The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0, XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
U11(mark(X1), X2, X3) →⁺ mark(U11(X1, X2, X3))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
active, U12, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U12 < proper
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(8) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U12, active, snd, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U12 < active
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U12 < proper
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

Induction Base:
U12(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))

Induction Step:
U12(gen_tt:mark:0':nil:ok3_0(+(1, +(n5_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →_R^Ω(1)
mark(U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(10) Complex Obligation (BEST)

(11) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
snd, active, splitAt, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
snd < active
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
snd < proper
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)

Induction Base:
snd(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
snd(gen_tt:mark:0':nil:ok3_0(+(1, +(n4339_0, 1)))) →_R^Ω(1)
mark(snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(13) Complex Obligation (BEST)

(14) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
splitAt, active, U22, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
splitAt < active
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
splitAt < proper
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)

Induction Base:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, +(n5294_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(16) Complex Obligation (BEST)

(17) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U22, active, U32, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U22 < active
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U22 < proper
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)

Induction Base:
U22(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U22(gen_tt:mark:0':nil:ok3_0(+(1, +(n8594_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(19) Complex Obligation (BEST)

(20) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U32, active, U42, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U32 < active
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U32 < proper
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)

Induction Base:
U32(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U32(gen_tt:mark:0':nil:ok3_0(+(1, +(n11999_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(22) Complex Obligation (BEST)

(23) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U42, active, head, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U42 < active
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U42 < proper
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(24) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)

Induction Base:
U42(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))

Induction Step:
U42(gen_tt:mark:0':nil:ok3_0(+(1, +(n15708_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →_R^Ω(1)
mark(U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(25) Complex Obligation (BEST)

(26) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
head, active, afterNth, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
head < active
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
head < proper
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)

Induction Base:
head(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
head(gen_tt:mark:0':nil:ok3_0(+(1, +(n22332_0, 1)))) →_R^Ω(1)
mark(head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(28) Complex Obligation (BEST)

(29) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
afterNth, active, U52, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
afterNth < active
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
afterNth < proper
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)

Induction Base:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, +(n24037_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(31) Complex Obligation (BEST)

(32) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U52, active, U62, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U52 < active
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U52 < proper
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(33) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)

Induction Base:
U52(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U52(gen_tt:mark:0':nil:ok3_0(+(1, +(n28857_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(34) Complex Obligation (BEST)

(35) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U62, active, U63, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U62 < active
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U62 < proper
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(36) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)

Induction Base:
U62(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))

Induction Step:
U62(gen_tt:mark:0':nil:ok3_0(+(1, +(n33782_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →_R^Ω(1)
mark(U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(37) Complex Obligation (BEST)

(38) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U63, active, U64, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U63 < active
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U63 < proper
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(39) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)

Induction Base:
U63(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))

Induction Step:
U63(gen_tt:mark:0':nil:ok3_0(+(1, +(n45673_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →_R^Ω(1)
mark(U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(40) Complex Obligation (BEST)

(41) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U64, active, pair, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U64 < active
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U64 < proper
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(42) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)

Induction Base:
U64(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U64(gen_tt:mark:0':nil:ok3_0(+(1, +(n58568_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(43) Complex Obligation (BEST)

(44) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
pair, active, cons, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
pair < active
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
pair < proper
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(45) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)

Induction Base:
pair(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
pair(gen_tt:mark:0':nil:ok3_0(+(1, +(n64797_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(46) Complex Obligation (BEST)

(47) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
cons, active, U72, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
cons < active
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
cons < proper
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(48) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)

Induction Base:
cons(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
cons(gen_tt:mark:0':nil:ok3_0(+(1, +(n71529_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(49) Complex Obligation (BEST)

(50) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U72, active, U82, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U72 < active
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U72 < proper
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(51) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)

Induction Base:
U72(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U72(gen_tt:mark:0':nil:ok3_0(+(1, +(n78366_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(52) Complex Obligation (BEST)

(53) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U82, active, fst, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U82 < active
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U82 < proper
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(54) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)

Induction Base:
U82(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))

