YES Problem: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Proof: DP Processor: DPs: a__primes#() -> a__from#(s(s(0()))) a__primes#() -> a__sieve#(a__from(s(s(0())))) a__from#(X) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve#(cons(X,Y)) -> mark#(X) mark#(primes()) -> a__primes#() mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) TDG Processor: DPs: a__primes#() -> a__from#(s(s(0()))) a__primes#() -> a__sieve#(a__from(s(s(0())))) a__from#(X) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve#(cons(X,Y)) -> mark#(X) mark#(primes()) -> a__primes#() mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) graph: a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) -> a__if#(false(),X,Y) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) -> a__if#(true(),X,Y) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(head(X)) -> a__head#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(head(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(sieve(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(primes()) -> a__primes#() a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(primes()) -> a__primes#() a__if#(false(),X,Y) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(head(X)) -> a__head#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(head(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(sieve(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(primes()) -> a__primes#() a__if#(true(),X,Y) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__if#(true(),X,Y) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(primes()) -> a__primes#() a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X2) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X2) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(head(X)) -> a__head#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(head(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(sieve(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(primes()) -> a__primes#() a__head#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__head#(cons(X,Y)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__head#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__head#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) -> a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) -> a__if#(true(),X,Y) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(tail(X)) -> a__tail#(mark(X)) -> a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(tail(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(tail(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(head(X)) -> a__head#(mark(X)) -> a__head#(cons(X,Y)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(head(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(head(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(head(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(primes()) -> a__primes#() -> a__primes#() -> a__sieve#(a__from(s(s(0())))) mark#(primes()) -> a__primes#() -> a__primes#() -> a__from#(s(s(0()))) mark#(sieve(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(sieve(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(sieve(X)) -> a__sieve#(mark(X)) -> a__sieve#(cons(X,Y)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) -> a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) -> a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) -> a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> a__head#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(primes()) -> a__primes#() mark#(filter(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(divides(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> a__head#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(primes()) -> a__primes#() mark#(divides(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(from(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(from(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(primes()) -> a__primes#() a__from#(X) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__from#(X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__from#(X) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__from#(X) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__from#(X) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) -> a__sieve#(cons(X,Y)) -> mark#(X) a__primes#() -> a__from#(s(s(0()))) -> a__from#(X) -> mark#(X) EDG Processor: DPs: