Termination w.r.t. Q proof of /tmp/tmpxvi0Ob/Ex26_Luc03b

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 QDP

↳5 QDPOrderProof (⇔)

↳6 QDP

↳7 QDPOrderProof (⇔)

↳8 QDP

↳9 DependencyGraphProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 QDPOrderProof (⇔)

↳16 QDP

↳17 DependencyGraphProof (⇔)

↳18 QDP

↳19 QDPOrderProof (⇔)

↳20 QDP

↳21 DependencyGraphProof (⇔)

↳22 QDP

↳23 QDPOrderProof (⇔)

↳24 QDP

↳25 QDPOrderProof (⇔)

↳26 QDP

↳27 DependencyGraphProof (⇔)

↳28 QDP

↳29 QDPOrderProof (⇔)

↳30 QDP

↳31 QDPOrderProof (⇔)

↳32 QDP

↳33 QDPOrderProof (⇔)

↳34 QDP

↳35 QDPOrderProof (⇔)

↳36 QDP

↳37 DependencyGraphProof (⇔)

↳38 TRUE

(0) Obligation:

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

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

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

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__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)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
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 1 SCC with 4 less nodes.

(4) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(s(X), cons(Y, Z)) → ACTIVATE(Z)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(SQR(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__terms(x₁)) = 0A + 0A · x₁

POL(TERMS(x₁)) = 0A + 0A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(n__add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(n__sqr(x₁)) = 0A + 0A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(DBL(x₁)) = 0A + 0A · x₁

POL(n__first(x₁, x₂)) = -I + 5A · x₁ + 1A · x₂

POL(FIRST(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(cons(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(sqr(x₁)) = 0A + 0A · x₁

POL(0) = 3A

POL(terms(x₁)) = 0A + 0A · x₁

POL(recip(x₁)) = -I + 0A · x₁

POL(n__s(x₁)) = 0A + 0A · x₁

POL(add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = -I + 0A · x₁

POL(first(x₁, x₂)) = -I + 5A · x₁ + 1A · x₂

POL(nil) = 3A

The following usable rules [FROCOS05] were oriented:

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

(6) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(7) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(s(X), cons(Y, Z)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(SQR(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__terms(x₁)) = 0A + 0A · x₁

POL(TERMS(x₁)) = -I + 0A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(n__add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(n__sqr(x₁)) = 0A + 0A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(DBL(x₁)) = 0A + 0A · x₁

POL(n__first(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(FIRST(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(cons(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(sqr(x₁)) = 0A + 0A · x₁

POL(0) = 1A

POL(terms(x₁)) = 0A + 0A · x₁

POL(recip(x₁)) = 0A + -I · x₁

POL(n__s(x₁)) = 0A + 0A · x₁

POL(add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = -I + 0A · x₁

POL(first(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(nil) = 2A

The following usable rules [FROCOS05] were oriented:

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

(8) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(10) Obligation:

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

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

The TRS R consists of the following rules:

terms(N) → cons(recip(sqr(N)), n__terms(n__s(N)))
sqr(0) → 0
sqr(s(X)) → s(n__add(n__sqr(activate(X)), n__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)
s(X) → n__s(X)
add(X1, X2) → n__add(X1, X2)
sqr(X) → n__sqr(X)
dbl(X) → n__dbl(X)
first(X1, X2) → n__first(X1, X2)
activate(n__terms(X)) → terms(activate(X))
activate(n__s(X)) → s(X)
activate(n__add(X1, X2)) → add(activate(X1), activate(X2))
activate(n__sqr(X)) → sqr(activate(X))
activate(n__dbl(X)) → dbl(activate(X))
activate(n__first(X1, X2)) → first(activate(X1), activate(X2))
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.

ACTIVATE(n__first(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__terms(x₁)) = 0A + 0A · x₁

POL(TERMS(x₁)) = 0A + 0A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(SQR(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(n__add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(n__sqr(x₁)) = 0A + 0A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(DBL(x₁)) = 0A + 0A · x₁

POL(n__first(x₁, x₂)) = 1A + 5A · x₁ + 0A · x₂

POL(sqr(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(terms(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(recip(x₁)) = -I + 2A · x₁

POL(n__s(x₁)) = 0A + 0A · x₁

POL(add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = -I + 0A · x₁

POL(first(x₁, x₂)) = 1A + 5A · x₁ + 0A · x₂

POL(nil) = 3A

The following usable rules [FROCOS05] were oriented:

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

(12) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__first(X1, X2)) → ACTIVATE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__terms(x₁)) = 1A + 0A · x₁

