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

(0) Obligation:

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


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: INNERMOST

(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)

The following defined symbols can occur below the 0th argument of top: proper, active
The following defined symbols can occur below the 0th argument of proper: proper, active
The following defined symbols can occur below the 0th argument of active: proper, active

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
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))
proper(U11(X1, X2, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(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(tail(X)) → tail(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))

(2) Obligation:

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


The TRS R consists of the following rules:

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

Rewrite Strategy: INNERMOST

(3) CpxTrsMatchBoundsTAProof (EQUIVALENT transformation)

A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 2.

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by:
final states : [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]
transitions:
ok0(0) → 0
active0(0) → 0
mark0(0) → 0
tt0() → 0
nil0() → 0
00() → 0
U620(0, 0, 0, 0) → 1
top0(0) → 2
U810(0, 0, 0) → 3
U320(0, 0) → 4
cons0(0, 0) → 5
tail0(0) → 6
snd0(0) → 7
U520(0, 0) → 8
pair0(0, 0) → 9
sel0(0, 0) → 10
splitAt0(0, 0) → 11
U420(0, 0, 0) → 12
proper0(0) → 13
U640(0, 0) → 14
U820(0, 0, 0) → 15
U110(0, 0, 0) → 16
U310(0, 0) → 17
head0(0) → 18
U610(0, 0, 0, 0) → 19
U630(0, 0, 0, 0) → 20
natsFrom0(0) → 21
fst0(0) → 22
afterNth0(0, 0) → 23
U210(0, 0) → 24
s0(0) → 25
U710(0, 0) → 26
take0(0, 0) → 27
U510(0, 0) → 28
U720(0, 0) → 29
U120(0, 0, 0) → 30
U220(0, 0) → 31
U410(0, 0, 0) → 32
U621(0, 0, 0, 0) → 33
ok1(33) → 1
active1(0) → 34
top1(34) → 2
U621(0, 0, 0, 0) → 35
mark1(35) → 1
U811(0, 0, 0) → 36
mark1(36) → 3
U321(0, 0) → 37
mark1(37) → 4
cons1(0, 0) → 38
ok1(38) → 5
U811(0, 0, 0) → 39
ok1(39) → 3
tail1(0) → 40
ok1(40) → 6
snd1(0) → 41
ok1(41) → 7
U521(0, 0) → 42
ok1(42) → 8
pair1(0, 0) → 43
mark1(43) → 9
U321(0, 0) → 44
ok1(44) → 4
sel1(0, 0) → 45
ok1(45) → 10
splitAt1(0, 0) → 46
mark1(46) → 11
sel1(0, 0) → 47
mark1(47) → 10
U421(0, 0, 0) → 48
mark1(48) → 12
tt1() → 49
ok1(49) → 13
nil1() → 50
ok1(50) → 13
U641(0, 0) → 51
mark1(51) → 14
U821(0, 0, 0) → 52
ok1(52) → 15
U111(0, 0, 0) → 53
mark1(53) → 16
U821(0, 0, 0) → 54
mark1(54) → 15
U311(0, 0) → 55
ok1(55) → 17
head1(0) → 56
mark1(56) → 18
U611(0, 0, 0, 0) → 57
ok1(57) → 19
U111(0, 0, 0) → 58
ok1(58) → 16
U421(0, 0, 0) → 59
ok1(59) → 12
U631(0, 0, 0, 0) → 60
ok1(60) → 20
natsFrom1(0) → 61
ok1(61) → 21
fst1(0) → 62
mark1(62) → 22
afterNth1(0, 0) → 63
