TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60002 ms.

The following DP Processors were used


Problem 1 remains open; application of the following processors failed [DependencyGraph (timeout)].

The following open problems remain:



Open Dependency Pair Problem 1

Dependency Pairs

proper#(cons(X1, X2))proper#(X1)proper#(U41(X1, X2))proper#(X1)
active#(U182(X1, X2))U182#(active(X1), X2)proper#(tail(X))proper#(X)
active#(U211(tt, XS))isLNat#(XS)proper#(U121(X))U121#(proper(X))
proper#(U161(X1, X2))proper#(X2)U132#(mark(X))U132#(X)
active#(U191(tt, XS))pair#(nil, XS)U51#(mark(X1), X2)U51#(X1, X2)
isNatural#(ok(X))isNatural#(X)active#(U221(tt, N, XS))isLNat#(XS)
proper#(U11(X1, X2, X3))proper#(X2)proper#(U222(X1, X2, X3))proper#(X2)
active#(U212(X1, X2))active#(X1)active#(U21(tt, X, Y))isLNat#(Y)
proper#(U222(X1, X2, X3))proper#(X3)U81#(mark(X))U81#(X)
splitAt#(X1, mark(X2))splitAt#(X1, X2)proper#(U101(X1, X2))U101#(proper(X1), proper(X2))
proper#(U211(X1, X2))proper#(X1)active#(afterNth(X1, X2))afterNth#(active(X1), X2)
active#(U12(X1, X2, X3))active#(X1)U172#(mark(X1), X2, X3)U172#(X1, X2, X3)
active#(U41(X1, X2))active#(X1)U203#(mark(X1), X2, X3, X4)U203#(X1, X2, X3, X4)
active#(isLNat(cons(V1, V2)))isNatural#(V1)active#(isLNat(afterNth(V1, V2)))isNatural#(V1)
proper#(U12(X1, X2, X3))proper#(X1)natsFrom#(mark(X))natsFrom#(X)
proper#(U21(X1, X2, X3))proper#(X3)active#(take(N, XS))isNatural#(N)
active#(splitAt(0, XS))isLNat#(XS)active#(U12(X1, X2, X3))U12#(active(X1), X2, X3)
active#(snd(X))snd#(active(X))active#(fst(pair(X, Y)))U21#(isLNat(X), X, Y)
active#(U52(X))active#(X)active#(U22(X1, X2))active#(X1)
proper#(U51(X1, X2))proper#(X2)proper#(U71(X))U71#(proper(X))
U191#(mark(X1), X2)U191#(X1, X2)isLNat#(ok(X))isLNat#(X)
active#(sel(X1, X2))sel#(X1, active(X2))active#(U41(tt, V2))isLNat#(V2)
proper#(U202(X1, X2, X3, X4))U202#(proper(X1), proper(X2), proper(X3), proper(X4))active#(afterNth(X1, X2))active#(X2)
proper#(U171(X1, X2, X3))proper#(X2)proper#(U21(X1, X2, X3))U21#(proper(X1), proper(X2), proper(X3))
active#(U172(tt, N, XS))afterNth#(N, XS)proper#(pair(X1, X2))pair#(proper(X1), proper(X2))
proper#(tail(X))tail#(proper(X))U22#(ok(X1), ok(X2))U22#(X1, X2)
active#(U151(tt, V2))U152#(isLNat(V2))active#(U161(X1, X2))active#(X1)
active#(U161(X1, X2))U161#(active(X1), X2)proper#(U21(X1, X2, X3))proper#(X2)
proper#(U212(X1, X2))proper#(X1)active#(U222(tt, N, XS))splitAt#(N, XS)
proper#(U181(X1, X2))proper#(X2)afterNth#(X1, mark(X2))afterNth#(X1, X2)
cons#(ok(X1), ok(X2))cons#(X1, X2)proper#(U182(X1, X2))proper#(X2)
proper#(U201(X1, X2, X3, X4))proper#(X2)active#(head(cons(N, XS)))isNatural#(N)
proper#(U51(X1, X2))U51#(proper(X1), proper(X2))proper#(isLNat(X))isLNat#(proper(X))
active#(sel(X1, X2))active#(X2)active#(U51(X1, X2))U51#(active(X1), X2)
proper#(afterNth(X1, X2))proper#(X1)proper#(U91(X))proper#(X)
active#(splitAt(s(N), cons(X, XS)))isNatural#(N)U71#(mark(X))U71#(X)
proper#(fst(X))fst#(proper(X))proper#(pair(X1, X2))proper#(X2)
active#(natsFrom(N))isNatural#(N)U52#(ok(X))U52#(X)
take#(X1, mark(X2))take#(X1, X2)proper#(U12(X1, X2, X3))proper#(X3)
U32#(ok(X1), ok(X2))U32#(X1, X2)U191#(ok(X1), ok(X2))U191#(X1, X2)
active#(U101(tt, V2))U102#(isLNat(V2))active#(U203(tt, N, X, XS))splitAt#(N, XS)
proper#(take(X1, X2))take#(proper(X1), proper(X2))active#(snd(X))active#(X)
active#(cons(X1, X2))active#(X1)sel#(mark(X1), X2)sel#(X1, X2)
active#(U51(tt, V2))isLNat#(V2)active#(pair(X1, X2))pair#(X1, active(X2))
active#(isLNat(fst(V1)))isPLNat#(V1)active#(U11(X1, X2, X3))U11#(active(X1), X2, X3)
U31#(ok(X1), ok(X2), ok(X3))U31#(X1, X2, X3)proper#(U61(X))U61#(proper(X))
