MAYBE Problem: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Proof: DP Processor: DPs: a__from#(X) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) a__2nd#(cons(X,XS)) -> a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) a__sel#(s(N),cons(X,XS)) -> mark#(N) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) 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#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) TDG Processor: DPs: a__from#(X) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) a__2nd#(cons(X,XS)) -> a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) a__sel#(s(N),cons(X,XS)) -> mark#(N) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) 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#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) graph: a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__sel#(0(),cons(X,XS)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) -> a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) -> a__sel#(s(N),cons(X,XS)) -> mark#(N) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) -> a__sel#(s(N),cons(X,XS)) -> mark#(XS) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) -> a__sel#(0(),cons(X,XS)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(sel(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(sel(X1,X2)) -> mark#(X2) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> mark#(X2) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(2nd(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(head(X)) -> a__head#(mark(X)) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(head(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> a__from#(mark(X)) a__sel#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(sel(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(sel(X1,X2)) -> mark#(X2) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(take(X1,X2)) -> mark#(X2) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(2nd(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(head(X)) -> a__head#(mark(X)) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(head(X)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(from(X)) -> a__from#(mark(X)) a__sel#(s(N),cons(X,XS)) -> mark#(N) -> mark#(from(X)) -> mark#(X) 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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(sel(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#(2nd(X)) -> a__2nd#(mark(X)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__take#(s(N),cons(X,XS)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__2nd#(cons(X,XS)) -> a__head#(mark(XS)) -> a__head#(cons(X,XS)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(sel(X1,X2)) -> mark#(X1) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(sel(X1,X2)) -> mark#(X2) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> mark#(X1) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(take(X1,X2)) -> mark#(X2) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(2nd(X)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(head(X)) -> a__head#(mark(X)) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(head(X)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> a__from#(mark(X)) a__2nd#(cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__head#(cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) a__head#(cons(X,XS)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2) a__head#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__head#(cons(X,XS)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__head#(cons(X,XS)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(head(X)) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__head#(cons(X,XS)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) -> a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) -> a__sel#(s(N),cons(X,XS)) -> mark#(N) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) -> a__sel#(s(N),cons(X,XS)) -> mark#(XS) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) -> a__sel#(0(),cons(X,XS)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(take(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> a__head