ok1(63) → 23
tail1(0) → 64
mark1(64) → 6
U211(0, 0) → 65
ok1(65) → 24
s1(0) → 66
ok1(66) → 25
U711(0, 0) → 67
ok1(67) → 26
01() → 68
ok1(68) → 13
afterNth1(0, 0) → 69
mark1(69) → 23
take1(0, 0) → 70
mark1(70) → 27
U511(0, 0) → 71
mark1(71) → 28
U721(0, 0) → 72
mark1(72) → 29
U121(0, 0, 0) → 73
ok1(73) → 30
U221(0, 0) → 74
mark1(74) → 31
U121(0, 0, 0) → 75
mark1(75) → 30
U511(0, 0) → 76
ok1(76) → 28
snd1(0) → 77
mark1(77) → 7
take1(0, 0) → 78
ok1(78) → 27
U721(0, 0) → 79
ok1(79) → 29
U211(0, 0) → 80
mark1(80) → 24
U521(0, 0) → 81
mark1(81) → 8
head1(0) → 82
ok1(82) → 18
pair1(0, 0) → 83
ok1(83) → 9
U631(0, 0, 0, 0) → 84
mark1(84) → 20
U611(0, 0, 0, 0) → 85
mark1(85) → 19
U641(0, 0) → 86
ok1(86) → 14
U711(0, 0) → 87
mark1(87) → 26
natsFrom1(0) → 88
mark1(88) → 21
s1(0) → 89
mark1(89) → 25
splitAt1(0, 0) → 90
ok1(90) → 11
fst1(0) → 91
ok1(91) → 22
U411(0, 0, 0) → 92
mark1(92) → 32
U221(0, 0) → 93
ok1(93) → 31
U311(0, 0) → 94
mark1(94) → 17
cons1(0, 0) → 95
mark1(95) → 5
proper1(0) → 96
top1(96) → 2
U411(0, 0, 0) → 97
ok1(97) → 32
ok1(33) → 33
ok1(33) → 35
mark1(35) → 33
mark1(35) → 35
mark1(36) → 36
mark1(36) → 39
mark1(37) → 37
mark1(37) → 44
ok1(38) → 38
ok1(38) → 95
ok1(39) → 36
ok1(39) → 39
ok1(40) → 40
ok1(40) → 64
ok1(41) → 41
ok1(41) → 77
ok1(42) → 42
ok1(42) → 81
mark1(43) → 43
mark1(43) → 83
ok1(44) → 37
ok1(44) → 44
ok1(45) → 45
ok1(45) → 47
mark1(46) → 46
mark1(46) → 90
mark1(47) → 45
mark1(47) → 47
mark1(48) → 48
mark1(48) → 59
ok1(49) → 96
ok1(50) → 96
mark1(51) → 51
mark1(51) → 86
ok1(52) → 52
ok1(52) → 54
mark1(53) → 53
mark1(53) → 58
mark1(54) → 52
mark1(54) → 54
ok1(55) → 55
ok1(55) → 94
mark1(56) → 56
mark1(56) → 82
ok1(57) → 57
ok1(57) → 85
ok1(58) → 53
ok1(58) → 58
ok1(59) → 48
ok1(59) → 59
ok1(60) → 60
ok1(60) → 84
ok1(61) → 61
ok1(61) → 88
mark1(62) → 62
mark1(62) → 91
ok1(63) → 63
ok1(63) → 69
mark1(64) → 40
mark1(64) → 64
ok1(65) → 65
ok1(65) → 80
ok1(66) → 66
ok1(66) → 89
ok1(67) → 67
ok1(67) → 87
ok1(68) → 96
mark1(69) → 63
mark1(69) → 69
mark1(70) → 70
mark1(70) → 78
mark1(71) → 71
mark1(71) → 76
mark1(72) → 72
mark1(72) → 79
ok1(73) → 73
ok1(73) → 75
mark1(74) → 74
mark1(74) → 93
mark1(75) → 73
mark1(75) → 75
ok1(76) → 71
ok1(76) → 76
mark1(77) → 41
mark1(77) → 77
ok1(78) → 70
ok1(78) → 78
ok1(79) → 72
ok1(79) → 79
mark1(80) → 65
mark1(80) → 80
mark1(81) → 42
mark1(81) → 81
ok1(82) → 56
ok1(82) → 82
ok1(83) → 43
ok1(83) → 83
mark1(84) → 60
mark1(84) → 84
mark1(85) → 57
mark1(85) → 85
ok1(86) → 51
ok1(86) → 86
mark1(87) → 67
mark1(87) → 87
mark1(88) → 61
mark1(88) → 88
mark1(89) → 66
mark1(89) → 89
ok1(90) → 46
ok1(90) → 90
ok1(91) → 62
ok1(91) → 91
mark1(92) → 92
mark1(92) → 97
ok1(93) → 74
ok1(93) → 93
mark1(94) → 55
mark1(94) → 94
mark1(95) → 38
mark1(95) → 95
ok1(97) → 92
ok1(97) → 97
active2(49) → 98
top2(98) → 2
active2(50) → 98
active2(68) → 98