Induction Step:
U82(gen_tt:mark:0':nil:ok3_0(+(1, +(n85507_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →_R^Ω(1)
mark(U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(55) Complex Obligation (BEST)

(56) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
fst, active, U11, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
fst < active
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
fst < proper
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(57) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)

Induction Base:
fst(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
fst(gen_tt:mark:0':nil:ok3_0(+(1, +(n97299_0, 1)))) →_R^Ω(1)
mark(fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(58) Complex Obligation (BEST)

(59) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U11, active, U21, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U11 < active
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U11 < proper
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(60) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)

Induction Base:
U11(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))

Induction Step:
U11(gen_tt:mark:0':nil:ok3_0(+(1, +(n100700_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →_R^Ω(1)
mark(U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(61) Complex Obligation (BEST)

(62) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U21, active, U31, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U21 < active
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U21 < proper
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(63) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)

Induction Base:
U21(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U21(gen_tt:mark:0':nil:ok3_0(+(1, +(n113408_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(64) Complex Obligation (BEST)

(65) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U31, active, natsFrom, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U31 < active
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U31 < proper
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(66) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)

Induction Base:
U31(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U31(gen_tt:mark:0':nil:ok3_0(+(1, +(n121863_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(67) Complex Obligation (BEST)

(68) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
natsFrom, active, s, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
natsFrom < active
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
natsFrom < proper
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(69) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)

Induction Base:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, +(n130622_0, 1)))) →_R^Ω(1)
mark(natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(70) Complex Obligation (BEST)

(71) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
s, active, U41, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
s < active
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
s < proper
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(72) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)

Induction Base:
s(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
s(gen_tt:mark:0':nil:ok3_0(+(1, +(n134623_0, 1)))) →_R^Ω(1)
mark(s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(73) Complex Obligation (BEST)

(74) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U41, active, U51, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U41 < active
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U41 < proper
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(75) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)

Induction Base:
U41(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))

Induction Step:
U41(gen_tt:mark:0':nil:ok3_0(+(1, +(n138725_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →_R^Ω(1)
mark(U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(76) Complex Obligation (BEST)

(77) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U51, active, U61, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U51 < active
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U51 < proper
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(78) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)

Induction Base:
U51(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U51(gen_tt:mark:0':nil:ok3_0(+(1, +(n153576_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(79) Complex Obligation (BEST)

(80) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U61, active, U71, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U61 < active
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U61 < proper
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(81) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)

Induction Base:
U61(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))

Induction Step:
U61(gen_tt:mark:0':nil:ok3_0(+(1, +(n163453_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) →_R^Ω(1)
mark(U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(82) Complex Obligation (BEST)

(83) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U71, active, U81, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U71 < active
U81 < active
sel < active
tail < active
take < active
active < top
U71 < proper
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(84) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)

Induction Base:
U71(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
U71(gen_tt:mark:0':nil:ok3_0(+(1, +(n185308_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(85) Complex Obligation (BEST)

(86) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
U81, active, sel, tail, take, proper, top

They will be analysed ascendingly in the following order:
U81 < active
sel < active
tail < active
take < active
active < top
U81 < proper
sel < proper
tail < proper
take < proper
proper < top

(87) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)

Induction Base:
U81(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))

Induction Step:
U81(gen_tt:mark:0':nil:ok3_0(+(1, +(n195989_0, 1))), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) →_R^Ω(1)
mark(U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(88) Complex Obligation (BEST)

(89) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
sel, active, tail, take, proper, top

They will be analysed ascendingly in the following order:
sel < active
tail < active
take < active
active < top
sel < proper
tail < proper
take < proper
proper < top

(90) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)

Induction Base:
sel(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
sel(gen_tt:mark:0':nil:ok3_0(+(1, +(n213113_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(91) Complex Obligation (BEST)

(92) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
tail, active, take, proper, top

They will be analysed ascendingly in the following order:
tail < active
take < active
active < top
tail < proper
take < proper
proper < top

(93) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n224699₀)

Induction Base:
tail(gen_tt:mark:0':nil:ok3_0(+(1, 0)))