proper#(U42(X))proper#(X)proper#(U51(X1, X2))proper#(X1)
splitAt#(ok(X1), ok(X2))splitAt#(X1, X2)proper#(U101(X1, X2))proper#(X1)
proper#(take(X1, X2))proper#(X1)proper#(U201(X1, X2, X3, X4))proper#(X3)
proper#(U222(X1, X2, X3))proper#(X1)U101#(mark(X1), X2)U101#(X1, X2)
U41#(mark(X1), X2)U41#(X1, X2)proper#(U203(X1, X2, X3, X4))proper#(X1)
active#(U203(X1, X2, X3, X4))active#(X1)U203#(ok(X1), ok(X2), ok(X3), ok(X4))U203#(X1, X2, X3, X4)
proper#(U151(X1, X2))proper#(X2)active#(U31(tt, N, XS))U32#(isLNat(XS), N)
proper#(take(X1, X2))proper#(X2)active#(take(X1, X2))active#(X1)
proper#(U171(X1, X2, X3))U171#(proper(X1), proper(X2), proper(X3))U102#(ok(X))U102#(X)
active#(isLNat(natsFrom(V1)))U71#(isNatural(V1))s#(mark(X))s#(X)
proper#(U182(X1, X2))U182#(proper(X1), proper(X2))U204#(mark(X1), X2)U204#(X1, X2)
active#(s(X))active#(X)active#(tail(X))active#(X)
U52#(mark(X))U52#(X)active#(U61(X))active#(X)
active#(U51(tt, V2))U52#(isLNat(V2))proper#(isNatural(X))proper#(X)
splitAt#(mark(X1), X2)splitAt#(X1, X2)active#(U161(tt, N))natsFrom#(s(N))
proper#(U111(X))proper#(X)proper#(U32(X1, X2))proper#(X2)
active#(take(X1, X2))take#(active(X1), X2)active#(U101(X1, X2))active#(X1)
active#(natsFrom(N))U161#(isNatural(N), N)active#(splitAt(X1, X2))splitAt#(active(X1), X2)
active#(U202(X1, X2, X3, X4))active#(X1)active#(splitAt(X1, X2))splitAt#(X1, active(X2))
active#(U42(X))U42#(active(X))active#(isLNat(take(V1, V2)))isNatural#(V1)
proper#(U201(X1, X2, X3, X4))proper#(X4)active#(isLNat(natsFrom(V1)))isNatural#(V1)
proper#(U91(X))U91#(proper(X))proper#(U142(X))U142#(proper(X))
U12#(mark(X1), X2, X3)U12#(X1, X2, X3)active#(take(X1, X2))active#(X2)
active#(tail(cons(N, XS)))isNatural#(N)pair#(X1, mark(X2))pair#(X1, X2)
U21#(mark(X1), X2, X3)U21#(X1, X2, X3)active#(U21(X1, X2, X3))U21#(active(X1), X2, X3)
proper#(U131(X1, X2))proper#(X1)proper#(U142(X))proper#(X)
active#(U31(X1, X2, X3))active#(X1)proper#(U131(X1, X2))proper#(X2)
U211#(mark(X1), X2)U211#(X1, X2)proper#(isNatural(X))isNatural#(proper(X))
U151#(mark(X1), X2)U151#(X1, X2)U22#(mark(X1), X2)U22#(X1, X2)
proper#(U201(X1, X2, X3, X4))U201#(proper(X1), proper(X2), proper(X3), proper(X4))U101#(ok(X1), ok(X2))U101#(X1, X2)
active#(afterNth(X1, X2))afterNth#(X1, active(X2))active#(U12(tt, N, XS))snd#(splitAt(N, XS))
proper#(U212(X1, X2))proper#(X2)top#(ok(X))active#(X)
U141#(ok(X1), ok(X2))U141#(X1, X2)active#(pair(X1, X2))active#(X1)
U142#(mark(X))U142#(X)proper#(isLNat(X))proper#(X)
proper#(U151(X1, X2))proper#(X1)proper#(U202(X1, X2, X3, X4))proper#(X4)
proper#(U81(X))U81#(proper(X))active#(U161(tt, N))cons#(N, natsFrom(s(N)))
active#(U172(X1, X2, X3))U172#(active(X1), X2, X3)U61#(mark(X))U61#(X)
pair#(ok(X1), ok(X2))pair#(X1, X2)active#(head(X))head#(active(X))
U201#(mark(X1), X2, X3, X4)U201#(X1, X2, X3, X4)proper#(U203(X1, X2, X3, X4))proper#(X2)
proper#(U11(X1, X2, X3))proper#(X1)active#(U32(X1, X2))active#(X1)
proper#(s(X))s#(proper(X))active#(U131(X1, X2))U131#(active(X1), X2)
active#(splitAt(X1, X2))active#(X1)active#(U31(tt, N, XS))isLNat#(XS)
proper#(U152(X))U152#(proper(X))top#(ok(X))top#(active(X))
active#(U91(X))active#(X)U172#(ok(X1), ok(X2), ok(X3))U172#(X1, X2, X3)
proper#(U102(X))proper#(X)U42#(mark(X))U42#(X)
proper#(U171(X1, X2, X3))proper#(X1)active#(U171(tt, N, XS))isLNat#(XS)
active#(pair(X1, X2))active#(X2)snd#(ok(X))snd#(X)
active#(head(X))active#(X)active#(U171(X1, X2, X3))active#(X1)
active#(U11(X1, X2, X3))active#(X1)U181#(mark(X1), X2)U181#(X1, X2)
proper#(U141(X1, X2))proper#(X2)proper#(U152(X))proper#(X)