(4) BOUNDS(1, n^1)

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

Converted Cpx (relative) TRS to CDT

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

U62(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U62(z0, z1, z2, z3))
U62(mark(z0), z1, z2, z3) → mark(U62(z0, z1, z2, z3))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U64(mark(z0), z1) → mark(U64(z0, z1))
U64(ok(z0), ok(z1)) → ok(U64(z0, z1))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U61(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U61(z0, z1, z2, z3))
U61(mark(z0), z1, z2, z3) → mark(U61(z0, z1, z2, z3))
U63(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U63(z0, z1, z2, z3))
U63(mark(z0), z1, z2, z3) → mark(U63(z0, z1, z2, z3))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
PROPER(tt) → c27
PROPER(nil) → c28
PROPER(0) → c29
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
TOP(ok(z0)) → c2(TOP(active(z0)))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
PROPER(tt) → c27
PROPER(nil) → c28
PROPER(0) → c29
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

U62, top, U81, U32, cons, tail, snd, U52, pair, sel, splitAt, U42, proper, U64, U82, U11, U31, head, U61, U63, natsFrom, fst, afterNth, U21, s, U71, take, U51, U72, U12, U22, U41

Defined Pair Symbols:

U62', TOP, U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', PROPER, U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41'

Compound Symbols:

c, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c27, c28, c29, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69

(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)

Removed 4 trailing nodes:

PROPER(nil) → c28
PROPER(0) → c29
PROPER(tt) → c27
TOP(ok(z0)) → c2(TOP(active(z0)))

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

U62(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U62(z0, z1, z2, z3))
U62(mark(z0), z1, z2, z3) → mark(U62(z0, z1, z2, z3))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U64(mark(z0), z1) → mark(U64(z0, z1))
U64(ok(z0), ok(z1)) → ok(U64(z0, z1))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U61(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U61(z0, z1, z2, z3))
U61(mark(z0), z1, z2, z3) → mark(U61(z0, z1, z2, z3))
U63(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U63(z0, z1, z2, z3))
U63(mark(z0), z1, z2, z3) → mark(U63(z0, z1, z2, z3))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
TOP(mark(z0)) → c3(TOP(proper(z0)), PROPER(z0))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

U62, top, U81, U32, cons, tail, snd, U52, pair, sel, splitAt, U42, proper, U64, U82, U11, U31, head, U61, U63, natsFrom, fst, afterNth, U21, s, U71, take, U51, U72, U12, U22, U41

Defined Pair Symbols:

U62', TOP, U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41'

Compound Symbols:

c, c1, c3, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69

(9) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID) transformation)

Removed 1 trailing tuple parts

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

U62(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U62(z0, z1, z2, z3))
U62(mark(z0), z1, z2, z3) → mark(U62(z0, z1, z2, z3))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)
U64(mark(z0), z1) → mark(U64(z0, z1))
U64(ok(z0), ok(z1)) → ok(U64(z0, z1))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U61(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U61(z0, z1, z2, z3))
U61(mark(z0), z1, z2, z3) → mark(U61(z0, z1, z2, z3))
U63(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U63(z0, z1, z2, z3))
U63(mark(z0), z1, z2, z3) → mark(U63(z0, z1, z2, z3))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

U62, top, U81, U32, cons, tail, snd, U52, pair, sel, splitAt, U42, proper, U64, U82, U11, U31, head, U61, U63, natsFrom, fst, afterNth, U21, s, U71, take, U51, U72, U12, U22, U41

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(11) CdtUsableRulesProof (EQUIVALENT transformation)

The following rules are not usable and were removed:

