WORST_CASE(?,O(n^2)) * Step 1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: D(*(x,y)) -> +(*(y,D(x)),*(x,D(y))) D(+(x,y)) -> +(D(x),D(y)) D(-(x,y)) -> -(D(x),D(y)) D(constant()) -> 0() D(t()) -> 1() - Signature: {D/1} / {*/2,+/2,-/2,0/0,1/0,constant/0,t/0} - Obligation: innermost runtime complexity wrt. defined symbols {D} and constructors {*,+,-,0,1,constant,t} + 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(*) = {2}, uargs(+) = {1,2}, uargs(-) = {1,2} Following symbols are considered usable: {D} TcT has computed the following interpretation: p(*) = 1 + x1 + x2 p(+) = 1 + x1 + x2 p(-) = 2 + x1 + x2 p(0) = 0 p(1) = 0 p(D) = 2*x1 + 2*x1^2 p(constant) = 1 p(t) = 1 Following rules are strictly oriented: D(*(x,y)) = 4 + 6*x + 4*x*y + 2*x^2 + 6*y + 2*y^2 > 3 + 3*x + 2*x^2 + 3*y + 2*y^2 = +(*(y,D(x)),*(x,D(y))) D(+(x,y)) = 4 + 6*x + 4*x*y + 2*x^2 + 6*y + 2*y^2 > 1 + 2*x + 2*x^2 + 2*y + 2*y^2 = +(D(x),D(y)) D(-(x,y)) = 12 + 10*x + 4*x*y + 2*x^2 + 10*y + 2*y^2 > 2 + 2*x + 2*x^2 + 2*y + 2*y^2 = -(D(x),D(y)) D(constant()) = 4 > 0 = 0() D(t()) = 4 > 0 = 1() Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^2))