WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(s(X)) -> f(X) g(cons(0(),Y)) -> g(Y) g(cons(s(X),Y)) -> s(X) h(cons(X,Y)) -> h(g(cons(X,Y))) - Signature: {f/1,g/1,h/1} / {0/0,cons/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,cons,s} + 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: uargs(h) = {1} Following symbols are considered usable: {f,g,h} TcT has computed the following interpretation: p(0) = [1] [1] p(cons) = [0 2] x1 + [0 2] x2 + [2] [0 1] [0 1] [0] p(f) = [0 2] x1 + [0] [0 1] [4] p(g) = [0 2] x1 + [1] [0 2] [0] p(h) = [4 0] x1 + [0] [1 0] [0] p(s) = [0 0] x1 + [6] [0 1] [4] Following rules are strictly oriented: f(s(X)) = [0 2] X + [8] [0 1] [8] > [0 2] X + [0] [0 1] [4] = f(X) g(cons(0(),Y)) = [0 2] Y + [3] [0 2] [2] > [0 2] Y + [1] [0 2] [0] = g(Y) g(cons(s(X),Y)) = [0 2] X + [0 2] Y + [9] [0 2] [0 2] [8] > [0 0] X + [6] [0 1] [4] = s(X) h(cons(X,Y)) = [0 8] X + [0 8] Y + [8] [0 2] [0 2] [2] > [0 8] X + [0 8] Y + [4] [0 2] [0 2] [1] = h(g(cons(X,Y))) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))