YES Problem: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Proof: DP Processor: DPs: a__oddNs#() -> a__pairNs#() a__oddNs#() -> a__incr#(a__pairNs()) a__incr#(cons(X,XS)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) a__repItems#(cons(X,XS)) -> mark#(X) mark#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(repItems(X)) -> mark#(X) mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) TDG Processor: DPs: a__oddNs#() -> a__pairNs#() a__oddNs#() -> a__incr#(a__pairNs()) a__incr#(cons(X,XS)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) a__repItems#(cons(X,XS)) -> mark#(X) mark#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) mark#(repItems(X)) -> mark#(X) mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) graph: a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(repItems(X)) -> a__repItems#(mark(X)) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__repItems#(cons(X,XS)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(pair(X1,X2)) -> mark#(X1) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(pair(X1,X2)) -> mark#(X2) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(repItems(X)) -> a__repItems#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(repItems(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(tail(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(zip(X1,X2)) -> mark#(X1) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(zip(X1,X2)) -> mark#(X2) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> mark#(X1) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> mark#(X2) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(oddNs()) -> a__oddNs#() a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(incr(X)) -> a__incr#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(incr(X)) -> mark#(X) a__tail#(cons(X,XS)) -> mark#(XS) -> mark#(pairNs()) -> a__pairNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(pair(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(pair(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(s(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(cons(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(repItems(X)) -> a__repItems#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(repItems(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(tail(X)) -> a__tail#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(tail(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(zip(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(zip(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(take(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(take(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(oddNs()) -> a__oddNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(incr(X)) -> a__incr#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(incr(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) -> mark#(pairNs()) -> a__pairNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(repItems(X)) -> a__repItems#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(repItems(X)) -> a__repItems#(mark(X)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() mark#(tail(X)) -> a__tail#(mark(X)) -> a__tail#(cons(X,XS)) -> mark#(XS) mark#(tail(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> 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#(repItems(X)) -> a__repItems#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) 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#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(tail(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(tail(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(tail(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(tail(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(tail(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(tail(X)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() mark#(repItems(X)) -> a__repItems#(mark(X)) -> a__repItems#(cons(X,XS)) -> mark#(X) mark#(repItems(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(repItems(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(repItems(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