WORST_CASE(?,O(n^2)) * Step 1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) sum(0()) -> 0() sum(s(x)) -> +(sum(x),s(x)) - Signature: {+/2,sum/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {+,sum} and constructors {0,s} + Applied Processor: NaturalPI {shape = Quadratic, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(quadratic): The following argument positions are considered usable: uargs(+) = {1}, uargs(s) = {1} Following symbols are considered usable: {+,sum} TcT has computed the following interpretation: p(+) = 4 + x1 + 2*x2 p(0) = 2 p(s) = 2 + x1 p(sum) = 3*x1 + x1^2 Following rules are strictly oriented: +(x,0()) = 8 + x > x = x +(x,s(y)) = 8 + x + 2*y > 6 + x + 2*y = s(+(x,y)) sum(0()) = 10 > 2 = 0() sum(s(x)) = 10 + 7*x + x^2 > 8 + 5*x + x^2 = +(sum(x),s(x)) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^2))