U62(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U62(z0, z1, z2, z3))
U62(mark(z0), z1, z2, z3) → mark(U62(z0, z1, z2, z3))
top(ok(z0)) → top(active(z0))
top(mark(z0)) → top(proper(z0))
U81(mark(z0), z1, z2) → mark(U81(z0, z1, z2))
U81(ok(z0), ok(z1), ok(z2)) → ok(U81(z0, z1, z2))
U32(mark(z0), z1) → mark(U32(z0, z1))
U32(ok(z0), ok(z1)) → ok(U32(z0, z1))
cons(ok(z0), ok(z1)) → ok(cons(z0, z1))
cons(mark(z0), z1) → mark(cons(z0, z1))
tail(ok(z0)) → ok(tail(z0))
tail(mark(z0)) → mark(tail(z0))
snd(ok(z0)) → ok(snd(z0))
snd(mark(z0)) → mark(snd(z0))
U52(ok(z0), ok(z1)) → ok(U52(z0, z1))
U52(mark(z0), z1) → mark(U52(z0, z1))
pair(mark(z0), z1) → mark(pair(z0, z1))
pair(z0, mark(z1)) → mark(pair(z0, z1))
pair(ok(z0), ok(z1)) → ok(pair(z0, z1))
sel(ok(z0), ok(z1)) → ok(sel(z0, z1))
sel(z0, mark(z1)) → mark(sel(z0, z1))
sel(mark(z0), z1) → mark(sel(z0, z1))
splitAt(mark(z0), z1) → mark(splitAt(z0, z1))
splitAt(z0, mark(z1)) → mark(splitAt(z0, z1))
splitAt(ok(z0), ok(z1)) → ok(splitAt(z0, z1))
U42(mark(z0), z1, z2) → mark(U42(z0, z1, z2))
U42(ok(z0), ok(z1), ok(z2)) → ok(U42(z0, z1, z2))
U64(mark(z0), z1) → mark(U64(z0, z1))
U64(ok(z0), ok(z1)) → ok(U64(z0, z1))
U82(ok(z0), ok(z1), ok(z2)) → ok(U82(z0, z1, z2))
U82(mark(z0), z1, z2) → mark(U82(z0, z1, z2))
U11(mark(z0), z1, z2) → mark(U11(z0, z1, z2))
U11(ok(z0), ok(z1), ok(z2)) → ok(U11(z0, z1, z2))
U31(ok(z0), ok(z1)) → ok(U31(z0, z1))
U31(mark(z0), z1) → mark(U31(z0, z1))
head(mark(z0)) → mark(head(z0))
head(ok(z0)) → ok(head(z0))
U61(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U61(z0, z1, z2, z3))
U61(mark(z0), z1, z2, z3) → mark(U61(z0, z1, z2, z3))
U63(ok(z0), ok(z1), ok(z2), ok(z3)) → ok(U63(z0, z1, z2, z3))
U63(mark(z0), z1, z2, z3) → mark(U63(z0, z1, z2, z3))
natsFrom(ok(z0)) → ok(natsFrom(z0))
natsFrom(mark(z0)) → mark(natsFrom(z0))
fst(mark(z0)) → mark(fst(z0))
fst(ok(z0)) → ok(fst(z0))
afterNth(ok(z0), ok(z1)) → ok(afterNth(z0, z1))
afterNth(z0, mark(z1)) → mark(afterNth(z0, z1))
afterNth(mark(z0), z1) → mark(afterNth(z0, z1))
U21(ok(z0), ok(z1)) → ok(U21(z0, z1))
U21(mark(z0), z1) → mark(U21(z0, z1))
s(ok(z0)) → ok(s(z0))
s(mark(z0)) → mark(s(z0))
U71(ok(z0), ok(z1)) → ok(U71(z0, z1))
U71(mark(z0), z1) → mark(U71(z0, z1))
take(mark(z0), z1) → mark(take(z0, z1))
take(ok(z0), ok(z1)) → ok(take(z0, z1))
take(z0, mark(z1)) → mark(take(z0, z1))
U51(mark(z0), z1) → mark(U51(z0, z1))
U51(ok(z0), ok(z1)) → ok(U51(z0, z1))
U72(mark(z0), z1) → mark(U72(z0, z1))
U72(ok(z0), ok(z1)) → ok(U72(z0, z1))
U12(ok(z0), ok(z1), ok(z2)) → ok(U12(z0, z1, z2))
U12(mark(z0), z1, z2) → mark(U12(z0, z1, z2))
U22(mark(z0), z1) → mark(U22(z0, z1))
U22(ok(z0), ok(z1)) → ok(U22(z0, z1))
U41(mark(z0), z1, z2) → mark(U41(z0, z1, z2))
U41(ok(z0), ok(z1), ok(z2)) → ok(U41(z0, z1, z2))

