Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

terms1(N) -> cons2(recip1(sqr1(N)), n__terms1(n__s1(N)))
sqr1(0) -> 0
sqr1(s1(X)) -> s1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
dbl1(0) -> 0
dbl1(s1(X)) -> s1(n__s1(n__dbl1(activate1(X))))
add2(0, X) -> X
add2(s1(X), Y) -> s1(n__add2(activate1(X), Y))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(activate1(X), activate1(Z)))
terms1(X) -> n__terms1(X)
s1(X) -> n__s1(X)
add2(X1, X2) -> n__add2(X1, X2)
sqr1(X) -> n__sqr1(X)
dbl1(X) -> n__dbl1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__terms1(X)) -> terms1(activate1(X))
activate1(n__s1(X)) -> s1(X)
activate1(n__add2(X1, X2)) -> add2(activate1(X1), activate1(X2))
activate1(n__sqr1(X)) -> sqr1(activate1(X))
activate1(n__dbl1(X)) -> dbl1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
activate1(X) -> X

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

terms1(N) -> cons2(recip1(sqr1(N)), n__terms1(n__s1(N)))
sqr1(0) -> 0
sqr1(s1(X)) -> s1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
dbl1(0) -> 0
dbl1(s1(X)) -> s1(n__s1(n__dbl1(activate1(X))))
add2(0, X) -> X
add2(s1(X), Y) -> s1(n__add2(activate1(X), Y))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(activate1(X), activate1(Z)))
terms1(X) -> n__terms1(X)
s1(X) -> n__s1(X)
add2(X1, X2) -> n__add2(X1, X2)
sqr1(X) -> n__sqr1(X)
dbl1(X) -> n__dbl1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__terms1(X)) -> terms1(activate1(X))
activate1(n__s1(X)) -> s1(X)
activate1(n__add2(X1, X2)) -> add2(activate1(X1), activate1(X2))
activate1(n__sqr1(X)) -> sqr1(activate1(X))
activate1(n__dbl1(X)) -> dbl1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
activate1(X) -> X

Q is empty.

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

ACTIVATE1(n__terms1(X)) -> TERMS1(activate1(X))
SQR1(s1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__sqr1(X)) -> SQR1(activate1(X))
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ACTIVATE1(n__add2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__s1(X)) -> S1(X)
TERMS1(N) -> SQR1(N)
ACTIVATE1(n__sqr1(X)) -> ACTIVATE1(X)
DBL1(s1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__dbl1(X)) -> DBL1(activate1(X))
ACTIVATE1(n__add2(X1, X2)) -> ADD2(activate1(X1), activate1(X2))
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X2)
SQR1(s1(X)) -> S1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
ACTIVATE1(n__dbl1(X)) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> ACTIVATE1(X)
ACTIVATE1(n__terms1(X)) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ADD2(s1(X), Y) -> S1(n__add2(activate1(X), Y))
ACTIVATE1(n__add2(X1, X2)) -> ACTIVATE1(X1)
DBL1(s1(X)) -> S1(n__s1(n__dbl1(activate1(X))))

The TRS R consists of the following rules:

terms1(N) -> cons2(recip1(sqr1(N)), n__terms1(n__s1(N)))
sqr1(0) -> 0
sqr1(s1(X)) -> s1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
dbl1(0) -> 0
dbl1(s1(X)) -> s1(n__s1(n__dbl1(activate1(X))))
add2(0, X) -> X
add2(s1(X), Y) -> s1(n__add2(activate1(X), Y))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(activate1(X), activate1(Z)))
terms1(X) -> n__terms1(X)
s1(X) -> n__s1(X)
add2(X1, X2) -> n__add2(X1, X2)
sqr1(X) -> n__sqr1(X)
dbl1(X) -> n__dbl1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__terms1(X)) -> terms1(activate1(X))
activate1(n__s1(X)) -> s1(X)
activate1(n__add2(X1, X2)) -> add2(activate1(X1), activate1(X2))
activate1(n__sqr1(X)) -> sqr1(activate1(X))
activate1(n__dbl1(X)) -> dbl1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
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__terms1(X)) -> TERMS1(activate1(X))
SQR1(s1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__sqr1(X)) -> SQR1(activate1(X))
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ACTIVATE1(n__add2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__s1(X)) -> S1(X)
TERMS1(N) -> SQR1(N)
ACTIVATE1(n__sqr1(X)) -> ACTIVATE1(X)
DBL1(s1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__dbl1(X)) -> DBL1(activate1(X))
ACTIVATE1(n__add2(X1, X2)) -> ADD2(activate1(X1), activate1(X2))
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X2)
SQR1(s1(X)) -> S1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
ACTIVATE1(n__dbl1(X)) -> ACTIVATE1(X)
ADD2(s1(X), Y) -> ACTIVATE1(X)
ACTIVATE1(n__terms1(X)) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ADD2(s1(X), Y) -> S1(n__add2(activate1(X), Y))
ACTIVATE1(n__add2(X1, X2)) -> ACTIVATE1(X1)
DBL1(s1(X)) -> S1(n__s1(n__dbl1(activate1(X))))

