Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar14/TRCSRSLASHEx2_Luc03b

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:

fst₂(0, Z) -> nil
fst₂(s₁(X), cons₂(Y, Z)) -> cons₂(Y, n__fst₂(activate₁(X), activate₁(Z)))
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(n__add₂(activate₁(X), Y))
len₁(nil) -> 0
len₁(cons₂(X, Z)) -> s₁(n__len₁(activate₁(Z)))
fst₂(X1, X2) -> n__fst₂(X1, X2)
from₁(X) -> n__from₁(X)
add₂(X1, X2) -> n__add₂(X1, X2)
len₁(X) -> n__len₁(X)
activate₁(n__fst₂(X1, X2)) -> fst₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(n__add₂(X1, X2)) -> add₂(X1, X2)
activate₁(n__len₁(X)) -> len₁(X)
activate₁(X) -> X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

fst₂(0, Z) -> nil
fst₂(s₁(X), cons₂(Y, Z)) -> cons₂(Y, n__fst₂(activate₁(X), activate₁(Z)))
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(n__add₂(activate₁(X), Y))
len₁(nil) -> 0
len₁(cons₂(X, Z)) -> s₁(n__len₁(activate₁(Z)))
fst₂(X1, X2) -> n__fst₂(X1, X2)
from₁(X) -> n__from₁(X)
add₂(X1, X2) -> n__add₂(X1, X2)
len₁(X) -> n__len₁(X)
activate₁(n__fst₂(X1, X2)) -> fst₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(n__add₂(X1, X2)) -> add₂(X1, X2)
activate₁(n__len₁(X)) -> len₁(X)
activate₁(X) -> X

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVATE₁(n__fst₂(X1, X2)) -> FST₂(X1, X2)
FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(X)
ACTIVATE₁(n__len₁(X)) -> LEN₁(X)
ADD₂(s₁(X), Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__add₂(X1, X2)) -> ADD₂(X1, X2)
ACTIVATE₁(n__from₁(X)) -> FROM₁(X)
LEN₁(cons₂(X, Z)) -> ACTIVATE₁(Z)
FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(Z)

The TRS R consists of the following rules:

fst₂(0, Z) -> nil
fst₂(s₁(X), cons₂(Y, Z)) -> cons₂(Y, n__fst₂(activate₁(X), activate₁(Z)))
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(n__add₂(activate₁(X), Y))
len₁(nil) -> 0
len₁(cons₂(X, Z)) -> s₁(n__len₁(activate₁(Z)))
fst₂(X1, X2) -> n__fst₂(X1, X2)
from₁(X) -> n__from₁(X)
add₂(X1, X2) -> n__add₂(X1, X2)
len₁(X) -> n__len₁(X)
activate₁(n__fst₂(X1, X2)) -> fst₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(n__add₂(X1, X2)) -> add₂(X1, X2)
activate₁(n__len₁(X)) -> len₁(X)
activate₁(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:

ACTIVATE₁(n__fst₂(X1, X2)) -> FST₂(X1, X2)
FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(X)
ACTIVATE₁(n__len₁(X)) -> LEN₁(X)
ADD₂(s₁(X), Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__add₂(X1, X2)) -> ADD₂(X1, X2)
ACTIVATE₁(n__from₁(X)) -> FROM₁(X)
LEN₁(cons₂(X, Z)) -> ACTIVATE₁(Z)
FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(Z)

The TRS R consists of the following rules:

fst₂(0, Z) -> nil
fst₂(s₁(X), cons₂(Y, Z)) -> cons₂(Y, n__fst₂(activate₁(X), activate₁(Z)))
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(n__add₂(activate₁(X), Y))
len₁(nil) -> 0
len₁(cons₂(X, Z)) -> s₁(n__len₁(activate₁(Z)))
fst₂(X1, X2) -> n__fst₂(X1, X2)
from₁(X) -> n__from₁(X)
add₂(X1, X2) -> n__add₂(X1, X2)
len₁(X) -> n__len₁(X)
activate₁(n__fst₂(X1, X2)) -> fst₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(n__add₂(X1, X2)) -> add₂(X1, X2)
activate₁(n__len₁(X)) -> len₁(X)
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(X)
ACTIVATE₁(n__fst₂(X1, X2)) -> FST₂(X1, X2)
ACTIVATE₁(n__len₁(X)) -> LEN₁(X)
ADD₂(s₁(X), Y) -> ACTIVATE₁(X)
ACTIVATE₁(n__add₂(X1, X2)) -> ADD₂(X1, X2)
FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(Z)
LEN₁(cons₂(X, Z)) -> ACTIVATE₁(Z)

The TRS R consists of the following rules:

fst₂(0, Z) -> nil
fst₂(s₁(X), cons₂(Y, Z)) -> cons₂(Y, n__fst₂(activate₁(X), activate₁(Z)))
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(n__add₂(activate₁(X), Y))
len₁(nil) -> 0
len₁(cons₂(X, Z)) -> s₁(n__len₁(activate₁(Z)))
fst₂(X1, X2) -> n__fst₂(X1, X2)
from₁(X) -> n__from₁(X)
add₂(X1, X2) -> n__add₂(X1, X2)
len₁(X) -> n__len₁(X)
activate₁(n__fst₂(X1, X2)) -> fst₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(n__add₂(X1, X2)) -> add₂(X1, X2)
activate₁(n__len₁(X)) -> len₁(X)
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be strictly oriented and are deleted.

FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(X)
ACTIVATE₁(n__fst₂(X1, X2)) -> FST₂(X1, X2)
ACTIVATE₁(n__len₁(X)) -> LEN₁(X)
ACTIVATE₁(n__add₂(X1, X2)) -> ADD₂(X1, X2)
FST₂(s₁(X), cons₂(Y, Z)) -> ACTIVATE₁(Z)
LEN₁(cons₂(X, Z)) -> ACTIVATE₁(Z)

The remaining pairs can at least by weakly be oriented.

ADD₂(s₁(X), Y) -> ACTIVATE₁(X)

Used ordering: Combined order from the following AFS and order.
FST₂(x1, x2) = FST₂(x1, x2)
s₁(x1) = s₁(x1)
cons₂(x1, x2) = cons₁(x2)
ACTIVATE₁(x1) = ACTIVATE₁(x1)
n__fst₂(x1, x2) = n__fst₂(x1, x2)
n__len₁(x1) = x1
LEN₁(x1) = LEN₁(x1)
ADD₂(x1, x2) = x1
n__add₂(x1, x2) = x1

Lexicographic Path Order [19].
Precedence:

[FST₂, s₁, cons₁, ACTIVATE_1] > LEN₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

ADD₂(s₁(X), Y) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

fst₂(0, Z) -> nil
fst₂(s₁(X), cons₂(Y, Z)) -> cons₂(Y, n__fst₂(activate₁(X), activate₁(Z)))
from₁(X) -> cons₂(X, n__from₁(s₁(X)))
add₂(0, X) -> X
add₂(s₁(X), Y) -> s₁(n__add₂(activate₁(X), Y))
len₁(nil) -> 0
len₁(cons₂(X, Z)) -> s₁(n__len₁(activate₁(Z)))
fst₂(X1, X2) -> n__fst₂(X1, X2)
from₁(X) -> n__from₁(X)
add₂(X1, X2) -> n__add₂(X1, X2)
len₁(X) -> n__len₁(X)
activate₁(n__fst₂(X1, X2)) -> fst₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(X)
activate₁(n__add₂(X1, X2)) -> add₂(X1, X2)
activate₁(n__len₁(X)) -> len₁(X)
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 0 SCCs with 1 less node.