WORST_CASE(?,O(n^1)) * Step 1: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: a__b() -> a() a__b() -> b() a__f(X1,X2,X3) -> f(X1,X2,X3) a__f(a(),X,X) -> a__f(X,a__b(),b()) mark(a()) -> a() mark(b()) -> a__b() mark(f(X1,X2,X3)) -> a__f(X1,mark(X2),X3) - Signature: {a__b/0,a__f/3,mark/1} / {a/0,b/0,f/3} - Obligation: innermost runtime complexity wrt. defined symbols {a__b,a__f,mark} and constructors {a,b,f} + Applied Processor: NaturalPI {shape = Linear, restrict = NoRestrict, uargs = UArgs, urules = URules, selector = Nothing} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(a__f) = {2} Following symbols are considered usable: {a__b,a__f,mark} TcT has computed the following interpretation: p(a) = 1 p(a__b) = 2 p(a__f) = 8 + 8*x1 + x2 + 8*x3 p(b) = 0 p(f) = 2 + x1 + x2 + x3 p(mark) = 10 + 8*x1 Following rules are strictly oriented: a__b() = 2 > 1 = a() a__b() = 2 > 0 = b() a__f(X1,X2,X3) = 8 + 8*X1 + X2 + 8*X3 > 2 + X1 + X2 + X3 = f(X1,X2,X3) a__f(a(),X,X) = 16 + 9*X > 10 + 8*X = a__f(X,a__b(),b()) mark(a()) = 18 > 1 = a() mark(b()) = 10 > 2 = a__b() mark(f(X1,X2,X3)) = 26 + 8*X1 + 8*X2 + 8*X3 > 18 + 8*X1 + 8*X2 + 8*X3 = a__f(X1,mark(X2),X3) Following rules are (at-least) weakly oriented: WORST_CASE(?,O(n^1))