MAYBE

Problem:
 ge(0(),0()) -> true()
 ge(s(x),0()) -> ge(x,0())
 ge(0(),s(0())) -> false()
 ge(0(),s(s(x))) -> ge(0(),s(x))
 ge(s(x),s(y)) -> ge(x,y)
 minus(0(),0()) -> 0()
 minus(0(),s(x)) -> minus(0(),x)
 minus(s(x),0()) -> s(minus(x,0()))
 minus(s(x),s(y)) -> minus(x,y)
 plus(0(),0()) -> 0()
 plus(0(),s(x)) -> s(plus(0(),x))
 plus(s(x),y) -> s(plus(x,y))
 div(x,y) -> ify(ge(y,s(0())),x,y)
 ify(false(),x,y) -> divByZeroError()
 ify(true(),x,y) -> if(ge(x,y),x,y)
 if(false(),x,y) -> 0()
 if(true(),x,y) -> s(div(minus(x,y),y))

Proof:
 DP Processor:
  DPs:
   ge#(s(x),0()) -> ge#(x,0())
   ge#(0(),s(s(x))) -> ge#(0(),s(x))
   ge#(s(x),s(y)) -> ge#(x,y)
   minus#(0(),s(x)) -> minus#(0(),x)
   minus#(s(x),0()) -> minus#(x,0())
   minus#(s(x),s(y)) -> minus#(x,y)
   plus#(0(),s(x)) -> plus#(0(),x)
   plus#(s(x),y) -> plus#(x,y)
   div#(x,y) -> ge#(y,s(0()))
   div#(x,y) -> ify#(ge(y,s(0())),x,y)
   ify#(true(),x,y) -> ge#(x,y)
   ify#(true(),x,y) -> if#(ge(x,y),x,y)
   if#(true(),x,y) -> minus#(x,y)
   if#(true(),x,y) -> div#(minus(x,y),y)
  TRS:
   ge(0(),0()) -> true()
   ge(s(x),0()) -> ge(x,0())
   ge(0(),s(0())) -> false()
   ge(0(),s(s(x))) -> ge(0(),s(x))
   ge(s(x),s(y)) -> ge(x,y)
   minus(0(),0()) -> 0()
   minus(0(),s(x)) -> minus(0(),x)
   minus(s(x),0()) -> s(minus(x,0()))
   minus(s(x),s(y)) -> minus(x,y)
   plus(0(),0()) -> 0()
   plus(0(),s(x)) -> s(plus(0(),x))
   plus(s(x),y) -> s(plus(x,y))
   div(x,y) -> ify(ge(y,s(0())),x,y)
   ify(false(),x,y) -> divByZeroError()
   ify(true(),x,y) -> if(ge(x,y),x,y)
   if(false(),x,y) -> 0()
   if(true(),x,y) -> s(div(minus(x,y),y))
  TDG Processor:
   DPs:
    ge#(s(x),0()) -> ge#(x,0())
    ge#(0(),s(s(x))) -> ge#(0(),s(x))
    ge#(s(x),s(y)) -> ge#(x,y)
    minus#(0(),s(x)) -> minus#(0(),x)
    minus#(s(x),0()) -> minus#(x,0())
    minus#(s(x),s(y)) -> minus#(x,y)
    plus#(0(),s(x)) -> plus#(0(),x)
    plus#(s(x),y) -> plus#(x,y)
    div#(x,y) -> ge#(y,s(0()))
    div#(x,y) -> ify#(ge(y,s(0())),x,y)
    ify#(true(),x,y) -> ge#(x,y)
    ify#(true(),x,y) -> if#(ge(x,y),x,y)
    if#(true(),x,y) -> minus#(x,y)
    if#(true(),x,y) -> div#(minus(x,y),y)
   TRS:
    ge(0(),0()) -> true()
    ge(s(x),0()) -> ge(x,0())
    ge(0(),s(0())) -> false()
    ge(0(),s(s(x))) -> ge(0(),s(x))
    ge(s(x),s(y)) -> ge(x,y)
    minus(0(),0()) -> 0()
    minus(0(),s(x)) -> minus(0(),x)
    minus(s(x),0()) -> s(minus(x,0()))
    minus(s(x),s(y)) -> minus(x,y)
    plus(0(),0()) -> 0()
    plus(0(),s(x)) -> s(plus(0(),x))
    plus(s(x),y) -> s(plus(x,y))
    div(x,y) -> ify(ge(y,s(0())),x,y)
