MAYBE

Problem:
 ge(x,0()) -> true()
 ge(0(),s(y)) -> false()
 ge(s(x),s(y)) -> ge(x,y)
 minus(x,0()) -> x
 minus(0(),y) -> 0()
 minus(s(x),s(y)) -> minus(x,y)
 id_inc(x) -> x
 id_inc(x) -> s(x)
 quot(x,y) -> div(x,y,0())
 div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
 if(false(),b,x,y,z) -> div_by_zero()
 if(true(),false(),x,y,z) -> z
 if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))

Proof:
 DP Processor:
  DPs:
   ge#(s(x),s(y)) -> ge#(x,y)
   minus#(s(x),s(y)) -> minus#(x,y)
   quot#(x,y) -> div#(x,y,0())
   div#(x,y,z) -> ge#(x,y)
   div#(x,y,z) -> ge#(y,s(0()))
   div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
   if#(true(),true(),x,y,z) -> id_inc#(z)
   if#(true(),true(),x,y,z) -> minus#(x,y)
   if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z))
  TRS:
   ge(x,0()) -> true()
   ge(0(),s(y)) -> false()
   ge(s(x),s(y)) -> ge(x,y)
   minus(x,0()) -> x
   minus(0(),y) -> 0()
   minus(s(x),s(y)) -> minus(x,y)
   id_inc(x) -> x
   id_inc(x) -> s(x)
   quot(x,y) -> div(x,y,0())
   div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
   if(false(),b,x,y,z) -> div_by_zero()
   if(true(),false(),x,y,z) -> z
   if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
  Usable Rule Processor:
   DPs:
    ge#(s(x),s(y)) -> ge#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y)
    quot#(x,y) -> div#(x,y,0())
    div#(x,y,z) -> ge#(x,y)
    div#(x,y,z) -> ge#(y,s(0()))
    div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
    if#(true(),true(),x,y,z) -> id_inc#(z)
    if#(true(),true(),x,y,z) -> minus#(x,y)
    if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z))
   TRS:
    f17(x,y) -> x
    f17(x,y) -> y
    ge(x,0()) -> true()
    ge(0(),s(y)) -> false()
    ge(s(x),s(y)) -> ge(x,y)
    id_inc(x) -> x
    id_inc(x) -> s(x)
    minus(x,0()) -> x
    minus(0(),y) -> 0()
    minus(s(x),s(y)) -> minus(x,y)
   EDG Processor:
    DPs:
     ge#(s(x),s(y)) -> ge#(x,y)
     minus#(s(x),s(y)) -> minus#(x,y)
     quot#(x,y) -> div#(x,y,0())
     div#(x,y,z) -> ge#(x,y)
     div#(x,y,z) -> ge#(y,s(0()))
     div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
     if#(true(),true(),x,y,z) -> id_inc#(z)
     if#(true(),true(),x,y,z) -> minus#(x,y)
     if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z))
    TRS:
     f17(x,y) -> x
     f17(x,y) -> y
     ge(x,0()) -> true()
     ge(0(),s(y)) -> false()
     ge(s(x),s(y)) -> ge(x,y)
     id_inc(x) -> x
     id_inc(x) -> s(x)
     minus(x,0()) -> x
     minus(0(),y) -> 0()
     minus(s(x),s(y)) -> minus(x,y)
    graph:
     if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z)) ->
     div#(x,y,z) -> ge#(x,y)
     if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z)) ->
     div#(x,y,z) -> ge#(y,s(0()))
     if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z)) ->
     div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
     if#(true(),true(),x,y,z) -> minus#(x,y) ->
     minus#(s(x),s(y)) -> minus#(x,y)
     div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z) ->
     if#(true(),true(),x,y,z) -> id_inc#(z)
     div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z) ->
     if#(true(),true(),x,y,z) -> minus#(x,y)
     div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z) ->
     if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z))
     div#(x,y,z) -> ge#(y,s(0())) -> ge#(s(x),s(y)) -> ge#(x,y)
     div#(x,y,z) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
     quot#(x,y) -> div#(x,y,0()) -> div#(x,y,z) -> ge#(x,y)
     quot#(x,y) -> div#(x,y,0()) -> div#(x,y,z) -> ge#(y,s(0()))
     quot#(x,y) -> div#(x,y,0()) ->
     div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
     minus#(s(x),s(y)) -> minus#(x,y) -> minus#(s(x),s(y)) -> minus#(x,y)
     ge#(s(x),s(y)) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    Restore Modifier:
     DPs:
      ge#(s(x),s(y)) -> ge#(x,y)
      minus#(s(x),s(y)) -> minus#(x,y)
      quot#(x,y) -> div#(x,y,0())
      div#(x,y,z) -> ge#(x,y)
      div#(x,y,z) -> ge#(y,s(0()))
      div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
      if#(true(),true(),x,y,z) -> id_inc#(z)
      if#(true(),true(),x,y,z) -> minus#(x,y)
      if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z))
     TRS:
      ge(x,0()) -> true()
      ge(0(),s(y)) -> false()
      ge(s(x),s(y)) -> ge(x,y)
      minus(x,0()) -> x
      minus(0(),y) -> 0()
      minus(s(x),s(y)) -> minus(x,y)
      id_inc(x) -> x
      id_inc(x) -> s(x)
      quot(x,y) -> div(x,y,0())
      div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
      if(false(),b,x,y,z) -> div_by_zero()
      if(true(),false(),x,y,z) -> z
      if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
     SCC Processor:
      #sccs: 3
      #rules: 4
      #arcs: 14/81
      DPs:
       if#(true(),true(),x,y,z) -> div#(minus(x,y),y,id_inc(z))
       div#(x,y,z) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
      TRS:
       ge(x,0()) -> true()
       ge(0(),s(y)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
       minus(x,0()) -> x
       minus(0(),y) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       id_inc(x) -> x
       id_inc(x) -> s(x)
       quot(x,y) -> div(x,y,0())
       div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
       if(false(),b,x,y,z) -> div_by_zero()
       if(true(),false(),x,y,z) -> z
       if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
      Open
      
      DPs:
       ge#(s(x),s(y)) -> ge#(x,y)
      TRS:
       ge(x,0()) -> true()
       ge(0(),s(y)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
       minus(x,0()) -> x
       minus(0(),y) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       id_inc(x) -> x
       id_inc(x) -> s(x)
       quot(x,y) -> div(x,y,0())
       div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
       if(false(),b,x,y,z) -> div_by_zero()
       if(true(),false(),x,y,z) -> z
       if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
      Open
      
      DPs:
       minus#(s(x),s(y)) -> minus#(x,y)
      TRS:
       ge(x,0()) -> true()
       ge(0(),s(y)) -> false()
       ge(s(x),s(y)) -> ge(x,y)
       minus(x,0()) -> x
       minus(0(),y) -> 0()
       minus(s(x),s(y)) -> minus(x,y)
       id_inc(x) -> x
       id_inc(x) -> s(x)
       quot(x,y) -> div(x,y,0())
       div(x,y,z) -> if(ge(y,s(0())),ge(x,y),x,y,z)
       if(false(),b,x,y,z) -> div_by_zero()
       if(true(),false(),x,y,z) -> z
       if(true(),true(),x,y,z) -> div(minus(x,y),y,id_inc(z))
      Open