WORST_CASE(?,O(n^1)) * Step 1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(s(x),s(y)) -> -(x,y) double(0()) -> 0() double(s(x)) -> s(s(double(x))) half(0()) -> 0() half(double(x)) -> x half(s(0())) -> 0() half(s(s(x))) -> s(half(x)) if(0(),y,z) -> y if(s(x),y,z) -> z - Signature: {-/2,double/1,half/1,if/3} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {-,double,half,if} and constructors {0,s} + 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(s) = {1} Following symbols are considered usable: {-,double,half,if} TcT has computed the following interpretation: p(-) = [5] x1 + [4] x2 + [0] p(0) = [1] p(double) = [8] x1 + [5] p(half) = [2] x1 + [3] p(if) = [6] x1 + [2] x2 + [2] x3 + [4] p(s) = [1] x1 + [3] Following rules are strictly oriented: -(x,0()) = [5] x + [4] > [1] x + [0] = x -(s(x),s(y)) = [5] x + [4] y + [27] > [5] x + [4] y + [0] = -(x,y) double(0()) = [13] > [1] = 0() double(s(x)) = [8] x + [29] > [8] x + [11] = s(s(double(x))) half(0()) = [5] > [1] = 0() half(double(x)) = [16] x + [13] > [1] x + [0] = x half(s(0())) = [11] > [1] = 0() half(s(s(x))) = [2] x + [15] > [2] x + [6] = s(half(x)) if(0(),y,z) = [2] y + [2] z + [10] > [1] y + [0] = y if(s(x),y,z) = [6] x + [2] y + [2] z + [22] > [1] z + [0] = z Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))