WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt} + 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(U12) = {2,3}, uargs(plus) = {1,2}, uargs(s) = {1} Following symbols are considered usable: {U11,U12,activate,plus} TcT has computed the following interpretation: p(0) = [8] p(U11) = [2] x1 + [2] x2 + [1] x3 + [5] p(U12) = [2] x1 + [2] x2 + [1] x3 + [0] p(activate) = [1] x1 + [1] p(plus) = [1] x1 + [2] x2 + [0] p(s) = [1] x1 + [14] p(tt) = [9] Following rules are strictly oriented: U11(tt(),M,N) = [2] M + [1] N + [23] > [2] M + [1] N + [21] = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = [2] M + [1] N + [18] > [2] M + [1] N + [17] = s(plus(activate(N),activate(M))) activate(X) = [1] X + [1] > [1] X + [0] = X plus(N,0()) = [1] N + [16] > [1] N + [0] = N plus(N,s(M)) = [2] M + [1] N + [28] > [2] M + [1] N + [23] = U11(tt(),M,N) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))