    ify(false(),x,y) -> divByZeroError()
    ify(true(),x,y) -> if(ge(x,y),x,y)
    if(false(),x,y) -> 0()
    if(true(),x,y) -> s(div(minus(x,y),y))
   graph:
    if#(true(),x,y) -> div#(minus(x,y),y) ->
    div#(x,y) -> ify#(ge(y,s(0())),x,y)
    if#(true(),x,y) -> div#(minus(x,y),y) ->
    div#(x,y) -> ge#(y,s(0()))
    if#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    if#(true(),x,y) -> minus#(x,y) ->
    minus#(s(x),0()) -> minus#(x,0())
    if#(true(),x,y) -> minus#(x,y) ->
    minus#(0(),s(x)) -> minus#(0(),x)
    ify#(true(),x,y) -> if#(ge(x,y),x,y) ->
    if#(true(),x,y) -> div#(minus(x,y),y)
    ify#(true(),x,y) -> if#(ge(x,y),x,y) ->
    if#(true(),x,y) -> minus#(x,y)
    ify#(true(),x,y) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    ify#(true(),x,y) -> ge#(x,y) -> ge#(0(),s(s(x))) -> ge#(0(),s(x))
    ify#(true(),x,y) -> ge#(x,y) -> ge#(s(x),0()) -> ge#(x,0())
    div#(x,y) -> ify#(ge(y,s(0())),x,y) ->
    ify#(true(),x,y) -> if#(ge(x,y),x,y)
    div#(x,y) -> ify#(ge(y,s(0())),x,y) -> ify#(true(),x,y) -> ge#(x,y)
    div#(x,y) -> ge#(y,s(0())) -> ge#(s(x),s(y)) -> ge#(x,y)
    div#(x,y) -> ge#(y,s(0())) -> ge#(0(),s(s(x))) -> ge#(0(),s(x))
    div#(x,y) -> ge#(y,s(0())) -> ge#(s(x),0()) -> ge#(x,0())
    plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y)
    plus#(s(x),y) -> plus#(x,y) -> plus#(0(),s(x)) -> plus#(0(),x)
    plus#(0(),s(x)) -> plus#(0(),x) -> plus#(s(x),y) -> plus#(x,y)
    plus#(0(),s(x)) -> plus#(0(),x) ->
    plus#(0(),s(x)) -> plus#(0(),x)
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(s(x),0()) -> minus#(x,0())
    minus#(s(x),s(y)) -> minus#(x,y) ->
    minus#(0(),s(x)) -> minus#(0(),x)
    minus#(s(x),0()) -> minus#(x,0()) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(s(x),0()) -> minus#(x,0()) ->
    minus#(s(x),0()) -> minus#(x,0())
    minus#(s(x),0()) -> minus#(x,0()) ->
    minus#(0(),s(x)) -> minus#(0(),x)
    minus#(0(),s(x)) -> minus#(0(),x) ->
    minus#(s(x),s(y)) -> minus#(x,y)
    minus#(0(),s(x)) -> minus#(0(),x) ->
    minus#(s(x),0()) -> minus#(x,0())
    minus#(0(),s(x)) -> minus#(0(),x) ->
    minus#(0(),s(x)) -> minus#(0(),x)
    ge#(s(x),s(y)) -> ge#(x,y) -> ge#(s(x),s(y)) -> ge#(x,y)
    ge#(s(x),s(y)) -> ge#(x,y) -> ge#(0(),s(s(x))) -> ge#(0(),s(x))
    ge#(s(x),s(y)) -> ge#(x,y) -> ge#(s(x),0()) -> ge#(x,0())
    ge#(s(x),0()) -> ge#(x,0()) -> ge#(s(x),s(y)) -> ge#(x,y)
    ge#(s(x),0()) -> ge#(x,0()) -> ge#(0(),s(s(x))) -> ge#(0(),s(x))
    ge#(s(x),0()) -> ge#(x,0()) -> ge#(s(x),0()) -> ge#(x,0())
    ge#(0(),s(s(x))) -> ge#(0(),s(x)) ->
