Termination w.r.t. Q proof of /tmp/tmpi_1Yfn/LISTUTILITIES_nosorts-noand

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, N, XS) → ACTIVATE(XS)
AFTERNTH(N, XS) → U11¹(tt, N, XS)
U42¹(tt, N, XS) → AFTERNTH(activate(N), activate(XS))
U63¹(tt, N, X, XS) → SPLITAT(activate(N), activate(XS))
TAIL(cons(N, XS)) → U71¹(tt, activate(XS))
U63¹(tt, N, X, XS) → ACTIVATE(XS)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U61¹(tt, N, X, XS) → ACTIVATE(XS)
U61¹(tt, N, X, XS) → U62¹(tt, activate(N), activate(X), activate(XS))
U82¹(tt, N, XS) → ACTIVATE(N)
U21¹(tt, X) → U22¹(tt, activate(X))
U82¹(tt, N, XS) → SPLITAT(activate(N), activate(XS))
U12¹(tt, N, XS) → ACTIVATE(XS)
U72¹(tt, XS) → ACTIVATE(XS)
SND(pair(X, Y)) → U51¹(tt, Y)
U81¹(tt, N, XS) → ACTIVATE(XS)
ACTIVATE(n__s(X)) → S(activate(X))
U42¹(tt, N, XS) → ACTIVATE(N)
U62¹(tt, N, X, XS) → ACTIVATE(XS)
U11¹(tt, N, XS) → ACTIVATE(N)
U63¹(tt, N, X, XS) → ACTIVATE(X)
U52¹(tt, Y) → ACTIVATE(Y)
TAKE(N, XS) → U81¹(tt, N, XS)
U63¹(tt, N, X, XS) → U64¹(splitAt(activate(N), activate(XS)), activate(X))
SEL(N, XS) → U41¹(tt, N, XS)
U64¹(pair(YS, ZS), X) → ACTIVATE(X)
FST(pair(X, Y)) → U21¹(tt, X)
U61¹(tt, N, X, XS) → ACTIVATE(X)
SPLITAT(s(N), cons(X, XS)) → ACTIVATE(XS)
U51¹(tt, Y) → ACTIVATE(Y)
HEAD(cons(N, XS)) → U31¹(tt, N)
U11¹(tt, N, XS) → U12¹(tt, activate(N), activate(XS))
U31¹(tt, N) → ACTIVATE(N)
U81¹(tt, N, XS) → U82¹(tt, activate(N), activate(XS))
U62¹(tt, N, X, XS) → ACTIVATE(N)
U82¹(tt, N, XS) → FST(splitAt(activate(N), activate(XS)))
U41¹(tt, N, XS) → ACTIVATE(XS)
U81¹(tt, N, XS) → ACTIVATE(N)
U12¹(tt, N, XS) → ACTIVATE(N)
TAIL(cons(N, XS)) → ACTIVATE(XS)
ACTIVATE(n__natsFrom(X)) → NATSFROM(activate(X))
U31¹(tt, N) → U32¹(tt, activate(N))
U12¹(tt, N, XS) → SND(splitAt(activate(N), activate(XS)))
U82¹(tt, N, XS) → ACTIVATE(XS)
U41¹(tt, N, XS) → ACTIVATE(N)
U71¹(tt, XS) → U72¹(tt, activate(XS))
U22¹(tt, X) → ACTIVATE(X)
U61¹(tt, N, X, XS) → ACTIVATE(N)
U51¹(tt, Y) → U52¹(tt, activate(Y))
U42¹(tt, N, XS) → HEAD(afterNth(activate(N), activate(XS)))
U21¹(tt, X) → ACTIVATE(X)
ACTIVATE(n__natsFrom(X)) → ACTIVATE(X)
U12¹(tt, N, XS) → SPLITAT(activate(N), activate(XS))
U71¹(tt, XS) → ACTIVATE(XS)
U32¹(tt, N) → ACTIVATE(N)
U63¹(tt, N, X, XS) → ACTIVATE(N)
U62¹(tt, N, X, XS) → ACTIVATE(X)
U42¹(tt, N, XS) → ACTIVATE(XS)
SPLITAT(s(N), cons(X, XS)) → U61¹(tt, N, X, activate(XS))
U41¹(tt, N, XS) → U42¹(tt, activate(N), activate(XS))
U62¹(tt, N, X, XS) → U63¹(tt, activate(N), activate(X), activate(XS))

