WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: f(x,h1(y,z)) -> h2(0(),x,h1(y,z)) f(j(x,y),y) -> g(f(x,k(y))) g(h2(x,y,h1(z,u))) -> h2(s(x),y,h1(z,u)) h2(x,j(y,h1(z,u)),h1(z,u)) -> h2(s(x),y,h1(s(z),u)) i(f(x,h(y))) -> y i(h2(s(x),y,h1(x,z))) -> z k(h(x)) -> h1(0(),x) k(h1(x,y)) -> h1(s(x),y) - Signature: {f/2,g/1,h2/3,i/1,k/1} / {0/0,h/1,h1/2,j/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f,g,h2,i,k} and constructors {0,h,h1,j,s} + Applied Processor: NaturalPI {shape = StronglyLinear, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(stronglyLinear): The following argument positions are considered usable: uargs(f) = {2}, uargs(g) = {1} Following symbols are considered usable: {f,g,h2,i,k} TcT has computed the following interpretation: p(0) = 4 p(f) = 1 + x1 + x2 p(g) = 1 + x1 p(h) = 4 + x1 p(h1) = x2 p(h2) = x2 + x3 p(i) = 1 + x1 p(j) = 4 + x1 p(k) = 1 + x1 p(s) = 0 Following rules are strictly oriented: f(x,h1(y,z)) = 1 + x + z > x + z = h2(0(),x,h1(y,z)) f(j(x,y),y) = 5 + x + y > 3 + x + y = g(f(x,k(y))) g(h2(x,y,h1(z,u))) = 1 + u + y > u + y = h2(s(x),y,h1(z,u)) h2(x,j(y,h1(z,u)),h1(z,u)) = 4 + u + y > u + y = h2(s(x),y,h1(s(z),u)) i(f(x,h(y))) = 6 + x + y > y = y i(h2(s(x),y,h1(x,z))) = 1 + y + z > z = z k(h(x)) = 5 + x > x = h1(0(),x) k(h1(x,y)) = 1 + y > y = h1(s(x),y) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))