a__primes#() -> a__from#(s(s(0()))) a__primes#() -> a__sieve#(a__from(s(s(0())))) a__from#(X) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve#(cons(X,Y)) -> mark#(X) mark#(primes()) -> a__primes#() mark#(sieve(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(head(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) graph: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(primes()) -> a__primes#() a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(sieve(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(head(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(head(X)) -> a__head#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(primes()) -> a__primes#() a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(primes()) -> a__primes#() a__if#(false(),X,Y) -> mark#(Y) -> mark#(sieve(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(head(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(head(X)) -> a__head#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(primes()) -> a__primes#() a__if#(true(),X,Y) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__if#(true(),X,Y) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__if#(true(),X,Y) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(primes()) -> a__primes#() a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(sieve(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(from(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(from(X)) -> a__from#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(head(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(head(X)) -> a__head#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X2) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(filter(X1,X2)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X2) a__tail#(cons(X,Y)) -> mark#(Y) -> mark#(divides(X1,X2)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(primes()) -> a__primes#() a__head#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__head#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__head#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__head#(cons(X,Y)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__head#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__head#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) -> a__if#(true(),X,Y) -> mark#(X) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) -> a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) -> a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(tail(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(tail(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(tail(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(head(X)) -> a__head#(mark(X)) -> a__head#(cons(X,Y)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(head(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(head(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(head(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(head(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(head(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(head(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(primes()) -> a__primes#() -> a__primes#() -> a__from#(s(s(0()))) mark#(primes()) -> a__primes#() -> a__primes#() -> a__sieve#(a__from(s(s(0())))) mark#(sieve(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() 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#(from(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(sieve(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(sieve(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(sieve(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(sieve(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(sieve(X)) -> a__sieve#(mark(X)) -> a__sieve#(cons(X,Y)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) -> a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) -> a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) -> a__filter#(s(s(X)),cons(Y,Z)) -> a__if#(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(primes()) -> a__primes#() mark#(filter(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> a__head#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(filter(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(filter(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(primes()) -> a__primes#() mark#(divides(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> a__head#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X2) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(primes()) -> a__primes#() mark#(divides(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(divides(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(primes()) -> a__primes#() 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#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() mark#(from(X)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(from(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(primes()) -> a__primes#() 