The TRS could not be proven terminating. The proof attempt took 60001 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (3655ms), SubtermCriterion (1ms), DependencyGraph (3030ms), PolynomialLinearRange4iUR (timeout), DependencyGraph (3008ms), PolynomialLinearRange8NegiUR (30037ms), ReductionPairSAT (timeout)].
| mark#(take(X1, X2)) | → | mark#(X1) | mark#(afterNth(X1, X2)) | → | a__afterNth#(mark(X1), mark(X2)) | |
| a__tail#(cons(N, XS)) | → | mark#(XS) | a__take#(N, XS) | → | mark#(XS) | |
| mark#(pair(X1, X2)) | → | mark#(X2) | a__take#(N, XS) | → | mark#(N) | |
| a__afterNth#(N, XS) | → | mark#(XS) | a__snd#(pair(X, Y)) | → | mark#(Y) | |
| a__U12#(pair(YS, ZS), X) | → | mark#(X) | a__U11#(tt, N, X, XS) | → | mark#(XS) | |
| a__and#(tt, X) | → | mark#(X) | a__fst#(pair(X, Y)) | → | mark#(X) | |
| a__U11#(tt, N, X, XS) | → | a__U12#(a__splitAt(mark(N), mark(XS)), X) | mark#(U11(X1, X2, X3, X4)) | → | a__U11#(mark(X1), X2, X3, X4) | |
| mark#(s(X)) | → | mark#(X) | mark#(sel(X1, X2)) | → | mark#(X1) | |
| a__splitAt#(s(N), cons(X, XS)) | → | a__U11#(tt, N, X, XS) | a__afterNth#(N, XS) | → | a__splitAt#(mark(N), mark(XS)) | |
| mark#(U12(X1, X2)) | → | mark#(X1) | a__sel#(N, XS) | → | a__head#(a__afterNth(mark(N), mark(XS))) | |
| a__sel#(N, XS) | → | mark#(N) | a__afterNth#(N, XS) | → | mark#(N) | |
| a__U12#(pair(YS, ZS), X) | → | mark#(ZS) | mark#(cons(X1, X2)) | → | mark#(X1) | |
| mark#(afterNth(X1, X2)) | → | mark#(X1) | mark#(and(X1, X2)) | → | mark#(X1) | |
| mark#(pair(X1, X2)) | → | mark#(X1) | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | |
| mark#(tail(X)) | → | mark#(X) | mark#(U12(X1, X2)) | → | a__U12#(mark(X1), X2) | |
| mark#(snd(X)) | → | a__snd#(mark(X)) | a__sel#(N, XS) | → | mark#(XS) | |
| mark#(afterNth(X1, X2)) | → | mark#(X2) | a__natsFrom#(N) | → | mark#(N) | |
| a__sel#(N, XS) | → | a__afterNth#(mark(N), mark(XS)) | mark#(head(X)) | → | mark#(X) | |
| mark#(splitAt(X1, X2)) | → | mark#(X1) | mark#(sel(X1, X2)) | → | mark#(X2) | |
| mark#(head(X)) | → | a__head#(mark(X)) | a__head#(cons(N, XS)) | → | mark#(N) | |
| mark#(fst(X)) | → | a__fst#(mark(X)) | mark#(tail(X)) | → | a__tail#(mark(X)) | |
| mark#(fst(X)) | → | mark#(X) | mark#(splitAt(X1, X2)) | → | mark#(X2) | |
| mark#(snd(X)) | → | mark#(X) | a__take#(N, XS) | → | a__fst#(a__splitAt(mark(N), mark(XS))) | |
| a__U11#(tt, N, X, XS) | → | mark#(N) | a__afterNth#(N, XS) | → | a__snd#(a__splitAt(mark(N), mark(XS))) | |
| mark#(natsFrom(X)) | → | mark#(X) | a__take#(N, XS) | → | a__splitAt#(mark(N), mark(XS)) | |
