MAYBE

Problem:
 le(0(),Y) -> true()
 le(s(X),0()) -> false()
 le(s(X),s(Y)) -> le(X,Y)
 minus(0(),Y) -> 0()
 minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
 ifMinus(true(),s(X),Y) -> 0()
 ifMinus(false(),s(X),Y) -> s(minus(X,Y))
 quot(0(),s(Y)) -> 0()
 quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))

Proof:
 DP Processor:
  DPs:
   le#(s(X),s(Y)) -> le#(X,Y)
   minus#(s(X),Y) -> le#(s(X),Y)
   minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
   ifMinus#(false(),s(X),Y) -> minus#(X,Y)
   quot#(s(X),s(Y)) -> minus#(X,Y)
   quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
  TRS:
   le(0(),Y) -> true()
   le(s(X),0()) -> false()
   le(s(X),s(Y)) -> le(X,Y)
   minus(0(),Y) -> 0()
   minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
   ifMinus(true(),s(X),Y) -> 0()
   ifMinus(false(),s(X),Y) -> s(minus(X,Y))
   quot(0(),s(Y)) -> 0()
   quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
  Usable Rule Processor:
   DPs:
    le#(s(X),s(Y)) -> le#(X,Y)
    minus#(s(X),Y) -> le#(s(X),Y)
    minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
    ifMinus#(false(),s(X),Y) -> minus#(X,Y)
    quot#(s(X),s(Y)) -> minus#(X,Y)
    quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
   TRS:
    f12(x,y) -> x
    f12(x,y) -> y
    le(s(X),0()) -> false()
    le(s(X),s(Y)) -> le(X,Y)
    le(0(),Y) -> true()
    minus(0(),Y) -> 0()
    minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
    ifMinus(true(),s(X),Y) -> 0()
    ifMinus(false(),s(X),Y) -> s(minus(X,Y))
   CDG Processor:
    DPs:
     le#(s(X),s(Y)) -> le#(X,Y)
     minus#(s(X),Y) -> le#(s(X),Y)
     minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
     ifMinus#(false(),s(X),Y) -> minus#(X,Y)
     quot#(s(X),s(Y)) -> minus#(X,Y)
     quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
    TRS:
     f12(x,y) -> x
     f12(x,y) -> y
     le(s(X),0()) -> false()
     le(s(X),s(Y)) -> le(X,Y)
     le(0(),Y) -> true()
     minus(0(),Y) -> 0()
     minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
     ifMinus(true(),s(X),Y) -> 0()
     ifMinus(false(),s(X),Y) -> s(minus(X,Y))
    graph:
     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),Y) -> le#(s(X),Y)
     quot#(s(X),s(Y)) -> minus#(X,Y) ->
     minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
     ifMinus#(false(),s(X),Y) -> minus#(X,Y) ->
     minus#(s(X),Y) -> le#(s(X),Y)
     ifMinus#(false(),s(X),Y) -> minus#(X,Y) ->
     minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
     minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y) ->
     ifMinus#(false(),s(X),Y) -> minus#(X,Y)
     minus#(s(X),Y) -> le#(s(X),Y) -> le#(s(X),s(Y)) -> le#(X,Y)
     le#(s(X),s(Y)) -> le#(X,Y) -> le#(s(X),s(Y)) -> le#(X,Y)
    Restore Modifier:
     DPs:
      le#(s(X),s(Y)) -> le#(X,Y)
      minus#(s(X),Y) -> le#(s(X),Y)
      minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
      ifMinus#(false(),s(X),Y) -> minus#(X,Y)
      quot#(s(X),s(Y)) -> minus#(X,Y)
      quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
     TRS:
      le(0(),Y) -> true()
      le(s(X),0()) -> false()
      le(s(X),s(Y)) -> le(X,Y)
      minus(0(),Y) -> 0()
      minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
      ifMinus(true(),s(X),Y) -> 0()
      ifMinus(false(),s(X),Y) -> s(minus(X,Y))
      quot(0(),s(Y)) -> 0()
      quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
     SCC Processor:
      #sccs: 3
      #rules: 4
      #arcs: 9/36
      DPs:
       quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y))
      TRS:
       le(0(),Y) -> true()
       le(s(X),0()) -> false()
       le(s(X),s(Y)) -> le(X,Y)
       minus(0(),Y) -> 0()
       minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
       ifMinus(true(),s(X),Y) -> 0()
       ifMinus(false(),s(X),Y) -> s(minus(X,Y))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
      Open
      
      DPs:
       minus#(s(X),Y) -> ifMinus#(le(s(X),Y),s(X),Y)
       ifMinus#(false(),s(X),Y) -> minus#(X,Y)
      TRS:
       le(0(),Y) -> true()
       le(s(X),0()) -> false()
       le(s(X),s(Y)) -> le(X,Y)
       minus(0(),Y) -> 0()
       minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
       ifMinus(true(),s(X),Y) -> 0()
       ifMinus(false(),s(X),Y) -> s(minus(X,Y))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
      Open
      
      DPs:
       le#(s(X),s(Y)) -> le#(X,Y)
      TRS:
       le(0(),Y) -> true()
       le(s(X),0()) -> false()
       le(s(X),s(Y)) -> le(X,Y)
       minus(0(),Y) -> 0()
       minus(s(X),Y) -> ifMinus(le(s(X),Y),s(X),Y)
       ifMinus(true(),s(X),Y) -> 0()
       ifMinus(false(),s(X),Y) -> s(minus(X,Y))
       quot(0(),s(Y)) -> 0()
       quot(s(X),s(Y)) -> s(quot(minus(X,Y),s(Y)))
      Open