WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: e(g(X)) -> e(X) f(a()) -> f(c(a())) f(a()) -> f(d(a())) f(c(X)) -> X f(c(a())) -> f(d(b())) f(c(b())) -> f(d(a())) f(d(X)) -> X - Signature: {e/1,f/1} / {a/0,b/0,c/1,d/1,g/1} - Obligation: innermost runtime complexity wrt. defined symbols {e,f} and constructors {a,b,c,d,g} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: none Following symbols are considered usable: {e,f} TcT has computed the following interpretation: p(a) = [4] [1] p(b) = [4] [1] p(c) = [1 2] x1 + [1] [0 0] [0] p(d) = [1 1] x1 + [1] [0 0] [0] p(e) = [1 0] x1 + [0] [0 0] [0] p(f) = [2 7] x1 + [0] [1 6] [2] p(g) = [1 4] x1 + [1] [0 0] [1] Following rules are strictly oriented: e(g(X)) = [1 4] X + [1] [0 0] [0] > [1 0] X + [0] [0 0] [0] = e(X) f(a()) = [15] [12] > [14] [9] = f(c(a())) f(a()) = [15] [12] > [12] [8] = f(d(a())) f(c(X)) = [2 4] X + [2] [1 2] [3] > [1 0] X + [0] [0 1] [0] = X f(c(a())) = [14] [9] > [12] [8] = f(d(b())) f(c(b())) = [14] [9] > [12] [8] = f(d(a())) f(d(X)) = [2 2] X + [2] [1 1] [3] > [1 0] X + [0] [0 1] [0] = X Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))