WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {a,c,f,g} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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__f) = {1}, uargs(g) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a) = [5] p(a__f) = [1] x1 + [2] p(c) = [1] p(f) = [1] x1 + [1] p(g) = [1] x1 + [3] p(mark) = [3] x1 + [0] Following rules are strictly oriented: a__f(X) = [1] X + [2] > [1] X + [1] = f(X) a__f(f(a())) = [8] > [1] = c(f(g(f(a())))) mark(a()) = [15] > [5] = a() mark(c(X)) = [3] > [1] = c(X) mark(f(X)) = [3] X + [3] > [3] X + [2] = a__f(mark(X)) mark(g(X)) = [3] X + [9] > [3] X + [3] = g(mark(X)) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))