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#(from(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(primes()) -> a__primes#() a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__sieve#(cons(X,Y)) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(primes()) -> a__primes#() a__from#(X) -> mark#(X) -> mark#(sieve(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(sieve(X)) -> a__sieve#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__from#(X) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__from#(X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__from#(X) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(filter(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(divides(X1,X2)) -> mark#(X1) a__primes#() -> a__sieve#(a__from(s(s(0())))) -> a__sieve#(cons(X,Y)) -> mark#(X) a__primes#() -> a__from#(s(s(0()))) -> a__from#(X) -> mark#(X) SCC Processor: #sccs: 1 #rules: 28 #arcs: 355/841 DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,Y)) -> mark#(X) mark#(head(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__primes#() -> a__from#(s(s(0()))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = 7x0 + 1, [a__sieve](x0) = x0 + 0, [filter](x0, x1) = x0 + x1, [a__head](x0) = 7x0 + 1, [0] = 0, [a__from](x0) = x0 + 0, [sieve](x0) = x0 + 0, [a__primes#] = 0, [a__tail#](x0) = x0, [tail](x0) = x0 + 0, [primes] = 0, [a__if#](x0, x1, x2) = x0 + x1 + x2, [a__from#](x0) = x0 + 0, [a__filter#](x0, x1) = x0 + x1 + 0, [mark](x0) = x0, [true] = 0, [a__primes] = 0, [cons](x0, x1) = x0 + x1 + 0, [if](x0, x1, x2) = x0 + x1 + x2, [a__head#](x0) = x0 + 0, [divides](x0, x1) = x0 + x1, [false] = 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__if](x0, x1, x2) = x0 + x1 + x2, [a__tail](x0) = x0 + 0, [a__filter](x0, x1) = x0 + x1, [from](x0) = x0 + 0, [s](x0) = x0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= Y + 0 = mark#(Y) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X + 0 = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y + 0 = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(tail(X)) = X + 0 >= X = a__tail#(mark(X)) a__tail#(cons(X,Y)) = X + Y + 0 >= Y + 0 = mark#(Y) mark#(tail(X)) = X + 0 >= X + 0 = mark#(X) mark#(head(X)) = 7X + 1 >= X + 0 = a__head#(mark(X)) a__head#(cons(X,Y)) = X + Y + 0 >= X + 0 = mark#(X) mark#(head(X)) = 7X + 1 >= X + 0 = mark#(X) mark#(from(X)) = X + 0 >= X + 0 = a__from#(mark(X)) a__from#(X) = X + 0 >= X + 0 = mark#(X) mark#(from(X)) = X + 0 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) a__sieve#(cons(X,Y)) = X + Y + 0 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = mark#(X) mark#(primes()) = 0 >= 0 = a__primes#() a__primes#() = 0 >= 0 = a__sieve#(a__from(s(s(0())))) a__primes#() = 0 >= 0 = a__from#(s(s(0()))) a__if#(true(),X,Y) = X + Y + 0 >= X + 0 = mark#(X) a__primes() = 0 >= 0 = a__sieve(a__from(s(s(0())))) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = 7X + 7Y + 7 >= X = mark(X) a__tail(cons(X,Y)) = X + Y + 0 >= Y = mark(Y) a__if(true(),X,Y) = X + Y + 0 >= X = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 0 >= 0 = a__primes() mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(head(X)) = 7X + 1 >= 7X + 1 = a__head(mark(X)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 >= X1 + X2 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + X2 >= X1 + X2 = divides(mark(X1),mark(X2)) a__primes() = 0 >= 0 = primes() a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__from(X) = X + 0 >= X + 0 = from(X) a__head(X) = 7X + 1 >= 7X + 1 = head(X) a__tail(X) = X + 0 >= X + 0 = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 >= X1 + X2 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__primes#() -> a__from#(s(s(0()))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) a__head#(cons(X,Y)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__primes#() -> a__from#(s(s(0()))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) SCC Processor: #sccs: 1 #rules: 25 #arcs: 354/676 DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__primes#() -> a__from#(s(s(0()))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Bounds Processor: bound: 0 enrichment: top-dp automaton: final states: {169} transitions: 00() -> 170* s0(171) -> 172* s0(170) -> 171* a__from{#,0}(172) -> 169* mark{#,0}(170) -> 169* mark{#,0}(172) -> 169* mark{#,0}(171) -> 169* problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,Y)) -> mark#(Y) mark#(tail(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = x0, [a__sieve](x0) = x0 + 0, [filter](x0, x1) = x0 + x1 + 0, [a__head](x0) = x0 + 0, [0] = 0, [a__from](x0) = x0 + 0, [sieve](x0) = x0 + 0, [a__primes#] = 0, [a__tail#](x0) = 7x0 + 0, [tail](x0) = 7x0 + 7, [primes] = 0, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__from#](x0) = x0 + 0, [a__filter#](x0, x1) = x0 + x1, [mark](x0) = x0 + 0, [true] = 0, [a__primes] = 0, [cons](x0, x1) = x0 + x1, [if](x0, x1, x2) = x0 + x1 + x2 + 0, [divides](x0, x1) = x0 + x1 + 0, [false] = 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = 7x0 + 7, [a__filter](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 0, [s](x0) = x0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z >= Y = mark#(Y) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z >= X = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 = mark#(X1) mark#(tail(X)) = 7X + 7 >= 7X + 7 = a__tail#(mark(X)) a__tail#(cons(X,Y)) = 7X + 7Y + 0 >= Y = mark#(Y) mark#(tail(X)) = 7X + 7 >= X = mark#(X) mark#(from(X)) = X + 0 >= X + 0 = a__from#(mark(X)) a__from#(X) = X + 0 >= X = mark#(X) mark#(from(X)) = X + 0 >= X = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) a__sieve#(cons(X,Y)) = X + Y + 0 >= X = mark#(X) mark#(sieve(X)) = X + 0 >= X = mark#(X) mark#(primes()) = 0 >= 0 = a__primes#() a__primes#() = 0 >= 0 = a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) = X + Y + 0 >= X = mark#(X) a__primes() = 0 >= 0 = a__sieve(a__from(s(s(0())))) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = X + Y + 0 >= X + 0 = mark(X) a__tail(cons(X,Y)) = 7X + 7Y + 7 >= Y + 0 = mark(Y) a__if(true(),X,Y) = X + Y + 0 >= X + 0 = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y + 0 = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 0 >= 0 = a__primes() mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = 7X + 7 >= 7X + 7 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 0 >= 0 = primes() a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__from(X) = X + 0 >= X + 0 = from(X) a__head(X) = X + 0 >= X = head(X) a__tail(X) = 7X + 7 >= 7X + 7 = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(tail(X)) -> a__tail#(mark(X)) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) SCC Processor: #sccs: 1 #rules: 21 #arcs: 283/484 DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) mark#(primes()) -> a__primes#() a__primes#() -> a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = x0, [a__sieve](x0) = x0 + 0, [filter](x0, x1) = x0 + x1, [a__head](x0) = x0 + 0, [0] = 4, [a__from](x0) = x0 + 0, [sieve](x0) = x0 + 0, [a__primes#] = 4, [tail](x0) = x0 + 4, [primes] = 5, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__from#](x0) = x0 + 0, [a__filter#](x0, x1) = x0 + x1 + 0, [mark](x0) = x0 + 0, [true] = 4, [a__primes] = 5, [cons](x0, x1) = x0 + x1 + 0, [if](x0, x1, x2) = x0 + x1 + x2 + 0, [divides](x0, x1) = x0 + x1, [false] = 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0 + 0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = x0 + 4, [a__filter](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 0, [s](x0) = x0 + 0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= Y + 0 = mark#(Y) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X + 0 = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y + 0 = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(from(X)) = X + 0 >= X + 0 = a__from#(mark(X)) a__from#(X) = X + 0 >= X + 0 = mark#(X) mark#(from(X)) = X + 0 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) a__sieve#(cons(X,Y)) = X + Y + 0 >= X + 0 = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = mark#(X) mark#(primes()) = 5 >= 4 = a__primes#() a__primes#() = 4 >= 4 = a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) = X + Y + 4 >= X + 0 = mark#(X) a__primes() = 5 >= 4 = a__sieve(a__from(s(s(0())))) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = X + Y + 0 >= X + 0 = mark(X) a__tail(cons(X,Y)) = X + Y + 4 >= Y + 0 = mark(Y) a__if(true(),X,Y) = X + Y + 4 >= X + 0 = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y + 0 = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 5 >= 5 = a__primes() mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = X + 4 >= X + 4 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 4 >= 4 = 0() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(true()) = 4 >= 4 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 5 >= 5 = primes() a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__from(X) = X + 0 >= X + 0 = from(X) a__head(X) = X + 0 >= X = head(X) a__tail(X) = X + 4 >= X + 4 = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 + 0 >= X1 + X2 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) a__primes#() -> a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) a__primes#() -> a__sieve#(a__from(s(s(0())))) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) SCC Processor: #sccs: 1 #rules: 19 #arcs: 218/400 DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(from(X)) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = x0 + 0, [a__sieve](x0) = x0 + 0, [filter](x0, x1) = x0 + x1 + 0, [a__head](x0) = x0 + 0, [0] = 4, [a__from](x0) = 1x0 + 2, [sieve](x0) = x0 + 0, [tail](x0) = x0, [primes] = 5, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__from#](x0) = x0 + 0, [a__filter#](x0, x1) = x0 + x1 + 0, [mark](x0) = x0 + 0, [true] = 0, [a__primes] = 5, [cons](x0, x1) = x0 + x1 + 0, [if](x0, x1, x2) = x0 + x1 + x2 + 0, [divides](x0, x1) = x0 + x1 + 0, [false] = 0, [a__sieve#](x0) = x0 + 0, [mark#](x0) = x0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = x0 + 0, [a__filter](x0, x1) = x0 + x1 + 0, [from](x0) = 1x0 + 2, [s](x0) = x0 + 0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= Y = mark#(Y) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X + 0 >= X = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 = mark#(X1) mark#(from(X)) = 1X + 2 >= X + 0 = a__from#(mark(X)) a__from#(X) = X + 0 >= X = mark#(X) mark#(from(X)) = 1X + 2 >= X = mark#(X) mark#(sieve(X)) = X + 0 >= X + 0 = a__sieve#(mark(X)) a__sieve#(cons(X,Y)) = X + Y + 0 >= X = mark#(X) mark#(sieve(X)) = X + 0 >= X = mark#(X) a__if#(true(),X,Y) = X + Y + 0 >= X = mark#(X) a__primes() = 5 >= 5 = a__sieve(a__from(s(s(0())))) a__from(X) = 1X + 2 >= 1X + 2 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = X + Y + 0 >= X + 0 = mark(X) a__tail(cons(X,Y)) = X + Y + 0 >= Y + 0 = mark(Y) a__if(true(),X,Y) = X + Y + 0 >= X + 0 = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y + 0 = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + Y + Z + 0 >= X + Y + Z + 0 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = X + Y + 0 >= X + Y + 0 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 5 >= 5 = a__primes() mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(from(X)) = 1X + 2 >= 1X + 2 = a__from(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 4 >= 4 = 0() mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 5 >= 5 = primes() a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__from(X) = 1X + 2 >= 1X + 2 = from(X) a__head(X) = X + 0 >= X + 0 = head(X) a__tail(X) = X + 0 >= X = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__from#(X) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__from#(X) -> mark#(X) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) SCC Processor: #sccs: 1 #rules: 16 #arcs: 201/289 DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(Y) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> a__sieve#(mark(X)) a__sieve#(cons(X,Y)) -> mark#(X) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = x0, [a__sieve](x0) = x0 + 0, [filter](x0, x1) = x0 + x1, [a__head](x0) = x0 + 0, [0] = 1, [a__from](x0) = 1x0 + 1, [sieve](x0) = x0, [tail](x0) = x0, [primes] = 3, [a__if#](x0, x1, x2) = 2x0 + 4x1 + 4x2 + 0, [a__filter#](x0, x1) = 4x0 + 4x1 + 0, [mark](x0) = x0 + 0, [true] = 2, [a__primes] = 3, [cons](x0, x1) = 1x0 + x1 + 1, [if](x0, x1, x2) = x0 + x1 + x2, [divides](x0, x1) = x0 + x1 + 0, [false] = 2, [a__sieve#](x0) = 4x0 + 0, [mark#](x0) = 4x0 + 4, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = x0 + 0, [a__filter](x0, x1) = x0 + x1 + 0, [from](x0) = 1x0 + 1, [s](x0) = x0 + 0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = 4X + 5Y + 4Z + 5 >= 4Y + 4 = mark#(Y) mark#(divides(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4 = mark#(X1) mark#(divides(X1,X2)) = 4X1 + 4X2 + 4 >= 4X2 + 4 = mark#(X2) mark#(cons(X1,X2)) = 5X1 + 4X2 + 5 >= 4X1 + 4 = mark#(X1) mark#(s(X)) = 4X + 4 >= 4X + 4 = mark#(X) mark#(filter(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) = 4X + 5Y + 4Z + 5 >= 4X + 4 = mark#(X) mark#(filter(X1,X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4 = mark#(X1) mark#(filter(X1,X2)) = 4X1 + 4X2 + 4 >= 4X2 + 4 = mark#(X2) mark#(if(X1,X2,X3)) = 4X1 + 4X2 + 4X3 + 4 >= 2X1 + 4X2 + 4X3 + 2 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = 4X + 4Y + 4 >= 4Y + 4 = mark#(Y) mark#(if(X1,X2,X3)) = 4X1 + 4X2 + 4X3 + 4 >= 4X1 + 4 = mark#(X1) mark#(sieve(X)) = 4X + 4 >= 4X + 4 = a__sieve#(mark(X)) a__sieve#(cons(X,Y)) = 5X + 4Y + 5 >= 4X + 4 = mark#(X) mark#(sieve(X)) = 4X + 4 >= 4X + 4 = mark#(X) a__if#(true(),X,Y) = 4X + 4Y + 4 >= 4X + 4 = mark#(X) a__primes() = 3 >= 2 = a__sieve(a__from(s(s(0())))) a__from(X) = 1X + 1 >= 1X + 1 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = 1X + Y + 1 >= X + 0 = mark(X) a__tail(cons(X,Y)) = 1X + Y + 1 >= Y + 0 = mark(Y) a__if(true(),X,Y) = X + Y + 2 >= X + 0 = mark(X) a__if(false(),X,Y) = X + Y + 2 >= Y + 0 = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + 1Y + Z + 1 >= X + 1Y + Z + 1 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = 1X + Y + 1 >= 1X + Y + 1 