MAYBE Problem: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) -> cons(mark(N),nats(s(N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes()) -> a__zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes() -> zprimes() Proof: DP Processor: DPs: a__filter#(cons(X,Y),s(N),M) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) a__nats#(N) -> mark#(N) a__zprimes#() -> a__nats#(s(s(0()))) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> a__nats#(mark(X)) mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) -> cons(mark(N),nats(s(N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes()) -> a__zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes() -> zprimes() CDG Processor: DPs: a__filter#(cons(X,Y),s(N),M) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) a__nats#(N) -> mark#(N) a__zprimes#() -> a__nats#(s(s(0()))) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> a__nats#(mark(X)) mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) -> cons(mark(N),nats(s(N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes()) -> a__zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes() -> zprimes() graph: a__zprimes#() -> a__nats#(s(s(0()))) -> a__nats#(N) -> mark#(N) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) -> a__sieve#(cons(s(N),Y)) -> mark#(N) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X3) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X2) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X1) a__nats#(N) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__nats#(N) -> mark#(N) -> mark#(sieve(X)) -> mark#(X) a__nats#(N) -> mark#(N) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__nats#(N) -> mark#(N) -> mark#(nats(X)) -> mark#(X) a__nats#(N) -> mark#(N) -> mark#(nats(X)) -> a__nats#(mark(X)) a__nats#(N) -> mark#(N) -> mark#(zprimes()) -> a__zprimes#() a__nats#(N) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__nats#(N) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X3) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X2) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> mark#(X1) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(sieve(X)) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(nats(X)) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(nats(X)) -> a__nats#(mark(X)) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(zprimes()) -> a__zprimes#() a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__sieve#(cons(s(N),Y)) -> mark#(N) -> mark#(s(X)) -> mark#(X) mark#(zprimes()) -> a__zprimes#() -> a__zprimes#() -> a__nats#(s(s(0()))) mark#(zprimes()) -> a__zprimes#() -> a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(nats(X)) -> a__nats#(mark(X)) -> a__nats#(N) -> mark#(N) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(nats(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(nats(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(nats(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(nats(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() mark#(nats(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(nats(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) -> a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() mark#(sieve(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(nats(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(nats(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(zprimes()) -> a__zprimes#() mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X3) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(nats(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(zprimes()) -> a__zprimes#() mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(nats(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(zprimes()) -> a__zprimes#() mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) -> a__filter#(cons(X,Y),s(N),M) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(nats(X)) -> a__nats#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X3) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X2) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> mark#(X1) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(nats(X)) -> mark#(X) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(nats(X)) -> a__nats#(mark(X)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(zprimes()) -> a__zprimes#() a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__filter#(cons(X,Y),s(N),M) -> mark#(X) -> mark#(s(X)) -> mark#(X) Restore Modifier: DPs: a__filter#(cons(X,Y),s(N),M) -> mark#(X) a__sieve#(cons(s(N),Y)) -> mark#(N) a__nats#(N) -> mark#(N) a__zprimes#() -> a__nats#(s(s(0()))) a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) mark#(filter(X1,X2,X3)) -> mark#(X3) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(nats(X)) -> mark#(X) mark#(nats(X)) -> a__nats#(mark(X)) mark#(zprimes()) -> a__zprimes#() mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) -> cons(mark(N),nats(s(N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes()) -> a__zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes() -> zprimes() SCC Processor: #sccs: 1 #rules: 16 #arcs: 117/256 DPs: a__zprimes#() -> a__nats#(s(s(0()))) a__nats#(N) -> mark#(N) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(zprimes()) -> a__zprimes#() a__zprimes#() -> a__sieve#(a__nats(s(s(0())))) a__sieve#(cons(s(N),Y)) -> mark#(N) mark#(nats(X)) -> a__nats#(mark(X)) mark#(nats(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2,X3)) -> a__filter#(mark(X1),mark(X2),mark(X3)) a__filter#(cons(X,Y),s(N),M) -> mark#(X) mark#(filter(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2,X3)) -> mark#(X2) mark#(filter(X1,X2,X3)) -> mark#(X3) TRS: a__filter(cons(X,Y),0(),M) -> cons(0(),filter(Y,M,M)) a__filter(cons(X,Y),s(N),M) -> cons(mark(X),filter(Y,N,M)) a__sieve(cons(0(),Y)) -> cons(0(),sieve(Y)) a__sieve(cons(s(N),Y)) -> cons(s(mark(N)),sieve(filter(Y,N,N))) a__nats(N) -> cons(mark(N),nats(s(N))) a__zprimes() -> a__sieve(a__nats(s(s(0())))) mark(filter(X1,X2,X3)) -> a__filter(mark(X1),mark(X2),mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes()) -> a__zprimes() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) a__filter(X1,X2,X3) -> filter(X1,X2,X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes() -> zprimes() Open