Termination w.r.t. Q proof of /tmp/tmpcjyMvs/Ex26_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:

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

                                                  ↳ NonMonReductionPairProof

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.
Using the following max-polynomial ordering, we can orient the general usable rules and all rules from P weakly and some rules from P strictly:
Polynomial interpretation [25,21]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(ADD(x₁, x₂)) = x₁ + x₂
POL(DBL(x₁)) = x₁
POL(FIRST(x₁, x₂)) = 1 + x₁ + x₂
POL(SQR(x₁)) = x₁
POL(TERMS(x₁)) = x₁
POL(activate(x₁)) = x₁
POL(add(x₁, x₂)) = max(0, x₁ + x₂)
POL(cons(x₁, x₂)) = x₂
POL(dbl(x₁)) = x₁
POL(first(x₁, x₂)) = max(0, 1 + x₁ + x₂)
POL(n__add(x₁, x₂)) = x₁ + x₂
POL(n__dbl(x₁)) = x₁
POL(n__first(x₁, x₂)) = 1 + x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__terms(x₁)) = x₁
POL(nil) = 0
POL(recip(x₁)) = 0
POL(s(x₁)) = x₁
POL(sqr(x₁)) = 0
POL(terms(x₁)) = x₁

The following pairs can be oriented strictly and are deleted.

FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The remaining pairs can at least be oriented weakly.

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)
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)
SQR(s(x0)) → DBL(x0)
ADD(s(X), Y) → ACTIVATE(X)

The following rules are usable:

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

↳ 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

                                                  ↳ NonMonReductionPairProof

                                                    ↳ 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__dbl(x0))) → DBL(dbl(x0))
ACTIVATE(n__first(X1, X2)) → FIRST(X1, X2)
ACTIVATE(n__terms(X)) → TERMS(X)
SQR(s(X)) → 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)
SQR(s(x0)) → SQR(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 1 less node.

↳ 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

                                                  ↳ NonMonReductionPairProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

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__terms(X)) → TERMS(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))
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)
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.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__dbl(X)) → DBL(X)
SQR(s(X)) → 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__s(x0))) → SQR(s(x0))
SQR(s(x0)) → SQR(x0)
ADD(s(X), Y) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Combined order from the following AFS and order.
ACTIVATE(x1) = x1
n__dbl(x1) = n__dbl(x1)
DBL(x1) = DBL(x1)
TERMS(x1) = x1
SQR(x1) = x1
s(x1) = s(x1)
dbl(x1) = dbl(x1)
n__terms(x1) = x1
n__add(x1, x2) = n__add(x1, x2)
add(x1, x2) = add(x1, x2)
n__s(x1) = n__s(x1)
ADD(x1, x2) = ADD(x1, x2)
activate(x1) = activate(x1)
n__first(x1, x2) = n__first(x1)
first(x1, x2) = first(x1)
terms(x1) = terms(x1)
sqr(x1) = sqr(x1)
0 = 0
cons(x1, x2) = cons
recip(x1) = recip(x1)
nil = nil

Recursive path order with status [2].
Quasi-Precedence:

[n_{_dbl₁}, dbl₁, sqr₁] > [n_{_add₂}, add₂, ADD₂] > [DBL₁, s₁, n_{_s₁}] > [activate₁, terms₁, cons] > first₁ > n_{_first₁}
[n_{_dbl₁}, dbl₁, sqr₁] > 0 > nil > n_{_first₁}
recip₁ > n_{_first₁}

Status:

DBL₁: multiset
n_{_dbl₁}: [1]
sqr₁: [1]
activate₁: [1]
n_{_s₁}: multiset
0: multiset
terms₁: [1]
add₂: multiset
n_{_add₂}: multiset
cons: []
n_{_first₁}: multiset
dbl₁: [1]
s₁: multiset
ADD₂: multiset
first₁: multiset
nil: multiset
recip₁: multiset

The following usable rules [17] were oriented:

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

↳ 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

                                                  ↳ NonMonReductionPairProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ QDPOrderProof

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

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

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

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 0 SCCs with 7 less nodes.