YES Problem: a__zeros() -> cons(0(),zeros()) a__tail(cons(X,XS)) -> mark(XS) mark(zeros()) -> a__zeros() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() a__zeros() -> zeros() a__tail(X) -> tail(X) Proof: DP Processor: DPs: a__tail#(cons(X,XS)) -> mark#(XS) mark#(zeros()) -> a__zeros#() mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) TRS: a__zeros() -> cons(0(),zeros()) a__tail(cons(X,XS)) -> mark(XS) mark(zeros()) -> a__zeros() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() a__zeros() -> zeros() a__tail(X) -> tail(X) Matrix Interpretation Processor: dim=1 usable rules: a__zeros() -> cons(0(),zeros()) a__tail(cons(X,XS)) -> mark(XS) mark(zeros()) -> a__zeros() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() a__zeros() -> zeros() a__tail(X) -> tail(X) interpretation: [mark#](x0) = x0 + 2, [a__tail#](x0) = x0 + 2, [a__zeros#] = 0, [tail](x0) = 4x0 + 2, [mark](x0) = 4x0 + 1, [a__tail](x0) = 4x0 + 2, [cons](x0, x1) = x0 + x1 + 1, [zeros] = 1, [0] = 0, [a__zeros] = 2 orientation: a__tail#(cons(X,XS)) = X + XS + 3 >= XS + 2 = mark#(XS) mark#(zeros()) = 3 >= 0 = a__zeros#() mark#(tail(X)) = 4X + 4 >= X + 2 = mark#(X) mark#(tail(X)) = 4X + 4 >= 4X + 3 = a__tail#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 3 >= X1 + 2 = mark#(X1) a__zeros() = 2 >= 2 = cons(0(),zeros()) a__tail(cons(X,XS)) = 4X + 4XS + 6 >= 4XS + 1 = mark(XS) mark(zeros()) = 5 >= 2 = a__zeros() mark(tail(X)) = 16X + 9 >= 16X + 6 = a__tail(mark(X)) mark(cons(X1,X2)) = 4X1 + 4X2 + 5 >= 4X1 + X2 + 2 = cons(mark(X1),X2) mark(0()) = 1 >= 0 = 0() a__zeros() = 2 >= 1 = zeros() a__tail(X) = 4X + 2 >= 4X + 2 = tail(X) problem: DPs: TRS: a__zeros() -> cons(0(),zeros()) a__tail(cons(X,XS)) -> mark(XS) mark(zeros()) -> a__zeros() mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() a__zeros() -> zeros() a__tail(X) -> tail(X) Qed