active#(U221(X1, X2, X3))active#(X1)proper#(U181(X1, X2))proper#(X1)
U132#(ok(X))U132#(X)proper#(fst(X))proper#(X)
U91#(ok(X))U91#(X)afterNth#(ok(X1), ok(X2))afterNth#(X1, X2)
active#(U151(X1, X2))active#(X1)U202#(mark(X1), X2, X3, X4)U202#(X1, X2, X3, X4)
active#(U141(tt, V2))isLNat#(V2)active#(splitAt(X1, X2))active#(X2)
active#(U172(tt, N, XS))head#(afterNth(N, XS))proper#(U31(X1, X2, X3))U31#(proper(X1), proper(X2), proper(X3))
active#(U211(X1, X2))active#(X1)active#(U181(X1, X2))active#(X1)
U121#(mark(X))U121#(X)proper#(U172(X1, X2, X3))U172#(proper(X1), proper(X2), proper(X3))
proper#(U81(X))proper#(X)proper#(U221(X1, X2, X3))proper#(X3)
active#(U182(X1, X2))active#(X1)active#(isLNat(tail(V1)))isLNat#(V1)
active#(U71(X))active#(X)proper#(U203(X1, X2, X3, X4))proper#(X3)
proper#(U52(X))proper#(X)proper#(U41(X1, X2))proper#(X2)
proper#(U171(X1, X2, X3))proper#(X3)active#(U101(tt, V2))isLNat#(V2)
active#(U51(X1, X2))active#(X1)proper#(U11(X1, X2, X3))proper#(X3)
active#(isNatural(head(V1)))U111#(isLNat(V1))active#(isNatural(sel(V1, V2)))U131#(isNatural(V1), V2)
active#(U222(X1, X2, X3))U222#(active(X1), X2, X3)proper#(U202(X1, X2, X3, X4))proper#(X2)
active#(U121(X))U121#(active(X))active#(U221(X1, X2, X3))U221#(active(X1), X2, X3)
active#(U142(X))U142#(active(X))active#(U211(X1, X2))U211#(active(X1), X2)
tail#(ok(X))tail#(X)proper#(U31(X1, X2, X3))proper#(X2)
active#(U204(X1, X2))active#(X1)active#(U181(tt, Y))isLNat#(Y)
active#(U101(X1, X2))U101#(active(X1), X2)active#(afterNth(N, XS))U11#(isNatural(N), N, XS)
active#(isPLNat(pair(V1, V2)))isLNat#(V1)proper#(U202(X1, X2, X3, X4))proper#(X3)
top#(mark(X))proper#(X)active#(isNatural(s(V1)))U121#(isNatural(V1))
active#(splitAt(0, XS))U191#(isLNat(XS), XS)active#(isLNat(cons(V1, V2)))U51#(isNatural(V1), V2)
proper#(U204(X1, X2))U204#(proper(X1), proper(X2))active#(U204(pair(YS, ZS), X))pair#(cons(X, YS), ZS)
active#(U161(tt, N))s#(N)active#(isPLNat(pair(V1, V2)))U141#(isLNat(V1), V2)
sel#(ok(X1), ok(X2))sel#(X1, X2)active#(U204(pair(YS, ZS), X))cons#(X, YS)
active#(U151(tt, V2))isLNat#(V2)active#(isPLNat(splitAt(V1, V2)))U151#(isNatural(V1), V2)
proper#(splitAt(X1, X2))proper#(X2)active#(isLNat(tail(V1)))U91#(isLNat(V1))
active#(isNatural(sel(V1, V2)))isNatural#(V1)active#(U204(X1, X2))U204#(active(X1), X2)
proper#(head(X))proper#(X)active#(U102(X))active#(X)
active#(U191(X1, X2))U191#(active(X1), X2)proper#(U181(X1, X2))U181#(proper(X1), proper(X2))
proper#(U191(X1, X2))proper#(X2)active#(isLNat(afterNth(V1, V2)))U41#(isNatural(V1), V2)
active#(sel(X1, X2))active#(X1)take#(ok(X1), ok(X2))take#(X1, X2)
active#(U201(tt, N, X, XS))isNatural#(X)proper#(U22(X1, X2))U22#(proper(X1), proper(X2))
active#(U221(tt, N, XS))U222#(isLNat(XS), N, XS)proper#(U212(X1, X2))U212#(proper(X1), proper(X2))
active#(U222(X1, X2, X3))active#(X1)active#(sel(N, XS))isNatural#(N)
active#(U52(X))U52#(active(X))active#(sel(X1, X2))sel#(active(X1), X2)
proper#(isPLNat(X))isPLNat#(proper(X))snd#(mark(X))snd#(X)
U222#(ok(X1), ok(X2), ok(X3))U222#(X1, X2, X3)active#(U202(tt, N, X, XS))U203#(isLNat(XS), N, X, XS)
active#(U181(X1, X2))U181#(active(X1), X2)active#(sel(N, XS))U171#(isNatural(N), N, XS)
active#(U41(tt, V2))U42#(isLNat(V2))proper#(U121(X))proper#(X)
active#(U102(X))U102#(active(X))U81#(ok(X))U81#(X)
proper#(isPLNat(X))proper#(X)U152#(mark(X))U152#(X)
U211#(ok(X1), ok(X2))U211#(X1, X2)proper#(U204(X1, X2))proper#(X2)
active#(U22(X1, X2))U22#(active(X1), X2)active#(U152(X))U152#(active(X))
active#(U132(X))U132#(active(X))active#(U202(tt, N, X, XS))isLNat#(XS)
U182#(mark(X1), X2)U182#(X1, X2)active#(U41(X1, X2))U41#(active(X1), X2)