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 3 >= 3 = a__primes() mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(from(X)) = 1X + 1 >= 1X + 1 = a__from(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 1 >= 1 = 0() mark(cons(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = cons(mark(X1),X2) mark(true()) = 2 >= 2 = true() mark(false()) = 2 >= 2 = false() mark(divides(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 3 >= 3 = primes() a__sieve(X) = X + 0 >= X = sieve(X) a__from(X) = 1X + 1 >= 1X + 1 = from(X) a__head(X) = X + 0 >= X = head(X) a__tail(X) = X + 0 >= X = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 + 0 >= X1 + X2 = filter(X1,X2) problem: DPs: mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> a__sieve#(mark(X)) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) SCC Processor: #sccs: 1 #rules: 12 #arcs: 148/169 DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(sieve(X)) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = x0, [a__sieve](x0) = 1x0 + 1, [filter](x0, x1) = x0 + x1 + 0, [a__head](x0) = x0 + 0, [0] = 2, [a__from](x0) = 1x0 + 4, [sieve](x0) = 1x0 + 1, [tail](x0) = x0, [primes] = 5, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__filter#](x0, x1) = x0 + x1 + 0, [mark](x0) = x0 + 0, [true] = 0, [a__primes] = 5, [cons](x0, x1) = 1x0 + x1 + 1, [if](x0, x1, x2) = x0 + x1 + x2 + 0, [divides](x0, x1) = x0 + x1 + 0, [false] = 0, [mark#](x0) = x0 + 0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = x0 + 0, [a__filter](x0, x1) = x0 + x1 + 0, [from](x0) = 1x0 + 4, [s](x0) = x0 + 0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X + 1Y + Z + 1 >= X + 0 = mark#(X) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y + 0 = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + 0 = mark#(X1) mark#(sieve(X)) = 1X + 1 >= X + 0 = mark#(X) a__if#(true(),X,Y) = X + Y + 0 >= X + 0 = mark#(X) a__primes() = 5 >= 5 = a__sieve(a__from(s(s(0())))) a__from(X) = 1X + 4 >= 1X + 4 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = 1X + Y + 1 >= X + 0 = mark(X) a__tail(cons(X,Y)) = 1X + Y + 1 >= Y + 0 = mark(Y) a__if(true(),X,Y) = X + Y + 0 >= X + 0 = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y + 0 = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + 1Y + Z + 1 >= X + 1Y + Z + 1 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = 2X + 1Y + 2 >= 1X + 1Y + 1 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 5 >= 5 = a__primes() mark(sieve(X)) = 1X + 1 >= 1X + 1 = a__sieve(mark(X)) mark(from(X)) = 1X + 4 >= 1X + 4 = a__from(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 2 >= 2 = 0() mark(cons(X1,X2)) = 1X1 + X2 + 1 >= 1X1 + X2 + 1 = cons(mark(X1),X2) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 5 >= 5 = primes() a__sieve(X) = 1X + 1 >= 1X + 1 = sieve(X) a__from(X) = 1X + 4 >= 1X + 4 = from(X) a__head(X) = X + 0 >= X = head(X) a__tail(X) = X + 0 >= X = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(divides(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = 1x0, [a__sieve](x0) = x0 + 0, [filter](x0, x1) = x0 + x1 + 0, [a__head](x0) = 1x0, [0] = 4, [a__from](x0) = 2x0 + 7, [sieve](x0) = x0 + 0, [tail](x0) = 3x0, [primes] = 7, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__filter#](x0, x1) = x0 + x1 + 0, [mark](x0) = x0, [true] = 0, [a__primes] = 7, [cons](x0, x1) = 2x0 + x1, [if](x0, x1, x2) = x0 + x1 + x2 + 0, [divides](x0, x1) = x0 + 1x1 + 0, [false] = 0, [mark#](x0) = x0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = 3x0, [a__filter](x0, x1) = x0 + x1 + 0, [from](x0) = 2x0 + 7, [s](x0) = x0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X + 2Y + Z + 0 >= X = mark#(X) mark#(divides(X1,X2)) = X1 + 1X2 + 0 >= X1 = mark#(X1) mark#(divides(X1,X2)) = X1 + 1X2 + 0 >= X2 = mark#(X2) mark#(s(X)) = X >= X = mark#(X) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(filter(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 = mark#(X1) a__if#(true(),X,Y) = X + Y + 0 >= X = mark#(X) a__primes() = 7 >= 7 = a__sieve(a__from(s(s(0())))) a__from(X) = 2X + 7 >= 2X + 7 = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = 3X + 1Y >= X = mark(X) a__tail(cons(X,Y)) = 5X + 3Y >= Y = mark(Y) a__if(true(),X,Y) = X + Y + 0 >= X = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = X + 2Y + Z + 0 >= X + 2Y + Z + 0 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = 2X + Y + 0 >= 2X + Y + 0 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 7 >= 7 = a__primes() mark(sieve(X)) = X + 0 >= X + 0 = a__sieve(mark(X)) mark(from(X)) = 2X + 7 >= 2X + 7 = a__from(mark(X)) mark(head(X)) = 1X >= 1X = a__head(mark(X)) mark(tail(X)) = 3X >= 3X = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 4 >= 4 = 0() mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + 1X2 + 0 >= X1 + 1X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 7 >= 7 = primes() a__sieve(X) = X + 0 >= X + 0 = sieve(X) a__from(X) = 2X + 7 >= 2X + 7 = from(X) a__head(X) = 1X >= 1X = head(X) a__tail(X) = 3X >= 3X = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = if(X1,X2,X3) a__filter(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) -> mark#(X1) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) interpretation: [head](x0) = 3x0, [a__sieve](x0) = 1x0 + 0, [filter](x0, x1) = 1x0 + x1 + 0, [a__head](x0) = 3x0, [0] = 0, [a__from](x0) = 1x0, [sieve](x0) = 1x0 + 0, [tail](x0) = x0 + 0, [primes] = 2, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 0, [a__filter#](x0, x1) = x0, [mark](x0) = x0, [true] = 0, [a__primes] = 2, [cons](x0, x1) = 1x0 + x1, [if](x0, x1, x2) = x0 + x1 + x2 + 0, [divides](x0, x1) = x0 + x1 + 0, [false] = 0, [mark#](x0) = x0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [a__tail](x0) = x0 + 0, [a__filter](x0, x1) = 1x0 + x1 + 0, [from](x0) = 1x0, [s](x0) = x0 orientation: a__filter#(s(s(X)),cons(Y,Z)) = X >= X = mark#(X) mark#(divides(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) mark#(filter(X1,X2)) = 1X1 + X2 + 0 >= X1 = a__filter#(mark(X1),mark(X2)) mark#(filter(X1,X2)) = 1X1 + X2 + 0 >= X1 = mark#(X1) mark#(filter(X1,X2)) = 1X1 + X2 + 0 >= X2 = mark#(X2) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) = X + Y + 0 >= Y = mark#(Y) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 = mark#(X1) a__if#(true(),X,Y) = X + Y + 0 >= X = mark#(X) a__primes() = 2 >= 2 = a__sieve(a__from(s(s(0())))) a__from(X) = 1X >= 1X = cons(mark(X),from(s(X))) a__head(cons(X,Y)) = 4X + 3Y >= X = mark(X) a__tail(cons(X,Y)) = 1X + Y + 0 >= Y = mark(Y) a__if(true(),X,Y) = X + Y + 0 >= X = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y = mark(Y) a__filter(s(s(X)),cons(Y,Z)) = 1X + 1Y + Z + 0 >= 1X + 1Y + Z + 0 = a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z),cons(Y,filter( X, sieve (Y)))) a__sieve(cons(X,Y)) = 2X + 1Y + 0 >= 1X + 1Y + 0 = cons(mark(X),filter(X,sieve(Y))) mark(primes()) = 2 >= 2 = a__primes() mark(sieve(X)) = 1X + 0 >= 1X + 0 = a__sieve(mark(X)) mark(from(X)) = 1X >= 1X = a__from(mark(X)) mark(head(X)) = 3X >= 3X = a__head(mark(X)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(filter(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = a__filter(mark(X1),mark(X2)) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(cons(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = cons(mark(X1),X2) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(divides(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = divides(mark(X1),mark(X2)) a__primes() = 2 >= 2 = primes() a__sieve(X) = 1X + 0 >= 1X + 0 = sieve(X) a__from(X) = 1X >= 1X = from(X) a__head(X) = 3X >= 3X = head(X) a__tail(X) = X + 0 >= X + 0 = tail(X) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = if(X1,X2,X3) a__filter(X1,X2) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = filter(X1,X2) problem: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Restore Modifier: DPs: a__filter#(s(s(X)),cons(Y,Z)) -> mark#(X) mark#(divides(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) SCC Processor: #sccs: 1 #rules: 7 #arcs: 93/64 DPs: mark#(divides(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(filter(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) Size-Change Termination Processor: DPs: TRS: a__primes() -> a__sieve(a__from(s(s(0())))) a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,Y)) -> mark(X) a__tail(cons(X,Y)) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__filter(s(s(X)),cons(Y,Z)) -> a__if(divides(s(s(mark(X))),mark(Y)),filter(s(s(X)),Z), cons(Y,filter(X,sieve(Y)))) a__sieve(cons(X,Y)) -> cons(mark(X),filter(X,sieve(Y))) mark(primes()) -> a__primes() mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(filter(X1,X2)) -> a__filter(mark(X1),mark(X2)) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(true()) -> true() mark(false()) -> false() mark(divides(X1,X2)) -> divides(mark(X1),mark(X2)) a__primes() -> primes() a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1,X2,X3) -> if(X1,X2,X3) a__filter(X1,X2) -> filter(X1,X2) The DP: mark#(divides(X1,X2)) -> mark#(X1) has the edges: 0 > 0 The DP: mark#(s(X)) -> mark#(X) has the edges: 0 > 0 The DP: mark#(filter(X1,X2)) -> mark#(X2) has the edges: 0 > 0 The DP: mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) has the edges: 0 > 2 0 > 1 The DP: a__if#(false(),X,Y) -> mark#(Y) has the edges: 2 >= 0 The DP: mark#(if(X1,X2,X3)) -> mark#(X1) has the edges: 0 > 0 The DP: a__if#(true(),X,Y) -> mark#(X) has the edges: 1 >= 0 Qed