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))
  TDG 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))
   graph:
    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) -> 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) -> ge#(x,y)
    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) -> 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)
    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) -> 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) -> if#(ge(y,s(0())),ge(x,y),x,y,z)
    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) -> ge#(x,y)
    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)
   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:
     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))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      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))
     Qed
    
    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))
    Subterm Criterion Processor:
     simple projection:
      pi(ge#) = 1
     problem:
      DPs:
       
      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))
     Qed