Termination w.r.t. Q proof of /tmp/tmpP4IRB0/ExSec4_2_DLMMU04

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

    ↳6 QDPOrderProof (⇔)

    ↳7 QDP

    ↳8 PisEmptyProof (⇔)

    ↳9 TRUE

  ↳10 QDP

    ↳11 QDPOrderProof (⇔)

    ↳12 QDP

    ↳13 PisEmptyProof (⇔)

    ↳14 TRUE

(0) Obligation:

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

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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.

(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:

SPLITAT(s(N), cons(X, XS)) → U(splitAt(N, activate(XS)), N, X, activate(XS))
SPLITAT(s(N), cons(X, XS)) → SPLITAT(N, activate(XS))
SPLITAT(s(N), cons(X, XS)) → ACTIVATE(XS)
U(pair(YS, ZS), N, X, XS) → ACTIVATE(X)
TAIL(cons(N, XS)) → ACTIVATE(XS)
SEL(N, XS) → HEAD(afterNth(N, XS))
SEL(N, XS) → AFTERNTH(N, XS)
TAKE(N, XS) → FST(splitAt(N, XS))
TAKE(N, XS) → SPLITAT(N, XS)
AFTERNTH(N, XS) → SND(splitAt(N, XS))
AFTERNTH(N, XS) → SPLITAT(N, XS)
ACTIVATE(n__natsFrom(X)) → NATSFROM(activate(X))
ACTIVATE(n__natsFrom(X)) → ACTIVATE(X)
ACTIVATE(n__s(X)) → S(activate(X))
ACTIVATE(n__s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 12 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = ACTIVATE(x1)
n__s(x1) = n__s(x1)
n__natsFrom(x1) = n__natsFrom(x1)
natsFrom(x1) = natsFrom(x1)
cons(x1, x2) = cons(x1, x2)
fst(x1) = x1
pair(x1, x2) = pair(x1, x2)
snd(x1) = x1
splitAt(x1, x2) = splitAt(x1, x2)
0 = 0
nil = nil
s(x1) = s(x1)
u(x1, x2, x3, x4) = u(x1, x2, x3, x4)
activate(x1) = activate(x1)
head(x1) = x1
tail(x1) = tail(x1)
sel(x1, x2) = sel(x1, x2)
afterNth(x1, x2) = afterNth(x1, x2)
take(x1, x2) = take(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

0 > [cons₂, pair₂]
0 > nil
sel₂ > [splitAt₂, u₄, afterNth₂] > nil
sel₂ > [splitAt₂, u₄, afterNth₂] > [activate₁, tail₁] > natsFrom₁ > [ACTIVATE₁, n_{_s₁}, n_{_natsFrom₁}, s₁]
sel₂ > [splitAt₂, u₄, afterNth₂] > [activate₁, tail₁] > natsFrom₁ > [cons₂, pair₂]
take₂ > [splitAt₂, u₄, afterNth₂] > nil
take₂ > [splitAt₂, u₄, afterNth₂] > [activate₁, tail₁] > natsFrom₁ > [ACTIVATE₁, n_{_s₁}, n_{_natsFrom₁}, s₁]
take₂ > [splitAt₂, u₄, afterNth₂] > [activate₁, tail₁] > natsFrom₁ > [cons₂, pair₂]

Status:

ACTIVATE₁: multiset
n_{_s₁}: multiset
n_{_natsFrom₁}: multiset
natsFrom₁: multiset
cons₂: multiset
pair₂: multiset
splitAt₂: [1,2]
0: multiset
nil: multiset
s₁: multiset
u₄: [2,1,4,3]
activate₁: multiset
tail₁: multiset
sel₂: multiset
afterNth₂: [1,2]
take₂: multiset

The following usable rules [FROCOS05] were oriented:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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

(7) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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.

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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

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

The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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.