    ge#(s(x),s(y)) -> ge#(x,y)
    ge#(0(),s(s(x))) -> ge#(0(),s(x)) ->
    ge#(0(),s(s(x))) -> ge#(0(),s(x))
    ge#(0(),s(s(x))) -> ge#(0(),s(x)) -> ge#(s(x),0()) -> ge#(x,0())
   SCC Processor:
    #sccs: 4
    #rules: 11
    #arcs: 37/196
    DPs:
     plus#(s(x),y) -> plus#(x,y)
     plus#(0(),s(x)) -> plus#(0(),x)
    TRS:
     ge(0(),0()) -> true()
     ge(s(x),0()) -> ge(x,0())
     ge(0(),s(0())) -> false()
     ge(0(),s(s(x))) -> ge(0(),s(x))
     ge(s(x),s(y)) -> ge(x,y)
     minus(0(),0()) -> 0()
     minus(0(),s(x)) -> minus(0(),x)
     minus(s(x),0()) -> s(minus(x,0()))
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),0()) -> 0()
     plus(0(),s(x)) -> s(plus(0(),x))
     plus(s(x),y) -> s(plus(x,y))
     div(x,y) -> ify(ge(y,s(0())),x,y)
     ify(false(),x,y) -> divByZeroError()
     ify(true(),x,y) -> if(ge(x,y),x,y)
     if(false(),x,y) -> 0()
     if(true(),x,y) -> s(div(minus(x,y),y))
    Subterm Criterion Processor:
     simple projection:
      pi(plus#) = 1
     problem:
      DPs:
       plus#(s(x),y) -> plus#(x,y)
      TRS:
       ge(0(),0()) -> true()
       ge(s(x),0()) -> ge(x,0())
       ge(0(),s(0())) -> false()
       ge(0(),s(s(x))) -> ge(0(),s(x))
       ge(s(x),s(y)) -> ge(x,y)
       minus(0(),0()) -> 0()
       minus(0(),s(x)) -> minus(0(),x)
       minus(s(x),0()) -> s(minus(x,0()))
       minus(s(x),s(y)) -> minus(x,y)
       plus(0(),0()) -> 0()
       plus(0(),s(x)) -> s(plus(0(),x))
       plus(s(x),y) -> s(plus(x,y))
       div(x,y) -> ify(ge(y,s(0())),x,y)
       ify(false(),x,y) -> divByZeroError()
       ify(true(),x,y) -> if(ge(x,y),x,y)
       if(false(),x,y) -> 0()
       if(true(),x,y) -> s(div(minus(x,y),y))
     Subterm Criterion Processor:
      simple projection:
       pi(plus#) = 0
      problem:
       DPs:
        
       TRS:
        ge(0(),0()) -> true()
        ge(s(x),0()) -> ge(x,0())
        ge(0(),s(0())) -> false()
        ge(0(),s(s(x))) -> ge(0(),s(x))
        ge(s(x),s(y)) -> ge(x,y)
        minus(0(),0()) -> 0()
        minus(0(),s(x)) -> minus(0(),x)
        minus(s(x),0()) -> s(minus(x,0()))
        minus(s(x),s(y)) -> minus(x,y)
        plus(0(),0()) -> 0()
        plus(0(),s(x)) -> s(plus(0(),x))
        plus(s(x),y) -> s(plus(x,y))
        div(x,y) -> ify(ge(y,s(0())),x,y)
        ify(false(),x,y) -> divByZeroError()
        ify(true(),x,y) -> if(ge(x,y),x,y)
        if(false(),x,y) -> 0()
        if(true(),x,y) -> s(div(minus(x,y),y))
      Qed
    
    DPs:
     if#(true(),x,y) -> div#(minus(x,y),y)
     div#(x,y) -> ify#(ge(y,s(0())),x,y)
     ify#(true(),x,y) -> if#(ge(x,y),x,y)
