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