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)
    Usable Rule Processor:
     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
     Arctic Interpretation Processor:
      dimension: 1
      usable rules:
       minus(x,x) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       minus(0(),x) -> 0()
       minus(x,0()) -> x
      interpretation:
       [div#](x0, x1) = x0 + 0,
       
       [s](x0) = 4x0 + 4,
       
       [0] = 0,
       
       [minus](x0, x1) = x0 + 0
      orientation:
       div#(s(x),s(y)) = 4x + 4 >= x + 0 = div#(minus(x,y),s(y))
       
       minus(x,x) = x + 0 >= 0 = 0()
       
       minus(s(x),s(y)) = 4x + 4 >= x + 0 = minus(x,y)
       
       minus(0(),x) = 0 >= 0 = 0()
       
       minus(x,0()) = x + 0 >= x = x
      problem:
       DPs:
        
       TRS:
        minus(x,x) -> 0()
        minus(s(x),s(y)) -> minus(x,y)
        minus(0(),x) -> 0()
        minus(x,0()) -> x
      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