#(mark(X)) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(sel(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(take(X1,X2)) -> mark#(X2) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(sel(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,XS)) -> a__head#(mark(XS)) mark#(2nd(X)) -> a__2nd#(mark(X)) -> a__2nd#(cons(X,XS)) -> mark#(XS) mark#(2nd(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(2nd(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2) mark#(2nd(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(2nd(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(2nd(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(2nd(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(2nd(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(head(X)) -> a__head#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(head(X)) -> mark#(X) mark#(2nd(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(2nd(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(head(X)) -> a__head#(mark(X)) -> a__head#(cons(X,XS)) -> mark#(X) 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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(head(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2) mark#(head(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(head(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(head(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(head(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(head(X)) -> mark#(X) -> mark#(2nd(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#(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#(s(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X2) -> mark#(sel(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#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(2nd(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> a__head#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(head(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) 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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> mark#(X1) -> mark#(sel(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#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(2nd(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> a__head#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(head(X)) -> mark#(X) mark#(take(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(take(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) 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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(sel(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#(2nd(X)) -> a__2nd#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(2nd(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#(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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(2nd(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)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) 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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(sel(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#(2nd(X)) -> a__2nd#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(2nd(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) 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#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(sel(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(take(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(2nd(X)) -> a__2nd#(mark(X)) a__from#(X) -> mark#(X) -> mark#(2nd(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) Arctic Interpretation Processor: dimension: 1 interpretation: [a__sel#](x0, x1) = x0 + x1 + 0, [a__take#](x0, x1) = x0 + x1, [a__2nd#](x0) = x0 + 0, [a__head#](x0) = x0, [mark#](x0) = x0, [a__from#](x0) = x0, [sel](x0, x1) = x0 + 5x1 + 0, [2nd](x0) = 4x0 + 0, [head](x0) = 4x0, [a__sel](x0, x1) = x0 + 5x1 + 0, [take](x0, x1) = x0 + x1, [nil] = 0, [a__take](x0, x1) = x0 + x1, [0] = 0, [a__2nd](x0) = 4x0 + 0, [a__head](x0) = 4x0, [cons](x0, x1) = x0 + x1, [from](x0) = x0, [s](x0) = x0, [mark](x0) = x0, [a__from](x0) = x0 