The TRS R consists of the following rules:

terms1(N) -> cons2(recip1(sqr1(N)), n__terms1(n__s1(N)))
sqr1(0) -> 0
sqr1(s1(X)) -> s1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
dbl1(0) -> 0
dbl1(s1(X)) -> s1(n__s1(n__dbl1(activate1(X))))
add2(0, X) -> X
add2(s1(X), Y) -> s1(n__add2(activate1(X), Y))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(activate1(X), activate1(Z)))
terms1(X) -> n__terms1(X)
s1(X) -> n__s1(X)
add2(X1, X2) -> n__add2(X1, X2)
sqr1(X) -> n__sqr1(X)
dbl1(X) -> n__dbl1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__terms1(X)) -> terms1(activate1(X))
activate1(n__s1(X)) -> s1(X)
activate1(n__add2(X1, X2)) -> add2(activate1(X1), activate1(X2))
activate1(n__sqr1(X)) -> sqr1(activate1(X))
activate1(n__dbl1(X)) -> dbl1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 4 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
QDP

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

ACTIVATE1(n__terms1(X)) -> TERMS1(activate1(X))
SQR1(s1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X1)
ACTIVATE1(n__sqr1(X)) -> SQR1(activate1(X))
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(X)
ACTIVATE1(n__add2(X1, X2)) -> ACTIVATE1(X2)
TERMS1(N) -> SQR1(N)
ACTIVATE1(n__sqr1(X)) -> ACTIVATE1(X)
DBL1(s1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__dbl1(X)) -> DBL1(activate1(X))
ACTIVATE1(n__add2(X1, X2)) -> ADD2(activate1(X1), activate1(X2))
ACTIVATE1(n__first2(X1, X2)) -> ACTIVATE1(X2)
ACTIVATE1(n__first2(X1, X2)) -> FIRST2(activate1(X1), activate1(X2))
ADD2(s1(X), Y) -> ACTIVATE1(X)
ACTIVATE1(n__dbl1(X)) -> ACTIVATE1(X)
ACTIVATE1(n__terms1(X)) -> ACTIVATE1(X)
FIRST2(s1(X), cons2(Y, Z)) -> ACTIVATE1(Z)
ACTIVATE1(n__add2(X1, X2)) -> ACTIVATE1(X1)

The TRS R consists of the following rules:

terms1(N) -> cons2(recip1(sqr1(N)), n__terms1(n__s1(N)))
sqr1(0) -> 0
sqr1(s1(X)) -> s1(n__add2(n__sqr1(activate1(X)), n__dbl1(activate1(X))))
dbl1(0) -> 0
dbl1(s1(X)) -> s1(n__s1(n__dbl1(activate1(X))))
add2(0, X) -> X
add2(s1(X), Y) -> s1(n__add2(activate1(X), Y))
first2(0, X) -> nil
first2(s1(X), cons2(Y, Z)) -> cons2(Y, n__first2(activate1(X), activate1(Z)))
terms1(X) -> n__terms1(X)
s1(X) -> n__s1(X)
add2(X1, X2) -> n__add2(X1, X2)
sqr1(X) -> n__sqr1(X)
dbl1(X) -> n__dbl1(X)
first2(X1, X2) -> n__first2(X1, X2)
activate1(n__terms1(X)) -> terms1(activate1(X))
activate1(n__s1(X)) -> s1(X)
activate1(n__add2(X1, X2)) -> add2(activate1(X1), activate1(X2))
activate1(n__sqr1(X)) -> sqr1(activate1(X))
activate1(n__dbl1(X)) -> dbl1(activate1(X))
activate1(n__first2(X1, X2)) -> first2(activate1(X1), activate1(X2))
activate1(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.