(repItems(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(repItems(X)) -> mark#(X) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(repItems(X)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) mark#(repItems(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(repItems(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(repItems(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(repItems(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(repItems(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(repItems(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(repItems(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(repItems(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(repItems(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() mark#(repItems(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(repItems(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(repItems(X)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) -> a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) -> a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(repItems(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(oddNs()) -> a__oddNs#() mark#(zip(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) -> mark#(pairNs()) -> a__pairNs#() mark#(zip(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() mark#(zip(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X1) -> mark#(pairNs()) -> a__pairNs#() mark#(pair(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(repItems(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(oddNs()) -> a__oddNs#() mark#(pair(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) -> mark#(pairNs()) -> a__pairNs#() mark#(pair(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() mark#(pair(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X1) -> mark#(pairNs()) -> a__pairNs#() mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) -> a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(repItems(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(tail(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X2) -> mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(incr(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(pairNs()) -> a__pairNs#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(tail(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() mark#(take(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(pairNs()) -> a__pairNs#() mark#(s(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> 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#(repItems(X)) -> a__repItems#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) 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#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pair(X1,X2)) -> 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#(repItems(X)) -> a__repItems#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(repItems(X)) -> mark#(X) 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#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(oddNs()) -> a__oddNs#() mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(incr(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) mark#(incr(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(incr(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) mark#(incr(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(incr(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(incr(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(incr(X)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) mark#(incr(X)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() mark#(incr(X)) -> a__incr#(mark(X)) -> a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() -> a__oddNs#() -> a__incr#(a__pairNs()) mark#(oddNs()) -> a__oddNs#() -> a__oddNs#() -> a__pairNs#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pair(X1,X2)) -> mark#(X2) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(repItems(X)) -> a__repItems#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(repItems(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> a__tail#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(tail(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(zip(X1,X2)) -> mark#(X2) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(oddNs()) -> a__oddNs#() a__incr#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(incr(X)) -> mark#(X) a__incr#(cons(X,XS)) -> mark#(X) -> mark#(pairNs()) -> a__pairNs#() a__oddNs#() -> a__incr#(a__pairNs()) -> a__incr#(cons(X,XS)) -> mark#(X) SCC