MAYBE

Problem:
 app(nil(),k) -> k
 app(l,nil()) -> l
 app(cons(x,l),k) -> cons(x,app(l,k))
 sum(cons(x,nil())) -> cons(x,nil())
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
 pred(cons(s(x),nil())) -> cons(x,nil())
 plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
 plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
 plus(zero(),y) -> y
 plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
 id(x) -> x
 if(true(),x,y) -> x
 if(false(),x,y) -> y
 not(x) -> if(x,false(),true())
 gt(s(x),zero()) -> true()
 gt(zero(),y) -> false()
 gt(s(x),s(y)) -> gt(x,y)

Proof:
 DP Processor:
  DPs:
   app#(cons(x,l),k) -> app#(l,k)
   sum#(cons(x,cons(y,l))) -> plus#(x,y)
   sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
   sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
   sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
   sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
   sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
   sum#(plus(cons(0(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
   plus#(s(x),s(y)) -> id#(y)
   plus#(s(x),s(y)) -> id#(x)
   plus#(s(x),s(y)) -> not#(gt(x,y))
   plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
   plus#(s(x),s(y)) -> gt#(x,y)
   plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
   plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
   plus#(s(x),x) -> id#(x)
   plus#(s(x),x) -> gt#(x,x)
   plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
   plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
   plus#(id(x),s(y)) -> gt#(s(y),y)
   plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
   plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
   not#(x) -> if#(x,false(),true())
   gt#(s(x),s(y)) -> gt#(x,y)
  TRS:
   app(nil(),k) -> k
   app(l,nil()) -> l
   app(cons(x,l),k) -> cons(x,app(l,k))
   sum(cons(x,nil())) -> cons(x,nil())
   sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
   sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
   sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
   pred(cons(s(x),nil())) -> cons(x,nil())
   plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
   plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
   plus(zero(),y) -> y
   plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
   id(x) -> x
   if(true(),x,y) -> x
   if(false(),x,y) -> y
   not(x) -> if(x,false(),true())
   gt(s(x),zero()) -> true()
   gt(zero(),y) -> false()
   gt(s(x),s(y)) -> gt(x,y)
  TDG Processor:
   DPs:
    app#(cons(x,l),k) -> app#(l,k)
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
    sum#(plus(cons(0(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),x) -> id#(x)
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    not#(x) -> if#(x,false(),true())
    gt#(s(x),s(y)) -> gt#(x,y)
   TRS:
    app(nil(),k) -> k
    app(l,nil()) -> l
    app(cons(x,l),k) -> cons(x,app(l,k))
    sum(cons(x,nil())) -> cons(x,nil())
    sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
    sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
    sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
    pred(cons(s(x),nil())) -> cons(x,nil())
    plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
    plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
    plus(zero(),y) -> y
    plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
    id(x) -> x
    if(true(),x,y) -> x
    if(false(),x,y) -> y
    not(x) -> if(x,false(),true())
    gt(s(x),zero()) -> true()
    gt(zero(),y) -> false()
    gt(s(x),s(y)) -> gt(x,y)
   graph:
    gt#(s(x),s(y)) -> gt#(x,y) -> gt#(s(x),s(y)) -> gt#(x,y)
    plus#(id(x),s(y)) -> gt#(s(y),y) ->
    gt#(s(x),s(y)) -> gt#(x,y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> gt#(x,x)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),x) -> id#(x)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> id#(x)
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y))) ->
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),s(y)) -> gt#(x,y) -> gt#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> not#(gt(x,y)) ->
    not#(x) -> if#(x,false(),true())
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),x) -> id#(x)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y))) ->
    plus#(s(x),s(y)) -> id#(y)
    plus#(s(x),x) -> gt#(x,x) ->
    gt#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> gt#(x,x)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),x) -> id#(x)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> id#(x)
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x)) ->
    plus#(s(x),s(y)) -> id#(y)
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(id(x),s(y)) -> if#(gt(s(y),y),y,s(y))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(id(x),s(y)) -> gt#(s(y),y)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),x) -> if#(gt(x,x),id(x),id(x))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),x) -> gt#(x,x)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),x) -> id#(x)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> if#(gt(x,y),x,y)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> gt#(x,y)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> if#(not(gt(x,y)),id(x),id(y))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> not#(gt(x,y))
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> id#(x)
    sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
    plus#(s(x),s(y)) -> id#(y)
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> pred#(sum(cons(s(x),cons(y,l))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
    sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
    sum#(cons(x,cons(y,l))) -> plus#(x,y)
    sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k)))) ->
    app#(cons(x,l),k) -> app#(l,k)
    app#(cons(x,l),k) -> app#(l,k) -> app#(cons(x,l),k) -> app#(l,k)
   SCC Processor:
    #sccs: 4
    #rules: 9
    #arcs: 91/576
    DPs:
     sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
     sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
     sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
     sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
    TRS:
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
     pred(cons(s(x),nil())) -> cons(x,nil())
     plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
     plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
     plus(zero(),y) -> y
     plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
     id(x) -> x
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     not(x) -> if(x,false(),true())
     gt(s(x),zero()) -> true()
     gt(zero(),y) -> false()
     gt(s(x),s(y)) -> gt(x,y)
    EDG Processor:
     DPs:
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
     TRS:
      app(nil(),k) -> k
      app(l,nil()) -> l
      app(cons(x,l),k) -> cons(x,app(l,k))
      sum(cons(x,nil())) -> cons(x,nil())
      sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
      sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
      sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
      pred(cons(s(x),nil())) -> cons(x,nil())
      plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
      plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
      plus(zero(),y) -> y
      plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
      id(x) -> x
      if(true(),x,y) -> x
      if(false(),x,y) -> y
      not(x) -> if(x,false(),true())
      gt(s(x),zero()) -> true()
      gt(zero(),y) -> false()
      gt(s(x),s(y)) -> gt(x,y)
     graph:
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
     SCC Processor:
      #sccs: 2
      #rules: 2
      #arcs: 7/16
      DPs:
       sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      TRS:
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
       sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
       pred(cons(s(x),nil())) -> cons(x,nil())
       plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
       plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
       plus(zero(),y) -> y
       plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
       id(x) -> x
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       not(x) -> if(x,false(),true())
       gt(s(x),zero()) -> true()
       gt(zero(),y) -> false()
       gt(s(x),s(y)) -> gt(x,y)
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [sum#](x0) = 4x0,
        