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

K tuples:none
Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TOP(mark(z0)) → c3(TOP(proper(z0)))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = x₁   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = 0   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

We considered the (Usable) Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = x₁   
POL(TOP(x₁)) = x₁   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = [1]   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = x₁ + x₂   
POL(CONS(x₁, x₂)) = x₁   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = x₁   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = x₁   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = x₁   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = [1]   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = x₁   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = x₂   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₁   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = x₁   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = x₁   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = x₂   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₁ + x₂   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = [2]x₂   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = [2]x₂   
POL(U21'(x₁, x₂)) = [2]x₂   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = [2]x₂   
POL(U71'(x₁, x₂)) = x₁   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = x₁   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(25) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = x₁   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(27) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = x₁   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = x₁   
POL(SEL(x₁, x₂)) = x₁   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = x₁   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = x₁   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₂   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = x₁   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = x₁   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(29) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = x₁   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = x₁   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = x₂   
POL(U21'(x₁, x₂)) = x₂   
POL(U22'(x₁, x₂)) = x₁   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = x₁   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = x₂   
POL(U71'(x₁, x₂)) = x₁   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = x₃   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(31) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = x₂   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = x₁   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₂   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = x₄   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = x₃   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(33) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

NATSFROM(ok(z0)) → c44(NATSFROM(z0))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = x₁   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(34) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(35) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U51'(mark(z0), z1) → c60(U51'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = x₁   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(36) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(37) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SEL(z0, mark(z1)) → c20(SEL(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = x₂   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = x₂   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = x₄   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(38) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(39) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = x₁   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = x₄   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(40) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SND(mark(z0)) → c13(SND(z0))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = x₁   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = x₄   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = x₂   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = x₂   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(42) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(43) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U64'(mark(z0), z1) → c30(U64'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = x₁   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(44) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(45) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U21'(mark(z0), z1) → c52(U21'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = x₁   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(46) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(47) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [1]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = x₂   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = x₃   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = x₁ + x₃   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = [1] + x₁   
POL(tt) = 0   

(48) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = x₂   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = x₂   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(50) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₁   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = x₁   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = x₁   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(52) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = x₁   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = x₁   
POL(U71'(x₁, x₂)) = x₂   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = x₃   
POL(U82'(x₁, x₂, x₃)) = x₃   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = [1] + x₁   
POL(nil) = 0   
POL(ok(x₁)) = x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(54) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = [2]x₂   
POL(U32'(x₁, x₂)) = [2]x₂   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = x₂   
POL(U61'(x₁, x₂, x₃, x₄)) = [2]x₂   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = x₃   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(56) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
SND(ok(z0)) → c12(SND(z0))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = x₁   
POL(CONS(x₁, x₂)) = x₁ + x₂   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₁   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = x₃   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(58) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

SND(ok(z0)) → c12(SND(z0))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
FST(ok(z0)) → c47(FST(z0))
S(ok(z0)) → c53(S(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

FST(ok(z0)) → c47(FST(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = x₁   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = x₂   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = x₂   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = x₁   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = x₁   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(60) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

SND(ok(z0)) → c12(SND(z0))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
S(ok(z0)) → c53(S(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
FST(ok(z0)) → c47(FST(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = [3]   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = x₃   
POL(U12'(x₁, x₂, x₃)) = [2]x₃   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = x₂   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = [2]x₂   
POL(U61'(x₁, x₂, x₃, x₄)) = [2]x₂   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = x₄   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = [2]x₂   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = [3]x₂   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [2] + x₁   
POL(proper(x₁)) = [3] + [2]x₁   
POL(tt) = 0   

