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(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)
activate(n__natsFrom(X)) → natsFrom(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(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)
activate(n__natsFrom(X)) → natsFrom(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)
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__natsFrom(X)) → NATSFROM(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)
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)
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(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)
activate(n__natsFrom(X)) → natsFrom(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)
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__natsFrom(X)) → NATSFROM(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)
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)
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(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)
activate(n__natsFrom(X)) → natsFrom(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 1 SCC with 54 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing

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(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U631(tt, N, X, XS) → SPLITAT(activate(N), activate(XS)) at position [0] we obtained the following new rules:

U631(tt, x0, y1, y2) → SPLITAT(x0, activate(y2))
U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(natsFrom(x0), activate(y2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing

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

U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(natsFrom(x0), activate(y2))
U631(tt, x0, y1, y2) → SPLITAT(x0, activate(y2))
U621(tt, N, X, XS) → U631(tt, activate(N), activate(X), activate(XS))
SPLITAT(s(N), cons(X, XS)) → U611(tt, N, 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(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U631(tt, x0, y1, y2) → SPLITAT(x0, activate(y2)) at position [1] we obtained the following new rules:

U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, natsFrom(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing

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

U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(natsFrom(x0), activate(y2))
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, natsFrom(x0))
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(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(natsFrom(x0), activate(y2)) at position [0] we obtained the following new rules:

U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(cons(x0, n__natsFrom(s(x0))), activate(y2))
U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(n__natsFrom(x0), activate(y2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof

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

U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(cons(x0, n__natsFrom(s(x0))), activate(y2))
U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, natsFrom(x0))
U631(tt, n__natsFrom(x0), y1, y2) → SPLITAT(n__natsFrom(x0), activate(y2))
U621(tt, N, X, XS) → U631(tt, activate(N), activate(X), activate(XS))
SPLITAT(s(N), cons(X, XS)) → U611(tt, N, 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(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)
activate(n__natsFrom(X)) → natsFrom(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 1 SCC with 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing

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

U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, natsFrom(x0))
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))
SPLITAT(s(N), cons(X, XS)) → U611(tt, N, 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(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)
activate(n__natsFrom(X)) → natsFrom(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, natsFrom(x0)) at position [1] we obtained the following new rules:

U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, n__natsFrom(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof

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

U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, n__natsFrom(x0))
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))
SPLITAT(s(N), cons(X, XS)) → U611(tt, N, X, 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))

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(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)
activate(n__natsFrom(X)) → natsFrom(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 1 SCC with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ QDPOrderProof

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

U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))
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))
SPLITAT(s(N), cons(X, XS)) → U611(tt, N, 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(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)
activate(n__natsFrom(X)) → natsFrom(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, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))
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))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\00/
·x1+
/00\
\00/
·x2

M( tt ) =
/0\
\0/

M( n__natsFrom(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( s(x1) ) =
/0\
\1/
+
/00\
\11/
·x1

M( activate(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( natsFrom(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

Tuple symbols:
M( U631(x1, ..., x4) ) = 0+
[0,0]
·x1+
[0,1]
·x2+
[0,0]
·x3+
[0,0]
·x4

M( U611(x1, ..., x4) ) = 0+
[0,0]
·x1+
[1,1]
·x2+
[0,0]
·x3+
[0,0]
·x4

M( SPLITAT(x1, x2) ) = 0+
[0,1]
·x1+
[0,0]
·x2

M( U621(x1, ..., x4) ) = 0+
[0,0]
·x1+
[0,1]
·x2+
[0,0]
·x3+
[0,0]
·x4


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ QDPOrderProof
                                    ↳ QDP
                                      ↳ DependencyGraphProof

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

U631(tt, y0, y1, x0) → SPLITAT(y0, x0)
U631(tt, y0, y1, n__natsFrom(x0)) → SPLITAT(y0, cons(x0, n__natsFrom(s(x0))))
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))

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(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)
activate(n__natsFrom(X)) → natsFrom(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 4 less nodes.