Termination w.r.t. Q proof of /tmp/tmpl9yU7d/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(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: 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)) = 1 + x₁
POL(natsFrom(x₁)) = 1 + x₁
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

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.