WORST_CASE(?,O(n^2)) * Step 1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: *(x,*(y,z)) -> *(otimes(x,y),z) *(x,oplus(y,z)) -> oplus(*(x,y),*(x,z)) *(+(x,y),z) -> oplus(*(x,z),*(y,z)) *(1(),y) -> y - Signature: {*/2} / {+/2,1/0,oplus/2,otimes/2} - Obligation: innermost runtime complexity wrt. defined symbols {*} and constructors {+,1,oplus,otimes} + 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(oplus) = {1,2} Following symbols are considered usable: {*} TcT has computed the following interpretation: p(*) = 1 + x1 + x1*x2 + 3*x2 p(+) = 3 + x1 + x2 p(1) = 1 p(oplus) = 1 + x1 + x2 p(otimes) = x1 + x2 Following rules are strictly oriented: *(x,*(y,z)) = 4 + 2*x + x*y + x*y*z + 3*x*z + 3*y + 3*y*z + 9*z > 1 + x + x*z + y + y*z + 3*z = *(otimes(x,y),z) *(x,oplus(y,z)) = 4 + 2*x + x*y + x*z + 3*y + 3*z > 3 + 2*x + x*y + x*z + 3*y + 3*z = oplus(*(x,y),*(x,z)) *(+(x,y),z) = 4 + x + x*z + y + y*z + 6*z > 3 + x + x*z + y + y*z + 6*z = oplus(*(x,z),*(y,z)) *(1(),y) = 2 + 4*y > y = y Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^2))