active#(U191(X1, X2))active#(X1)proper#(U202(X1, X2, X3, X4))proper#(X1)
active#(afterNth(N, XS))isNatural#(N)active#(isLNat(fst(V1)))U61#(isPLNat(V1))
afterNth#(mark(X1), X2)afterNth#(X1, X2)proper#(U203(X1, X2, X3, X4))proper#(X4)
active#(U91(X))U91#(active(X))proper#(U32(X1, X2))proper#(X1)
active#(U21(tt, X, Y))U22#(isLNat(Y), X)proper#(U222(X1, X2, X3))U222#(proper(X1), proper(X2), proper(X3))
active#(U61(X))U61#(active(X))active#(pair(X1, X2))pair#(active(X1), X2)
U201#(ok(X1), ok(X2), ok(X3), ok(X4))U201#(X1, X2, X3, X4)U11#(mark(X1), X2, X3)U11#(X1, X2, X3)
U182#(ok(X1), ok(X2))U182#(X1, X2)tail#(mark(X))tail#(X)
U131#(ok(X1), ok(X2))U131#(X1, X2)active#(U81(X))active#(X)
active#(U21(X1, X2, X3))active#(X1)proper#(pair(X1, X2))proper#(X1)
proper#(s(X))proper#(X)active#(U132(X))active#(X)
active#(U131(X1, X2))active#(X1)active#(U201(tt, N, X, XS))U202#(isNatural(X), N, X, XS)
active#(snd(pair(X, Y)))isLNat#(X)proper#(U221(X1, X2, X3))proper#(X1)
U171#(ok(X1), ok(X2), ok(X3))U171#(X1, X2, X3)proper#(natsFrom(X))natsFrom#(proper(X))
proper#(U32(X1, X2))U32#(proper(X1), proper(X2))proper#(U131(X1, X2))U131#(proper(X1), proper(X2))
proper#(U203(X1, X2, X3, X4))U203#(proper(X1), proper(X2), proper(X3), proper(X4))take#(mark(X1), X2)take#(X1, X2)
active#(U212(X1, X2))U212#(active(X1), X2)U221#(mark(X1), X2, X3)U221#(X1, X2, X3)
proper#(U132(X))proper#(X)active#(natsFrom(X))active#(X)
active#(U203(tt, N, X, XS))U204#(splitAt(N, XS), X)proper#(sel(X1, X2))sel#(proper(X1), proper(X2))
U204#(ok(X1), ok(X2))U204#(X1, X2)U41#(ok(X1), ok(X2))U41#(X1, X2)
active#(U211(tt, XS))U212#(isLNat(XS), XS)fst#(mark(X))fst#(X)
U152#(ok(X))U152#(X)active#(isPLNat(splitAt(V1, V2)))isNatural#(V1)
U221#(ok(X1), ok(X2), ok(X3))U221#(X1, X2, X3)U142#(ok(X))U142#(X)
active#(head(cons(N, XS)))U31#(isNatural(N), N, XS)active#(natsFrom(X))natsFrom#(active(X))
U171#(mark(X1), X2, X3)U171#(X1, X2, X3)U212#(mark(X1), X2)U212#(X1, X2)
proper#(U61(X))proper#(X)active#(fst(X))fst#(active(X))
U111#(ok(X))U111#(X)proper#(U41(X1, X2))U41#(proper(X1), proper(X2))
proper#(U11(X1, X2, X3))U11#(proper(X1), proper(X2), proper(X3))active#(fst(pair(X, Y)))isLNat#(X)
U91#(mark(X))U91#(X)proper#(U132(X))U132#(proper(X))
active#(U201(X1, X2, X3, X4))U201#(active(X1), X2, X3, X4)U212#(ok(X1), ok(X2))U212#(X1, X2)
proper#(afterNth(X1, X2))afterNth#(proper(X1), proper(X2))active#(U151(X1, X2))U151#(active(X1), X2)
proper#(splitAt(X1, X2))proper#(X1)U21#(ok(X1), ok(X2), ok(X3))U21#(X1, X2, X3)
proper#(U71(X))proper#(X)proper#(U211(X1, X2))proper#(X2)
proper#(U52(X))U52#(proper(X))U161#(ok(X1), ok(X2))U161#(X1, X2)
proper#(U221(X1, X2, X3))proper#(X2)proper#(snd(X))snd#(proper(X))
active#(U222(tt, N, XS))fst#(splitAt(N, XS))U102#(mark(X))U102#(X)
active#(U111(X))U111#(active(X))U202#(ok(X1), ok(X2), ok(X3), ok(X4))U202#(X1, X2, X3, X4)
proper#(U42(X))U42#(proper(X))U121#(ok(X))U121#(X)
active#(isLNat(take(V1, V2)))U101#(isNatural(V1), V2)active#(U142(X))active#(X)
proper#(U191(X1, X2))proper#(X1)sel#(X1, mark(X2))sel#(X1, X2)
head#(ok(X))head#(X)active#(U11(tt, N, XS))U12#(isLNat(XS), N, XS)
U11#(ok(X1), ok(X2), ok(X3))U11#(X1, X2, X3)proper#(U221(X1, X2, X3))U221#(proper(X1), proper(X2), proper(X3))
pair#(mark(X1), X2)pair#(X1, X2)active#(U171(tt, N, XS))U172#(isLNat(XS), N, XS)
cons#(mark(X1), X2)cons#(X1, X2)proper#(U201(X1, X2, X3, X4))proper#(X1)
active#(U121(X))active#(X)U111#(mark(X))U111#(X)
proper#(U172(X1, X2, X3))proper#(X3)active#(U31(X1, X2, X3))U31#(active(X1), X2, X3)
proper#(U21(X1, X2, X3))proper#(X1)natsFrom#(ok(X))natsFrom#(X)
proper#(sel(X1, X2))proper#(X2)U42#(ok(X))U42#(X)