Processor: #sccs: 1 #rules: 24 #arcs: 315/676 DPs: a__repItems#(cons(X,XS)) -> mark#(X) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(tail(X)) -> mark#(X) mark#(tail(X)) -> a__tail#(mark(X)) a__tail#(cons(X,XS)) -> mark#(XS) mark#(repItems(X)) -> mark#(X) mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__repItems#](x0) = 4x0 + 5, [a__tail#](x0) = 1x0 + 0, [a__zip#](x0, x1) = x0 + x1 + 0, [a__take#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [a__oddNs#] = 5, [tail](x0) = 1x0 + 1, [pairNs] = 5, [repItems](x0) = 4x0 + 5, [a__repItems](x0) = 4x0 + 5, [a__tail](x0) = 1x0 + 1, [zip](x0, x1) = x0 + x1, [pair](x0, x1) = x0 + x1 + 0, [a__zip](x0, x1) = x0 + x1 + 0, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [a__take](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__incr](x0) = x0 + 0, [a__oddNs] = 5, [cons](x0, x1) = x0 + x1 + 0, [incr](x0) = x0 + 0, [oddNs] = 5, [0] = 1, [a__pairNs] = 5 orientation: a__repItems#(cons(X,XS)) = 4X + 4XS + 5 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + XS + 0 >= X + 0 = mark#(X) mark#(oddNs()) = 5 >= 5 = a__oddNs#() a__oddNs#() = 5 >= 5 = a__incr#(a__pairNs()) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) = N + X + XS + 0 >= X + 0 = mark#(X) mark#(zip(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(zip(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(zip(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS + 0 >= Y + 0 = mark#(Y) mark#(tail(X)) = 1X + 1 >= X + 0 = mark#(X) mark#(tail(X)) = 1X + 1 >= 1X + 1 = a__tail#(mark(X)) a__tail#(cons(X,XS)) = 1X + 1XS + 1 >= XS + 0 = mark#(XS) mark#(repItems(X)) = 4X + 5 >= X + 0 = mark#(X) mark#(repItems(X)) = 4X + 5 >= 4X + 5 = a__repItems#(mark(X)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(pair(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(pair(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS + 0 >= X + 0 = mark#(X) a__pairNs() = 5 >= 5 = cons(0(),incr(oddNs())) a__oddNs() = 5 >= 5 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS + 1 >= 0 = nil() a__take(s(N),cons(X,XS)) = N + X + XS + 0 >= N + X + XS + 0 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = XS + 0 >= 0 = nil() a__zip(X,nil()) = X + 0 >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS + 0 >= X + XS + Y + YS + 0 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = 1X + 1XS + 1 >= XS + 0 = mark(XS) a__repItems(nil()) = 5 >= 0 = nil() a__repItems(cons(X,XS)) = 4X + 4XS + 5 >= X + 4XS + 5 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 5 >= 5 = a__pairNs() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(oddNs()) = 5 >= 5 = a__oddNs() mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = 1X + 1 >= 1X + 1 = a__tail(mark(X)) mark(repItems(X)) = 4X + 5 >= 4X + 5 = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 1 >= 1 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = pair(mark(X1),mark(X2)) a__pairNs() = 5 >= 5 = pairNs() a__incr(X) = X + 0 >= X + 0 = incr(X) a__oddNs() = 5 >= 5 = oddNs() a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__zip(X1,X2) = X1 + X2 + 0 >= X1 + X2 = zip(X1,X2) a__tail(X) = 1X + 1 >= 1X + 1 = tail(X) a__repItems(X) = 4X + 5 >= 4X + 5 = repItems(X) problem: DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(tail(X)) -> a__tail#(mark(X)) mark#(repItems(X)) -> a__repItems#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) SCC Processor: #sccs: 1 #rules: 18 #arcs: 297/400 DPs: a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) mark#(pair(X1,X2)) -> mark#(X1) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zip#](x0, x1) = 2x0 + x1 + 0, [a__take#](x0, x1) = x1, [mark#](x0) = x0, [a__incr#](x0) = x0, [a__oddNs#] = 