The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

U11¹(tt, N, XS) → ACTIVATE(XS)
AFTERNTH(N, XS) → U11¹(tt, N, XS)
U42¹(tt, N, XS) → AFTERNTH(activate(N), activate(XS))
U63¹(tt, N, X, XS) → SPLITAT(activate(N), activate(XS))
TAIL(cons(N, XS)) → U71¹(tt, activate(XS))
U63¹(tt, N, X, XS) → ACTIVATE(XS)
ACTIVATE(n__s(X)) → ACTIVATE(X)
U61¹(tt, N, X, XS) → ACTIVATE(XS)
U61¹(tt, N, X, XS) → U62¹(tt, activate(N), activate(X), activate(XS))
U82¹(tt, N, XS) → ACTIVATE(N)
U21¹(tt, X) → U22¹(tt, activate(X))
U82¹(tt, N, XS) → SPLITAT(activate(N), activate(XS))
U12¹(tt, N, XS) → ACTIVATE(XS)
U72¹(tt, XS) → ACTIVATE(XS)
SND(pair(X, Y)) → U51¹(tt, Y)
U81¹(tt, N, XS) → ACTIVATE(XS)
ACTIVATE(n__s(X)) → S(activate(X))
U42¹(tt, N, XS) → ACTIVATE(N)
U62¹(tt, N, X, XS) → ACTIVATE(XS)
U11¹(tt, N, XS) → ACTIVATE(N)
U63¹(tt, N, X, XS) → ACTIVATE(X)
U52¹(tt, Y) → ACTIVATE(Y)
TAKE(N, XS) → U81¹(tt, N, XS)
U63¹(tt, N, X, XS) → U64¹(splitAt(activate(N), activate(XS)), activate(X))
SEL(N, XS) → U41¹(tt, N, XS)
U64¹(pair(YS, ZS), X) → ACTIVATE(X)
FST(pair(X, Y)) → U21¹(tt, X)
U61¹(tt, N, X, XS) → ACTIVATE(X)
SPLITAT(s(N), cons(X, XS)) → ACTIVATE(XS)
U51¹(tt, Y) → ACTIVATE(Y)
HEAD(cons(N, XS)) → U31¹(tt, N)
U11¹(tt, N, XS) → U12¹(tt, activate(N), activate(XS))
U31¹(tt, N) → ACTIVATE(N)
U81¹(tt, N, XS) → U82¹(tt, activate(N), activate(XS))
U62¹(tt, N, X, XS) → ACTIVATE(N)
U82¹(tt, N, XS) → FST(splitAt(activate(N), activate(XS)))
U41¹(tt, N, XS) → ACTIVATE(XS)
U81¹(tt, N, XS) → ACTIVATE(N)
U12¹(tt, N, XS) → ACTIVATE(N)
TAIL(cons(N, XS)) → ACTIVATE(XS)
ACTIVATE(n__natsFrom(X)) → NATSFROM(activate(X))
U31¹(tt, N) → U32¹(tt, activate(N))
U12¹(tt, N, XS) → SND(splitAt(activate(N), activate(XS)))
U82¹(tt, N, XS) → ACTIVATE(XS)
U41¹(tt, N, XS) → ACTIVATE(N)
U71¹(tt, XS) → U72¹(tt, activate(XS))
U22¹(tt, X) → ACTIVATE(X)
U61¹(tt, N, X, XS) → ACTIVATE(N)
U51¹(tt, Y) → U52¹(tt, activate(Y))
U42¹(tt, N, XS) → HEAD(afterNth(activate(N), activate(XS)))
U21¹(tt, X) → ACTIVATE(X)
ACTIVATE(n__natsFrom(X)) → ACTIVATE(X)
U12¹(tt, N, XS) → SPLITAT(activate(N), activate(XS))
U71¹(tt, XS) → ACTIVATE(XS)
U32¹(tt, N) → ACTIVATE(N)
U63¹(tt, N, X, XS) → ACTIVATE(N)
U62¹(tt, N, X, XS) → ACTIVATE(X)
U42¹(tt, N, XS) → ACTIVATE(XS)
SPLITAT(s(N), cons(X, XS)) → U61¹(tt, N, X, activate(XS))
U41¹(tt, N, XS) → U42¹(tt, activate(N), activate(XS))
U62¹(tt, N, X, XS) → U63¹(tt, activate(N), activate(X), activate(XS))