active#(U181(tt, Y))U182#(isLNat(Y), Y)active#(isLNat(snd(V1)))U81#(isPLNat(V1))
U51#(ok(X1), ok(X2))U51#(X1, X2)proper#(U161(X1, X2))proper#(X1)
head#(mark(X))head#(X)proper#(U12(X1, X2, X3))proper#(X2)
proper#(U12(X1, X2, X3))U12#(proper(X1), proper(X2), proper(X3))active#(U131(tt, V2))U132#(isLNat(V2))
isPLNat#(ok(X))isPLNat#(X)active#(take(N, XS))U221#(isNatural(N), N, XS)
active#(s(X))s#(active(X))proper#(U31(X1, X2, X3))proper#(X1)
s#(ok(X))s#(X)active#(splitAt(s(N), cons(X, XS)))U201#(isNatural(N), N, X, XS)
proper#(sel(X1, X2))proper#(X1)proper#(U172(X1, X2, X3))proper#(X1)
active#(U32(X1, X2))U32#(active(X1), X2)active#(take(X1, X2))take#(X1, active(X2))
active#(isNatural(s(V1)))isNatural#(V1)proper#(U141(X1, X2))U141#(proper(X1), proper(X2))
active#(U141(X1, X2))U141#(active(X1), X2)U61#(ok(X))U61#(X)
active#(U171(X1, X2, X3))U171#(active(X1), X2, X3)proper#(natsFrom(X))proper#(X)
U71#(ok(X))U71#(X)proper#(U161(X1, X2))U161#(proper(X1), proper(X2))
U181#(ok(X1), ok(X2))U181#(X1, X2)active#(U81(X))U81#(active(X))
active#(U12(tt, N, XS))splitAt#(N, XS)active#(cons(X1, X2))cons#(active(X1), X2)
active#(tail(X))tail#(active(X))U131#(mark(X1), X2)U131#(X1, X2)
active#(U201(X1, X2, X3, X4))active#(X1)active#(U202(X1, X2, X3, X4))U202#(active(X1), X2, X3, X4)
top#(mark(X))top#(proper(X))active#(U203(X1, X2, X3, X4))U203#(active(X1), X2, X3, X4)
active#(isNatural(head(V1)))isLNat#(V1)proper#(cons(X1, X2))proper#(X2)
proper#(U22(X1, X2))proper#(X2)active#(U172(X1, X2, X3))active#(X1)
proper#(afterNth(X1, X2))proper#(X2)proper#(snd(X))proper#(X)
proper#(head(X))head#(proper(X))U12#(ok(X1), ok(X2), ok(X3))U12#(X1, X2, X3)
active#(U71(X))U71#(active(X))proper#(splitAt(X1, X2))splitAt#(proper(X1), proper(X2))
U222#(mark(X1), X2, X3)U222#(X1, X2, X3)proper#(U211(X1, X2))U211#(proper(X1), proper(X2))
active#(U11(tt, N, XS))isLNat#(XS)U161#(mark(X1), X2)U161#(X1, X2)
active#(U42(X))active#(X)proper#(U141(X1, X2))proper#(X1)
proper#(U151(X1, X2))U151#(proper(X1), proper(X2))proper#(U111(X))U111#(proper(X))
proper#(U22(X1, X2))proper#(X1)U151#(ok(X1), ok(X2))U151#(X1, X2)
active#(isLNat(snd(V1)))isPLNat#(V1)active#(afterNth(X1, X2))active#(X1)
proper#(U101(X1, X2))proper#(X2)active#(U111(X))active#(X)
proper#(U31(X1, X2, X3))proper#(X3)active#(U141(tt, V2))U142#(isLNat(V2))
active#(U131(tt, V2))isLNat#(V2)proper#(U102(X))U102#(proper(X))
active#(snd(pair(X, Y)))U181#(isLNat(X), Y)active#(fst(X))active#(X)
active#(U152(X))active#(X)fst#(ok(X))fst#(X)
U141#(mark(X1), X2)U141#(X1, X2)proper#(cons(X1, X2))cons#(proper(X1), proper(X2))
active#(U141(X1, X2))active#(X1)U32#(mark(X1), X2)U32#(X1, X2)
proper#(U182(X1, X2))proper#(X1)active#(tail(cons(N, XS)))U211#(isNatural(N), XS)
proper#(U191(X1, X2))U191#(proper(X1), proper(X2))U31#(mark(X1), X2, X3)U31#(X1, X2, X3)
proper#(U204(X1, X2))proper#(X1)proper#(U172(X1, X2, X3))proper#(X2)

Rewrite Rules

active(U101(tt, V2))mark(U102(isLNat(V2)))active(U102(tt))mark(tt)
active(U11(tt, N, XS))mark(U12(isLNat(XS), N, XS))active(U111(tt))mark(tt)
active(U12(tt, N, XS))mark(snd(splitAt(N, XS)))active(U121(tt))mark(tt)
active(U131(tt, V2))mark(U132(isLNat(V2)))active(U132(tt))mark(tt)
active(U141(tt, V2))mark(U142(isLNat(V2)))active(U142(tt))mark(tt)
active(U151(tt, V2))mark(U152(isLNat(V2)))active(U152(tt))mark(tt)
active(U161(tt, N))mark(cons(N, natsFrom(s(N))))active(U171(tt, N, XS))mark(U172(isLNat(XS), N, XS))
active(U172(tt, N, XS))mark(head(afterNth(N, XS)))active(U181(tt, Y))mark(U182(isLNat(Y), Y))
active(U182(tt, Y))mark(Y)active(U191(tt, XS))mark(pair(nil, XS))