0, [tail](x0) = x0, [pairNs] = 0, [repItems](x0) = x0, [a__repItems](x0) = x0, [a__tail](x0) = x0, [zip](x0, x1) = 2x0 + x1 + 0, [pair](x0, x1) = 1x0 + x1 + 0, [a__zip](x0, x1) = 2x0 + x1 + 0, [take](x0, x1) = x0 + x1, [nil] = 0, [a__take](x0, x1) = x0 + x1, [s](x0) = x0, [mark](x0) = x0, [a__incr](x0) = x0, [a__oddNs] = 0, [cons](x0, x1) = x0 + x1, [incr](x0) = x0, [oddNs] = 0, [0] = 0, [a__pairNs] = 0 orientation: a__zip#(cons(X,XS),cons(Y,YS)) = 2X + 2XS + Y + YS + 0 >= Y = mark#(Y) mark#(incr(X)) = X >= X = mark#(X) mark#(incr(X)) = X >= X = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + XS >= X = mark#(X) mark#(oddNs()) = 0 >= 0 = a__oddNs#() a__oddNs#() = 0 >= 0 = a__incr#(a__pairNs()) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X2 = a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) = X + XS >= X = mark#(X) mark#(zip(X1,X2)) = 2X1 + X2 + 0 >= X2 = mark#(X2) mark#(zip(X1,X2)) = 2X1 + X2 + 0 >= X1 = mark#(X1) mark#(zip(X1,X2)) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = a__zip#(mark(X1),mark(X2)) a__zip#(cons(X,XS),cons(Y,YS)) = 2X + 2XS + Y + YS + 0 >= X = mark#(X) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) mark#(pair(X1,X2)) = 1X1 + X2 + 0 >= X2 = mark#(X2) mark#(pair(X1,X2)) = 1X1 + X2 + 0 >= X1 = mark#(X1) a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS >= X + XS = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS + 0 >= 0 = nil() a__take(s(N),cons(X,XS)) = N + X + XS >= N + X + XS = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = XS + 2 >= 0 = nil() a__zip(X,nil()) = 2X + 0 >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 2X + 2XS + Y + YS + 0 >= 1X + 2XS + Y + YS + 0 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = X + XS >= XS = mark(XS) a__repItems(nil()) = 0 >= 0 = nil() a__repItems(cons(X,XS)) = X + XS >= X + XS = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = X >= X = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(repItems(X)) = X >= X = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = X >= X = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__zip(X1,X2) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = zip(X1,X2) a__tail(X) = X >= X = tail(X) a__repItems(X) = X >= X = repItems(X) problem: DPs: a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zip#](x0, x1) = x1 + 0, [a__take#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0, [a__oddNs#] = 0, [tail](x0) = x0 + 1, [pairNs] = 0, [repItems](x0) = 2x0, [a__repItems](x0) = 2x0, [a__tail](x0) = x0 + 1, [zip](x0, x1) = 6x0 + x1, [pair](x0, x1) = x1, [a__zip](x0, x1) = 6x0 + x1, [take](x0, x1) = x0 + 5x1 + 1, [nil] = 0, [a__take](x0, x1) = x0 + 5x1 + 1, [s](x0) = x0 + 0, [mark](x0) = x0, [a__incr](x0) = x0 + 0, [a__oddNs] = 0, [cons](x0, x1) = x0 + x1 + 0, [incr](x0) = x0 + 0, [oddNs] = 0, [0] = 0, [a__pairNs] = 0 orientation: a__zip#(cons(X,XS),cons(Y,YS)) = Y + YS + 0 >= Y + 0 = mark#(Y) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + XS + 0 >= X + 0 = mark#(X) mark#(oddNs()) = 0 >= 0 = a__oddNs#() a__oddNs#() = 0 >= 0 = a__incr#(a__pairNs()) mark#(take(X1,X2)) = X1 + 5X2 + 1 >= X2 + 0 = mark#(X2) mark#(take(X1,X2)) = X1 + 5X2 + 1 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = X1 + 5X2 + 1 >= X1 + X2 + 0 = a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) = N + X + XS + 0 >= X + 0 = mark#(X) mark#(zip(X1,X2)) = 6X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(zip(X1,X2)) = 6X1 + X2 + 0 >= X2 + 0 = a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(pair(X1,X2)) = X2 + 0 >= X2 + 0 = mark#(X2) a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = 5XS + 1 >= 0 = nil() a__take(s(N),cons(X,XS)) = N + 5X + 5XS + 5 >= N + X + 5XS + 1 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = XS + 6 >= 0 = nil() a__zip(X,nil()) = 6X + 0 >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 6X + 6XS + Y + YS + 6 >= 6XS + Y + YS + 0 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = X + XS + 1 >= XS = mark(XS) a__repItems(nil()) = 2 >= 0 = nil() a__repItems(cons(X,XS)) = 2X + 2XS + 2 >= X + 2XS + 0 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = X1 + 5X2 + 1 >= X1 + 