WORST_CASE(?,O(n^2)) * Step 1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(a()) -> b() f(c()) -> d() f(g(x,y)) -> g(f(x),f(y)) f(h(x,y)) -> g(h(y,f(x)),h(x,f(y))) g(x,x) -> h(e(),x) - Signature: {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2} - Obligation: innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h} + Applied Processor: NaturalPI {shape = Quadratic, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(quadratic): The following argument positions are considered usable: uargs(g) = {1,2}, uargs(h) = {2} Following symbols are considered usable: {f,g} TcT has computed the following interpretation: p(a) = 3 p(b) = 6 p(c) = 3 p(d) = 4 p(e) = 0 p(f) = 2*x1 + x1^2 p(g) = 3 + x1 + x2 p(h) = 2 + x1 + x2 Following rules are strictly oriented: f(a()) = 15 > 6 = b() f(c()) = 15 > 4 = d() f(g(x,y)) = 15 + 8*x + 2*x*y + x^2 + 8*y + y^2 > 3 + 2*x + x^2 + 2*y + y^2 = g(f(x),f(y)) f(h(x,y)) = 8 + 6*x + 2*x*y + x^2 + 6*y + y^2 > 7 + 3*x + x^2 + 3*y + y^2 = g(h(y,f(x)),h(x,f(y))) g(x,x) = 3 + 2*x > 2 + x = h(e(),x) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^2))