YES 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))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(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))))) plus#(s(x),y) -> plus#(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))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(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))))) plus#(s(x),y) -> plus#(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))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) graph: plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y) sum#(cons(x,cons(y,l))) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y) 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#(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#(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) EDG 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))))) plus#(s(x),y) -> plus#(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))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) graph: plus#(s(x),y) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y) sum#(cons(x,cons(y,l))) -> plus#(x,y) -> plus#(s(x),y) -> plus#(x,y) sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) -> sum#(cons(x,cons(y,l))) -> plus#(x,y) 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))) -> plus#(x,y) 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))) -> plus#(x,y) 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)))) -> 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)))) -> app#(cons(x,l),k) -> app#(l,k) app#(cons(x,l),k) -> app#(l,k) -> app#(cons(x,l),k) -> app#(l,k) CDG 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))))) plus#(s(x),y) -> plus#(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))))) plus(0(),y) -> y plus(s(x),y) -> s(plus(x,y)) graph: 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#(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) SCC Processor: #sccs: 0 #rules: 0 #arcs: 4/49