The TRS could not be proven terminating. The proof attempt took 60000 ms.
Problem 1 remains open; application of the following processors failed [DependencyGraph (28377ms), SubtermCriterion (2ms), DependencyGraph (26728ms), PolynomialLinearRange4iUR (timeout)].
| mark#(U101(X1, X2, X3)) | → | mark#(X1) | a__U91#(tt, XS) | → | mark#(XS) | |
| mark#(pair(X1, X2)) | → | mark#(X2) | a__U21#(tt, X) | → | mark#(X) | |
| a__isNatural#(head(V1)) | → | a__isLNat#(V1) | a__isPLNat#(pair(V1, V2)) | → | a__and#(a__isLNat(V1), isLNat(V2)) | |
| a__and#(tt, X) | → | mark#(X) | a__U11#(tt, N, XS) | → | a__snd#(a__splitAt(mark(N), mark(XS))) | |
| mark#(s(X)) | → | mark#(X) | mark#(sel(X1, X2)) | → | mark#(X1) | |
| a__isLNat#(tail(V1)) | → | a__isLNat#(V1) | a__tail#(cons(N, XS)) | → | a__U91#(a__and(a__isNatural(N), isLNat(XS)), XS) | |
| a__snd#(pair(X, Y)) | → | a__isLNat#(X) | a__U51#(tt, N, XS) | → | mark#(XS) | |
| mark#(cons(X1, X2)) | → | mark#(X1) | mark#(afterNth(X1, X2)) | → | mark#(X1) | |
| a__splitAt#(0, XS) | → | a__isLNat#(XS) | mark#(U101(X1, X2, X3)) | → | a__U101#(mark(X1), X2, X3) | |
| mark#(and(X1, X2)) | → | mark#(X1) | a__U101#(tt, N, XS) | → | mark#(N) | |
| mark#(pair(X1, X2)) | → | mark#(X1) | mark#(U81(X1, X2, X3, X4)) | → | a__U81#(mark(X1), X2, X3, X4) | |
| a__splitAt#(s(N), cons(X, XS)) | → | a__U81#(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) | mark#(and(X1, X2)) | → | a__and#(mark(X1), X2) | |
| mark#(tail(X)) | → | mark#(X) | a__head#(cons(N, XS)) | → | a__isNatural#(N) | |
| mark#(afterNth(X1, X2)) | → | mark#(X2) | mark#(U11(X1, X2, X3)) | → | mark#(X1) | |
| a__U31#(tt, N) | → | mark#(N) | a__splitAt#(s(N), cons(X, XS)) | → | a__and#(a__isNatural(N), and(isNatural(X), isLNat(XS))) | |
| a__U11#(tt, N, XS) | → | mark#(XS) | a__afterNth#(N, XS) | → | a__isNatural#(N) | |
| a__U101#(tt, N, XS) | → | a__splitAt#(mark(N), mark(XS)) | mark#(U51(X1, X2, X3)) | → | mark#(X1) | |
| mark#(head(X)) | → | mark#(X) | mark#(splitAt(X1, X2)) | → | mark#(X1) | |
| mark#(isPLNat(X)) | → | a__isPLNat#(X) | mark#(isNatural(X)) | → | a__isNatural#(X) | |
| mark#(head(X)) | → | a__head#(mark(X)) | a__isLNat#(cons(V1, V2)) | → | a__and#(a__isNatural(V1), isLNat(V2)) | |
| mark#(fst(X)) | → | a__fst#(mark(X)) | mark#(U41(X1, X2)) | → | mark#(X1) | |
| a__U51#(tt, N, XS) | → | mark#(N) | a__tail#(cons(N, XS)) | → | a__isNatural#(N) | |
| mark#(fst(X)) | → | mark#(X) | a__U101#(tt, N, XS) | → | a__fst#(a__splitAt(mark(N), mark(XS))) | |
| a__fst#(pair(X, Y)) | → | a__U21#(a__and(a__isLNat(X), isLNat(Y)), X) | mark#(U82(X1, X2)) | → | mark#(X1) | |
| a__sel#(N, XS) | → | a__U51#(a__and(a__isNatural(N), isLNat(XS)), N, XS) | a__splitAt#(s(N), cons(X, XS)) | → | a__isNatural#(N) | |
| mark#(snd(X)) | → | mark#(X) | a__isLNat#(cons(V1, V2)) | → | a__isNatural#(V1) | |
| mark#(natsFrom(X)) | → | mark#(X) | a__head#(cons(N, XS)) | → | a__and#(a__isNatural(N), isLNat(XS)) | |
