WORST_CASE(?,O(n^2)) * Step 1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(0(),y) -> 0() *(s(x),y) -> +(y,*(x,y)) -(x,0()) -> x -(0(),y) -> 0() -(s(x),s(y)) -> -(x,y) exp(x,0()) -> s(0()) exp(x,s(y)) -> *(x,exp(x,y)) - Signature: {*/2,-/2,exp/2} / {+/2,0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,-,exp} and constructors {+,0,s} + 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(+) = {2} Following symbols are considered usable: {*,-,exp} TcT has computed the following interpretation: p(*) = 2 + x1 + x2 p(+) = x2 p(-) = x1 + x1^2 + x2 + x2^2 p(0) = 3 p(exp) = x1*x2 + x2^2 p(s) = 2 + x1 Following rules are strictly oriented: *(0(),y) = 5 + y > 3 = 0() *(s(x),y) = 4 + x + y > 2 + x + y = +(y,*(x,y)) -(x,0()) = 12 + x + x^2 > x = x -(0(),y) = 12 + y + y^2 > 3 = 0() -(s(x),s(y)) = 12 + 5*x + x^2 + 5*y + y^2 > x + x^2 + y + y^2 = -(x,y) exp(x,0()) = 9 + 3*x > 5 = s(0()) exp(x,s(y)) = 4 + 2*x + x*y + 4*y + y^2 > 2 + x + x*y + y^2 = *(x,exp(x,y)) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^2))