orientation: a__from#(X) = X >= X = mark#(X) a__head#(cons(X,XS)) = X + XS >= X = mark#(X) a__2nd#(cons(X,XS)) = X + XS + 0 >= XS = mark#(XS) a__2nd#(cons(X,XS)) = X + XS + 0 >= XS = a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) = N + X + XS >= X = mark#(X) a__sel#(0(),cons(X,XS)) = X + XS + 0 >= X = mark#(X) a__sel#(s(N),cons(X,XS)) = N + X + XS + 0 >= XS = mark#(XS) a__sel#(s(N),cons(X,XS)) = N + X + XS + 0 >= N = mark#(N) a__sel#(s(N),cons(X,XS)) = N + X + XS + 0 >= N + XS + 0 = a__sel#(mark(N),mark(XS)) mark#(from(X)) = X >= X = mark#(X) mark#(from(X)) = X >= X = a__from#(mark(X)) mark#(head(X)) = 4X >= X = mark#(X) mark#(head(X)) = 4X >= X = a__head#(mark(X)) mark#(2nd(X)) = 4X + 0 >= X = mark#(X) mark#(2nd(X)) = 4X + 0 >= X + 0 = a__2nd#(mark(X)) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) = X1 + 5X2 + 0 >= X2 = mark#(X2) mark#(sel(X1,X2)) = X1 + 5X2 + 0 >= X1 = mark#(X1) mark#(sel(X1,X2)) = X1 + 5X2 + 0 >= X1 + X2 + 0 = a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__from(X) = X >= X = cons(mark(X),from(s(X))) a__head(cons(X,XS)) = 4X + 4XS >= X = mark(X) a__2nd(cons(X,XS)) = 4X + 4XS + 0 >= 4XS = a__head(mark(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__sel(0(),cons(X,XS)) = 5X + 5XS + 0 >= X = mark(X) a__sel(s(N),cons(X,XS)) = N + 5X + 5XS + 0 >= N + 5XS + 0 = a__sel(mark(N),mark(XS)) mark(from(X)) = X >= X = a__from(mark(X)) mark(head(X)) = 4X >= 4X = a__head(mark(X)) mark(2nd(X)) = 4X + 0 >= 4X + 0 = a__2nd(mark(X)) mark(take(X1,X2)) = X1 + X2 >= X1 + X2 = a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) = X1 + 5X2 + 0 >= X1 + 5X2 + 0 = a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(nil()) = 0 >= 0 = nil() a__from(X) = X >= X = from(X) a__head(X) = 4X >= 4X = head(X) a__2nd(X) = 4X + 0 >= 4X + 0 = 2nd(X) a__take(X1,X2) = X1 + X2 >= X1 + X2 = take(X1,X2) a__sel(X1,X2) = X1 + 5X2 + 0 >= X1 + 5X2 + 0 = sel(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) a__2nd#(cons(X,XS)) -> a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) a__sel#(s(N),cons(X,XS)) -> mark#(N) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__sel#](x0, x1) = x0 + x1 + 6, [a__take#](x0, x1) = x0 + x1 + 0, [a__2nd#](x0) = x0 + 6, [a__head#](x0) = x0 + 6, [mark#](x0) = x0, [a__from#](x0) = x0 + 1, [sel](x0, x1) = 3x0 + x1 + 6, [2nd](x0) = 3x0 + 6, [head](x0) = 3x0 + 6, [a__sel](x0, x1) = 3x0 + x1 + 6, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [a__take](x0, x1) = x0 + x1 + 0, [0] = 0, [a__2nd](x0) = 3x0 + 6, [a__head](x0) = 3x0 + 6, [cons](x0, x1) = x0 + x1, [from](x0) = x0 + 1, [s](x0) = x0, [mark](x0) = x0 + 0, [a__from](x0) = x0 + 1 orientation: a__from#(X) = X + 1 >= X = mark#(X) a__head#(cons(X,XS)) = X + XS + 6 >= X = mark#(X) a__2nd#(cons(X,XS)) = X + XS + 6 >= XS = mark#(XS) a__2nd#(cons(X,XS)) = X + XS + 6 >= XS + 6 = a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) = N + X + XS + 0 >= X = mark#(X) a__sel#(0(),cons(X,XS)) = X + XS + 6 >= X = mark#(X) a__sel#(s(N),cons(X,XS)) = N + X + XS + 6 >= XS = mark#(XS) a__sel#(s(N),cons(X,XS)) = N + X + XS + 6 >= N = mark#(N) a__sel#(s(N),cons(X,XS)) = N + X + XS + 6 >= N + XS + 6 = a__sel#(mark(N),mark(XS)) mark#(from(X)) = X + 1 >= X = mark#(X) mark#(from(X)) = X + 1 >= X + 1 = a__from#(mark(X)) mark#(2nd(X)) = 3X + 6 >= X + 6 = a__2nd#(mark(X)) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) = 3X1 + X2 + 6 >= X1 = mark#(X1) mark#(sel(X1,X2)) = 3X1 + X2 + 6 >= X1 + X2 + 6 = a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__from(X) = X + 1 >= X + 1 = cons(mark(X),from(s(X))) a__head(cons(X,XS)) = 3X + 3XS + 6 >= X + 0 = mark(X) a__2nd(cons(X,XS)) = 3X + 3XS + 6 >= 3XS + 6 = a__head(mark(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__sel(0(),cons(X,XS)) = X + XS + 6 >= X + 0 = mark(X) a__sel(s(N),cons(X,XS)) = 3N + X + XS + 6 >= 3N + XS + 6 = a__sel(mark(N),mark(XS)) mark(from(X)) = X + 1 >= X + 1 = a__from(mark(X)) mark(head(X)) = 3X + 6 >= 3X + 6 = a__head(mark(X)) mark(2nd(X)) = 3X + 6 >= 3X + 6 = a__2nd(mark(X)) mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) = 3X1 + X2 + 6 >= 3X1 + X2 + 6 = a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(nil()) = 0 >= 0 = nil() a__from(X) = X + 1 >= X + 1 = from(X) a__head(X) = 3X + 6 >= 3X + 6 = head(X) a__2nd(X) = 3X + 6 >= 3X + 6 = 2nd(X) a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__sel(X1,X2) = 3X1 + X2 + 6 >= 3X1 + X2 + 6 = sel(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__head#(cons(X,XS)) -> mark#(X) a__2nd#(cons(X,XS)) -> mark#(XS) a__2nd#(cons(X,XS)) -> a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) -> mark#(X) a__sel#(0(),cons(X,XS)) -> mark#(X) a__sel#(s(N),cons(X,XS)) -> mark#(XS) a__sel#(s(N),cons(X,XS)) -> mark#(N) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__sel#](x0, x1) = x0 + x1, [a__take#](x0, x1) = x1, [a__2nd#](x0) = 4x0 + 6, [a__head#](x0) = x0, [mark#](x0) = x0, [a__from#](x0) = 1x0, [sel](x0, x1) = x0 + 4x1 + 0, [2nd](x0) = 7x0 + 6, [head](x0) = 4x0, [a__sel](x0, x1) = x0 + 4x1 + 0, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [a__take](x0, x1) = x0 + x1 + 0, [0] = 0, [a__2nd](x0) = 7x0 + 6, [a__head](x0) = 4x0, [cons](x0, x1) = 2x0 + x1, [from](x0) = 6x0, [s](x0) = x0, [mark](x0) = x0, [a__from](x0) = 6x0 orientation: a__from#(X) = 1X >= X = mark#(X) a__head#(cons(X,XS)) = 2X + XS >= X = mark#(X) a__2nd#(cons(X,XS)) = 6X + 4XS + 6 >= XS = mark#(XS) a__2nd#(cons(X,XS)) = 6X + 4XS + 6 >= XS = a__head#(mark(XS)) a__take#(s(N),cons(X,XS)) = 2X + XS >= X = mark#(X) a__sel#(0(),cons(X,XS)) = 2X + XS + 0 >= X = mark#(X) a__sel#(s(N),cons(X,XS)) = N + 2X + XS >= XS = mark#(XS) a__sel#(s(N),cons(X,XS)) = N + 2X + XS >= N = mark#(N) a__sel#(s(N),cons(X,XS)) = N + 2X + XS >= N + XS = a__sel#(mark(N),mark(XS)) mark#(from(X)) = 6X >= X = mark#(X) mark#(from(X)) = 6X >= 1X = a__from#(mark(X)) mark#(2nd(X)) = 7X + 6 >= 4X + 6 = a__2nd#(mark(X)) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 = a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) = X1 + 4X2 + 0 >= X1 + X2 = a__sel#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = 2X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__from(X) = 6X >= 6X = cons(mark(X),from(s(X))) a__head(cons(X,XS)) = 6X + 4XS >= X = mark(X) a__2nd(cons(X,XS)) = 9X + 7XS + 6 >= 4XS = a__head(mark(XS)) a__take(0(),XS) = XS + 0 >= 0 = nil() a__take(s(N),cons(X,XS)) = N + 2X + XS + 0 >= N + 2X + XS + 0 = cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) = 6X + 4XS + 0 >= X = mark(X) a__sel(s(N),cons(X,XS)) = N + 6X + 4XS + 0 >= N + 4XS + 0 = a__sel(mark(N),mark(XS)) mark(from(X)) = 6X >= 6X = a__from(mark(X)) mark(head(X)) = 4X >= 4X = a__head(mark(X)) mark(2nd(X)) = 7X + 6 >= 7X + 6 = a__2nd(mark(X)) mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) = 2X1 + X2 >= 2X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(nil()) = 0 >= 0 = nil() a__from(X) = 6X >= 6X = from(X) a__head(X) = 4X >= 4X = head(X) a__2nd(X) = 7X + 6 >= 7X + 6 = 2nd(X) a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__sel(X1,X2) = X1 + 4X2 + 0 >= X1 + 4X2 + 0 = sel(X1,X2) problem: DPs: a__sel#(s(N),cons(X,XS)) -> mark#(XS) a__sel#(s(N),cons(X,XS)) -> mark#(N) a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) mark#(2nd(X)) -> a__2nd#(mark(X)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(take(X1,X2)) -> a__take#(mark(X1),mark(X2)) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) SCC Processor: #sccs: 1 #rules: 7 #arcs: 238/81 DPs: a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) a__sel#(s(N),cons(X,XS)) -> mark#(XS) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) -> mark#(N) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__sel#](x0, x1) = x0 + 6x1 + 0, [mark#](x0) = x0, [sel](x0, x1) = 1x0 + 6x1 + 6, [2nd](x0) = x0, [head](x0) = x0, [a__sel](x0, x1) = 1x0 + 6x1 + 6, [take](x0, x1) = x0 + x1, [nil] = 0, [a__take](x0, x1) = x0 + x1 + 0, [0] = 1, [a__2nd](x0) = x0 + 0, [a__head](x0) = x0 + 0, [cons](x0, x1) = x0 + x1 + 0, [from](x0) = x0, [s](x0) = x0, [mark](x0) = x0 + 0, [a__from](x0) = x0 + 0 orientation: a__sel#(s(N),cons(X,XS)) = N + 6X + 6XS + 6 >= N + 6XS + 6 = a__sel#(mark(N),mark(XS)) a__sel#(s(N),cons(X,XS)) = N + 6X + 6XS + 6 >= XS = mark#(XS) mark#(take(X1,X2)) = X1 + X2 >= X2 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(sel(X1,X2)) = 