active(U201(tt, N, X, XS))mark(U202(isNatural(X), N, X, XS))active(U202(tt, N, X, XS))mark(U203(isLNat(XS), N, X, XS))
active(U203(tt, N, X, XS))mark(U204(splitAt(N, XS), X))active(U204(pair(YS, ZS), X))mark(pair(cons(X, YS), ZS))
active(U21(tt, X, Y))mark(U22(isLNat(Y), X))active(U211(tt, XS))mark(U212(isLNat(XS), XS))
active(U212(tt, XS))mark(XS)active(U22(tt, X))mark(X)
active(U221(tt, N, XS))mark(U222(isLNat(XS), N, XS))active(U222(tt, N, XS))mark(fst(splitAt(N, XS)))
active(U31(tt, N, XS))mark(U32(isLNat(XS), N))active(U32(tt, N))mark(N)
active(U41(tt, V2))mark(U42(isLNat(V2)))active(U42(tt))mark(tt)
active(U51(tt, V2))mark(U52(isLNat(V2)))active(U52(tt))mark(tt)
active(U61(tt))mark(tt)active(U71(tt))mark(tt)
active(U81(tt))mark(tt)active(U91(tt))mark(tt)
active(afterNth(N, XS))mark(U11(isNatural(N), N, XS))active(fst(pair(X, Y)))mark(U21(isLNat(X), X, Y))
active(head(cons(N, XS)))mark(U31(isNatural(N), N, XS))active(isLNat(nil))mark(tt)
active(isLNat(afterNth(V1, V2)))mark(U41(isNatural(V1), V2))active(isLNat(cons(V1, V2)))mark(U51(isNatural(V1), V2))
active(isLNat(fst(V1)))mark(U61(isPLNat(V1)))active(isLNat(natsFrom(V1)))mark(U71(isNatural(V1)))
active(isLNat(snd(V1)))mark(U81(isPLNat(V1)))active(isLNat(tail(V1)))mark(U91(isLNat(V1)))
active(isLNat(take(V1, V2)))mark(U101(isNatural(V1), V2))active(isNatural(0))mark(tt)
active(isNatural(head(V1)))mark(U111(isLNat(V1)))active(isNatural(s(V1)))mark(U121(isNatural(V1)))
active(isNatural(sel(V1, V2)))mark(U131(isNatural(V1), V2))active(isPLNat(pair(V1, V2)))mark(U141(isLNat(V1), V2))
active(isPLNat(splitAt(V1, V2)))mark(U151(isNatural(V1), V2))active(natsFrom(N))mark(U161(isNatural(N), N))
active(sel(N, XS))mark(U171(isNatural(N), N, XS))active(snd(pair(X, Y)))mark(U181(isLNat(X), Y))
active(splitAt(0, XS))mark(U191(isLNat(XS), XS))active(splitAt(s(N), cons(X, XS)))mark(U201(isNatural(N), N, X, XS))
active(tail(cons(N, XS)))mark(U211(isNatural(N), XS))active(take(N, XS))mark(U221(isNatural(N), N, XS))
active(U101(X1, X2))U101(active(X1), X2)active(U102(X))U102(active(X))
active(U11(X1, X2, X3))U11(active(X1), X2, X3)active(U12(X1, X2, X3))U12(active(X1), X2, X3)
active(U111(X))U111(active(X))active(snd(X))snd(active(X))
active(splitAt(X1, X2))splitAt(active(X1), X2)active(splitAt(X1, X2))splitAt(X1, active(X2))
active(U121(X))U121(active(X))active(U131(X1, X2))U131(active(X1), X2)
active(U132(X))U132(active(X))active(U141(X1, X2))U141(active(X1), X2)
active(U142(X))U142(active(X))active(U151(X1, X2))U151(active(X1), X2)
active(U152(X))U152(active(X))active(U161(X1, X2))U161(active(X1), X2)
active(cons(X1, X2))cons(active(X1), X2)active(natsFrom(X))natsFrom(active(X))
active(s(X))s(active(X))active(U171(X1, X2, X3))U171(active(X1), X2, X3)
active(U172(X1, X2, X3))U172(active(X1), X2, X3)active(head(X))head(active(X))
active(afterNth(X1, X2))afterNth(active(X1), X2)active(afterNth(X1, X2))afterNth(X1, active(X2))
active(U181(X1, X2))U181(active(X1), X2)active(U182(X1, X2))U182(active(X1), X2)
active(U191(X1, X2))U191(active(X1), X2)active(pair(X1, X2))pair(active(X1), X2)
active(pair(X1, X2))pair(X1, active(X2))active(U201(X1, X2, X3, X4))U201(active(X1), X2, X3, X4)
active(U202(X1, X2, X3, X4))U202(active(X1), X2, X3, X4)active(U203(X1, X2, X3, X4))U203(active(X1), X2, X3, X4)
active(U204(X1, X2))U204(active(X1), X2)active(U21(X1, X2, X3))U21(active(X1), X2, X3)
active(U22(X1, X2))U22(active(X1), X2)active(U211(X1, X2))U211(active(X1), X2)
active(U212(X1, X2))U212(active(X1), X2)active(U221(X1, X2, X3))U221(active(X1), X2, X3)
active(U222(X1, X2, X3))U222(active(X1), X2, X3)active(fst(X))fst(active(X))