Induction Step:
tail(gen_tt:mark:0':nil:ok3_0(+(1, +(n224699_0, 1)))) →_R^Ω(1)
mark(tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0)))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(94) Complex Obligation (BEST)

(95) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n224699₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
take, active, proper, top

They will be analysed ascendingly in the following order:
take < active
active < top
take < proper
proper < top

(96) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n229998₀)

Induction Base:
take(gen_tt:mark:0':nil:ok3_0(+(1, 0)), gen_tt:mark:0':nil:ok3_0(b))

Induction Step:
take(gen_tt:mark:0':nil:ok3_0(+(1, +(n229998_0, 1))), gen_tt:mark:0':nil:ok3_0(b)) →_R^Ω(1)
mark(take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b))) →_IH
mark(*4_0)

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(97) Complex Obligation (BEST)

(98) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n224699₀)
take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n229998₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top

(99) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n224699₀)
take(gen_tt:mark:0':nil:ok3_0(+(1, n229998_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n229998₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(100) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)
tail(gen_tt:mark:0':nil:ok3_0(+(1, n224699_0))) → *4_0, rt ∈ Ω(n224699₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(101) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)
sel(gen_tt:mark:0':nil:ok3_0(+(1, n213113_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n213113₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(102) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)
U81(gen_tt:mark:0':nil:ok3_0(+(1, n195989_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n195989₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(103) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)
U71(gen_tt:mark:0':nil:ok3_0(+(1, n185308_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n185308₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(104) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)
U61(gen_tt:mark:0':nil:ok3_0(+(1, n163453_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n163453₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(105) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)
U51(gen_tt:mark:0':nil:ok3_0(+(1, n153576_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n153576₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(106) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)
U41(gen_tt:mark:0':nil:ok3_0(+(1, n138725_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n138725₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(107) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)
s(gen_tt:mark:0':nil:ok3_0(+(1, n134623_0))) → *4_0, rt ∈ Ω(n134623₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(108) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)
natsFrom(gen_tt:mark:0':nil:ok3_0(+(1, n130622_0))) → *4_0, rt ∈ Ω(n130622₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(109) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)
U31(gen_tt:mark:0':nil:ok3_0(+(1, n121863_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n121863₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(110) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)
U21(gen_tt:mark:0':nil:ok3_0(+(1, n113408_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n113408₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(111) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)
U11(gen_tt:mark:0':nil:ok3_0(+(1, n100700_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n100700₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(112) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)
fst(gen_tt:mark:0':nil:ok3_0(+(1, n97299_0))) → *4_0, rt ∈ Ω(n97299₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(113) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)
U82(gen_tt:mark:0':nil:ok3_0(+(1, n85507_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n85507₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(114) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)
U72(gen_tt:mark:0':nil:ok3_0(+(1, n78366_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n78366₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(115) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)
cons(gen_tt:mark:0':nil:ok3_0(+(1, n71529_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n71529₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(116) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)
pair(gen_tt:mark:0':nil:ok3_0(+(1, n64797_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n64797₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(117) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)
U64(gen_tt:mark:0':nil:ok3_0(+(1, n58568_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n58568₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(118) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)
U63(gen_tt:mark:0':nil:ok3_0(+(1, n45673_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n45673₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(119) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)
U62(gen_tt:mark:0':nil:ok3_0(+(1, n33782_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c), gen_tt:mark:0':nil:ok3_0(d)) → *4_0, rt ∈ Ω(n33782₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(120) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)
U52(gen_tt:mark:0':nil:ok3_0(+(1, n28857_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n28857₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(121) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)
afterNth(gen_tt:mark:0':nil:ok3_0(+(1, n24037_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n24037₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(122) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)
head(gen_tt:mark:0':nil:ok3_0(+(1, n22332_0))) → *4_0, rt ∈ Ω(n22332₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(123) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)
U42(gen_tt:mark:0':nil:ok3_0(+(1, n15708_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n15708₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(124) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)
U32(gen_tt:mark:0':nil:ok3_0(+(1, n11999_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n11999₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(125) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)
U22(gen_tt:mark:0':nil:ok3_0(+(1, n8594_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n8594₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(126) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)
splitAt(gen_tt:mark:0':nil:ok3_0(+(1, n5294_0)), gen_tt:mark:0':nil:ok3_0(b)) → *4_0, rt ∈ Ω(n5294₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(127) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)
snd(gen_tt:mark:0':nil:ok3_0(+(1, n4339_0))) → *4_0, rt ∈ Ω(n4339₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.

