MAYBE Problem: primes() -> sieve(from(s(s(0())))) from(X) -> cons(X,n__from(n__s(X))) head(cons(X,Y)) -> X tail(cons(X,Y)) -> activate(Y) if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y)))) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) from(X) -> n__from(X) s(X) -> n__s(X) filter(X1,X2) -> n__filter(X1,X2) cons(X1,X2) -> n__cons(X1,X2) sieve(X) -> n__sieve(X) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__sieve(X)) -> sieve(activate(X)) activate(X) -> X Proof: DP Processor: DPs: primes#() -> s#(0()) primes#() -> s#(s(0())) primes#() -> from#(s(s(0()))) primes#() -> sieve#(from(s(s(0())))) from#(X) -> cons#(X,n__from(n__s(X))) tail#(cons(X,Y)) -> activate#(Y) if#(true(),X,Y) -> activate#(X) if#(false(),X,Y) -> activate#(Y) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) filter#(s(s(X)),cons(Y,Z)) -> if#(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y)))) sieve#(cons(X,Y)) -> activate#(Y) sieve#(cons(X,Y)) -> cons#(X,n__filter(X,n__sieve(activate(Y)))) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) TRS: primes() -> sieve(from(s(s(0())))) from(X) -> cons(X,n__from(n__s(X))) head(cons(X,Y)) -> X tail(cons(X,Y)) -> activate(Y) if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y)))) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) from(X) -> n__from(X) s(X) -> n__s(X) filter(X1,X2) -> n__filter(X1,X2) cons(X1,X2) -> n__cons(X1,X2) sieve(X) -> n__sieve(X) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__sieve(X)) -> sieve(activate(X)) activate(X) -> X Usable Rule Processor: DPs: primes#() -> s#(0()) primes#() -> s#(s(0())) primes#() -> from#(s(s(0()))) primes#() -> sieve#(from(s(s(0())))) from#(X) -> cons#(X,n__from(n__s(X))) tail#(cons(X,Y)) -> activate#(Y) if#(true(),X,Y) -> activate#(X) if#(false(),X,Y) -> activate#(Y) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) filter#(s(s(X)),cons(Y,Z)) -> if#(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y)))) sieve#(cons(X,Y)) -> activate#(Y) sieve#(cons(X,Y)) -> cons#(X,n__filter(X,n__sieve(activate(Y)))) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) TRS: f29(x,y) -> x f29(x,y) -> y s(X) -> n__s(X) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__sieve(X)) -> sieve(activate(X)) activate(X) -> X filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y)))) filter(X1,X2) -> n__filter(X1,X2) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) sieve(X) -> n__sieve(X) CDG Processor: DPs: primes#() -> s#(0()) primes#() -> s#(s(0())) primes#() -> from#(s(s(0()))) primes#() -> sieve#(from(s(s(0())))) from#(X) -> cons#(X,n__from(n__s(X))) tail#(cons(X,Y)) -> activate#(Y) if#(true(),X,Y) -> activate#(X) if#(false(),X,Y) -> activate#(Y) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) filter#(s(s(X)),cons(Y,Z)) -> if#(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons (Y,n__filter(X,n__sieve(Y)))) sieve#(cons(X,Y)) -> activate#(Y) sieve#(cons(X,Y)) -> cons#(X,n__filter(X,n__sieve(activate(Y)))) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) TRS: f29(x,y) -> x f29(x,y) -> y s(X) -> n__s(X) from(X) -> cons(X,n__from(n__s(X))) from(X) -> n__from(X) cons(X1,X2) -> n__cons(X1,X2) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__sieve(X)) -> sieve(activate(X)) activate(X) -> X filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons(Y,n__filter(X,n__sieve(Y)))) filter(X1,X2) -> n__filter(X1,X2) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) sieve(X) -> n__sieve(X) graph: filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> activate#(X) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__from(X)) -> from#(activate(X)) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> activate#(X) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__s(X)) -> s#(activate(X)) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__filter(X1,X2)) -> activate#(X2) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__filter(X1,X2)) -> activate#(X1) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__cons(X1,X2)) -> activate#(X1) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__sieve(X)) -> activate#(X) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) -> activate#(n__sieve(X)) -> sieve#(activate(X)) if#(false(),X,Y) -> activate#(Y) -> activate#(n__from(X)) -> activate#(X) if#(false(),X,Y) -> activate#(Y) -> activate#(n__from(X)) -> from#(activate(X)) if#(false(),X,Y) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) if#(false(),X,Y) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) if#(false(),X,Y) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> activate#(X2) if#(false(),X,Y) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> activate#(X1) if#(false(),X,Y) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) if#(false(),X,Y) -> activate#(Y) -> activate#(n__cons(X1,X2)) -> activate#(X1) if#(false(),X,Y) -> activate#(Y) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) if#(false(),X,Y) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) if#(false(),X,Y) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) if#(true(),X,Y) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) if#(true(),X,Y) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) if#(true(),X,Y) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) if#(true(),X,Y) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) if#(true(),X,Y) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X2) if#(true(),X,Y) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X1) if#(true(),X,Y) -> activate#(X) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) if#(true(),X,Y) -> activate#(X) -> activate#(n__cons(X1,X2)) -> activate#(X1) if#(true(),X,Y) -> activate#(X) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) if#(true(),X,Y) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) if#(true(),X,Y) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> activate#(X) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__cons(X1,X2)) -> activate#(X1) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__sieve(X)) -> sieve#(activate(X)) -> sieve#(cons(X,Y)) -> activate#(Y) activate#(n__sieve(X)) -> sieve#(activate(X)) -> sieve#(cons(X,Y)) -> cons#(X,n__filter(X,n__sieve(activate(Y)))) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) -> filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) -> filter#(s(s(X)),cons(Y,Z)) -> if#(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons (Y,n__filter(X,n__sieve(Y)))) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> activate#(X) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> activate#(X) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__filter(X1,X2)) -> activate#(X2) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> activate#(X) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> activate#(X) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__filter(X1,X2)) -> activate#(X1) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__from(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__from(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__from(X)) -> activate#(X) -> activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__from(X)) -> activate#(X) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__from(X)) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) activate#(n__from(X)) -> from#(activate(X)) -> from#(X) -> cons#(X,n__from(n__s(X))) activate#(n__s(X)) -> activate#(X) -> activate#(n__from(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__s(X)) -> s#(activate(X)) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__s(X)) -> activate#(X) -> activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__s(X)) -> activate#(X) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__s(X)) -> activate#(X) -> activate#(n__sieve(X)) -> activate#(X) activate#(n__s(X)) -> activate#(X) -> activate#(n__sieve(X)) -> sieve#(activate(X)) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__from(X)) -> activate#(X) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__from(X)) -> from#(activate(X)) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> activate#(X2) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> activate#(X1) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__cons(X1,X2)) -> activate#(X1) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) tail#(cons(X,Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__from(X)) -> activate#(X) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__from(X)) -> from#(activate(X)) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__s(X)) -> activate#(X) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__s(X)) -> s#(activate(X)) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> activate#(X2) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> activate#(X1) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__cons(X1,X2)) -> activate#(X1) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> activate#(X) sieve#(cons(X,Y)) -> activate#(Y) -> activate#(n__sieve(X)) -> sieve#(activate(X)) primes#() -> sieve#(from(s(s(0())))) -> sieve#(cons(X,Y)) -> activate#(Y) primes#() -> sieve#(from(s(s(0())))) -> sieve#(cons(X,Y)) -> cons#(X,n__filter(X,n__sieve(activate(Y)))) primes#() -> from#(s(s(0()))) -> from#(X) -> cons#(X,n__from(n__s(X))) Restore Modifier: DPs: primes#() -> s#(0()) primes#() -> s#(s(0())) primes#() -> from#(s(s(0()))) primes#() -> sieve#(from(s(s(0())))) from#(X) -> cons#(X,n__from(n__s(X))) tail#(cons(X,Y)) -> activate#(Y) if#(true(),X,Y) -> activate#(X) if#(false(),X,Y) -> activate#(Y) filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) filter#(s(s(X)),cons(Y,Z)) -> if#(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons (Y,n__filter(X,n__sieve(Y)))) sieve#(cons(X,Y)) -> activate#(Y) sieve#(cons(X,Y)) -> cons#(X,n__filter(X,n__sieve(activate(Y)))) activate#(n__from(X)) -> activate#(X) activate#(n__from(X)) -> from#(activate(X)) activate#(n__s(X)) -> activate#(X) activate#(n__s(X)) -> s#(activate(X)) activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__cons(X1,X2)) -> cons#(activate(X1),X2) activate#(n__sieve(X)) -> activate#(X) activate#(n__sieve(X)) -> sieve#(activate(X)) TRS: primes() -> sieve(from(s(s(0())))) from(X) -> cons(X,n__from(n__s(X))) head(cons(X,Y)) -> X tail(cons(X,Y)) -> activate(Y) if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons (Y,n__filter(X,n__sieve(Y)))) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) from(X) -> n__from(X) s(X) -> n__s(X) filter(X1,X2) -> n__filter(X1,X2) cons(X1,X2) -> n__cons(X1,X2) sieve(X) -> n__sieve(X) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__sieve(X)) -> sieve(activate(X)) activate(X) -> X SCC Processor: #sccs: 1 #rules: 10 #arcs: 129/529 DPs: filter#(s(s(X)),cons(Y,Z)) -> activate#(Z) activate#(n__sieve(X)) -> sieve#(activate(X)) sieve#(cons(X,Y)) -> activate#(Y) activate#(n__sieve(X)) -> activate#(X) activate#(n__cons(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> filter#(activate(X1),activate(X2)) activate#(n__filter(X1,X2)) -> activate#(X1) activate#(n__filter(X1,X2)) -> activate#(X2) activate#(n__s(X)) -> activate#(X) activate#(n__from(X)) -> activate#(X) TRS: primes() -> sieve(from(s(s(0())))) from(X) -> cons(X,n__from(n__s(X))) head(cons(X,Y)) -> X tail(cons(X,Y)) -> activate(Y) if(true(),X,Y) -> activate(X) if(false(),X,Y) -> activate(Y) filter(s(s(X)),cons(Y,Z)) -> if(divides(s(s(X)),Y),n__filter(n__s(n__s(X)),activate(Z)),n__cons (Y,n__filter(X,n__sieve(Y)))) sieve(cons(X,Y)) -> cons(X,n__filter(X,n__sieve(activate(Y)))) from(X) -> n__from(X) s(X) -> n__s(X) filter(X1,X2) -> n__filter(X1,X2) cons(X1,X2) -> n__cons(X1,X2) sieve(X) -> n__sieve(X) activate(n__from(X)) -> from(activate(X)) activate(n__s(X)) -> s(activate(X)) activate(n__filter(X1,X2)) -> filter(activate(X1),activate(X2)) activate(n__cons(X1,X2)) -> cons(activate(X1),X2) activate(n__sieve(X)) -> sieve(activate(X)) activate(X) -> X Open