(62) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

SND(ok(z0)) → c12(SND(z0))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
S(ok(z0)) → c53(S(z0))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
FST(ok(z0)) → c47(FST(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

S(ok(z0)) → c53(S(z0))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = x₁   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = x₂   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = x₄   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = 0   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(64) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

SND(ok(z0)) → c12(SND(z0))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
FST(ok(z0)) → c47(FST(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
S(ok(z0)) → c53(S(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

SND(ok(z0)) → c12(SND(z0))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = x₁   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = x₂   
POL(SND(x₁)) = x₁   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = 0   
POL(U12'(x₁, x₂, x₃)) = 0   
POL(U21'(x₁, x₂)) = 0   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = x₁   
POL(U41'(x₁, x₂, x₃)) = 0   
POL(U42'(x₁, x₂, x₃)) = 0   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = 0   
POL(U61'(x₁, x₂, x₃, x₄)) = x₃   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = 0   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = 0   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = x₂   
POL(U82'(x₁, x₂, x₃)) = 0   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = x₁   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(66) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:

U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))

K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
FST(ok(z0)) → c47(FST(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
S(ok(z0)) → c53(S(z0))
SND(ok(z0)) → c12(SND(z0))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))

We considered the (Usable) Rules:none
And the Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(0) = 0   
POL(AFTERNTH(x₁, x₂)) = 0   
POL(CONS(x₁, x₂)) = 0   
POL(FST(x₁)) = 0   
POL(HEAD(x₁)) = 0   
POL(NATSFROM(x₁)) = 0   
POL(PAIR(x₁, x₂)) = 0   
POL(S(x₁)) = 0   
POL(SEL(x₁, x₂)) = 0   
POL(SND(x₁)) = 0   
POL(SPLITAT(x₁, x₂)) = 0   
POL(TAIL(x₁)) = 0   
POL(TAKE(x₁, x₂)) = 0   
POL(TOP(x₁)) = 0   
POL(U11'(x₁, x₂, x₃)) = [2]x₃   
POL(U12'(x₁, x₂, x₃)) = [2]x₂ + [2]x₃   
POL(U21'(x₁, x₂)) = [2]x₂   
POL(U22'(x₁, x₂)) = 0   
POL(U31'(x₁, x₂)) = 0   
POL(U32'(x₁, x₂)) = 0   
POL(U41'(x₁, x₂, x₃)) = [2]x₃   
POL(U42'(x₁, x₂, x₃)) = x₃   
POL(U51'(x₁, x₂)) = 0   
POL(U52'(x₁, x₂)) = [2]x₂   
POL(U61'(x₁, x₂, x₃, x₄)) = 0   
POL(U62'(x₁, x₂, x₃, x₄)) = 0   
POL(U63'(x₁, x₂, x₃, x₄)) = [2]x₂   
POL(U64'(x₁, x₂)) = 0   
POL(U71'(x₁, x₂)) = [2]x₂   
POL(U72'(x₁, x₂)) = 0   
POL(U81'(x₁, x₂, x₃)) = x₃   
POL(U82'(x₁, x₂, x₃)) = [3]x₂   
POL(c(x₁)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c11(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c15(x₁)) = x₁   
POL(c16(x₁)) = x₁   
POL(c17(x₁)) = x₁   
POL(c18(x₁)) = x₁   
POL(c19(x₁)) = x₁   
POL(c20(x₁)) = x₁   
POL(c21(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c23(x₁)) = x₁   
POL(c24(x₁)) = x₁   
POL(c25(x₁)) = x₁   
POL(c26(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c30(x₁)) = x₁   
POL(c31(x₁)) = x₁   
POL(c32(x₁)) = x₁   
POL(c33(x₁)) = x₁   
POL(c34(x₁)) = x₁   
POL(c35(x₁)) = x₁   
POL(c36(x₁)) = x₁   
POL(c37(x₁)) = x₁   
POL(c38(x₁)) = x₁   
POL(c39(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c40(x₁)) = x₁   
POL(c41(x₁)) = x₁   
POL(c42(x₁)) = x₁   
POL(c43(x₁)) = x₁   
POL(c44(x₁)) = x₁   
POL(c45(x₁)) = x₁   
POL(c46(x₁)) = x₁   
POL(c47(x₁)) = x₁   
POL(c48(x₁)) = x₁   
POL(c49(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(c50(x₁)) = x₁   
POL(c51(x₁)) = x₁   
POL(c52(x₁)) = x₁   
POL(c53(x₁)) = x₁   
POL(c54(x₁)) = x₁   
POL(c55(x₁)) = x₁   
POL(c56(x₁)) = x₁   
POL(c57(x₁)) = x₁   
POL(c58(x₁)) = x₁   
POL(c59(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c60(x₁)) = x₁   
POL(c61(x₁)) = x₁   
POL(c62(x₁)) = x₁   
POL(c63(x₁)) = x₁   
POL(c64(x₁)) = x₁   
POL(c65(x₁)) = x₁   
POL(c66(x₁)) = x₁   
POL(c67(x₁)) = x₁   
POL(c68(x₁)) = x₁   
POL(c69(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(mark(x₁)) = 0   
POL(nil) = 0   
POL(ok(x₁)) = [1] + x₁   
POL(proper(x₁)) = 0   
POL(tt) = 0   