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SPLITAT(x1, x2) = SPLITAT(x1, x2)
s(x1) = s(x1)
cons(x1, x2) = cons(x1, x2)
activate(x1) = activate(x1)
natsFrom(x1) = natsFrom(x1)
n__natsFrom(x1) = n__natsFrom(x1)
n__s(x1) = n__s(x1)
fst(x1) = x1
pair(x1, x2) = pair(x1, x2)
snd(x1) = x1
splitAt(x1, x2) = splitAt(x1, x2)
0 = 0
nil = nil
u(x1, x2, x3, x4) = u(x1, x3)
head(x1) = x1
tail(x1) = tail(x1)
sel(x1, x2) = sel(x1, x2)
afterNth(x1, x2) = afterNth(x1, x2)
take(x1, x2) = take(x1, x2)

Recursive path order with status [RPO].
Quasi-Precedence:

SPLITAT₂ > activate₁ > natsFrom₁ > [s₁, n_{_s₁}]
SPLITAT₂ > activate₁ > natsFrom₁ > cons₂
SPLITAT₂ > activate₁ > natsFrom₁ > n_{_natsFrom₁}
0 > pair₂ > activate₁ > natsFrom₁ > [s₁, n_{_s₁}]
0 > pair₂ > activate₁ > natsFrom₁ > cons₂
0 > pair₂ > activate₁ > natsFrom₁ > n_{_natsFrom₁}
0 > nil
tail₁ > activate₁ > natsFrom₁ > [s₁, n_{_s₁}]
tail₁ > activate₁ > natsFrom₁ > cons₂
tail₁ > activate₁ > natsFrom₁ > n_{_natsFrom₁}
[sel₂, afterNth₂] > splitAt₂ > nil
[sel₂, afterNth₂] > splitAt₂ > u₂ > pair₂ > activate₁ > natsFrom₁ > [s₁, n_{_s₁}]
[sel₂, afterNth₂] > splitAt₂ > u₂ > pair₂ > activate₁ > natsFrom₁ > cons₂
[sel₂, afterNth₂] > splitAt₂ > u₂ > pair₂ > activate₁ > natsFrom₁ > n_{_natsFrom₁}
take₂ > splitAt₂ > nil
take₂ > splitAt₂ > u₂ > pair₂ > activate₁ > natsFrom₁ > [s₁, n_{_s₁}]
take₂ > splitAt₂ > u₂ > pair₂ > activate₁ > natsFrom₁ > cons₂
take₂ > splitAt₂ > u₂ > pair₂ > activate₁ > natsFrom₁ > n_{_natsFrom₁}

Status:

SPLITAT₂: [1,2]
s₁: multiset
cons₂: [1,2]
activate₁: multiset
natsFrom₁: multiset
n_{_natsFrom₁}: multiset
n_{_s₁}: multiset
pair₂: multiset
splitAt₂: [1,2]
0: multiset
nil: multiset
u₂: multiset
tail₁: multiset
sel₂: multiset
afterNth₂: multiset
take₂: [1,2]

The following usable rules [FROCOS05] were oriented:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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

(12) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

natsFrom(N) → cons(N, n__natsFrom(n__s(N)))
fst(pair(XS, YS)) → XS
snd(pair(XS, YS)) → YS
splitAt(0, XS) → pair(nil, XS)
splitAt(s(N), cons(X, XS)) → u(splitAt(N, activate(XS)), N, X, activate(XS))
u(pair(YS, ZS), N, X, XS) → pair(cons(activate(X), YS), ZS)
head(cons(N, XS)) → N
tail(cons(N, XS)) → activate(XS)
sel(N, XS) → head(afterNth(N, XS))
take(N, XS) → fst(splitAt(N, XS))
afterNth(N, XS) → snd(splitAt(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.

(13) 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) Complex Obligation (AND)

(5) Obligation:

(6) QDPOrderProof (EQUIVALENT transformation)

(7) Obligation:

(8) PisEmptyProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) PisEmptyProof (EQUIVALENT transformation)

(14) TRUE