MAYBE

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

Proof:
 DP Processor:
  DPs:
   minus#(s(x),s(y)) -> minus#(x,y)
   le#(s(x),s(y)) -> le#(x,y)
   quot#(x,y) -> le#(y,x)
   quot#(x,y) -> le#(y,0())
   quot#(x,y) -> minus#(x,y)
   quot#(x,y) -> if_quot#(minus(x,y),y,le(y,0()),le(y,x))
   if_quot#(x,y,false(),true()) -> quot#(x,y)
  TRS:
   minus(x,x) -> 0()
   minus(0(),x) -> 0()
   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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
   if_quot(x,y,true(),z) -> divByZeroError()
   if_quot(x,y,false(),true()) -> s(quot(x,y))
   if_quot(x,y,false(),false()) -> 0()
  TDG Processor:
   DPs:
    minus#(s(x),s(y)) -> minus#(x,y)
    le#(s(x),s(y)) -> le#(x,y)
    quot#(x,y) -> le#(y,x)
    quot#(x,y) -> le#(y,0())
    quot#(x,y) -> minus#(x,y)
    quot#(x,y) -> if_quot#(minus(x,y),y,le(y,0()),le(y,x))
    if_quot#(x,y,false(),true()) -> quot#(x,y)
   TRS:
    minus(x,x) -> 0()
    minus(0(),x) -> 0()
    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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
    if_quot(x,y,true(),z) -> divByZeroError()
    if_quot(x,y,false(),true()) -> s(quot(x,y))
    if_quot(x,y,false(),false()) -> 0()
   graph:
    if_quot#(x,y,false(),true()) -> quot#(x,y) ->
    quot#(x,y) -> if_quot#(minus(x,y),y,le(y,0()),le(y,x))
    if_quot#(x,y,false(),true()) -> quot#(x,y) ->
    quot#(x,y) -> minus#(x,y)
    if_quot#(x,y,false(),true()) -> quot#(x,y) ->
    quot#(x,y) -> le#(y,0())
    if_quot#(x,y,false(),true()) -> quot#(x,y) ->
    quot#(x,y) -> le#(y,x)
    quot#(x,y) -> if_quot#(minus(x,y),y,le(y,0()),le(y,x)) ->
    if_quot#(x,y,false(),true()) -> quot#(x,y)
    quot#(x,y) -> le#(y,0()) -> le#(s(x),s(y)) -> le#(x,y)
    quot#(x,y) -> le#(y,x) -> le#(s(x),s(y)) -> le#(x,y)
    quot#(x,y) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(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)
   SCC Processor:
    #sccs: 3
    #rules: 4
    #arcs: 10/49
    DPs:
     if_quot#(x,y,false(),true()) -> quot#(x,y)
     quot#(x,y) -> if_quot#(minus(x,y),y,le(y,0()),le(y,x))
    TRS:
     minus(x,x) -> 0()
     minus(0(),x) -> 0()
     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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
     if_quot(x,y,true(),z) -> divByZeroError()
     if_quot(x,y,false(),true()) -> s(quot(x,y))
     if_quot(x,y,false(),false()) -> 0()
    Open
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     minus(x,x) -> 0()
     minus(0(),x) -> 0()
     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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
     if_quot(x,y,true(),z) -> divByZeroError()
     if_quot(x,y,false(),true()) -> s(quot(x,y))
     if_quot(x,y,false(),false()) -> 0()
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       minus(x,x) -> 0()
       minus(0(),x) -> 0()
       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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
       if_quot(x,y,true(),z) -> divByZeroError()
       if_quot(x,y,false(),true()) -> s(quot(x,y))
       if_quot(x,y,false(),false()) -> 0()
     Qed
    
    DPs:
     le#(s(x),s(y)) -> le#(x,y)
    TRS:
     minus(x,x) -> 0()
     minus(0(),x) -> 0()
     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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
     if_quot(x,y,true(),z) -> divByZeroError()
     if_quot(x,y,false(),true()) -> s(quot(x,y))
     if_quot(x,y,false(),false()) -> 0()
    Subterm Criterion Processor:
     simple projection:
      pi(le#) = 1
     problem:
      DPs:
       
      TRS:
       minus(x,x) -> 0()
       minus(0(),x) -> 0()
       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,y) -> if_quot(minus(x,y),y,le(y,0()),le(y,x))
       if_quot(x,y,true(),z) -> divByZeroError()
       if_quot(x,y,false(),true()) -> s(quot(x,y))
       if_quot(x,y,false(),false()) -> 0()
     Qed