    TRS:
     ge(0(),0()) -> true()
     ge(s(x),0()) -> ge(x,0())
     ge(0(),s(0())) -> false()
     ge(0(),s(s(x))) -> ge(0(),s(x))
     ge(s(x),s(y)) -> ge(x,y)
     minus(0(),0()) -> 0()
     minus(0(),s(x)) -> minus(0(),x)
     minus(s(x),0()) -> s(minus(x,0()))
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),0()) -> 0()
     plus(0(),s(x)) -> s(plus(0(),x))
     plus(s(x),y) -> s(plus(x,y))
     div(x,y) -> ify(ge(y,s(0())),x,y)
     ify(false(),x,y) -> divByZeroError()
     ify(true(),x,y) -> if(ge(x,y),x,y)
     if(false(),x,y) -> 0()
     if(true(),x,y) -> s(div(minus(x,y),y))
    Open
    
    DPs:
     minus#(0(),s(x)) -> minus#(0(),x)
     minus#(s(x),0()) -> minus#(x,0())
     minus#(s(x),s(y)) -> minus#(x,y)
    TRS:
     ge(0(),0()) -> true()
     ge(s(x),0()) -> ge(x,0())
     ge(0(),s(0())) -> false()
     ge(0(),s(s(x))) -> ge(0(),s(x))
     ge(s(x),s(y)) -> ge(x,y)
     minus(0(),0()) -> 0()
     minus(0(),s(x)) -> minus(0(),x)
     minus(s(x),0()) -> s(minus(x,0()))
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),0()) -> 0()
     plus(0(),s(x)) -> s(plus(0(),x))
     plus(s(x),y) -> s(plus(x,y))
     div(x,y) -> ify(ge(y,s(0())),x,y)
     ify(false(),x,y) -> divByZeroError()
     ify(true(),x,y) -> if(ge(x,y),x,y)
     if(false(),x,y) -> 0()
     if(true(),x,y) -> s(div(minus(x,y),y))
    Subterm Criterion Processor:
     simple projection:
      pi(minus#) = 1
     problem:
      DPs:
       minus#(s(x),0()) -> minus#(x,0())
      TRS:
       ge(0(),0()) -> true()
       ge(s(x),0()) -> ge(x,0())
       ge(0(),s(0())) -> false()
       ge(0(),s(s(x))) -> ge(0(),s(x))
       ge(s(x),s(y)) -> ge(x,y)
       minus(0(),0()) -> 0()
       minus(0(),s(x)) -> minus(0(),x)
       minus(s(x),0()) -> s(minus(x,0()))
       minus(s(x),s(y)) -> minus(x,y)
       plus(0(),0()) -> 0()
       plus(0(),s(x)) -> s(plus(0(),x))
       plus(s(x),y) -> s(plus(x,y))
       div(x,y) -> ify(ge(y,s(0())),x,y)
       ify(false(),x,y) -> divByZeroError()
       ify(true(),x,y) -> if(ge(x,y),x,y)
       if(false(),x,y) -> 0()
       if(true(),x,y) -> s(div(minus(x,y),y))
     Subterm Criterion Processor:
      simple projection:
       pi(minus#) = 0
      problem:
       DPs:
        
       TRS:
        ge(0(),0()) -> true()
        ge(s(x),0()) -> ge(x,0())
        ge(0(),s(0())) -> false()
        ge(0(),s(s(x))) -> ge(0(),s(x))
        ge(s(x),s(y)) -> ge(x,y)
        minus(0(),0()) -> 0()
        minus(0(),s(x)) -> minus(0(),x)
        minus(s(x),0()) -> s(minus(x,0()))
        minus(s(x),s(y)) -> minus(x,y)