| a__U82#(pair(YS, ZS), X) | → | mark#(X) | a__snd#(pair(X, Y)) | → | a__U61#(a__and(a__isLNat(X), isLNat(Y)), Y) | |
| mark#(U51(X1, X2, X3)) | → | a__U51#(mark(X1), X2, X3) | a__afterNth#(N, XS) | → | a__U11#(a__and(a__isNatural(N), isLNat(XS)), N, XS) | |
| a__U11#(tt, N, XS) | → | mark#(N) | a__isLNat#(take(V1, V2)) | → | a__isNatural#(V1) | |
| mark#(take(X1, X2)) | → | a__take#(mark(X1), mark(X2)) | a__afterNth#(N, XS) | → | a__and#(a__isNatural(N), isLNat(XS)) | |
| mark#(splitAt(X1, X2)) | → | a__splitAt#(mark(X1), mark(X2)) | a__isLNat#(fst(V1)) | → | a__isPLNat#(V1) | |
| mark#(take(X1, X2)) | → | mark#(X1) | mark#(afterNth(X1, X2)) | → | a__afterNth#(mark(X1), mark(X2)) | |
| a__sel#(N, XS) | → | a__isNatural#(N) | a__take#(N, XS) | → | a__isNatural#(N) | |
| a__U81#(tt, N, X, XS) | → | a__splitAt#(mark(N), mark(XS)) | mark#(U91(X1, X2)) | → | a__U91#(mark(X1), X2) | |
| a__U11#(tt, N, XS) | → | a__splitAt#(mark(N), mark(XS)) | a__U61#(tt, Y) | → | mark#(Y) | |
| a__U71#(tt, XS) | → | mark#(XS) | mark#(U41(X1, X2)) | → | a__U41#(mark(X1), X2) | |
| a__isLNat#(snd(V1)) | → | a__isPLNat#(V1) | a__head#(cons(N, XS)) | → | a__U31#(a__and(a__isNatural(N), isLNat(XS)), N) | |
| mark#(U31(X1, X2)) | → | mark#(X1) | mark#(U71(X1, X2)) | → | a__U71#(mark(X1), X2) | |
| a__isNatural#(s(V1)) | → | a__isNatural#(V1) | a__isPLNat#(splitAt(V1, V2)) | → | a__isNatural#(V1) | |
| a__U81#(tt, N, X, XS) | → | a__U82#(a__splitAt(mark(N), mark(XS)), X) | a__U51#(tt, N, XS) | → | a__head#(a__afterNth(mark(N), mark(XS))) | |
| a__U82#(pair(YS, ZS), X) | → | mark#(ZS) | a__snd#(pair(X, Y)) | → | a__and#(a__isLNat(X), isLNat(Y)) | |
| a__isLNat#(natsFrom(V1)) | → | a__isNatural#(V1) | a__fst#(pair(X, Y)) | → | a__isLNat#(X) | |
| mark#(isLNat(X)) | → | a__isLNat#(X) | a__tail#(cons(N, XS)) | → | a__and#(a__isNatural(N), isLNat(XS)) | |
| a__isNatural#(sel(V1, V2)) | → | a__and#(a__isNatural(V1), isLNat(V2)) | a__fst#(pair(X, Y)) | → | a__and#(a__isLNat(X), isLNat(Y)) | |
| mark#(U71(X1, X2)) | → | mark#(X1) | mark#(U82(X1, X2)) | → | a__U82#(mark(X1), X2) | |
| mark#(U61(X1, X2)) | → | a__U61#(mark(X1), X2) | a__sel#(N, XS) | → | a__and#(a__isNatural(N), isLNat(XS)) | |
| mark#(snd(X)) | → | a__snd#(mark(X)) | mark#(U11(X1, X2, X3)) | → | a__U11#(mark(X1), X2, X3) | |
| a__splitAt#(0, XS) | → | a__U71#(a__isLNat(XS), XS) | mark#(U61(X1, X2)) | → | mark#(X1) | |
| mark#(U91(X1, X2)) | → | mark#(X1) | a__isPLNat#(splitAt(V1, V2)) | → | a__and#(a__isNatural(V1), isLNat(V2)) | |
| mark#(U21(X1, X2)) | → | mark#(X1) | mark#(U31(X1, X2)) | → | a__U31#(mark(X1), X2) | |
| a__isLNat#(afterNth(V1, V2)) | → | a__and#(a__isNatural(V1), isLNat(V2)) | a__U41#(tt, N) | → | mark#(N) | |
| a__take#(N, XS) | → | a__U101#(a__and(a__isNatural(N), isLNat(XS)), N, XS) | a__U81#(tt, N, X, XS) | → | mark#(XS) | |