        [false] = 1,
        
        [true] = 0,
        
        [zero] = 0,
        
        [id](x0) = 2x0,
        
        [not](x0) = 2x0 + 2,
        
        [if](x0, x1, x2) = 2x1 + x2,
        
        [gt](x0, x1) = 1,
        
        [pred](x0) = x0,
        
        [s](x0) = 0,
        
        [0] = 0,
        
        [plus](x0, x1) = 2x1 + 7,
        
        [sum](x0) = 1,
        
        [cons](x0, x1) = x1 + 1,
        
        [app](x0, x1) = 4x0 + 2x1,
        
        [nil] = 0
       orientation:
        sum#(app(l,cons(x,cons(y,k)))) = 8k + 16l + 16 >= 16l + 8 = sum#(app(l,sum(cons(x,cons(y,k)))))
        
        app(nil(),k) = 2k >= k = k
        
        app(l,nil()) = 4l >= l = l
        
        app(cons(x,l),k) = 2k + 4l + 4 >= 2k + 4l + 1 = cons(x,app(l,k))
        
        sum(cons(x,nil())) = 1 >= 1 = cons(x,nil())
        
        sum(cons(x,cons(y,l))) = 1 >= 1 = sum(cons(plus(x,y),l))
        
        sum(app(l,cons(x,cons(y,k)))) = 1 >= 1 = sum(app(l,sum(cons(x,cons(y,k)))))
        
        sum(plus(cons(0(),x),cons(y,l))) = 1 >= 1 = pred(sum(cons(s(x),cons(y,l))))
        
        pred(cons(s(x),nil())) = 1 >= 1 = cons(x,nil())
        