(128) Obligation:

TRS:
Rules:
active(U11(tt, N, XS)) → mark(U12(tt, N, XS))
active(U12(tt, N, XS)) → mark(snd(splitAt(N, XS)))
active(U21(tt, X)) → mark(U22(tt, X))
active(U22(tt, X)) → mark(X)
active(U31(tt, N)) → mark(U32(tt, N))
active(U32(tt, N)) → mark(N)
active(U41(tt, N, XS)) → mark(U42(tt, N, XS))
active(U42(tt, N, XS)) → mark(head(afterNth(N, XS)))
active(U51(tt, Y)) → mark(U52(tt, Y))
active(U52(tt, Y)) → mark(Y)
active(U61(tt, N, X, XS)) → mark(U62(tt, N, X, XS))
active(U62(tt, N, X, XS)) → mark(U63(tt, N, X, XS))
active(U63(tt, N, X, XS)) → mark(U64(splitAt(N, XS), X))
active(U64(pair(YS, ZS), X)) → mark(pair(cons(X, YS), ZS))
active(U71(tt, XS)) → mark(U72(tt, XS))
active(U72(tt, XS)) → mark(XS)
active(U81(tt, N, XS)) → mark(U82(tt, N, XS))
active(U82(tt, N, XS)) → mark(fst(splitAt(N, XS)))
active(afterNth(N, XS)) → mark(U11(tt, N, XS))
active(fst(pair(X, Y))) → mark(U21(tt, X))
active(head(cons(N, XS))) → mark(U31(tt, N))
active(natsFrom(N)) → mark(cons(N, natsFrom(s(N))))
active(sel(N, XS)) → mark(U41(tt, N, XS))
active(snd(pair(X, Y))) → mark(U51(tt, Y))
active(splitAt(0', XS)) → mark(pair(nil, XS))
active(splitAt(s(N), cons(X, XS))) → mark(U61(tt, N, X, XS))
active(tail(cons(N, XS))) → mark(U71(tt, XS))
active(take(N, XS)) → mark(U81(tt, N, XS))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(snd(X)) → snd(active(X))
active(splitAt(X1, X2)) → splitAt(active(X1), X2)
active(splitAt(X1, X2)) → splitAt(X1, active(X2))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X1, X2)) → U32(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3, X4)) → U61(active(X1), X2, X3, X4)
active(U62(X1, X2, X3, X4)) → U62(active(X1), X2, X3, X4)
active(U63(X1, X2, X3, X4)) → U63(active(X1), X2, X3, X4)
active(U64(X1, X2)) → U64(active(X1), X2)
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(cons(X1, X2)) → cons(active(X1), X2)
active(U71(X1, X2)) → U71(active(X1), X2)
active(U72(X1, X2)) → U72(active(X1), X2)
active(U81(X1, X2, X3)) → U81(active(X1), X2, X3)
active(U82(X1, X2, X3)) → U82(active(X1), X2, X3)
active(fst(X)) → fst(active(X))
active(natsFrom(X)) → natsFrom(active(X))
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(tail(X)) → tail(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
U11(mark(X1), X2, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
snd(mark(X)) → mark(snd(X))
splitAt(mark(X1), X2) → mark(splitAt(X1, X2))
splitAt(X1, mark(X2)) → mark(splitAt(X1, X2))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X1), X2) → mark(U32(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3, X4) → mark(U61(X1, X2, X3, X4))
U62(mark(X1), X2, X3, X4) → mark(U62(X1, X2, X3, X4))
U63(mark(X1), X2, X3, X4) → mark(U63(X1, X2, X3, X4))
U64(mark(X1), X2) → mark(U64(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
U71(mark(X1), X2) → mark(U71(X1, X2))
U72(mark(X1), X2) → mark(U72(X1, X2))
U81(mark(X1), X2, X3) → mark(U81(X1, X2, X3))
U82(mark(X1), X2, X3) → mark(U82(X1, X2, X3))
fst(mark(X)) → mark(fst(X))
natsFrom(mark(X)) → mark(natsFrom(X))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
tail(mark(X)) → mark(tail(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(snd(X)) → snd(proper(X))
proper(splitAt(X1, X2)) → splitAt(proper(X1), proper(X2))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X1, X2)) → U32(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(head(X)) → head(proper(X))
proper(afterNth(X1, X2)) → afterNth(proper(X1), proper(X2))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3, X4)) → U61(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U62(X1, X2, X3, X4)) → U62(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U63(X1, X2, X3, X4)) → U63(proper(X1), proper(X2), proper(X3), proper(X4))
proper(U64(X1, X2)) → U64(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(U71(X1, X2)) → U71(proper(X1), proper(X2))
proper(U72(X1, X2)) → U72(proper(X1), proper(X2))
