MAYBE

Problem:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 quot(0(),s(y)) -> 0()
 quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
 plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> minus#(x,y)
   quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
   plus#(s(x),y) -> plus#(x,y)
   plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
   plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
  TRS:
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   quot(0(),s(y)) -> 0()
   quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
   plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
  Usable Rule Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> minus#(x,y)
    quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
    plus#(s(x),y) -> plus#(x,y)
    plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
    plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
   TRS:
    f8(x,y) -> x
    f8(x,y) -> y
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    plus(0(),y) -> y
    plus(s(x),y) -> s(plus(x,y))
    plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
    plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
   TDG Processor:
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
     quot#(s(x),s(y)) -> minus#(x,y)
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
     plus#(s(x),y) -> plus#(x,y)
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
    TRS:
     f8(x,y) -> x
     f8(x,y) -> y
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),y) -> y
     plus(s(x),y) -> s(plus(x,y))
     plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
     plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
    graph:
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0()))) ->
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0()))) ->
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0()))) ->
     plus#(s(x),y) -> plus#(x,y)
     plus#(s(x),y) -> plus#(x,y) ->
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
     plus#(s(x),y) -> plus#(x,y) ->
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
     plus#(s(x),y) -> plus#(x,y) ->
     plus#(s(x),y) -> plus#(x,y)
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
     plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
     plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
     plus#(s(x),y) -> plus#(x,y)
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
     quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
     quot#(s(x),s(y)) -> minus#(x,y)
     quot#(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)
    CDG Processor:
     DPs:
      minus#(s(x),s(y)) -> minus#(x,y)
      quot#(s(x),s(y)) -> minus#(x,y)
      quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
      plus#(s(x),y) -> plus#(x,y)
      plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
      plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
     TRS:
      f8(x,y) -> x
      f8(x,y) -> y
      minus(x,0()) -> x
      minus(s(x),s(y)) -> minus(x,y)
      plus(0(),y) -> y
      plus(s(x),y) -> s(plus(x,y))
      plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
      plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
     graph:
      plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0()))) ->
      plus#(s(x),y) -> plus#(x,y)
      plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
      plus#(s(x),y) -> plus#(x,y) ->
      plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
      plus#(s(x),y) -> plus#(x,y) ->
      plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
      plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
      plus#(s(x),y) -> plus#(x,y)
      plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
      plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
      plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0()))) ->
      plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
      quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
      quot#(s(x),s(y)) -> minus#(x,y)
      quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ->
      quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
      quot#(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)
     Restore Modifier:
      DPs:
       minus#(s(x),s(y)) -> minus#(x,y)
       quot#(s(x),s(y)) -> minus#(x,y)
       quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
       plus#(s(x),y) -> plus#(x,y)
       plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
       plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       quot(0(),s(y)) -> 0()
       quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
       plus(0(),y) -> y
       plus(s(x),y) -> s(plus(x,y))
       plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
       plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
      SCC Processor:
       #sccs: 3
       #rules: 5
       #arcs: 11/36
       DPs:
        quot#(s(x),s(y)) -> quot#(minus(x,y),s(y))
       TRS:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        plus(0(),y) -> y
        plus(s(x),y) -> s(plus(x,y))
        plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
        plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
       Open
       
       DPs:
        minus#(s(x),s(y)) -> minus#(x,y)
       TRS:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        plus(0(),y) -> y
        plus(s(x),y) -> s(plus(x,y))
        plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
        plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
       Open
       
       DPs:
        plus#(plus(x,s(0())),plus(y,s(s(z)))) -> plus#(plus(y,s(s(z))),plus(x,s(0())))
        plus#(s(x),y) -> plus#(x,y)
        plus#(minus(x,s(0())),minus(y,s(s(z)))) -> plus#(minus(y,s(s(z))),minus(x,s(0())))
       TRS:
        minus(x,0()) -> x
        minus(s(x),s(y)) -> minus(x,y)
        quot(0(),s(y)) -> 0()
        quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
        plus(0(),y) -> y
        plus(s(x),y) -> s(plus(x,y))
        plus(minus(x,s(0())),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0())))
        plus(plus(x,s(0())),plus(y,s(s(z)))) -> plus(plus(y,s(s(z))),plus(x,s(0())))
       Open