        plus(s(x),s(y)) = 7 >= 0 = s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
        
        plus(s(x),x) = 2x + 7 >= 7 = plus(if(gt(x,x),id(x),id(x)),s(x))
        
        plus(zero(),y) = 2y + 7 >= y = y
        
        plus(id(x),s(y)) = 7 >= 0 = s(plus(x,if(gt(s(y),y),y,s(y))))
        
        id(x) = 2x >= x = x
        
        if(true(),x,y) = 2x + y >= x = x
        
        if(false(),x,y) = 2x + y >= y = y
        
        not(x) = 2x + 2 >= 2 = if(x,false(),true())
        
        gt(s(x),zero()) = 1 >= 0 = true()
        
        gt(zero(),y) = 1 >= 1 = false()
        
        gt(s(x),s(y)) = 1 >= 1 = gt(x,y)
       problem:
        DPs:
         
        TRS:
         app(nil(),k) -> k
         app(l,nil()) -> l
         app(cons(x,l),k) -> cons(x,app(l,k))
         sum(cons(x,nil())) -> cons(x,nil())
         sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
         sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
         sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
         pred(cons(s(x),nil())) -> cons(x,nil())
         plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
         plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
         plus(zero(),y) -> y
         plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
         id(x) -> x
         if(true(),x,y) -> x
         if(false(),x,y) -> y
         not(x) -> if(x,false(),true())
         gt(s(x),zero()) -> true()
         gt(zero(),y) -> false()
         gt(s(x),s(y)) -> gt(x,y)
       Qed
      
      DPs:
       sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      TRS:
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
       sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
       pred(cons(s(x),nil())) -> cons(x,nil())
       plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
       plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
       plus(zero(),y) -> y
       plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
       id(x) -> x
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       not(x) -> if(x,false(),true())
       gt(s(x),zero()) -> true()
       gt(zero(),y) -> false()
       gt(s(x),s(y)) -> gt(x,y)
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [sum#](x0) = x0,
        
        [false] = 0,
        
        [true] = 0,
        
        [zero] = 0,
        
        [id](x0) = 3x0,
        
        [not](x0) = 0,
        
        [if](x0, x1, x2) = 2x1 + x2,
        
        [gt](x0, x1) = 4,
        
        [pred](x0) = 6,
        
        [s](x0) = 0,
        
        [0] = 0,
        
        [plus](x0, x1) = x1 + 3,
        
        [sum](x0) = 6,
        
        [cons](x0, x1) = x1 + 1,
        
        [app](x0, x1) = 4x0 + 2x1 + 6,
        
        [nil] = 5
       orientation:
        sum#(cons(x,cons(y,l))) = l + 2 >= l + 1 = sum#(cons(plus(x,y),l))
        
        app(nil(),k) = 2k + 26 >= k = k
        
        app(l,nil()) = 4l + 16 >= l = l
        
        app(cons(x,l),k) = 2k + 4l + 10 >= 2k + 4l + 7 = cons(x,app(l,k))
        
        sum(cons(x,nil())) = 6 >= 6 = cons(x,nil())
        
        sum(cons(x,cons(y,l))) = 6 >= 6 = sum(cons(plus(x,y),l))
        
        sum(app(l,cons(x,cons(y,k)))) = 6 >= 6 = sum(app(l,sum(cons(x,cons(y,k)))))
        
        sum(plus(cons(0(),x),cons(y,l))) = 6 >= 6 = pred(sum(cons(s(x),cons(y,l))))
        
        pred(cons(s(x),nil())) = 6 >= 6 = cons(x,nil())
        
        plus(s(x),s(y)) = 3 >= 0 = s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
        
        plus(s(x),x) = x + 3 >= 3 = plus(if(gt(x,x),id(x),id(x)),s(x))
        
        plus(zero(),y) = y + 3 >= y = y
        
        plus(id(x),s(y)) = 3 >= 0 = s(plus(x,if(gt(s(y),y),y,s(y))))
        
        id(x) = 3x >= x = x
        
        if(true(),x,y) = 2x + y >= x = x
        
        if(false(),x,y) = 2x + y >= y = y
        
        not(x) = 0 >= 0 = if(x,false(),true())
        
        gt(s(x),zero()) = 4 >= 0 = true()
        
        gt(zero(),y) = 4 >= 0 = false()
        
        gt(s(x),s(y)) = 4 >= 4 = gt(x,y)
       problem:
        DPs:
         
