MAYBE

Problem:
 division(x,y) -> div(x,y,0())
 div(x,y,z) -> if(lt(x,y),x,y,inc(z))
 if(true(),x,y,z) -> z
 if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
 minus(x,0()) -> x
 minus(s(x),s(y)) -> minus(x,y)
 lt(x,0()) -> false()
 lt(0(),s(y)) -> true()
 lt(s(x),s(y)) -> lt(x,y)
 inc(0()) -> s(0())
 inc(s(x)) -> s(inc(x))

Proof:
 DP Processor:
  DPs:
   division#(x,y) -> div#(x,y,0())
   div#(x,y,z) -> inc#(z)
   div#(x,y,z) -> lt#(x,y)
   div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
   if#(false(),x,s(y),z) -> minus#(x,s(y))
   if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z)
   minus#(s(x),s(y)) -> minus#(x,y)
   lt#(s(x),s(y)) -> lt#(x,y)
   inc#(s(x)) -> inc#(x)
  TRS:
   division(x,y) -> div(x,y,0())
   div(x,y,z) -> if(lt(x,y),x,y,inc(z))
   if(true(),x,y,z) -> z
   if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
   minus(x,0()) -> x
   minus(s(x),s(y)) -> minus(x,y)
   lt(x,0()) -> false()
   lt(0(),s(y)) -> true()
   lt(s(x),s(y)) -> lt(x,y)
   inc(0()) -> s(0())
   inc(s(x)) -> s(inc(x))
  TDG Processor:
   DPs:
    division#(x,y) -> div#(x,y,0())
    div#(x,y,z) -> inc#(z)
    div#(x,y,z) -> lt#(x,y)
    div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
    if#(false(),x,s(y),z) -> minus#(x,s(y))
    if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z)
    minus#(s(x),s(y)) -> minus#(x,y)
    lt#(s(x),s(y)) -> lt#(x,y)
    inc#(s(x)) -> inc#(x)
   TRS:
    division(x,y) -> div(x,y,0())
    div(x,y,z) -> if(lt(x,y),x,y,inc(z))
    if(true(),x,y,z) -> z
    if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
    minus(x,0()) -> x
    minus(s(x),s(y)) -> minus(x,y)
    lt(x,0()) -> false()
    lt(0(),s(y)) -> true()
    lt(s(x),s(y)) -> lt(x,y)
    inc(0()) -> s(0())
    inc(s(x)) -> s(inc(x))
   graph:
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    if#(false(),x,s(y),z) -> minus#(x,s(y)) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z) ->
    div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
    if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z) ->
    div#(x,y,z) -> lt#(x,y)
    if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z) ->
    div#(x,y,z) -> inc#(z)
    lt#(s(x),s(y)) -> lt#(x,y) -> lt#(s(x),s(y)) -> lt#(x,y)
    inc#(s(x)) -> inc#(x) -> inc#(s(x)) -> inc#(x)
    div#(x,y,z) -> if#(lt(x,y),x,y,inc(z)) ->
    if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z)
    div#(x,y,z) -> if#(lt(x,y),x,y,inc(z)) ->
    if#(false(),x,s(y),z) -> minus#(x,s(y))
    div#(x,y,z) -> lt#(x,y) -> lt#(s(x),s(y)) -> lt#(x,y)
    div#(x,y,z) -> inc#(z) -> inc#(s(x)) -> inc#(x)
    division#(x,y) -> div#(x,y,0()) ->
    div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
    division#(x,y) -> div#(x,y,0()) -> div#(x,y,z) -> lt#(x,y)
    division#(x,y) -> div#(x,y,0()) -> div#(x,y,z) -> inc#(z)
   SCC Processor:
    #sccs: 4
    #rules: 5
    #arcs: 14/81
    DPs:
     if#(false(),x,s(y),z) -> div#(minus(x,s(y)),s(y),z)
     div#(x,y,z) -> if#(lt(x,y),x,y,inc(z))
    TRS:
     division(x,y) -> div(x,y,0())
     div(x,y,z) -> if(lt(x,y),x,y,inc(z))
     if(true(),x,y,z) -> z
     if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     lt(x,0()) -> false()
     lt(0(),s(y)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
     inc(0()) -> s(0())
     inc(s(x)) -> s(inc(x))
    Open
    
    DPs:
     lt#(s(x),s(y)) -> lt#(x,y)
    TRS:
     division(x,y) -> div(x,y,0())
     div(x,y,z) -> if(lt(x,y),x,y,inc(z))
     if(true(),x,y,z) -> z
     if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     lt(x,0()) -> false()
     lt(0(),s(y)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
     inc(0()) -> s(0())
     inc(s(x)) -> s(inc(x))
    Subterm Criterion Processor:
     simple projection:
      pi(lt#) = 1
     problem:
      DPs:
       
      TRS:
       division(x,y) -> div(x,y,0())
       div(x,y,z) -> if(lt(x,y),x,y,inc(z))
       if(true(),x,y,z) -> z
       if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       lt(x,0()) -> false()
       lt(0(),s(y)) -> true()
       lt(s(x),s(y)) -> lt(x,y)
       inc(0()) -> s(0())
       inc(s(x)) -> s(inc(x))
     Qed
    
    DPs:
     inc#(s(x)) -> inc#(x)
    TRS:
     division(x,y) -> div(x,y,0())
     div(x,y,z) -> if(lt(x,y),x,y,inc(z))
     if(true(),x,y,z) -> z
     if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     lt(x,0()) -> false()
     lt(0(),s(y)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
     inc(0()) -> s(0())
     inc(s(x)) -> s(inc(x))
    Subterm Criterion Processor:
     simple projection:
      pi(inc#) = 0
     problem:
      DPs:
       
      TRS:
       division(x,y) -> div(x,y,0())
       div(x,y,z) -> if(lt(x,y),x,y,inc(z))
       if(true(),x,y,z) -> z
       if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       lt(x,0()) -> false()
       lt(0(),s(y)) -> true()
       lt(s(x),s(y)) -> lt(x,y)
       inc(0()) -> s(0())
       inc(s(x)) -> s(inc(x))
     Qed
    
    DPs:
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     division(x,y) -> div(x,y,0())
     div(x,y,z) -> if(lt(x,y),x,y,inc(z))
     if(true(),x,y,z) -> z
     if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
     minus(x,0()) -> x
     minus(s(x),s(y)) -> minus(x,y)
     lt(x,0()) -> false()
     lt(0(),s(y)) -> true()
     lt(s(x),s(y)) -> lt(x,y)
     inc(0()) -> s(0())
     inc(s(x)) -> s(inc(x))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       
      TRS:
       division(x,y) -> div(x,y,0())
       div(x,y,z) -> if(lt(x,y),x,y,inc(z))
       if(true(),x,y,z) -> z
       if(false(),x,s(y),z) -> div(minus(x,s(y)),s(y),z)
       minus(x,0()) -> x
       minus(s(x),s(y)) -> minus(x,y)
       lt(x,0()) -> false()
       lt(0(),s(y)) -> true()
       lt(s(x),s(y)) -> lt(x,y)
       inc(0()) -> s(0())
       inc(s(x)) -> s(inc(x))
     Qed