WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(X) -> g(h(f(X))) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(X) mark(h(X)) -> h(mark(X)) - Signature: {a__f/1,mark/1} / {f/1,g/1,h/1} - Obligation: innermost runtime complexity wrt. defined symbols {a__f,mark} and constructors {f,g,h} + Applied Processor: NaturalPI {shape = Linear, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(a__f) = {1}, uargs(h) = {1} Following symbols are considered usable: {a__f,mark} TcT has computed the following interpretation: p(a__f) = 7 + x1 p(f) = 1 + x1 p(g) = 2 p(h) = 2 + x1 p(mark) = 9 + 8*x1 Following rules are strictly oriented: a__f(X) = 7 + X > 1 + X = f(X) a__f(X) = 7 + X > 2 = g(h(f(X))) mark(f(X)) = 17 + 8*X > 16 + 8*X = a__f(mark(X)) mark(g(X)) = 25 > 2 = g(X) mark(h(X)) = 25 + 8*X > 11 + 8*X = h(mark(X)) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))