1X1 + 6X2 + 6 >= X1 + 6X2 + 6 = a__sel#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) = N + 6X + 6XS + 6 >= N = mark#(N) mark#(s(X)) = X >= X = mark#(X) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__head(cons(X,XS)) = X + XS + 0 >= X + 0 = mark(X) a__2nd(cons(X,XS)) = X + XS + 0 >= XS + 0 = a__head(mark(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__sel(0(),cons(X,XS)) = 6X + 6XS + 6 >= X + 0 = mark(X) a__sel(s(N),cons(X,XS)) = 1N + 6X + 6XS + 6 >= 1N + 6XS + 6 = a__sel(mark(N),mark(XS)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(head(X)) = X + 0 >= X + 0 = a__head(mark(X)) mark(2nd(X)) = X + 0 >= X + 0 = a__2nd(mark(X)) mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) = 1X1 + 6X2 + 6 >= 1X1 + 6X2 + 6 = a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 1 >= 1 = 0() mark(nil()) = 0 >= 0 = nil() a__from(X) = X + 0 >= X = from(X) a__head(X) = X + 0 >= X = head(X) a__2nd(X) = X + 0 >= X = 2nd(X) a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 = take(X1,X2) a__sel(X1,X2) = 1X1 + 6X2 + 6 >= 1X1 + 6X2 + 6 = sel(X1,X2) problem: DPs: a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(sel(X1,X2)) -> a__sel#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) -> mark#(N) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Arctic Interpretation Processor: dimension: 1 interpretation: [a__sel#](x0, x1) = x0 + x1, [mark#](x0) = x0 + 0, [sel](x0, x1) = 1x0 + 1x1 + 0, [2nd](x0) = 2x0, [head](x0) = x0, [a__sel](x0, x1) = 1x0 + 1x1 + 0, [take](x0, x1) = x0 + x1 + 0, [nil] = 0, [a__take](x0, x1) = x0 + x1 + 0, [0] = 0, [a__2nd](x0) = 2x0, [a__head](x0) = x0, [cons](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 0, [s](x0) = x0 + 0, [mark](x0) = x0, [a__from](x0) = x0 + 0 orientation: a__sel#(s(N),cons(X,XS)) = N + X + XS + 0 >= N + XS = a__sel#(mark(N),mark(XS)) mark#(take(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = mark#(X2) mark#(take(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(sel(X1,X2)) = 1X1 + 1X2 + 0 >= X1 + X2 = a__sel#(mark(X1),mark(X2)) a__sel#(s(N),cons(X,XS)) = N + X + XS + 0 >= N + 0 = mark#(N) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__head(cons(X,XS)) = X + XS + 0 >= X = mark(X) a__2nd(cons(X,XS)) = 2X + 2XS + 2 >= XS = a__head(mark(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__sel(0(),cons(X,XS)) = 1X + 1XS + 1 >= X = mark(X) a__sel(s(N),cons(X,XS)) = 1N + 1X + 1XS + 1 >= 1N + 1XS + 0 = a__sel(mark(N),mark(XS)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(head(X)) = X >= X = a__head(mark(X)) mark(2nd(X)) = 2X >= 2X = a__2nd(mark(X)) mark(take(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) = 1X1 + 1X2 + 0 >= 1X1 + 1X2 + 0 = a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() mark(nil()) = 0 >= 0 = nil() a__from(X) = X + 0 >= X + 0 = from(X) a__head(X) = X >= X = head(X) a__2nd(X) = 2X >= 2X = 2nd(X) a__take(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = take(X1,X2) a__sel(X1,X2) = 1X1 + 1X2 + 0 >= 1X1 + 1X2 + 0 = sel(X1,X2) problem: DPs: a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) a__sel#(s(N),cons(X,XS)) -> mark#(N) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) SCC Processor: #sccs: 2 #rules: 4 #arcs: 26/25 DPs: a__sel#(s(N),cons(X,XS)) -> a__sel#(mark(N),mark(XS)) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Open DPs: mark#(take(X1,X2)) -> mark#(X2) mark#(take(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Subterm Criterion Processor: simple projection: pi(mark#) = 0 problem: DPs: TRS: a__from(X) -> cons(mark(X),from(s(X))) a__head(cons(X,XS)) -> mark(X) a__2nd(cons(X,XS)) -> a__head(mark(XS)) a__take(0(),XS) -> nil() a__take(s(N),cons(X,XS)) -> cons(mark(X),take(N,XS)) a__sel(0(),cons(X,XS)) -> mark(X) a__sel(s(N),cons(X,XS)) -> a__sel(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(2nd(X)) -> a__2nd(mark(X)) mark(take(X1,X2)) -> a__take(mark(X1),mark(X2)) mark(sel(X1,X2)) -> a__sel(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() mark(nil()) -> nil() a__from(X) -> from(X) a__head(X) -> head(X) a__2nd(X) -> 2nd(X) a__take(X1,X2) -> take(X1,X2) a__sel(X1,X2) -> sel(X1,X2) Qed