Termination w.r.t. Q proof of /tmp/tmp09r4Su/LISTUTILITIES_nosorts-noand

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 PisEmptyProof (⇔)

↳8 TRUE

(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 54 less nodes.

(4) Obligation:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U62¹(tt, N, X, XS) → U63¹(tt, activate(N), activate(X), activate(XS))
U63¹(tt, N, X, XS) → SPLITAT(activate(N), 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 remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
U62¹(x1, x2, x3, x4) = U62¹(x2)
U63¹(x1, x2, x3, x4) = U63¹(x2)
activate(x1) = activate(x1)
SPLITAT(x1, x2) = x1
s(x1) = s(x1)
U61¹(x1, x2, x3, x4) = U61¹(x2)
n__natsFrom(x1) = n__natsFrom
natsFrom(x1) = natsFrom
cons(x1, x2) = cons

Lexicographic Path Order [LPO].
Precedence:

s₁ > U61^1₁ > U62^1₁ > U63^1₁ > activate₁ > natsFrom > n_{_natsFrom}
s₁ > U61^1₁ > U62^1₁ > U63^1₁ > activate₁ > natsFrom > cons

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

Q DP problem:
P is empty.
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.

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) PisEmptyProof (EQUIVALENT transformation)

(8) TRUE