The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 55 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__natsFrom(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVATE(n__natsFrom(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → ACTIVATE(X)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ACTIVATE(n__natsFrom(X)) → ACTIVATE(X)
The graph contains the following edges 1 > 1
ACTIVATE(n__s(X)) → ACTIVATE(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

U63¹(tt, N, X, XS) → SPLITAT(activate(N), activate(XS))
U61¹(tt, N, X, XS) → U62¹(tt, activate(N), activate(X), activate(XS))
SPLITAT(s(N), cons(X, XS)) → U61¹(tt, N, X, activate(XS))
U62¹(tt, N, X, XS) → U63¹(tt, activate(N), activate(X), activate(XS))

The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

SPLITAT(s(N), cons(X, XS)) → U61¹(tt, N, X, activate(XS))

The remaining pairs can at least be oriented weakly.

U63¹(tt, N, X, XS) → SPLITAT(activate(N), activate(XS))
U61¹(tt, N, X, XS) → U62¹(tt, activate(N), activate(X), activate(XS))
U62¹(tt, N, X, XS) → U63¹(tt, activate(N), activate(X), activate(XS))

Used ordering: Polynomial interpretation [25]:

POL(SPLITAT(x₁, x₂)) = x₁
POL(U61¹(x₁, x₂, x₃, x₄)) = x₂
POL(U62¹(x₁, x₂, x₃, x₄)) = x₂
POL(U63¹(x₁, x₂, x₃, x₄)) = x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = 0
POL(n__natsFrom(x₁)) = 0
POL(n__s(x₁)) = 1 + x₁
POL(natsFrom(x₁)) = 0
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

The following usable rules [17] were oriented:

activate(X) → X
activate(n__s(X)) → s(activate(X))
activate(n__natsFrom(X)) → natsFrom(activate(X))
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
s(X) → n__s(X)
natsFrom(X) → n__natsFrom(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

U63¹(tt, N, X, XS) → SPLITAT(activate(N), activate(XS))
U62¹(tt, N, X, XS) → U63¹(tt, activate(N), activate(X), activate(XS))
U61¹(tt, N, X, XS) → U62¹(tt, activate(N), activate(X), activate(XS))

The TRS R consists of the following rules:

U11(tt, N, XS) → U12(tt, activate(N), activate(XS))
U12(tt, N, XS) → snd(splitAt(activate(N), activate(XS)))
U21(tt, X) → U22(tt, activate(X))
U22(tt, X) → activate(X)
U31(tt, N) → U32(tt, activate(N))
U32(tt, N) → activate(N)
U41(tt, N, XS) → U42(tt, activate(N), activate(XS))
U42(tt, N, XS) → head(afterNth(activate(N), activate(XS)))
U51(tt, Y) → U52(tt, activate(Y))
U52(tt, Y) → activate(Y)
U61(tt, N, X, XS) → U62(tt, activate(N), activate(X), activate(XS))
U62(tt, N, X, XS) → U63(tt, activate(N), activate(X), activate(XS))
U63(tt, N, X, XS) → U64(splitAt(activate(N), activate(XS)), activate(X))
U64(pair(YS, ZS), X) → pair(cons(activate(X), YS), ZS)
U71(tt, XS) → U72(tt, activate(XS))
U72(tt, XS) → activate(XS)
U81(tt, N, XS) → U82(tt, activate(N), activate(XS))
U82(tt, N, XS) → fst(splitAt(activate(N), activate(XS)))
afterNth(N, XS) → U11(tt, N, XS)
fst(pair(X, Y)) → U21(tt, X)
head(cons(N, XS)) → U31(tt, N)
natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
sel(N, XS) → U41(tt, N, XS)
snd(pair(X, Y)) → U51(tt, Y)
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → U61(tt, N, X, activate(XS))
tail(cons(N, XS)) → U71(tt, activate(XS))
take(N, XS) → U81(tt, N, XS)
natsFrom(X) → n__natsFrom(X)
s(X) → n__s(X)
activate(n__natsFrom(X)) → natsFrom(activate(X))
activate(n__s(X)) → s(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.