MAYBE

Problem:
 plus(0(),Y) -> Y
 plus(s(X),Y) -> s(plus(X,Y))
 min(X,0()) -> X
 min(s(X),s(Y)) -> min(X,Y)
 min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
 quot(0(),s(Y)) -> 0()
 quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))

Proof:
 DP Processor:
  DPs:
   plus#(s(X),Y) -> plus#(X,Y)
   min#(s(X),s(Y)) -> min#(X,Y)
   min#(min(X,Y),Z()) -> plus#(Y,Z())
   min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
   quot#(s(X),s(Y)) -> min#(X,Y)
   quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
  TRS:
   plus(0(),Y) -> Y
   plus(s(X),Y) -> s(plus(X,Y))
   min(X,0()) -> X
   min(s(X),s(Y)) -> min(X,Y)
   min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
   quot(0(),s(Y)) -> 0()
   quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
  Usable Rule Processor:
   DPs:
    plus#(s(X),Y) -> plus#(X,Y)
    min#(s(X),s(Y)) -> min#(X,Y)
    min#(min(X,Y),Z()) -> plus#(Y,Z())
    min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
    quot#(s(X),s(Y)) -> min#(X,Y)
    quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
   TRS:
    f9(x,y) -> x
    f9(x,y) -> y
    plus(0(),Y) -> Y
    plus(s(X),Y) -> s(plus(X,Y))
    min(X,0()) -> X
    min(s(X),s(Y)) -> min(X,Y)
    min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
   CDG Processor:
    DPs:
     plus#(s(X),Y) -> plus#(X,Y)
     min#(s(X),s(Y)) -> min#(X,Y)
     min#(min(X,Y),Z()) -> plus#(Y,Z())
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
     quot#(s(X),s(Y)) -> min#(X,Y)
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
    TRS:
     f9(x,y) -> x
     f9(x,y) -> y
     plus(0(),Y) -> Y
     plus(s(X),Y) -> s(plus(X,Y))
     min(X,0()) -> X
     min(s(X),s(Y)) -> min(X,Y)
     min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
    graph:
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) ->
     quot#(s(X),s(Y)) -> min#(X,Y)
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) ->
     quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
     quot#(s(X),s(Y)) -> min#(X,Y) -> min#(s(X),s(Y)) -> min#(X,Y)
     quot#(s(X),s(Y)) -> min#(X,Y) ->
     min#(min(X,Y),Z()) -> plus#(Y,Z())
     quot#(s(X),s(Y)) -> min#(X,Y) ->
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z())) ->
     min#(s(X),s(Y)) -> min#(X,Y)
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z())) ->
     min#(min(X,Y),Z()) -> plus#(Y,Z())
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z())) ->
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
     min#(min(X,Y),Z()) -> plus#(Y,Z()) -> plus#(s(X),Y) -> plus#(X,Y)
     min#(s(X),s(Y)) -> min#(X,Y) -> min#(s(X),s(Y)) -> min#(X,Y)
     min#(s(X),s(Y)) -> min#(X,Y) -> min#(min(X,Y),Z()) -> plus#(Y,Z())
     min#(s(X),s(Y)) -> min#(X,Y) ->
     min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
     plus#(s(X),Y) -> plus#(X,Y) -> plus#(s(X),Y) -> plus#(X,Y)
    Restore Modifier:
     DPs:
      plus#(s(X),Y) -> plus#(X,Y)
      min#(s(X),s(Y)) -> min#(X,Y)
      min#(min(X,Y),Z()) -> plus#(Y,Z())
      min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
      quot#(s(X),s(Y)) -> min#(X,Y)
      quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
     TRS:
      plus(0(),Y) -> Y
      plus(s(X),Y) -> s(plus(X,Y))
      min(X,0()) -> X
      min(s(X),s(Y)) -> min(X,Y)
      min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
      quot(0(),s(Y)) -> 0()
      quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
     SCC Processor:
      #sccs: 3
      #rules: 4
      #arcs: 13/36
      DPs:
       quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y))
      TRS:
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
       min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
      Open
      
      DPs:
       min#(min(X,Y),Z()) -> min#(X,plus(Y,Z()))
       min#(s(X),s(Y)) -> min#(X,Y)
      TRS:
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
       min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
      Open
      
      DPs:
       plus#(s(X),Y) -> plus#(X,Y)
      TRS:
       plus(0(),Y) -> Y
       plus(s(X),Y) -> s(plus(X,Y))
       min(X,0()) -> X
       min(s(X),s(Y)) -> min(X,Y)
       min(min(X,Y),Z()) -> min(X,plus(Y,Z()))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
      Open