| mark#(sel(X1, X2)) | → | a__sel#(mark(X1), mark(X2)) | mark#(take(X1, X2)) | → | a__take#(mark(X1), mark(X2)) | |
| mark#(splitAt(X1, X2)) | → | a__splitAt#(mark(X1), mark(X2)) | mark#(take(X1, X2)) | → | mark#(X2) | |
| a__splitAt#(0, XS) | → | mark#(XS) | mark#(U11(X1, X2, X3, X4)) | → | mark#(X1) | |
| a__U11#(tt, N, X, XS) | → | a__splitAt#(mark(N), mark(XS)) | mark#(natsFrom(X)) | → | a__natsFrom#(mark(X)) |
| a__U11(tt, N, X, XS) | → | a__U12(a__splitAt(mark(N), mark(XS)), X) | a__U12(pair(YS, ZS), X) | → | pair(cons(mark(X), YS), mark(ZS)) | |
| a__afterNth(N, XS) | → | a__snd(a__splitAt(mark(N), mark(XS))) | a__and(tt, X) | → | mark(X) | |
| a__fst(pair(X, Y)) | → | mark(X) | a__head(cons(N, XS)) | → | mark(N) | |
| a__natsFrom(N) | → | cons(mark(N), natsFrom(s(N))) | a__sel(N, XS) | → | a__head(a__afterNth(mark(N), mark(XS))) | |
| a__snd(pair(X, Y)) | → | mark(Y) | a__splitAt(0, XS) | → | pair(nil, mark(XS)) | |
| a__splitAt(s(N), cons(X, XS)) | → | a__U11(tt, N, X, XS) | a__tail(cons(N, XS)) | → | mark(XS) | |
| a__take(N, XS) | → | a__fst(a__splitAt(mark(N), mark(XS))) | mark(U11(X1, X2, X3, X4)) | → | a__U11(mark(X1), X2, X3, X4) | |
| mark(U12(X1, X2)) | → | a__U12(mark(X1), X2) | mark(splitAt(X1, X2)) | → | a__splitAt(mark(X1), mark(X2)) | |
| mark(afterNth(X1, X2)) | → | a__afterNth(mark(X1), mark(X2)) | mark(snd(X)) | → | a__snd(mark(X)) | |
| mark(and(X1, X2)) | → | a__and(mark(X1), X2) | mark(fst(X)) | → | a__fst(mark(X)) | |
| mark(head(X)) | → | a__head(mark(X)) | mark(natsFrom(X)) | → | a__natsFrom(mark(X)) | |
| mark(sel(X1, X2)) | → | a__sel(mark(X1), mark(X2)) | mark(tail(X)) | → | a__tail(mark(X)) | |
| mark(take(X1, X2)) | → | a__take(mark(X1), mark(X2)) | mark(tt) | → | tt | |
| mark(pair(X1, X2)) | → | pair(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__U11(X1, X2, X3, X4) | → | U11(X1, X2, X3, X4) | |
| a__U12(X1, X2) | → | U12(X1, X2) | a__splitAt(X1, X2) | → | splitAt(X1, X2) | |
| a__afterNth(X1, X2) | → | afterNth(X1, X2) | a__snd(X) | → | snd(X) | |
| a__and(X1, X2) | → | and(X1, X2) | a__fst(X) | → | fst(X) | |
| a__head(X) | → | head(X) | a__natsFrom(X) | → | natsFrom(X) | |
| a__sel(X1, X2) | → | sel(X1, X2) | a__tail(X) | → | tail(X) | |
| a__take(X1, X2) | → | take(X1, X2) |
Termination of terms over the following signature is verified: a__take, pair, natsFrom, a__natsFrom, a__and, fst, a__fst, head, a__head, a__snd, cons, snd, a__afterNth, mark, tail, splitAt, and, a__tail, 0, s, tt, a__splitAt, take, a__sel, U11, a__U12, U12, a__U11, afterNth, sel, nil