MAYBE

Problem:
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 le(0(),y) -> true()
 le(s(x),0()) -> false()
 le(s(x),s(y)) -> le(x,y)
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
 if_quot(true(),x,y) -> s(quot(minus(x,y),y))
 if_quot(false(),x,y) -> 0()

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   le#(s(x),s(y)) -> le#(x,y)
   quot#(x,s(y)) -> le#(s(y),x)
   quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y))
   if_quot#(true(),x,y) -> minus#(x,y)
   if_quot#(true(),x,y) -> quot#(minus(x,y),y)
  TRS:
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   le(0(),y) -> true()
   le(s(x),0()) -> false()
   le(s(x),s(y)) -> le(x,y)
   quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
   if_quot(true(),x,y) -> s(quot(minus(x,y),y))
   if_quot(false(),x,y) -> 0()
  Usable Rule Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    le#(s(x),s(y)) -> le#(x,y)
    quot#(x,s(y)) -> le#(s(y),x)
    quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y))
    if_quot#(true(),x,y) -> minus#(x,y)
    if_quot#(true(),x,y) -> quot#(minus(x,y),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(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
   EDG Processor:
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
     le#(s(x),s(y)) -> le#(x,y)
     quot#(x,s(y)) -> le#(s(y),x)
     quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y))
     if_quot#(true(),x,y) -> minus#(x,y)
     if_quot#(true(),x,y) -> quot#(minus(x,y),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(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
    graph:
     if_quot#(true(),x,y) -> quot#(minus(x,y),y) ->
     quot#(x,s(y)) -> le#(s(y),x)
     if_quot#(true(),x,y) -> quot#(minus(x,y),y) ->
     quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y))
     if_quot#(true(),x,y) -> minus#(x,y) ->
     minus#(s(x),s(y)) -> minus#(x,y)
     quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y)) ->
     if_quot#(true(),x,y) -> minus#(x,y)
     quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y)) ->
     if_quot#(true(),x,y) -> quot#(minus(x,y),y)
     quot#(x,s(y)) -> le#(s(y),x) -> le#(s(x),s(y)) -> le#(x,y)
     le#(s(x),s(y)) -> le#(x,y) -> le#(s(x),s(y)) -> le#(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)
      le#(s(x),s(y)) -> le#(x,y)
      quot#(x,s(y)) -> le#(s(y),x)
      quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y))
      if_quot#(true(),x,y) -> minus#(x,y)
      if_quot#(true(),x,y) -> quot#(minus(x,y),y)
     TRS:
      minus(x,0()) -> x
      minus(s(x),s(y)) -> minus(x,y)
      le(0(),y) -> true()
      le(s(x),0()) -> false()
      le(s(x),s(y)) -> le(x,y)
      quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
      if_quot(true(),x,y) -> s(quot(minus(x,y),y))
      if_quot(false(),x,y) -> 0()
     SCC Processor:
      #sccs: 3
      #rules: 4
      #arcs: 8/36
      DPs:
       if_quot#(true(),x,y) -> quot#(minus(x,y),y)
       quot#(x,s(y)) -> if_quot#(le(s(y),x),x,s(y))
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
       if_quot(true(),x,y) -> s(quot(minus(x,y),y))
       if_quot(false(),x,y) -> 0()
      Open
      
      DPs:
       le#(s(x),s(y)) -> le#(x,y)
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
       if_quot(true(),x,y) -> s(quot(minus(x,y),y))
       if_quot(false(),x,y) -> 0()
      Open
      
      DPs:
       minus#(s(x),s(y)) -> minus#(x,y)
      TRS:
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       le(0(),y) -> true()
       le(s(x),0()) -> false()
       le(s(x),s(y)) -> le(x,y)
       quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
       if_quot(true(),x,y) -> s(quot(minus(x,y),y))
       if_quot(false(),x,y) -> 0()
      Open