(68) Obligation:

Complexity Dependency Tuples Problem
Rules:

proper(tt) → ok(tt)
proper(nil) → ok(nil)
proper(0) → ok(0)

Tuples:

U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
TAIL(ok(z0)) → c10(TAIL(z0))
TAIL(mark(z0)) → c11(TAIL(z0))
SND(ok(z0)) → c12(SND(z0))
SND(mark(z0)) → c13(SND(z0))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
FST(mark(z0)) → c46(FST(z0))
FST(ok(z0)) → c47(FST(z0))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
S(ok(z0)) → c53(S(z0))
S(mark(z0)) → c54(S(z0))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
TOP(mark(z0)) → c3(TOP(proper(z0)))

S tuples:none
K tuples:

TOP(mark(z0)) → c3(TOP(proper(z0)))
TAKE(mark(z0), z1) → c57(TAKE(z0, z1))
U62'(mark(z0), z1, z2, z3) → c1(U62'(z0, z1, z2, z3))
CONS(mark(z0), z1) → c9(CONS(z0, z1))
PAIR(mark(z0), z1) → c16(PAIR(z0, z1))
AFTERNTH(z0, mark(z1)) → c49(AFTERNTH(z0, z1))
AFTERNTH(mark(z0), z1) → c50(AFTERNTH(z0, z1))
U72'(mark(z0), z1) → c62(U72'(z0, z1))
U52'(mark(z0), z1) → c15(U52'(z0, z1))
U31'(mark(z0), z1) → c37(U31'(z0, z1))
NATSFROM(mark(z0)) → c45(NATSFROM(z0))
TAKE(z0, mark(z1)) → c59(TAKE(z0, z1))
PAIR(ok(z0), ok(z1)) → c18(PAIR(z0, z1))
SPLITAT(ok(z0), ok(z1)) → c24(SPLITAT(z0, z1))
U31'(ok(z0), ok(z1)) → c36(U31'(z0, z1))
U64'(ok(z0), ok(z1)) → c31(U64'(z0, z1))
U82'(ok(z0), ok(z1), ok(z2)) → c32(U82'(z0, z1, z2))
U82'(mark(z0), z1, z2) → c33(U82'(z0, z1, z2))
HEAD(mark(z0)) → c38(HEAD(z0))
HEAD(ok(z0)) → c39(HEAD(z0))
U21'(ok(z0), ok(z1)) → c51(U21'(z0, z1))
U71'(ok(z0), ok(z1)) → c55(U71'(z0, z1))
U71'(mark(z0), z1) → c56(U71'(z0, z1))
TAKE(ok(z0), ok(z1)) → c58(TAKE(z0, z1))
U12'(ok(z0), ok(z1), ok(z2)) → c64(U12'(z0, z1, z2))
TAIL(mark(z0)) → c11(TAIL(z0))
U61'(mark(z0), z1, z2, z3) → c41(U61'(z0, z1, z2, z3))
SEL(mark(z0), z1) → c21(SEL(z0, z1))
SPLITAT(mark(z0), z1) → c22(SPLITAT(z0, z1))
U63'(mark(z0), z1, z2, z3) → c43(U63'(z0, z1, z2, z3))
FST(mark(z0)) → c46(FST(z0))
S(mark(z0)) → c54(S(z0))
U12'(mark(z0), z1, z2) → c65(U12'(z0, z1, z2))
U41'(mark(z0), z1, z2) → c68(U41'(z0, z1, z2))