5X2 + 1 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 6X1 + X2 >= 6X1 + X2 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = X + 1 >= X + 1 = a__tail(mark(X)) mark(repItems(X)) = 2X >= 2X = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = X2 >= X2 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = X + 0 >= X + 0 = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = X1 + 5X2 + 1 >= X1 + 5X2 + 1 = take(X1,X2) a__zip(X1,X2) = 6X1 + X2 >= 6X1 + X2 = zip(X1,X2) a__tail(X) = X + 1 >= X + 1 = tail(X) a__repItems(X) = 2X >= 2X = repItems(X) problem: DPs: a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zip#](x0, x1) = x0 + x1 + 0, [a__take#](x0, x1) = x0 + x1, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [a__oddNs#] = 2, [tail](x0) = x0, [pairNs] = 2, [repItems](x0) = x0 + 3, [a__repItems](x0) = x0 + 3, [a__tail](x0) = x0, [zip](x0, x1) = x0 + x1, [pair](x0, x1) = x1, [a__zip](x0, x1) = x0 + x1, [take](x0, x1) = 2x0 + 1x1, [nil] = 0, [a__take](x0, x1) = 2x0 + 1x1, [s](x0) = x0 + 0, [mark](x0) = x0, [a__incr](x0) = x0 + 0, [a__oddNs] = 2, [cons](x0, x1) = x0 + x1, [incr](x0) = x0 + 0, [oddNs] = 2, [0] = 0, [a__pairNs] = 2 orientation: a__zip#(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS + 0 >= Y + 0 = mark#(Y) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + XS + 0 >= X + 0 = mark#(X) mark#(oddNs()) = 2 >= 2 = a__oddNs#() a__oddNs#() = 2 >= 2 = a__incr#(a__pairNs()) mark#(take(X1,X2)) = 2X1 + 1X2 + 0 >= X1 + 0 = mark#(X1) mark#(take(X1,X2)) = 2X1 + 1X2 + 0 >= X1 + X2 = a__take#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) = N + X + XS + 0 >= X + 0 = mark#(X) mark#(zip(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(zip(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(pair(X1,X2)) = X2 + 0 >= X2 + 0 = mark#(X2) a__pairNs() = 2 >= 2 = cons(0(),incr(oddNs())) a__oddNs() = 2 >= 2 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = 1XS + 2 >= 0 = nil() a__take(s(N),cons(X,XS)) = 2N + 1X + 1XS + 2 >= 2N + X + 1XS = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = XS + 0 >= 0 = nil() a__zip(X,nil()) = X + 0 >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS >= XS + Y + YS = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = X + XS >= XS = mark(XS) a__repItems(nil()) = 3 >= 0 = nil() a__repItems(cons(X,XS)) = X + XS + 3 >= X + XS + 3 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 2 >= 2 = a__pairNs() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(oddNs()) = 2 >= 2 = a__oddNs() mark(take(X1,X2)) = 2X1 + 1X2 >= 2X1 + 1X2 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = X1 + X2 >= X1 + X2 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = X >= X = a__tail(mark(X)) mark(repItems(X)) = X + 3 >= X + 3 = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = X2 >= X2 = pair(mark(X1),mark(X2)) a__pairNs() = 2 >= 2 = pairNs() a__incr(X) = X + 0 >= X + 0 = incr(X) a__oddNs() = 2 >= 2 = oddNs() a__take(X1,X2) = 2X1 + 1X2 >= 2X1 + 1X2 = take(X1,X2) a__zip(X1,X2) = X1 + X2 >= X1 + X2 = zip(X1,X2) a__tail(X) = X >= X = tail(X) a__repItems(X) = X + 3 >= X + 3 = repItems(X) problem: DPs: a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X1) a__take#(s(N),cons(X,XS)) -> mark#(X) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) SCC Processor: #sccs: 1 #rules: 12 #arcs: 175/169 DPs: a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X1) mark#(zip(X1,X2)) -> mark#(X2) mark#(zip(X1,X2)) -> a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) mark#(pair(X1,X2)) -> mark#(X2) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__zip#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [a__oddNs#] = 6, [tail](x0) = x0 + 0, [pairNs] = 6, [repItems](x0) = x0, [a__repItems](x0) = x0 + 0, [a__tail](x0) = x0 + 0, [zip](x0, x1) = 4x0 + 2x1 + 4, [pair](x0, x1) = 4x0 + 2x1 + 4, [a__zip](x0, x1) = 4x0 + 2x1 + 4, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [a__take](x0, x1) = x0 + x1 + 0, [s](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__incr](x0) = x0 + 0, [a__oddNs] = 6, [cons](x0, x1) = x0 + x1 + 0, [incr](x0) = x0, [oddNs] = 6, [0] = 0, [a__pairNs] = 6 