WORST_CASE(?,O(n^2)) * Step 1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, 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(log) = {1}, uargs(quot) = {1}, uargs(s) = {1} Following symbols are considered usable: {log,min,quot} TcT has computed the following interpretation: p(0) = [1] [0] p(log) = [2 0] x1 + [1] [0 1] [0] p(min) = [1 0] x1 + [1] [0 1] [0] p(quot) = [1 1] x1 + [1] [0 1] [0] p(s) = [1 2] x1 + [1] [0 1] [2] Following rules are strictly oriented: log(s(0())) = [5] [2] > [1] [0] = 0() log(s(s(X))) = [2 8] X + [13] [0 1] [4] > [2 8] X + [10] [0 1] [4] = s(log(s(quot(X,s(s(0())))))) min(X,0()) = [1 0] X + [1] [0 1] [0] > [1 0] X + [0] [0 1] [0] = X min(s(X),s(Y)) = [1 2] X + [2] [0 1] [2] > [1 0] X + [1] [0 1] [0] = min(X,Y) quot(0(),s(Y)) = [2] [0] > [1] [0] = 0() quot(s(X),s(Y)) = [1 3] X + [4] [0 1] [2] > [1 3] X + [3] [0 1] [2] = s(quot(min(X,Y),s(Y))) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^2))