WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(+(x,y)) -> *(f(x),f(y)) f(+(x,s(0()))) -> +(s(s(0())),f(x)) f(0()) -> s(0()) f(s(0())) -> *(s(s(0())),f(0())) f(s(0())) -> s(s(0())) - Signature: {f/1} / {*/2,+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} 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(*) = {1,2}, uargs(+) = {2} Following symbols are considered usable: {f} TcT has computed the following interpretation: p(*) = x1 + x2 p(+) = 4 + x1 + x2 p(0) = 1 p(f) = 5 + 2*x1 p(s) = 6 Following rules are strictly oriented: f(+(x,y)) = 13 + 2*x + 2*y > 10 + 2*x + 2*y = *(f(x),f(y)) f(+(x,s(0()))) = 25 + 2*x > 15 + 2*x = +(s(s(0())),f(x)) f(0()) = 7 > 6 = s(0()) f(s(0())) = 17 > 13 = *(s(s(0())),f(0())) f(s(0())) = 17 > 6 = s(s(0())) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))