WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: sum(0()) -> 0() sum(s(x)) -> +(sum(x),s(x)) sum1(0()) -> 0() sum1(s(x)) -> s(+(sum1(x),+(x,x))) - Signature: {sum/1,sum1/1} / {+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {sum,sum1} and constructors {+,0,s} + Applied Processor: NaturalPI {shape = Linear, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(+) = {1}, uargs(s) = {1} Following symbols are considered usable: {sum,sum1} TcT has computed the following interpretation: p(+) = 4 + x1 p(0) = 2 p(s) = 4 + x1 p(sum) = 1 + 5*x1 p(sum1) = 5*x1 Following rules are strictly oriented: sum(0()) = 11 > 2 = 0() sum(s(x)) = 21 + 5*x > 5 + 5*x = +(sum(x),s(x)) sum1(0()) = 10 > 2 = 0() sum1(s(x)) = 20 + 5*x > 8 + 5*x = s(+(sum1(x),+(x,x))) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))