| a__isNatural#(sel(V1, V2)) | → | a__isNatural#(V1) | mark#(sel(X1, X2)) | → | mark#(X2) | |
| mark#(U81(X1, X2, X3, X4)) | → | mark#(X1) | a__isLNat#(afterNth(V1, V2)) | → | a__isNatural#(V1) | |
| mark#(tail(X)) | → | a__tail#(mark(X)) | a__U101#(tt, N, XS) | → | mark#(XS) | |
| a__natsFrom#(N) | → | a__U41#(a__isNatural(N), N) | mark#(splitAt(X1, X2)) | → | mark#(X2) | |
| a__U81#(tt, N, X, XS) | → | mark#(N) | a__take#(N, XS) | → | a__and#(a__isNatural(N), isLNat(XS)) | |
| a__U51#(tt, N, XS) | → | a__afterNth#(mark(N), mark(XS)) | a__natsFrom#(N) | → | a__isNatural#(N) | |
| mark#(sel(X1, X2)) | → | a__sel#(mark(X1), mark(X2)) | mark#(U21(X1, X2)) | → | a__U21#(mark(X1), X2) | |
| mark#(take(X1, X2)) | → | mark#(X2) | a__isPLNat#(pair(V1, V2)) | → | a__isLNat#(V1) | |
| a__isLNat#(take(V1, V2)) | → | a__and#(a__isNatural(V1), isLNat(V2)) | mark#(natsFrom(X)) | → | a__natsFrom#(mark(X)) |
| a__U101(tt, N, XS) | → | a__fst(a__splitAt(mark(N), mark(XS))) | a__U11(tt, N, XS) | → | a__snd(a__splitAt(mark(N), mark(XS))) | |
| a__U21(tt, X) | → | mark(X) | a__U31(tt, N) | → | mark(N) | |
| a__U41(tt, N) | → | cons(mark(N), natsFrom(s(N))) | a__U51(tt, N, XS) | → | a__head(a__afterNth(mark(N), mark(XS))) | |
| a__U61(tt, Y) | → | mark(Y) | a__U71(tt, XS) | → | pair(nil, mark(XS)) | |
| a__U81(tt, N, X, XS) | → | a__U82(a__splitAt(mark(N), mark(XS)), X) | a__U82(pair(YS, ZS), X) | → | pair(cons(mark(X), YS), mark(ZS)) | |
| a__U91(tt, XS) | → | mark(XS) | a__afterNth(N, XS) | → | a__U11(a__and(a__isNatural(N), isLNat(XS)), N, XS) | |
| a__and(tt, X) | → | mark(X) | a__fst(pair(X, Y)) | → | a__U21(a__and(a__isLNat(X), isLNat(Y)), X) | |
| a__head(cons(N, XS)) | → | a__U31(a__and(a__isNatural(N), isLNat(XS)), N) | a__isLNat(nil) | → | tt | |
| a__isLNat(afterNth(V1, V2)) | → | a__and(a__isNatural(V1), isLNat(V2)) | a__isLNat(cons(V1, V2)) | → | a__and(a__isNatural(V1), isLNat(V2)) | |
| a__isLNat(fst(V1)) | → | a__isPLNat(V1) | a__isLNat(natsFrom(V1)) | → | a__isNatural(V1) | |
| a__isLNat(snd(V1)) | → | a__isPLNat(V1) | a__isLNat(tail(V1)) | → | a__isLNat(V1) | |
| a__isLNat(take(V1, V2)) | → | a__and(a__isNatural(V1), isLNat(V2)) | a__isNatural(0) | → | tt | |
| a__isNatural(head(V1)) | → | a__isLNat(V1) | a__isNatural(s(V1)) | → | a__isNatural(V1) | |
| a__isNatural(sel(V1, V2)) | → | a__and(a__isNatural(V1), isLNat(V2)) | a__isPLNat(pair(V1, V2)) | → | a__and(a__isLNat(V1), isLNat(V2)) | |
| a__isPLNat(splitAt(V1, V2)) | → | a__and(a__isNatural(V1), isLNat(V2)) | a__natsFrom(N) | → | a__U41(a__isNatural(N), N) | |
| a__sel(N, XS) | → | a__U51(a__and(a__isNatural(N), isLNat(XS)), N, XS) | a__snd(pair(X, Y)) | → | a__U61(a__and(a__isLNat(X), isLNat(Y)), Y) | |
| a__splitAt(0, XS) | → | a__U71(a__isLNat(XS), XS) | a__splitAt(s(N), cons(X, XS)) | → | a__U81(a__and(a__isNatural(N), and(isNatural(X), isLNat(XS))), N, X, XS) | |