        TRS:
         app(nil(),k) -> k
         app(l,nil()) -> l
         app(cons(x,l),k) -> cons(x,app(l,k))
         sum(cons(x,nil())) -> cons(x,nil())
         sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
         sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
         sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
         pred(cons(s(x),nil())) -> cons(x,nil())
         plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
         plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
         plus(zero(),y) -> y
         plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
         id(x) -> x
         if(true(),x,y) -> x
         if(false(),x,y) -> y
         not(x) -> if(x,false(),true())
         gt(s(x),zero()) -> true()
         gt(zero(),y) -> false()
         gt(s(x),s(y)) -> gt(x,y)
       Qed
    
    DPs:
     app#(cons(x,l),k) -> app#(l,k)
    TRS:
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
     pred(cons(s(x),nil())) -> cons(x,nil())
     plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
     plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
     plus(zero(),y) -> y
     plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
     id(x) -> x
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     not(x) -> if(x,false(),true())
     gt(s(x),zero()) -> true()
     gt(zero(),y) -> false()
     gt(s(x),s(y)) -> gt(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(app#) = 0
     problem:
      DPs:
       
      TRS:
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
       sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
       pred(cons(s(x),nil())) -> cons(x,nil())
       plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
       plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
       plus(zero(),y) -> y
       plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
       id(x) -> x
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       not(x) -> if(x,false(),true())
       gt(s(x),zero()) -> true()
       gt(zero(),y) -> false()
       gt(s(x),s(y)) -> gt(x,y)
     Qed
    
    DPs:
     plus#(id(x),s(y)) -> plus#(x,if(gt(s(y),y),y,s(y)))
     plus#(s(x),s(y)) -> plus#(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))
     plus#(s(x),x) -> plus#(if(gt(x,x),id(x),id(x)),s(x))
    TRS:
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
     pred(cons(s(x),nil())) -> cons(x,nil())
     plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
     plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
     plus(zero(),y) -> y
     plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
     id(x) -> x
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     not(x) -> if(x,false(),true())
     gt(s(x),zero()) -> true()
     gt(zero(),y) -> false()
     gt(s(x),s(y)) -> gt(x,y)
    Open
    
    DPs:
     gt#(s(x),s(y)) -> gt#(x,y)
    TRS:
     app(nil(),k) -> k
     app(l,nil()) -> l
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
     pred(cons(s(x),nil())) -> cons(x,nil())
     plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
     plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
     plus(zero(),y) -> y
     plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
     id(x) -> x
     if(true(),x,y) -> x
     if(false(),x,y) -> y
     not(x) -> if(x,false(),true())
     gt(s(x),zero()) -> true()
     gt(zero(),y) -> false()
     gt(s(x),s(y)) -> gt(x,y)
    Subterm Criterion Processor:
     simple projection:
      pi(gt#) = 1
     problem:
      DPs:
       
      TRS:
       app(nil(),k) -> k
       app(l,nil()) -> l
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
       sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
       pred(cons(s(x),nil())) -> cons(x,nil())
       plus(s(x),s(y)) -> s(s(plus(if(gt(x,y),x,y),if(not(gt(x,y)),id(x),id(y)))))
       plus(s(x),x) -> plus(if(gt(x,x),id(x),id(x)),s(x))
       plus(zero(),y) -> y
       plus(id(x),s(y)) -> s(plus(x,if(gt(s(y),y),y,s(y))))
       id(x) -> x
       if(true(),x,y) -> x
       if(false(),x,y) -> y
       not(x) -> if(x,false(),true())
       gt(s(x),zero()) -> true()
       gt(zero(),y) -> false()
       gt(s(x),s(y)) -> gt(x,y)
     Qed