WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__tail,a__zeros,mark} and constructors {0,cons,tail ,zeros} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__tail) = {1}, uargs(cons) = {1} Following symbols are considered usable: {a__tail,a__zeros,mark} TcT has computed the following interpretation: p(0) = [0] [0] p(a__tail) = [1 0] x1 + [4] [0 1] [6] p(a__zeros) = [3] [4] p(cons) = [1 0] x1 + [0 1] x2 + [2] [0 1] [0 1] [4] p(mark) = [0 1] x1 + [4] [0 1] [4] p(tail) = [0 0] x1 + [0] [0 1] [6] p(zeros) = [1] [0] Following rules are strictly oriented: a__tail(X) = [1 0] X + [4] [0 1] [6] > [0 0] X + [0] [0 1] [6] = tail(X) a__tail(cons(X,XS)) = [1 0] X + [0 1] XS + [6] [0 1] [0 1] [10] > [0 1] XS + [4] [0 1] [4] = mark(XS) a__zeros() = [3] [4] > [2] [4] = cons(0(),zeros()) a__zeros() = [3] [4] > [1] [0] = zeros() mark(0()) = [4] [4] > [0] [0] = 0() mark(cons(X1,X2)) = [0 1] X1 + [0 1] X2 + [8] [0 1] [0 1] [8] > [0 1] X1 + [0 1] X2 + [6] [0 1] [0 1] [8] = cons(mark(X1),X2) mark(tail(X)) = [0 1] X + [10] [0 1] [10] > [0 1] X + [8] [0 1] [10] = a__tail(mark(X)) mark(zeros()) = [4] [4] > [3] [4] = a__zeros() Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))