MAYBE

Problem:
 minus(x,x) -> 0()
 minus(s(x),s(y)) -> minus(x,y)
 minus(0(),x) -> 0()
 minus(x,0()) -> x
 div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
 div(0(),s(y)) -> 0()
 f(x,0(),b) -> x
 f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   div#(s(x),s(y)) -> minus#(x,y)
   div#(s(x),s(y)) -> div#(minus(x,y),s(y))
   f#(x,s(y),b) -> minus#(s(y),s(0()))
   f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b)
   f#(x,s(y),b) -> div#(f(x,minus(s(y),s(0())),b),b)
  TRS:
   minus(x,x) -> 0()
   minus(s(x),s(y)) -> minus(x,y)
   minus(0(),x) -> 0()
   minus(x,0()) -> x
   div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
   div(0(),s(y)) -> 0()
   f(x,0(),b) -> x
   f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
  TDG Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    div#(s(x),s(y)) -> minus#(x,y)
    div#(s(x),s(y)) -> div#(minus(x,y),s(y))
    f#(x,s(y),b) -> minus#(s(y),s(0()))
    f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b)
    f#(x,s(y),b) -> div#(f(x,minus(s(y),s(0())),b),b)
   TRS:
    minus(x,x) -> 0()
    minus(s(x),s(y)) -> minus(x,y)
    minus(0(),x) -> 0()
    minus(x,0()) -> x
    div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
    div(0(),s(y)) -> 0()
    f(x,0(),b) -> x
    f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
   graph:
    f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b) ->
    f#(x,s(y),b) -> div#(f(x,minus(s(y),s(0())),b),b)
    f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b) ->
    f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b)
    f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b) ->
    f#(x,s(y),b) -> minus#(s(y),s(0()))
    f#(x,s(y),b) -> div#(f(x,minus(s(y),s(0())),b),b) ->
    div#(s(x),s(y)) -> div#(minus(x,y),s(y))
    f#(x,s(y),b) -> div#(f(x,minus(s(y),s(0())),b),b) ->
    div#(s(x),s(y)) -> minus#(x,y)
    f#(x,s(y),b) -> minus#(s(y),s(0())) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    div#(s(x),s(y)) -> div#(minus(x,y),s(y)) ->
    div#(s(x),s(y)) -> div#(minus(x,y),s(y))
    div#(s(x),s(y)) -> div#(minus(x,y),s(y)) ->
    div#(s(x),s(y)) -> minus#(x,y)
    div#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
   SCC Processor:
    #sccs: 3
    #rules: 3
    #arcs: 10/36
    DPs:
     f#(x,s(y),b) -> f#(x,minus(s(y),s(0())),b)
    TRS:
     minus(x,x) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     minus(0(),x) -> 0()
     minus(x,0()) -> x
     div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
     div(0(),s(y)) -> 0()
     f(x,0(),b) -> x
     f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
    Open
    
    DPs:
     div#(s(x),s(y)) -> div#(minus(x,y),s(y))
    TRS:
     minus(x,x) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     minus(0(),x) -> 0()
     minus(x,0()) -> x
     div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
     div(0(),s(y)) -> 0()
     f(x,0(),b) -> x
     f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [div#](x0, x1) = 3x0 + 0,
      
      [f](x0, x1, x2) = 4x0 + 2x1 + x2 + 6,
      
      [div](x0, x1) = x0 + 0,
      
      [s](x0) = 1x0 + 3,
      
      [0] = 0,
      
      [minus](x0, x1) = x0 + 1
     orientation:
      div#(s(x),s(y)) = 4x + 6 >= 3x + 4 = div#(minus(x,y),s(y))
      
      minus(x,x) = x + 1 >= 0 = 0()
      
      minus(s(x),s(y)) = 1x + 3 >= x + 1 = minus(x,y)
      
      minus(0(),x) = 1 >= 0 = 0()
      
      minus(x,0()) = x + 1 >= x = x
      
      div(s(x),s(y)) = 1x + 3 >= 1x + 3 = s(div(minus(x,y),s(y)))
      
      div(0(),s(y)) = 0 >= 0 = 0()
      
      f(x,0(),b) = b + 4x + 6 >= x = x
      
      f(x,s(y),b) = b + 4x + 3y + 6 >= b + 4x + 3y + 6 = div(f(x,minus(s(y),s(0())),b),b)
     problem:
      DPs:
       
      TRS:
       minus(x,x) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       minus(0(),x) -> 0()
       minus(x,0()) -> x
       div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
       div(0(),s(y)) -> 0()
       f(x,0(),b) -> x
       f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
     Qed
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     minus(x,x) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
     minus(0(),x) -> 0()
     minus(x,0()) -> x
     div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
     div(0(),s(y)) -> 0()
     f(x,0(),b) -> x
     f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       minus(x,x) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       minus(0(),x) -> 0()
       minus(x,0()) -> x
       div(s(x),s(y)) -> s(div(minus(x,y),s(y)))
       div(0(),s(y)) -> 0()
       f(x,0(),b) -> x
       f(x,s(y),b) -> div(f(x,minus(s(y),s(0())),b),b)
     Qed