proper(U81(X1, X2, X3)) → U81(proper(X1), proper(X2), proper(X3))
proper(U82(X1, X2, X3)) → U82(proper(X1), proper(X2), proper(X3))
proper(fst(X)) → fst(proper(X))
proper(natsFrom(X)) → natsFrom(proper(X))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0') → ok(0')
proper(nil) → ok(nil)
proper(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
U11(ok(X1), ok(X2), ok(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
snd(ok(X)) → ok(snd(X))
splitAt(ok(X1), ok(X2)) → ok(splitAt(X1, X2))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X1), ok(X2)) → ok(U32(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
head(ok(X)) → ok(head(X))
afterNth(ok(X1), ok(X2)) → ok(afterNth(X1, X2))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U61(X1, X2, X3, X4))
U62(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U62(X1, X2, X3, X4))
U63(ok(X1), ok(X2), ok(X3), ok(X4)) → ok(U63(X1, X2, X3, X4))
U64(ok(X1), ok(X2)) → ok(U64(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
U71(ok(X1), ok(X2)) → ok(U71(X1, X2))
U72(ok(X1), ok(X2)) → ok(U72(X1, X2))
U81(ok(X1), ok(X2), ok(X3)) → ok(U81(X1, X2, X3))
U82(ok(X1), ok(X2), ok(X3)) → ok(U82(X1, X2, X3))
fst(ok(X)) → ok(fst(X))
natsFrom(ok(X)) → ok(natsFrom(X))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
tail(ok(X)) → ok(tail(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Types:
active :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U11 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
tt :: tt:mark:0':nil:ok
mark :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
U12 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
snd :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
splitAt :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U21 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U22 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U31 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U32 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U41 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U42 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
head :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
afterNth :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U51 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U52 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U61 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U62 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U63 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U64 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
pair :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
cons :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U71 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U72 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U81 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
U82 :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
fst :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
natsFrom :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
s :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
sel :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
0' :: tt:mark:0':nil:ok
nil :: tt:mark:0':nil:ok
tail :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
take :: tt:mark:0':nil:ok → tt:mark:0':nil:ok → tt:mark:0':nil:ok
proper :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
ok :: tt:mark:0':nil:ok → tt:mark:0':nil:ok
top :: tt:mark:0':nil:ok → top
hole_tt:mark:0':nil:ok1_0 :: tt:mark:0':nil:ok
hole_top2_0 :: top
gen_tt:mark:0':nil:ok3_0 :: Nat → tt:mark:0':nil:ok

Lemmas:
U12(gen_tt:mark:0':nil:ok3_0(+(1, n5_0)), gen_tt:mark:0':nil:ok3_0(b), gen_tt:mark:0':nil:ok3_0(c)) → *4_0, rt ∈ Ω(n5₀)

Generator Equations:
gen_tt:mark:0':nil:ok3_0(0) ⇔ tt
gen_tt:mark:0':nil:ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':nil:ok3_0(x))

No more defined symbols left to analyse.