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