TAIL(ok(z0)) → c10(TAIL(z0))
SEL(ok(z0), ok(z1)) → c19(SEL(z0, z1))
U22'(ok(z0), ok(z1)) → c67(U22'(z0, z1))
U41'(ok(z0), ok(z1), ok(z2)) → c69(U41'(z0, z1, z2))
U81'(ok(z0), ok(z1), ok(z2)) → c5(U81'(z0, z1, z2))
U61'(ok(z0), ok(z1), ok(z2), ok(z3)) → c40(U61'(z0, z1, z2, z3))
NATSFROM(ok(z0)) → c44(NATSFROM(z0))
U51'(mark(z0), z1) → c60(U51'(z0, z1))
SEL(z0, mark(z1)) → c20(SEL(z0, z1))
U32'(mark(z0), z1) → c6(U32'(z0, z1))
U32'(ok(z0), ok(z1)) → c7(U32'(z0, z1))
SND(mark(z0)) → c13(SND(z0))
U64'(mark(z0), z1) → c30(U64'(z0, z1))
U21'(mark(z0), z1) → c52(U21'(z0, z1))
U81'(mark(z0), z1, z2) → c4(U81'(z0, z1, z2))
PAIR(z0, mark(z1)) → c17(PAIR(z0, z1))
SPLITAT(z0, mark(z1)) → c23(SPLITAT(z0, z1))
U11'(mark(z0), z1, z2) → c34(U11'(z0, z1, z2))
U22'(mark(z0), z1) → c66(U22'(z0, z1))
U42'(mark(z0), z1, z2) → c25(U42'(z0, z1, z2))
U52'(ok(z0), ok(z1)) → c14(U52'(z0, z1))
U62'(ok(z0), ok(z1), ok(z2), ok(z3)) → c(U62'(z0, z1, z2, z3))
CONS(ok(z0), ok(z1)) → c8(CONS(z0, z1))
U11'(ok(z0), ok(z1), ok(z2)) → c35(U11'(z0, z1, z2))
AFTERNTH(ok(z0), ok(z1)) → c48(AFTERNTH(z0, z1))
FST(ok(z0)) → c47(FST(z0))
U51'(ok(z0), ok(z1)) → c61(U51'(z0, z1))
U72'(ok(z0), ok(z1)) → c63(U72'(z0, z1))
U63'(ok(z0), ok(z1), ok(z2), ok(z3)) → c42(U63'(z0, z1, z2, z3))
S(ok(z0)) → c53(S(z0))
SND(ok(z0)) → c12(SND(z0))
U42'(ok(z0), ok(z1), ok(z2)) → c26(U42'(z0, z1, z2))

Defined Rule Symbols:

proper

Defined Pair Symbols:

U62', U81', U32', CONS, TAIL, SND, U52', PAIR, SEL, SPLITAT, U42', U64', U82', U11', U31', HEAD, U61', U63', NATSFROM, FST, AFTERNTH, U21', S, U71', TAKE, U51', U72', U12', U22', U41', TOP

Compound Symbols:

c, c1, c4, c5, c6, c7, c8, c9, c10, c11, c12, c13, c14, c15, c16, c17, c18, c19, c20, c21, c22, c23, c24, c25, c26, c30, c31, c32, c33, c34, c35, c36, c37, c38, c39, c40, c41, c42, c43, c44, c45, c46, c47, c48, c49, c50, c51, c52, c53, c54, c55, c56, c57, c58, c59, c60, c61, c62, c63, c64, c65, c66, c67, c68, c69, c3

(69) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)

The set S is empty

(70) BOUNDS(1, 1)