POL(TERMS(x₁)) = -I + 0A · x₁

POL(activate(x₁)) = 0A + 0A · x₁

POL(SQR(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(n__add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(n__sqr(x₁)) = -I + 0A · x₁

POL(n__dbl(x₁)) = 0A + 0A · x₁

POL(DBL(x₁)) = -I + 0A · x₁

POL(n__first(x₁, x₂)) = 2A + -I · x₁ + 1A · x₂

POL(sqr(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(terms(x₁)) = 1A + 0A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(recip(x₁)) = 0A + -I · x₁

POL(n__s(x₁)) = -I + 0A · x₁

POL(add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = 0A + 0A · x₁

POL(first(x₁, x₂)) = 2A + -I · x₁ + 1A · x₂

POL(nil) = 1A

The following usable rules [FROCOS05] were oriented:

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

(14) Obligation:

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

ACTIVATE(n__terms(X)) → TERMS(activate(X))
TERMS(N) → SQR(N)
SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TERMS(N) → SQR(N)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__terms(x₁)) = 0A + 2A · x₁

POL(TERMS(x₁)) = 0A + 2A · x₁

POL(activate(x₁)) = -I + 0A · x₁

POL(SQR(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(n__add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(n__sqr(x₁)) = 0A + 0A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(DBL(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(terms(x₁)) = 0A + 2A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(recip(x₁)) = -I + 0A · x₁

POL(n__s(x₁)) = 0A + 0A · x₁

POL(add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = -I + 0A · x₁

POL(n__first(x₁, x₂)) = 2A + -I · x₁ + -I · x₂

POL(first(x₁, x₂)) = 2A + -I · x₁ + -I · x₂

POL(nil) = 1A

The following usable rules [FROCOS05] were oriented:

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

(16) Obligation:

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

ACTIVATE(n__terms(X)) → TERMS(activate(X))
SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(18) Obligation:

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

ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → SQR(activate(X))
SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__sqr(X)) → SQR(activate(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 1A + 0A · x₁

POL(n__terms(x₁)) = -I + 0A · x₁

POL(n__add(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 1A + 0A · x₁

POL(n__sqr(x₁)) = 1A + 1A · x₁

POL(SQR(x₁)) = 0A + 0A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(DBL(x₁)) = 1A + 0A · x₁

POL(sqr(x₁)) = 1A + 1A · x₁

POL(0) = 0A

POL(terms(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(recip(x₁)) = -I + 0A · x₁

POL(n__s(x₁)) = 1A + 0A · x₁

POL(add(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = 0A + 0A · x₁

POL(n__first(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(first(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

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

(20) Obligation:

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

ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
SQR(s(X)) → ACTIVATE(X)
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(22) Obligation:

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

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__sqr(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__sqr(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

POL(n__terms(x₁)) = -I + 0A · x₁

POL(n__sqr(x₁)) = 1A + 1A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(DBL(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = 1A + 1A · x₁

POL(0) = 0A

POL(terms(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(recip(x₁)) = 0A + -I · x₁

POL(n__s(x₁)) = -I + 0A · x₁

POL(add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = 0A + 0A · x₁

POL(n__first(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(first(x₁, x₂)) = 2A + 0A · x₁ + 0A · x₂

POL(nil) = 2A

The following usable rules [FROCOS05] were oriented:

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

(24) Obligation:

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

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__dbl(X)) → DBL(activate(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ACTIVATE(x₁)) = 1A + 0A · x₁

POL(n__add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(ADD(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 1A + 0A · x₁

POL(n__terms(x₁)) = -I + 0A · x₁

POL(n__dbl(x₁)) = 1A + 1A · x₁

POL(DBL(x₁)) = -I + 0A · x₁

POL(sqr(x₁)) = 1A + 1A · x₁

POL(0) = 3A

POL(terms(x₁)) = 0A + 0A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(recip(x₁)) = 0A + -I · x₁

POL(n__s(x₁)) = 1A + 0A · x₁

POL(n__sqr(x₁)) = 1A + 1A · x₁

POL(add(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(dbl(x₁)) = 1A + 1A · x₁

POL(n__first(x₁, x₂)) = 4A + 0A · x₁ + -I · x₂

POL(first(x₁, x₂)) = 4A + 0A · x₁ + -I · x₂

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

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

(26) Obligation:

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

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(28) Obligation:

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

ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__terms(X)) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__terms(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ADD(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(s(x₁)) = 1A + 0A · x₁

POL(ACTIVATE(x₁)) = -I + 0A · x₁

POL(n__terms(x₁)) = 0A + 5A · x₁

POL(n__add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(n__dbl(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = 0A + 1A · x₁