orientation: a__zip#(cons(X,XS),cons(Y,YS)) = X + XS + Y + YS + 0 >= Y + 0 = mark#(Y) mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + XS + 0 >= X + 0 = mark#(X) mark#(oddNs()) = 6 >= 6 = a__oddNs#() a__oddNs#() = 6 >= 6 = a__incr#(a__pairNs()) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(zip(X1,X2)) = 4X1 + 2X2 + 4 >= X2 + 0 = mark#(X2) mark#(zip(X1,X2)) = 4X1 + 2X2 + 4 >= X1 + X2 + 0 = a__zip#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) mark#(pair(X1,X2)) = 4X1 + 2X2 + 4 >= X2 + 0 = mark#(X2) a__pairNs() = 6 >= 6 = cons(0(),incr(oddNs())) a__oddNs() = 6 >= 6 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS + 0 >= 0 = nil() a__take(s(N),cons(X,XS)) = N + X + XS + 0 >= N + X + XS + 0 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = 2XS + 4 >= 0 = nil() a__zip(X,nil()) = 4X + 4 >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 4X + 4XS + 2Y + 2YS + 4 >= 4X + 4XS + 2Y + 2YS + 4 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = X + XS + 0 >= XS + 0 = mark(XS) a__repItems(nil()) = 0 >= 0 = nil() a__repItems(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 6 >= 6 = a__pairNs() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(oddNs()) = 6 >= 6 = a__oddNs() mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(repItems(X)) = X + 0 >= X + 0 = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = pair(mark(X1),mark(X2)) a__pairNs() = 6 >= 6 = pairNs() a__incr(X) = X + 0 >= X = incr(X) a__oddNs() = 6 >= 6 = oddNs() a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__zip(X1,X2) = 4X1 + 2X2 + 4 >= 4X1 + 2X2 + 4 = zip(X1,X2) a__tail(X) = X + 0 >= X + 0 = tail(X) a__repItems(X) = X + 0 >= X = repItems(X) problem: DPs: a__zip#(cons(X,XS),cons(Y,YS)) -> mark#(Y) mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) SCC Processor: #sccs: 1 #rules: 8 #arcs: 76/81 DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(take(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0 + 0, [a__incr#](x0) = x0 + 0, [a__oddNs#] = 0, [tail](x0) = x0 + 0, [pairNs] = 0, [repItems](x0) = 5x0 + 0, [a__repItems](x0) = 5x0 + 0, [a__tail](x0) = x0 + 0, [zip](x0, x1) = 5x0 + 3x1 + 0, [pair](x0, x1) = 5x0, [a__zip](x0, x1) = 5x0 + 3x1 + 0, [take](x0, x1) = 4x0 + x1 + 1, [nil] = 2, [a__take](x0, x1) = 4x0 + x1 + 1, [s](x0) = x0 + 0, [mark](x0) = x0, [a__incr](x0) = x0 + 0, [a__oddNs] = 0, [cons](x0, x1) = x0 + x1 + 0, [incr](x0) = x0 + 0, [oddNs] = 0, [0] = 0, [a__pairNs] = 0 orientation: mark#(incr(X)) = X + 0 >= X + 0 = mark#(X) mark#(incr(X)) = X + 0 >= X + 0 = a__incr#(mark(X)) a__incr#(cons(X,XS)) = X + XS + 0 >= X + 0 = mark#(X) mark#(oddNs()) = 0 >= 0 = a__oddNs#() a__oddNs#() = 0 >= 0 = a__incr#(a__pairNs()) mark#(take(X1,X2)) = 4X1 + X2 + 1 >= X1 + 0 = mark#(X1) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__pairNs() = 0 >= 0 = cons(0(),incr(oddNs())) a__oddNs() = 0 >= 0 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = X + XS + 0 >= X + XS + 0 = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS + 4 >= 2 = nil() a__take(s(N),cons(X,XS)) = 4N + X + XS + 4 >= 4N + X + XS + 1 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = 3XS + 7 >= 2 = nil() a__zip(X,nil()) = 5X + 5 >= 2 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 5X + 5XS + 3Y + 3YS + 5 >= 5X + 5XS + 3YS + 0 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = X + XS + 0 >= XS = mark(XS) a__repItems(nil()) = 7 >= 2 = nil() a__repItems(cons(X,XS)) = 5X + 5XS + 5 >= X + 5XS + 0 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 0 >= 0 = a__pairNs() mark(incr(X)) = X + 0 >= X + 0 = a__incr(mark(X)) mark(oddNs()) = 0 >= 0 = a__oddNs() mark(take(X1,X2)) = 4X1 + X2 + 1 >= 4X1 + X2 + 1 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 5X1 + 3X2 + 0 >= 5X1 + 3X2 + 0 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = X + 0 >= X + 0 = a__tail(mark(X)) mark(repItems(X)) = 5X + 0 >= 5X + 0 = a__repItems(mark(X)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(nil()) = 2 >= 2 = nil() mark(pair(X1,X2)) = 5X1 >= 5X1 = pair(mark(X1),mark(X2)) a__pairNs() = 0 >= 0 = pairNs() a__incr(X) = X + 0 >= X + 0 = incr(X) a__oddNs() = 0 >= 0 = oddNs() a__take(X1,X2) = 4X1 + X2 + 1 >= 4X1 + X2 + 1 = take(X1,X2) a__zip(X1,X2) = 5X1 + 3X2 + 0 >= 5X1 + 3X2 + 0 = zip(X1,X2) a__tail(X) = X + 0 >= X + 0 = tail(X) a__repItems(X) = 5X + 0 >= 5X + 0 = repItems(X) problem: DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) a__incr#(cons(X,XS)) -> mark#(X) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Arctic Interpretation Processor: dimension: 1 interpretation: [mark#](x0) = x0, [a__incr#](x0) = x0, [a__oddNs#] = 1, [tail](x0) = 1x0 + 0, [pairNs] = 1, [repItems](x0) = 2x0 + 7, [a__repItems](x0) = 2x0 + 7, [a__tail](x0) = 1x0 + 0, [zip](x0, x1) = 2x0 + 4x1 + 0, [pair](x0, x1) = x1 + 3, [a__zip](x0, x1) = 2x0 + 4x1 + 0, [take](x0, x1) = 2x0 + x1, [nil] = 0, [a__take](x0, x1) = 2x0 + x1, [s](x0) = x0, [mark](x0) = x0, [a__incr](x0) = x0, [a__oddNs] = 1, [cons](x0, x1) = 1x0 + x1 + 0, [incr](x0) = x0, [oddNs] = 1, [0] = 0, [a__pairNs] = 1 orientation: mark#(incr(X)) = X >= X = mark#(X) mark#(incr(X)) = X >= X = a__incr#(mark(X)) a__incr#(cons(X,XS)) = 1X + XS + 0 >= X = mark#(X) mark#(oddNs()) = 1 >= 1 = a__oddNs#() a__oddNs#() = 1 >= 1 = a__incr#(a__pairNs()) mark#(cons(X1,X2)) = 1X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__pairNs() = 1 >= 1 = cons(0(),incr(oddNs())) a__oddNs() = 1 >= 1 = a__incr(a__pairNs()) a__incr(cons(X,XS)) = 1X + XS + 0 >= 1X + XS + 0 = cons(s(mark(X)),incr(XS)) a__take(0(),XS) = XS + 2 >= 0 = nil() a__take(s(N),cons(X,XS)) = 2N + 1X + XS + 0 >= 2N + 1X + XS + 0 = cons(mark(X),take(N,XS)) a__zip(nil(),XS) = 4XS + 2 >= 0 = nil() a__zip(X,nil()) = 2X + 4 >= 0 = nil() a__zip(cons(X,XS),cons(Y,YS)) = 3X + 2XS + 5Y + 4YS + 4 >= 2XS + 1Y + 4YS + 4 = cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) = 2X + 1XS + 1 >= XS = mark(XS) a__repItems(nil()) = 7 >= 0 = nil() a__repItems(cons(X,XS)) = 3X + 2XS + 7 >= 1X + 2XS + 7 = cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) = 1 >= 1 = a__pairNs() mark(incr(X)) = X >= X = a__incr(mark(X)) mark(oddNs()) = 1 >= 1 = a__oddNs() mark(take(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) = 2X1 + 4X2 + 0 >= 2X1 + 4X2 + 0 = a__zip(mark(X1),mark(X2)) mark(tail(X)) = 1X + 0 >= 1X + 0 = a__tail(mark(X)) mark(repItems(X)) = 2X + 7 >= 2X + 7 = a__repItems(mark(X)) mark(cons(X1,X2)) = 1X1 + X2 + 0 >= 1X1 + X2 + 0 = cons(mark(X1),X2) mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(mark(X)) mark(nil()) = 0 >= 0 = nil() mark(pair(X1,X2)) = X2 + 3 >= X2 + 3 = pair(mark(X1),mark(X2)) a__pairNs() = 1 >= 1 = pairNs() a__incr(X) = X >= X = incr(X) a__oddNs() = 1 >= 1 = oddNs() a__take(X1,X2) = 2X1 + X2 >= 2X1 + X2 = take(X1,X2) a__zip(X1,X2) = 2X1 + 4X2 + 0 >= 2X1 + 4X2 + 0 = zip(X1,X2) a__tail(X) = 1X + 0 >= 1X + 0 = tail(X) a__repItems(X) = 2X + 7 >= 2X + 7 = repItems(X) problem: DPs: mark#(incr(X)) -> mark#(X) mark#(incr(X)) -> a__incr#(mark(X)) mark#(oddNs()) -> a__oddNs#() a__oddNs#() -> a__incr#(a__pairNs()) mark#(s(X)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) SCC Processor: #sccs: 1 #rules: 2 #arcs: 33/25 DPs: mark#(s(X)) -> mark#(X) mark#(incr(X)) -> mark#(X) TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__pairNs() -> cons(0(),incr(oddNs())) a__oddNs() -> a__incr(a__pairNs()) a__incr(cons(X,XS)) -> cons(s(mark(X)),incr(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__zip(nil(),XS) -> nil() a__zip(X,nil()) -> nil() a__zip(cons(X,XS),cons(Y,YS)) -> cons(pair(mark(X),mark(Y)),zip(XS,YS)) a__tail(cons(X,XS)) -> mark(XS) a__repItems(nil()) -> nil() a__repItems(cons(X,XS)) -> cons(mark(X),cons(X,repItems(XS))) mark(pairNs()) -> a__pairNs() mark(incr(X)) -> a__incr(mark(X)) mark(oddNs()) -> a__oddNs() mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(zip(X1,X2)) -> a__zip(mark(X1),mark(X2)) mark(tail(X)) -> a__tail(mark(X)) mark(repItems(X)) -> a__repItems(mark(X)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(0()) -> 0() mark(s(X)) -> s(mark(X)) mark(nil()) -> nil() mark(pair(X1,X2)) -> pair(mark(X1),mark(X2)) a__pairNs() -> pairNs() a__incr(X) -> incr(X) a__oddNs() -> oddNs() a__take(X1,X2) -> take(X1,X2) a__zip(X1,X2) -> zip(X1,X2) a__tail(X) -> tail(X) a__repItems(X) -> repItems(X) Qed