        plus(0(),0()) -> 0()
        plus(0(),s(x)) -> s(plus(0(),x))
        plus(s(x),y) -> s(plus(x,y))
        div(x,y) -> ify(ge(y,s(0())),x,y)
        ify(false(),x,y) -> divByZeroError()
        ify(true(),x,y) -> if(ge(x,y),x,y)
        if(false(),x,y) -> 0()
        if(true(),x,y) -> s(div(minus(x,y),y))
      Qed
    
    DPs:
     ge#(s(x),0()) -> ge#(x,0())
     ge#(0(),s(s(x))) -> ge#(0(),s(x))
     ge#(s(x),s(y)) -> ge#(x,y)
    TRS:
     ge(0(),0()) -> true()
     ge(s(x),0()) -> ge(x,0())
     ge(0(),s(0())) -> false()
     ge(0(),s(s(x))) -> ge(0(),s(x))
     ge(s(x),s(y)) -> ge(x,y)
     minus(0(),0()) -> 0()
     minus(0(),s(x)) -> minus(0(),x)
     minus(s(x),0()) -> s(minus(x,0()))
     minus(s(x),s(y)) -> minus(x,y)
     plus(0(),0()) -> 0()
     plus(0(),s(x)) -> s(plus(0(),x))
     plus(s(x),y) -> s(plus(x,y))
     div(x,y) -> ify(ge(y,s(0())),x,y)
     ify(false(),x,y) -> divByZeroError()
     ify(true(),x,y) -> if(ge(x,y),x,y)
     if(false(),x,y) -> 0()
     if(true(),x,y) -> s(div(minus(x,y),y))
    Subterm Criterion Processor:
     simple projection:
      pi(ge#) = 1
     problem:
      DPs:
       ge#(s(x),0()) -> ge#(x,0())
      TRS:
       ge(0(),0()) -> true()
       ge(s(x),0()) -> ge(x,0())
       ge(0(),s(0())) -> false()
       ge(0(),s(s(x))) -> ge(0(),s(x))
       ge(s(x),s(y)) -> ge(x,y)
       minus(0(),0()) -> 0()
       minus(0(),s(x)) -> minus(0(),x)
       minus(s(x),0()) -> s(minus(x,0()))
       minus(s(x),s(y)) -> minus(x,y)
       plus(0(),0()) -> 0()
       plus(0(),s(x)) -> s(plus(0(),x))
       plus(s(x),y) -> s(plus(x,y))
       div(x,y) -> ify(ge(y,s(0())),x,y)
       ify(false(),x,y) -> divByZeroError()
       ify(true(),x,y) -> if(ge(x,y),x,y)
       if(false(),x,y) -> 0()
       if(true(),x,y) -> s(div(minus(x,y),y))
     Subterm Criterion Processor:
      simple projection:
       pi(ge#) = 0
      problem:
       DPs:
        
       TRS:
        ge(0(),0()) -> true()
        ge(s(x),0()) -> ge(x,0())
        ge(0(),s(0())) -> false()
        ge(0(),s(s(x))) -> ge(0(),s(x))
        ge(s(x),s(y)) -> ge(x,y)
        minus(0(),0()) -> 0()
        minus(0(),s(x)) -> minus(0(),x)
        minus(s(x),0()) -> s(minus(x,0()))
        minus(s(x),s(y)) -> minus(x,y)
        plus(0(),0()) -> 0()
        plus(0(),s(x)) -> s(plus(0(),x))
        plus(s(x),y) -> s(plus(x,y))
        div(x,y) -> ify(ge(y,s(0())),x,y)
        ify(false(),x,y) -> divByZeroError()
        ify(true(),x,y) -> if(ge(x,y),x,y)
        if(false(),x,y) -> 0()
        if(true(),x,y) -> s(div(minus(x,y),y))
      Qed