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
   geq(X,n__0()) -> true()
   geq(n__0(),n__s(Y)) -> false()
   geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
   div(0(),n__s(Y)) -> 0()
   div(s(X),n__s(Y)) ->
   if(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),n__0())
   if(true(),X,Y) -> activate(X)
   if(false(),X,Y) -> activate(Y)
   0() -> n__0()
   s(X) -> n__s(X)
   activate(n__0()) -> 0()
   activate(n__s(X)) -> s(X)
   activate(X) -> X
  Arctic Interpretation Processor:
   dimension: 1
   usable rules:
    minus(n__0(),Y) -> 0()
    minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
    geq(X,n__0()) -> true()
    geq(n__0(),n__s(Y)) -> false()
    geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
    0() -> n__0()
    s(X) -> n__s(X)
    activate(n__0()) -> 0()
    activate(n__s(X)) -> s(X)
    activate(X) -> X
   interpretation:
    [s#](x0) = 0,
    
    [if#](x0, x1, x2) = x0 + 1x2 + 4,
    
    [div#](x0, x1) = x0 + 0,
    
    [geq#](x0, x1) = 3x0 + 2,
    
    [activate#](x0) = 2,
    
    [0#] = 1,
    
    [minus#](x0, x1) = 2x0 + 0,
    
    [if](x0, x1, x2) = 1x0 + x1 + 9,
    
    [s](x0) = 5x0 + 5,
    
    [div](x0, x1) = x0 + 0,
    
    [false] = 0,
    
    [true] = 0,
    
    [geq](x0, x1) = 3x0 + 4,
    
    [activate](x0) = 3x0 + 2,
    
    [n__s](x0) = 4x0 + 3,
    
    [0] = 2,
    
    [minus](x0, x1) = 3x0 + 0,
    
    [n__0] = 0
   orientation:
    minus#(n__0(),Y) = 2 >= 1 = 0#()
    
    minus#(n__s(X),n__s(Y)) = 6X + 5 >= 2 = activate#(Y)
    
    minus#(n__s(X),n__s(Y)) = 6X + 5 >= 2 = activate#(X)
    
    minus#(n__s(X),n__s(Y)) = 6X + 5 >= 5X + 4 = minus#(activate(X),activate(Y))
    
    geq#(n__s(X),n__s(Y)) = 7X + 6 >= 2 = activate#(Y)
    
    geq#(n__s(X),n__s(Y)) = 7X + 6 >= 2 = activate#(X)
    
    geq#(n__s(X),n__s(Y)) = 7X + 6 >= 6X + 5 = geq#(activate(X),activate(Y))
    
    div#(s(X),n__s(Y)) = 5X + 5 >= 2X + 0 = minus#(X,activate(Y))
    
    div#(s(X),n__s(Y)) = 5X + 5 >= 3X + 0 = div#(minus(X,activate(Y)),n__s(activate(Y)))
    
    div#(s(X),n__s(Y)) = 5X + 5 >= 2 = activate#(Y)
    
    div#(s(X),n__s(Y)) = 5X + 5 >= 3X + 2 = geq#(X,activate(Y))
    
    div#(s(X),n__s(Y)) = 5X + 5 >= 3X + 4 = if#(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),
    n__0())
    
    if#(true(),X,Y) = 1Y + 4 >= 2 = activate#(X)
    
    if#(false(),X,Y) = 1Y + 4 >= 2 = activate#(Y)
    
    activate#(n__0()) = 2 >= 1 = 0#()
    
    activate#(n__s(X)) = 2 >= 0 = s#(X)
    
    minus(n__0(),Y) = 3 >= 2 = 0()
    
    minus(n__s(X),n__s(Y)) = 7X + 6 >= 6X + 5 = minus(activate(X),activate(Y))
    
    geq(X,n__0()) = 3X + 4 >= 0 = true()
    
    geq(n__0(),n__s(Y)) = 4 >= 0 = false()
    
    geq(n__s(X),n__s(Y)) = 7X + 6 >= 6X + 5 = geq(activate(X),activate(Y))
    
    div(0(),n__s(Y)) = 2 >= 2 = 0()
    
    div(s(X),n__s(Y)) = 5X + 5 >= 7X + 9 = if(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),
   n__0())
    
    if(true(),X,Y) = X + 9 >= 3X + 2 = activate(X)
    
    if(false(),X,Y) = X + 9 >= 3Y + 2 = activate(Y)
    
    0() = 2 >= 0 = n__0()
    
    s(X) = 5X + 5 >= 4X + 3 = n__s(X)
    
    activate(n__0()) = 3 >= 2 = 0()
    
    activate(n__s(X)) = 7X + 6 >= 5X + 5 = s(X)
    
    activate(X) = 3X + 2 >= X = X
   problem:
    DPs:
     
    TRS:
     minus(n__0(),Y) -> 0()
     minus(n__s(X),n__s(Y)) -> minus(activate(X),activate(Y))
     geq(X,n__0()) -> true()
     geq(n__0(),n__s(Y)) -> false()
     geq(n__s(X),n__s(Y)) -> geq(activate(X),activate(Y))
     div(0(),n__s(Y)) -> 0()
     div(s(X),n__s(Y)) ->
     if(geq(X,activate(Y)),n__s(div(minus(X,activate(Y)),n__s(activate(Y)))),n__0())
     if(true(),X,Y) -> activate(X)
     if(false(),X,Y) -> activate(Y)
     0() -> n__0()
     s(X) -> n__s(X)
     activate(n__0()) -> 0()
     activate(n__s(X)) -> s(X)
     activate(X) -> X
   Qed