* Step 1: Sum WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M)) nats(N) -> cons(N,nats(s(N))) sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) sieve(cons(s(N),Y)) -> cons(s(N),sieve(filter(Y,N,N))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) filter(cons(X,Y),s(N),M) -> cons(X,filter(Y,N,M)) nats(N) -> cons(N,nats(s(N))) sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) sieve(cons(s(N),Y)) -> cons(s(N),sieve(filter(Y,N,N))) zprimes() -> sieve(nats(s(s(0())))) - Signature: {filter/3,nats/1,sieve/1,zprimes/0} / {0/0,cons/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {filter,nats,sieve,zprimes} and constructors {0,cons,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: filter(y,z,u){y -> cons(x,y),z -> s(z)} = filter(cons(x,y),s(z),u) ->^+ cons(x,filter(y,z,u)) = C[filter(y,z,u) = filter(y,z,u){}] WORST_CASE(Omega(n^1),?)