POL(0) = 0A

POL(terms(x₁)) = 0A + 5A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(recip(x₁)) = 0A + 0A · x₁

POL(n__s(x₁)) = 1A + 0A · x₁

POL(n__sqr(x₁)) = 0A + 1A · x₁

POL(dbl(x₁)) = 0A + 0A · x₁

POL(add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(n__first(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(first(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

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

(30) Obligation:

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

ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__add(X1, X2)) → ACTIVATE(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ADD(x₁, x₂)) = -I + 0A · x₁ + 1A · x₂

POL(s(x₁)) = 5A + 0A · x₁

POL(ACTIVATE(x₁)) = 0A + 0A · x₁

POL(n__add(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(activate(x₁)) = 0A + 0A · x₁

POL(n__dbl(x₁)) = -I + 0A · x₁

POL(sqr(x₁)) = 1A + 1A · x₁

POL(0) = 3A

POL(terms(x₁)) = 0A + -I · x₁

POL(cons(x₁, x₂)) = -I + 0A · x₁ + -I · x₂

POL(recip(x₁)) = 0A + -I · x₁

POL(n__terms(x₁)) = 0A + -I · x₁

POL(n__s(x₁)) = 5A + 0A · x₁

POL(n__sqr(x₁)) = 1A + 1A · x₁

POL(dbl(x₁)) = -I + 0A · x₁

POL(add(x₁, x₂)) = 1A + 0A · x₁ + 1A · x₂

POL(n__first(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(first(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(nil) = 3A

The following usable rules [FROCOS05] were oriented:

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

(32) Obligation:

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

ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

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

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

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(ADD(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

POL(ACTIVATE(x₁)) = -I + 0A · x₁

POL(n__add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(activate(x₁)) = -I + 0A · x₁

POL(n__dbl(x₁)) = 0A + 1A · x₁

POL(sqr(x₁)) = 0A + 1A · x₁

POL(0) = 0A

POL(terms(x₁)) = 5A + 0A · x₁

POL(cons(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(recip(x₁)) = 0A + -I · x₁

POL(n__terms(x₁)) = 5A + 0A · x₁

POL(n__s(x₁)) = 0A + 0A · x₁

POL(n__sqr(x₁)) = 0A + 1A · x₁

POL(dbl(x₁)) = 0A + 1A · x₁

POL(add(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(n__first(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(first(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

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

(34) Obligation:

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

ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)

The TRS R consists of the following rules:

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

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

(35) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ADD(s(X), Y) → ACTIVATE(X)
ACTIVATE(n__add(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ADD(x1, x2) = ADD(x1, x2)
s(x1) = s(x1)
ACTIVATE(x1) = x1
n__add(x1, x2) = n__add(x1, x2)
activate(x1) = x1
sqr(x1) = sqr(x1)
0 = 0
terms(x1) = terms
cons(x1, x2) = x2
recip(x1) = x1
n__terms(x1) = n__terms
n__s(x1) = n__s(x1)
n__sqr(x1) = n__sqr(x1)
n__dbl(x1) = n__dbl(x1)
dbl(x1) = dbl(x1)
add(x1, x2) = add(x1, x2)
n__first(x1, x2) = x2
first(x1, x2) = x2
nil = nil

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

[sqr₁, n_{_sqr₁}] > [ADD₂, n_{_add₂}, add₂] > [s₁, n_{_s₁}, nil]
[sqr₁, n_{_sqr₁}] > 0 > [s₁, n_{_s₁}, nil]
[sqr₁, n_{_sqr₁}] > [n_{_dbl₁}, dbl₁] > [s₁, n_{_s₁}, nil]
[terms, n_{_terms}] > [s₁, n_{_s₁}, nil]

Status:

n_{_dbl₁}: multiset
sqr₁: [1]
n_{_s₁}: multiset
n_{_terms}: multiset
0: multiset
add₂: [1,2]
n_{_add₂}: [1,2]
dbl₁: multiset
n_{_sqr₁}: [1]
s₁: multiset
ADD₂: [1,2]
nil: multiset
terms: multiset

The following usable rules [FROCOS05] were oriented:

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

(36) Obligation:

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

ACTIVATE(n__add(X1, X2)) → ADD(activate(X1), activate(X2))

The TRS R consists of the following rules:

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

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyGraphProof (EQUIVALENT transformation)

(4) Obligation:

(5) QDPOrderProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) DependencyGraphProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) DependencyGraphProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) QDPOrderProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) DependencyGraphProof (EQUIVALENT transformation)

(38) TRUE