active(U31(X1, X2, X3))U31(active(X1), X2, X3)active(U32(X1, X2))U32(active(X1), X2)
active(U41(X1, X2))U41(active(X1), X2)active(U42(X))U42(active(X))
active(U51(X1, X2))U51(active(X1), X2)active(U52(X))U52(active(X))
active(U61(X))U61(active(X))active(U71(X))U71(active(X))
active(U81(X))U81(active(X))active(U91(X))U91(active(X))
active(tail(X))tail(active(X))active(take(X1, X2))take(active(X1), X2)
active(take(X1, X2))take(X1, active(X2))active(sel(X1, X2))sel(active(X1), X2)
active(sel(X1, X2))sel(X1, active(X2))U101(mark(X1), X2)mark(U101(X1, X2))
U102(mark(X))mark(U102(X))U11(mark(X1), X2, X3)mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3)mark(U12(X1, X2, X3))U111(mark(X))mark(U111(X))
snd(mark(X))mark(snd(X))splitAt(mark(X1), X2)mark(splitAt(X1, X2))
splitAt(X1, mark(X2))mark(splitAt(X1, X2))U121(mark(X))mark(U121(X))
U131(mark(X1), X2)mark(U131(X1, X2))U132(mark(X))mark(U132(X))
U141(mark(X1), X2)mark(U141(X1, X2))U142(mark(X))mark(U142(X))
U151(mark(X1), X2)mark(U151(X1, X2))U152(mark(X))mark(U152(X))
U161(mark(X1), X2)mark(U161(X1, X2))cons(mark(X1), X2)mark(cons(X1, X2))
natsFrom(mark(X))mark(natsFrom(X))s(mark(X))mark(s(X))
U171(mark(X1), X2, X3)mark(U171(X1, X2, X3))U172(mark(X1), X2, X3)mark(U172(X1, X2, X3))
head(mark(X))mark(head(X))afterNth(mark(X1), X2)mark(afterNth(X1, X2))
afterNth(X1, mark(X2))mark(afterNth(X1, X2))U181(mark(X1), X2)mark(U181(X1, X2))
U182(mark(X1), X2)mark(U182(X1, X2))U191(mark(X1), X2)mark(U191(X1, X2))
pair(mark(X1), X2)mark(pair(X1, X2))pair(X1, mark(X2))mark(pair(X1, X2))
U201(mark(X1), X2, X3, X4)mark(U201(X1, X2, X3, X4))U202(mark(X1), X2, X3, X4)mark(U202(X1, X2, X3, X4))
U203(mark(X1), X2, X3, X4)mark(U203(X1, X2, X3, X4))U204(mark(X1), X2)mark(U204(X1, X2))
U21(mark(X1), X2, X3)mark(U21(X1, X2, X3))U22(mark(X1), X2)mark(U22(X1, X2))
U211(mark(X1), X2)mark(U211(X1, X2))U212(mark(X1), X2)mark(U212(X1, X2))
U221(mark(X1), X2, X3)mark(U221(X1, X2, X3))U222(mark(X1), X2, X3)mark(U222(X1, X2, X3))
fst(mark(X))mark(fst(X))U31(mark(X1), X2, X3)mark(U31(X1, X2, X3))
U32(mark(X1), X2)mark(U32(X1, X2))U41(mark(X1), X2)mark(U41(X1, X2))
U42(mark(X))mark(U42(X))U51(mark(X1), X2)mark(U51(X1, X2))
U52(mark(X))mark(U52(X))U61(mark(X))mark(U61(X))
U71(mark(X))mark(U71(X))U81(mark(X))mark(U81(X))
U91(mark(X))mark(U91(X))tail(mark(X))mark(tail(X))
take(mark(X1), X2)mark(take(X1, X2))take(X1, mark(X2))mark(take(X1, X2))
sel(mark(X1), X2)mark(sel(X1, X2))sel(X1, mark(X2))mark(sel(X1, X2))
proper(U101(X1, X2))U101(proper(X1), proper(X2))proper(tt)ok(tt)
proper(U102(X))U102(proper(X))proper(isLNat(X))isLNat(proper(X))
proper(U11(X1, X2, X3))U11(proper(X1), proper(X2), proper(X3))proper(U12(X1, X2, X3))U12(proper(X1), proper(X2), proper(X3))
proper(U111(X))U111(proper(X))proper(snd(X))snd(proper(X))
proper(splitAt(X1, X2))splitAt(proper(X1), proper(X2))proper(U121(X))U121(proper(X))
proper(U131(X1, X2))U131(proper(X1), proper(X2))proper(U132(X))U132(proper(X))
proper(U141(X1, X2))U141(proper(X1), proper(X2))proper(U142(X))U142(proper(X))
proper(U151(X1, X2))U151(proper(X1), proper(X2))proper(U152(X))U152(proper(X))
proper(U161(X1, X2))U161(proper(X1), proper(X2))proper(cons(X1, X2))cons(proper(X1), proper(X2))
proper(natsFrom(X))natsFrom(proper(X))proper(s(X))s(proper(X))
proper(U171(X1, X2, X3))U171(proper(X1), proper(X2), proper(X3))proper(U172(X1, X2, X3))U172(proper(X1), proper(X2), proper(X3))
proper(head(X))head(proper(X))proper(afterNth(X1, X2))afterNth(proper(X1), proper(X2))
proper(U181(X1, X2))U181(proper(X1), proper(X2))proper(U182(X1, X2))U182(proper(X1), proper(X2))
proper(U191(X1, X2))U191(proper(X1), proper(X2))proper(pair(X1, X2))pair(proper(X1), proper(X2))
