YES Problem: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) Proof: DP Processor: DPs: a__app#(nil(),YS) -> mark#(YS) a__app#(cons(X,XS),YS) -> mark#(X) a__from#(X) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(prefix(X)) -> mark#(X) mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) TDG Processor: DPs: a__app#(nil(),YS) -> mark#(YS) a__app#(cons(X,XS),YS) -> mark#(X) a__from#(X) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(prefix(X)) -> mark#(X) mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) graph: a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(prefix(X)) -> a__prefix#(mark(X)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(prefix(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(zWadr(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(zWadr(X1,X2)) -> mark#(X2) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(app(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(app(X1,X2)) -> mark#(X2) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(prefix(X)) -> a__prefix#(mark(X)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(prefix(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X2) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X2) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) -> a__app#(cons(X,XS),YS) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) -> a__app#(nil(),YS) -> mark#(YS) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(prefix(X)) -> a__prefix#(mark(X)) a__from#(X) -> mark#(X) -> mark#(prefix(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X2) mark#(prefix(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(prefix(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(prefix(X)) -> mark#(X) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(prefix(X)) -> mark#(X) -> mark#(prefix(X)) -> mark#(X) mark#(prefix(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(prefix(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(prefix(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(prefix(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(prefix(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(prefix(X)) -> mark#(X) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(prefix(X)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X1) mark#(prefix(X)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) -> a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) -> a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) -> a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(prefix(X)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(app(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> mark#(X2) -> mark#(app(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(prefix(X)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> mark#(X2) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(prefix(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(prefix(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(app(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> mark#(X2) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(app(X1,X2)) -> mark#(X2) -> mark#(prefix(X)) -> mark#(X) mark#(app(X1,X2)) -> mark#(X2) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(app(X1,X2)) -> mark#(X2) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> mark#(X2) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(app(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(app(X1,X2)) -> mark#(X2) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(app(X1,X2)) -> mark#(X2) -> mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> mark#(X2) -> mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(app(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> mark#(X1) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(app(X1,X2)) -> mark#(X1) -> mark#(prefix(X)) -> mark#(X) mark#(app(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(app(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(app(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(app(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(app(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) -> a__app#(cons(X,XS),YS) -> mark#(X) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) -> a__app#(nil(),YS) -> mark#(YS) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(prefix(X)) -> a__prefix#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(prefix(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zWadr(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(app(X1,X2)) -> mark#(X2) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(prefix(X)) -> a__prefix#(mark(X)) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(prefix(X)) -> mark#(X) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X1) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(zWadr(X1,X2)) -> mark#(X2) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X1) a__app#(cons(X,XS),YS) -> mark#(X) -> mark#(app(X1,X2)) -> mark#(X2) a__app#(nil(),YS) -> mark#(YS) -> mark#(s(X)) -> mark#(X) a__app#(nil(),YS) -> mark#(YS) -> mark#(cons(X1,X2)) -> mark#(X1) a__app#(nil(),YS) -> mark#(YS) -> mark#(prefix(X)) -> a__prefix#(mark(X)) a__app#(nil(),YS) -> mark#(YS) -> mark#(prefix(X)) -> mark#(X) a__app#(nil(),YS) -> mark#(YS) -> mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__app#(nil(),YS) -> mark#(YS) -> mark#(zWadr(X1,X2)) -> mark#(X1) a__app#(nil(),YS) -> mark#(YS) -> mark#(zWadr(X1,X2)) -> mark#(X2) a__app#(nil(),YS) -> mark#(YS) -> mark#(from(X)) -> a__from#(mark(X)) a__app#(nil(),YS) -> mark#(YS) -> mark#(from(X)) -> mark#(X) a__app#(nil(),YS) -> mark#(YS) -> mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__app#(nil(),YS) -> mark#(YS) -> mark#(app(X1,X2)) -> mark#(X1) a__app#(nil(),YS) -> mark#(YS) -> mark#(app(X1,X2)) -> mark#(X2) SCC Processor: #sccs: 1 #rules: 17 #arcs: 164/324 DPs: a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__app#(nil(),YS) -> mark#(YS) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) mark#(prefix(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) a__app#(cons(X,XS),YS) -> mark#(X) TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zWadr#](x0, x1) = x0 + x1 + 0, [a__from#](x0) = 2x0 + 0, [mark#](x0) = x0 + 0, [a__app#](x0, x1) = x0 + x1 + 0, [prefix](x0) = 3x0 + 4, [a__prefix](x0) = 3x0 + 4, [zWadr](x0, x1) = x0 + x1 + 0, [a__zWadr](x0, x1) = x0 + x1 + 0, [from](x0) = 2x0 + 2, [s](x0) = 2x0 + 2, [a__from](x0) = 2x0 + 2, [app](x0, x1) = x0 + x1, [cons](x0, x1) = x0 + 2, [mark](x0) = x0 + 0, [a__app](x0, x1) = x0 + x1 + 0, [nil] = 0 orientation: a__zWadr#(cons(X,XS),cons(Y,YS)) = X + Y + 2 >= Y + 0 = mark#(Y) mark#(app(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(app(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(app(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__app#(mark(X1),mark(X2)) a__app#(nil(),YS) = YS + 0 >= YS + 0 = mark#(YS) mark#(from(X)) = 2X + 2 >= X + 0 = mark#(X) mark#(from(X)) = 2X + 2 >= 2X + 2 = a__from#(mark(X)) a__from#(X) = 2X + 0 >= X + 0 = mark#(X) mark#(zWadr(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(zWadr(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(zWadr(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) = X + Y + 2 >= X + 0 = mark#(X) mark#(prefix(X)) = 3X + 4 >= X + 0 = mark#(X) mark#(cons(X1,X2)) = X1 + 2 >= X1 + 0 = mark#(X1) mark#(s(X)) = 2X + 2 >= X + 0 = mark#(X) a__zWadr#(cons(X,XS),cons(Y,YS)) = X + Y + 2 >= X + Y + 2 = a__app#(mark(Y),cons(mark(X),nil())) a__app#(cons(X,XS),YS) = X + YS + 2 >= X + 0 = mark#(X) a__app(nil(),YS) = YS + 0 >= YS + 0 = mark(YS) a__app(cons(X,XS),YS) = X + YS + 2 >= X + 2 = cons(mark(X),app(XS,YS)) a__from(X) = 2X + 2 >= X + 2 = cons(mark(X),from(s(X))) a__zWadr(nil(),YS) = YS + 0 >= 0 = nil() a__zWadr(XS,nil()) = XS + 0 >= 0 = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = X + Y + 2 >= X + Y + 2 = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) = 3L + 4 >= 2 = cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__app(mark(X1),mark(X2)) mark(from(X)) = 2X + 2 >= 2X + 2 = a__from(mark(X)) mark(zWadr(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) = 3X + 4 >= 3X + 4 = a__prefix(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + 2 >= X1 + 2 = cons(mark(X1),X2) mark(s(X)) = 2X + 2 >= 2X + 2 = s(mark(X)) a__app(X1,X2) = X1 + X2 + 0 >= X1 + X2 = app(X1,X2) a__from(X) = 2X + 2 >= 2X + 2 = from(X) a__zWadr(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = zWadr(X1,X2) a__prefix(X) = 3X + 4 >= 3X + 4 = prefix(X) problem: DPs: a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(app(X1,X2)) -> mark#(X2) mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__app#(nil(),YS) -> mark#(YS) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> mark#(X1) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) a__app#(cons(X,XS),YS) -> mark#(X) TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zWadr#](x0, x1) = x0 + x1, [a__from#](x0) = x0 + 4, [mark#](x0) = x0, [a__app#](x0, x1) = x0 + x1 + 0, [prefix](x0) = 0, [a__prefix](x0) = 0, [zWadr](x0, x1) = 7x0 + x1 + 0, [a__zWadr](x0, x1) = 7x0 + x1 + 0, [from](x0) = x0 + 5, [s](x0) = 0, [a__from](x0) = x0 + 5, [app](x0, x1) = x0 + 7x1 + 0, [cons](x0, x1) = x0 + 0, [mark](x0) = x0, [a__app](x0, x1) = x0 + 7x1 + 0, [nil] = 0 orientation: a__zWadr#(cons(X,XS),cons(Y,YS)) = X + Y + 0 >= Y = mark#(Y) mark#(app(X1,X2)) = X1 + 7X2 + 0 >= X2 = mark#(X2) mark#(app(X1,X2)) = X1 + 7X2 + 0 >= X1 = mark#(X1) mark#(app(X1,X2)) = X1 + 7X2 + 0 >= X1 + X2 + 0 = a__app#(mark(X1),mark(X2)) a__app#(nil(),YS) = YS + 0 >= YS = mark#(YS) mark#(from(X)) = X + 5 >= X + 4 = a__from#(mark(X)) a__from#(X) = X + 4 >= X = mark#(X) mark#(zWadr(X1,X2)) = 7X1 + X2 + 0 >= X2 = mark#(X2) mark#(zWadr(X1,X2)) = 7X1 + X2 + 0 >= X1 = mark#(X1) mark#(zWadr(X1,X2)) = 7X1 + X2 + 0 >= X1 + X2 = a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) = X + Y + 0 >= X = mark#(X) mark#(cons(X1,X2)) = X1 + 0 >= X1 = mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) = X + Y + 0 >= X + Y + 0 = a__app#(mark(Y),cons(mark(X),nil())) a__app#(cons(X,XS),YS) = X + YS + 0 >= X = mark#(X) a__app(nil(),YS) = 7YS + 0 >= YS = mark(YS) a__app(cons(X,XS),YS) = X + 7YS + 0 >= X + 0 = cons(mark(X),app(XS,YS)) a__from(X) = X + 5 >= X + 0 = cons(mark(X),from(s(X))) a__zWadr(nil(),YS) = YS + 7 >= 0 = nil() a__zWadr(XS,nil()) = 7XS + 0 >= 0 = nil() a__zWadr(cons(X,XS),cons(Y,YS)) = 7X + Y + 7 >= 7X + Y + 7 = cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) = 0 >= 0 = cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) = X1 + 7X2 + 0 >= X1 + 7X2 + 0 = a__app(mark(X1),mark(X2)) mark(from(X)) = X + 5 >= X + 5 = a__from(mark(X)) mark(zWadr(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) = 0 >= 0 = a__prefix(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + 0 >= X1 + 0 = cons(mark(X1),X2) mark(s(X)) = 0 >= 0 = s(mark(X)) a__app(X1,X2) = X1 + 7X2 + 0 >= X1 + 7X2 + 0 = app(X1,X2) a__from(X) = X + 5 >= X + 5 = from(X) a__zWadr(X1,X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = zWadr(X1,X2) a__prefix(X) = 0 >= 0 = prefix(X) problem: DPs: a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(app(X1,X2)) -> mark#(X1) mark#(app(X1,X2)) -> a__app#(mark(X1),mark(X2)) a__app#(nil(),YS) -> mark#(YS) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(zWadr(X1,X2)) -> mark#(X2) mark#(zWadr(X1,X2)) -> a__zWadr#(mark(X1),mark(X2)) a__zWadr#(cons(X,XS),cons(Y,YS)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) a__zWadr#(cons(X,XS),cons(Y,YS)) -> a__app#(mark(Y),cons(mark(X),nil())) a__app#(cons(X,XS),YS) -> mark#(X) TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) KBO Processor: argument filtering: pi(nil) = [] pi(a__app) = [0,1] pi(mark) = [0] pi(cons) = 0 pi(app) = [0,1] pi(a__from) = [0] pi(s) = 0 pi(from) = [0] pi(a__zWadr) = [0,1] pi(zWadr) = [0,1] pi(a__prefix) = [0] pi(prefix) = [0] pi(a__app#) = [0,1] pi(mark#) = [0] pi(a__from#) = [0] pi(a__zWadr#) = [0,1] weight function: w0 = 1 w(a__zWadr#) = w(a__from#) = w(prefix) = w(a__prefix) = w(zWadr) = w( a__zWadr) = w(from) = w(s) = w(a__from) = w(nil) = 1 w(mark#) = w(a__app#) = w(app) = w(cons) = w(mark) = w(a__app) = 0 precedence: mark# ~ mark > a__prefix ~ a__zWadr ~ a__from ~ a__app > a__zWadr# ~ a__from# ~ a__app# ~ prefix ~ zWadr ~ from ~ s ~ app ~ cons ~ nil problem: DPs: mark#(cons(X1,X2)) -> mark#(X1) TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__app(nil(),YS) -> mark(YS) a__app(cons(X,XS),YS) -> cons(mark(X),app(XS,YS)) a__from(X) -> cons(mark(X),from(s(X))) a__zWadr(nil(),YS) -> nil() a__zWadr(XS,nil()) -> nil() a__zWadr(cons(X,XS),cons(Y,YS)) -> cons(a__app(mark(Y),cons(mark(X),nil())),zWadr(XS,YS)) a__prefix(L) -> cons(nil(),zWadr(L,prefix(L))) mark(app(X1,X2)) -> a__app(mark(X1),mark(X2)) mark(from(X)) -> a__from(mark(X)) mark(zWadr(X1,X2)) -> a__zWadr(mark(X1),mark(X2)) mark(prefix(X)) -> a__prefix(mark(X)) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) a__app(X1,X2) -> app(X1,X2) a__from(X) -> from(X) a__zWadr(X1,X2) -> zWadr(X1,X2) a__prefix(X) -> prefix(X) Qed