| a__tail(cons(N, XS)) | → | a__U91(a__and(a__isNatural(N), isLNat(XS)), XS) | a__take(N, XS) | → | a__U101(a__and(a__isNatural(N), isLNat(XS)), N, XS) | |
| mark(U101(X1, X2, X3)) | → | a__U101(mark(X1), X2, X3) | mark(fst(X)) | → | a__fst(mark(X)) | |
| mark(splitAt(X1, X2)) | → | a__splitAt(mark(X1), mark(X2)) | mark(U11(X1, X2, X3)) | → | a__U11(mark(X1), X2, X3) | |
| mark(snd(X)) | → | a__snd(mark(X)) | mark(U21(X1, X2)) | → | a__U21(mark(X1), X2) | |
| mark(U31(X1, X2)) | → | a__U31(mark(X1), X2) | mark(U41(X1, X2)) | → | a__U41(mark(X1), X2) | |
| mark(natsFrom(X)) | → | a__natsFrom(mark(X)) | mark(U51(X1, X2, X3)) | → | a__U51(mark(X1), X2, X3) | |
| mark(head(X)) | → | a__head(mark(X)) | mark(afterNth(X1, X2)) | → | a__afterNth(mark(X1), mark(X2)) | |
| mark(U61(X1, X2)) | → | a__U61(mark(X1), X2) | mark(U71(X1, X2)) | → | a__U71(mark(X1), X2) | |
| mark(U81(X1, X2, X3, X4)) | → | a__U81(mark(X1), X2, X3, X4) | mark(U82(X1, X2)) | → | a__U82(mark(X1), X2) | |
| mark(U91(X1, X2)) | → | a__U91(mark(X1), X2) | mark(and(X1, X2)) | → | a__and(mark(X1), X2) | |
| mark(isNatural(X)) | → | a__isNatural(X) | mark(isLNat(X)) | → | a__isLNat(X) | |
| mark(isPLNat(X)) | → | a__isPLNat(X) | mark(tail(X)) | → | a__tail(mark(X)) | |
| mark(take(X1, X2)) | → | a__take(mark(X1), mark(X2)) | mark(sel(X1, X2)) | → | a__sel(mark(X1), mark(X2)) | |
| mark(tt) | → | tt | mark(cons(X1, X2)) | → | cons(mark(X1), X2) | |
| mark(s(X)) | → | s(mark(X)) | mark(pair(X1, X2)) | → | pair(mark(X1), mark(X2)) | |
| mark(nil) | → | nil | mark(0) | → | 0 | |
| a__U101(X1, X2, X3) | → | U101(X1, X2, X3) | a__fst(X) | → | fst(X) | |
| a__splitAt(X1, X2) | → | splitAt(X1, X2) | a__U11(X1, X2, X3) | → | U11(X1, X2, X3) | |
| a__snd(X) | → | snd(X) | a__U21(X1, X2) | → | U21(X1, X2) | |
| a__U31(X1, X2) | → | U31(X1, X2) | a__U41(X1, X2) | → | U41(X1, X2) | |
| a__natsFrom(X) | → | natsFrom(X) | a__U51(X1, X2, X3) | → | U51(X1, X2, X3) | |
| a__head(X) | → | head(X) | a__afterNth(X1, X2) | → | afterNth(X1, X2) | |
| a__U61(X1, X2) | → | U61(X1, X2) | a__U71(X1, X2) | → | U71(X1, X2) | |
| a__U81(X1, X2, X3, X4) | → | U81(X1, X2, X3, X4) | a__U82(X1, X2) | → | U82(X1, X2) | |
| a__U91(X1, X2) | → | U91(X1, X2) | a__and(X1, X2) | → | and(X1, X2) | |
| a__isNatural(X) | → | isNatural(X) | a__isLNat(X) | → | isLNat(X) | |
| a__isPLNat(X) | → | isPLNat(X) | a__tail(X) | → | tail(X) | |
| a__take(X1, X2) | → | take(X1, X2) | a__sel(X1, X2) | → | sel(X1, X2) |
Termination of terms over the following signature is verified: natsFrom, a__natsFrom, fst, U61, a__isPLNat, a__U41, a__isNatural, a__U71, head, U21, mark, isPLNat, tail, U71, and, 0, isLNat, a__U82, a__U31, a__sel, a__U81, U31, a__U51, a__take, pair, isNatural, a__and, a__U91, U41, U91, a__fst, a__isLNat, a__head, a__U101, a__snd, a__U21, cons, snd, a__afterNth, splitAt, a__tail, s, U51, a__splitAt, tt, U82, take, U81, U11, a__U11, afterNth, a__U61, sel, nil, U101