proper(nil)ok(nil)proper(U201(X1, X2, X3, X4))U201(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U202(X1, X2, X3, X4))U202(proper(X1), proper(X2), proper(X3), proper(X4))proper(isNatural(X))isNatural(proper(X))
proper(U203(X1, X2, X3, X4))U203(proper(X1), proper(X2), proper(X3), proper(X4))proper(U204(X1, X2))U204(proper(X1), proper(X2))
proper(U21(X1, X2, X3))U21(proper(X1), proper(X2), proper(X3))proper(U22(X1, X2))U22(proper(X1), proper(X2))
proper(U211(X1, X2))U211(proper(X1), proper(X2))proper(U212(X1, X2))U212(proper(X1), proper(X2))
proper(U221(X1, X2, X3))U221(proper(X1), proper(X2), proper(X3))proper(U222(X1, X2, X3))U222(proper(X1), proper(X2), proper(X3))
proper(fst(X))fst(proper(X))proper(U31(X1, X2, X3))U31(proper(X1), proper(X2), proper(X3))
proper(U32(X1, X2))U32(proper(X1), proper(X2))proper(U41(X1, X2))U41(proper(X1), proper(X2))
proper(U42(X))U42(proper(X))proper(U51(X1, X2))U51(proper(X1), proper(X2))
proper(U52(X))U52(proper(X))proper(U61(X))U61(proper(X))
proper(U71(X))U71(proper(X))proper(U81(X))U81(proper(X))
proper(U91(X))U91(proper(X))proper(isPLNat(X))isPLNat(proper(X))
proper(tail(X))tail(proper(X))proper(take(X1, X2))take(proper(X1), proper(X2))
proper(0)ok(0)proper(sel(X1, X2))sel(proper(X1), proper(X2))
U101(ok(X1), ok(X2))ok(U101(X1, X2))U102(ok(X))ok(U102(X))
isLNat(ok(X))ok(isLNat(X))U11(ok(X1), ok(X2), ok(X3))ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3))ok(U12(X1, X2, X3))U111(ok(X))ok(U111(X))
snd(ok(X))ok(snd(X))splitAt(ok(X1), ok(X2))ok(splitAt(X1, X2))
U121(ok(X))ok(U121(X))U131(ok(X1), ok(X2))ok(U131(X1, X2))
U132(ok(X))ok(U132(X))U141(ok(X1), ok(X2))ok(U141(X1, X2))
U142(ok(X))ok(U142(X))U151(ok(X1), ok(X2))ok(U151(X1, X2))
U152(ok(X))ok(U152(X))U161(ok(X1), ok(X2))ok(U161(X1, X2))
cons(ok(X1), ok(X2))ok(cons(X1, X2))natsFrom(ok(X))ok(natsFrom(X))
s(ok(X))ok(s(X))U171(ok(X1), ok(X2), ok(X3))ok(U171(X1, X2, X3))
U172(ok(X1), ok(X2), ok(X3))ok(U172(X1, X2, X3))head(ok(X))ok(head(X))
afterNth(ok(X1), ok(X2))ok(afterNth(X1, X2))U181(ok(X1), ok(X2))ok(U181(X1, X2))
U182(ok(X1), ok(X2))ok(U182(X1, X2))U191(ok(X1), ok(X2))ok(U191(X1, X2))
pair(ok(X1), ok(X2))ok(pair(X1, X2))U201(ok(X1), ok(X2), ok(X3), ok(X4))ok(U201(X1, X2, X3, X4))
U202(ok(X1), ok(X2), ok(X3), ok(X4))ok(U202(X1, X2, X3, X4))isNatural(ok(X))ok(isNatural(X))
U203(ok(X1), ok(X2), ok(X3), ok(X4))ok(U203(X1, X2, X3, X4))U204(ok(X1), ok(X2))ok(U204(X1, X2))
U21(ok(X1), ok(X2), ok(X3))ok(U21(X1, X2, X3))U22(ok(X1), ok(X2))ok(U22(X1, X2))
U211(ok(X1), ok(X2))ok(U211(X1, X2))U212(ok(X1), ok(X2))ok(U212(X1, X2))
U221(ok(X1), ok(X2), ok(X3))ok(U221(X1, X2, X3))U222(ok(X1), ok(X2), ok(X3))ok(U222(X1, X2, X3))
fst(ok(X))ok(fst(X))U31(ok(X1), ok(X2), ok(X3))ok(U31(X1, X2, X3))
U32(ok(X1), ok(X2))ok(U32(X1, X2))U41(ok(X1), ok(X2))ok(U41(X1, X2))
U42(ok(X))ok(U42(X))U51(ok(X1), ok(X2))ok(U51(X1, X2))
U52(ok(X))ok(U52(X))U61(ok(X))ok(U61(X))
U71(ok(X))ok(U71(X))U81(ok(X))ok(U81(X))
U91(ok(X))ok(U91(X))isPLNat(ok(X))ok(isPLNat(X))
tail(ok(X))ok(tail(X))take(ok(X1), ok(X2))ok(take(X1, X2))
sel(ok(X1), ok(X2))ok(sel(X1, X2))top(mark(X))top(proper(X))
top(ok(X))top(active(X))

Original Signature

Termination of terms over the following signature is verified: natsFrom, U161, U204, U203, fst, U111, U61, U211, U212, ok, proper, head, U21, U22, U151, mark, isPLNat, U152, tail, U71, U191, 0, U121, U221, isLNat, U222, active, U31, U32, U181, U182, pair, isNatural, U132, U131, U42, U41, U91, top, cons, snd, U171, U172, splitAt, U201, U142, U202, U141, s, U51, tt, take, U81, U52, U11, U12, afterNth, U102, sel, U101, nil