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:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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:

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(X)) → S(n__add(sqr(activate(X)), dbl(activate(X))))
SQR(s(X)) → DBL(activate(X))
ACTIVATE(n__terms(X)) → TERMS(X)
ACTIVATE(n__s(X)) → S(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → SQR(activate(X))
DBL(s(X)) → S(n__s(n__dbl(activate(X))))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → S(n__add(activate(X), Y))
ADD(s(X), Y) → ACTIVATE(X)
TERMS(N) → S(N)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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:

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(X)) → S(n__add(sqr(activate(X)), dbl(activate(X))))
SQR(s(X)) → DBL(activate(X))
ACTIVATE(n__terms(X)) → TERMS(X)
ACTIVATE(n__s(X)) → S(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → SQR(activate(X))
DBL(s(X)) → S(n__s(n__dbl(activate(X))))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → S(n__add(activate(X), Y))
ADD(s(X), Y) → ACTIVATE(X)
TERMS(N) → S(N)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing

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

ACTIVATE(n__dbl(X)) → DBL(X)
DBL(s(X)) → ACTIVATE(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(X)) → DBL(activate(X))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → ACTIVATE(X)
SQR(s(X)) → SQR(activate(X))
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SQR(s(X)) → SQR(activate(X)) at position [0] we obtained the following new rules:

SQR(s(x0)) → SQR(x0)
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__s(x0))) → SQR(s(x0))
SQR(s(n__terms(x0))) → SQR(terms(x0))
SQR(s(n__first(x0, x1))) → SQR(first(x0, x1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing

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

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(X)) → DBL(activate(X))
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__terms(x0))) → SQR(terms(x0))
SQR(s(n__s(x0))) → SQR(s(x0))
SQR(s(n__first(x0, x1))) → SQR(first(x0, x1))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
SQR(s(x0)) → SQR(x0)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SQR(s(X)) → DBL(activate(X)) at position [0] we obtained the following new rules:

SQR(s(n__first(x0, x1))) → DBL(first(x0, x1))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(x0)) → DBL(x0)
SQR(s(n__terms(x0))) → DBL(terms(x0))
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing

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

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → ACTIVATE(X)
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__first(x0, x1))) → DBL(first(x0, x1))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
SQR(s(n__terms(x0))) → SQR(terms(x0))
SQR(s(n__first(x0, x1))) → SQR(first(x0, x1))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
SQR(s(x0)) → SQR(x0)
SQR(s(x0)) → DBL(x0)
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SQR(s(n__terms(x0))) → SQR(terms(x0)) at position [0] we obtained the following new rules:

SQR(s(n__terms(x0))) → SQR(cons(recip(sqr(x0)), n__terms(s(x0))))
SQR(s(n__terms(x0))) → SQR(n__terms(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof

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

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
SQR(s(n__terms(x0))) → SQR(n__terms(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__first(x0, x1))) → DBL(first(x0, x1))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
SQR(s(n__first(x0, x1))) → SQR(first(x0, x1))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
SQR(s(x0)) → SQR(x0)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
SQR(s(n__terms(x0))) → SQR(cons(recip(sqr(x0)), n__terms(s(x0))))
ADD(s(X), Y) → ACTIVATE(X)
SQR(s(x0)) → DBL(x0)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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:

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__first(x0, x1))) → DBL(first(x0, x1))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
SQR(s(n__first(x0, x1))) → SQR(first(x0, x1))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
SQR(s(x0)) → SQR(x0)
SQR(s(x0)) → DBL(x0)
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SQR(s(n__first(x0, x1))) → SQR(first(x0, x1)) at position [0] we obtained the following new rules:

SQR(s(n__first(x0, x1))) → SQR(n__first(x0, x1))
SQR(s(n__first(s(x0), cons(x1, x2)))) → SQR(cons(x1, n__first(activate(x0), activate(x2))))
SQR(s(n__first(0, x0))) → SQR(nil)



↳ 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:

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
SQR(s(n__first(0, x0))) → SQR(nil)
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(n__first(x0, x1))) → SQR(n__first(x0, x1))
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → ACTIVATE(X)
SQR(s(n__first(s(x0), cons(x1, x2)))) → SQR(cons(x1, n__first(activate(x0), activate(x2))))
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__first(x0, x1))) → DBL(first(x0, x1))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
SQR(s(x0)) → SQR(x0)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)
SQR(s(x0)) → DBL(x0)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing

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

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__first(x0, x1))) → DBL(first(x0, x1))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
SQR(s(x0)) → SQR(x0)
SQR(s(x0)) → DBL(x0)
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SQR(s(n__first(x0, x1))) → DBL(first(x0, x1)) at position [0] we obtained the following new rules:

SQR(s(n__first(s(x0), cons(x1, x2)))) → DBL(cons(x1, n__first(activate(x0), activate(x2))))
SQR(s(n__first(x0, x1))) → DBL(n__first(x0, x1))
SQR(s(n__first(0, x0))) → DBL(nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof

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

SQR(s(n__first(s(x0), cons(x1, x2)))) → DBL(cons(x1, n__first(activate(x0), activate(x2))))
ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__terms(X)) → TERMS(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → ACTIVATE(X)
SQR(s(n__first(x0, x1))) → DBL(n__first(x0, x1))
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
SQR(s(x0)) → SQR(x0)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)
SQR(s(x0)) → DBL(x0)
SQR(s(n__first(0, x0))) → DBL(nil)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing

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

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(X)) → ACTIVATE(X)
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
SQR(s(n__terms(x0))) → DBL(terms(x0))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
SQR(s(x0)) → SQR(x0)
SQR(s(x0)) → DBL(x0)
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SQR(s(n__terms(x0))) → DBL(terms(x0)) at position [0] we obtained the following new rules:

SQR(s(n__terms(x0))) → DBL(n__terms(x0))
SQR(s(n__terms(x0))) → DBL(cons(recip(sqr(x0)), n__terms(s(x0))))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ Narrowing
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ DependencyGraphProof
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof

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

SQR(s(n__terms(x0))) → DBL(n__terms(x0))
ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(n__terms(x0))) → DBL(cons(recip(sqr(x0)), n__terms(s(x0))))
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
SQR(s(x0)) → SQR(x0)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
ADD(s(X), Y) → ACTIVATE(X)
SQR(s(x0)) → DBL(x0)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
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
                            ↳ QDP
                              ↳ DependencyGraphProof
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP

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

ACTIVATE(n__dbl(X)) → DBL(X)
TERMS(N) → SQR(N)
SQR(s(n__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
SQR(s(n__add(x0, x1))) → DBL(add(x0, x1))
SQR(s(n__s(x0))) → DBL(s(x0))
SQR(s(n__dbl(x0))) → SQR(dbl(x0))
SQR(s(n__add(x0, x1))) → SQR(add(x0, x1))
DBL(s(X)) → ACTIVATE(X)
SQR(s(n__s(x0))) → SQR(s(x0))
ACTIVATE(n__add(X1, X2)) → ADD(X1, X2)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)
SQR(s(x0)) → SQR(x0)
SQR(s(x0)) → DBL(x0)
ADD(s(X), Y) → ACTIVATE(X)

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(sqr(activate(X)), dbl(activate(X))))
dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(activate(X), activate(Z)))
terms(X) → n__terms(X)
add(X1, X2) → n__add(X1, X2)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(X) → X

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