Termination w.r.t. Q proof of /tmp/tmp3gJ0MQ/Ex2_Luc02a

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 QDPOrderProof (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 QDP

↳7 QDPOrderProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 DependencyGraphProof (⇔)

↳12 QDP

↳13 QDPOrderProof (⇔)

↳14 QDP

↳15 QDPOrderProof (⇔)

↳16 QDP

↳17 QDPOrderProof (⇔)

↳18 QDP

↳19 QDPOrderProof (⇔)

↳20 QDP

↳21 QDPOrderProof (⇔)

↳22 QDP

↳23 QDPOrderProof (⇔)

↳24 QDP

↳25 QDPOrderProof (⇔)

↳26 QDP

↳27 DependencyGraphProof (⇔)

↳28 AND

  ↳29 QDP

    ↳30 QDPOrderProof (⇔)

    ↳31 QDP

    ↳32 QDPOrderProof (⇔)

    ↳33 QDP

    ↳34 QDPOrderProof (⇔)

    ↳35 QDP

    ↳36 QDPOrderProof (⇔)

    ↳37 QDP

    ↳38 PisEmptyProof (⇔)

    ↳39 TRUE

  ↳40 QDP

    ↳41 QDPOrderProof (⇔)

    ↳42 QDP

    ↳43 PisEmptyProof (⇔)

    ↳44 TRUE

(0) Obligation:

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

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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:

A__TERMS(N) → A__SQR(mark(N))
A__TERMS(N) → MARK(N)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
A__SQR(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(terms(X)) → A__TERMS(mark(X))
MARK(terms(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(recip(X)) → MARK(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__FIRST(s(X), cons(Y, Z)) → MARK(Y)

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

POL(A__TERMS(x₁)) = 0A + 1A · x₁

POL(A__SQR(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

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

POL(a__sqr(x₁)) = 0A + 0A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

POL(A__DBL(x₁)) = 0A + 0A · x₁

POL(0) = 0A

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

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

POL(terms(x₁)) = 0A + 1A · x₁

POL(sqr(x₁)) = 0A + 0A · x₁

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

POL(dbl(x₁)) = 0A + 0A · x₁

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

POL(recip(x₁)) = -I + 0A · x₁

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

POL(a__terms(x₁)) = 0A + 1A · x₁

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

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__first(X1, X2) → first(X1, X2)
a__dbl(X) → dbl(X)
a__add(X1, X2) → add(X1, X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(4) Obligation:

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

A__TERMS(N) → A__SQR(mark(N))
A__TERMS(N) → MARK(N)
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
A__SQR(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
MARK(terms(X)) → A__TERMS(mark(X))
MARK(terms(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(recip(X)) → MARK(X)
MARK(s(X)) → MARK(X)

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
MARK(terms(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(recip(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__TERMS(N) → MARK(N)

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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.

MARK(terms(X)) → MARK(X)
A__TERMS(N) → MARK(N)

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

POL(A__SQR(x₁)) = 3A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

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

POL(a__sqr(x₁)) = 3A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(a__dbl(x₁)) = 3A + 0A · x₁

POL(0) = 0A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(terms(x₁)) = 3A + 1A · x₁

POL(A__TERMS(x₁)) = 3A + 1A · x₁

POL(sqr(x₁)) = 3A + 0A · x₁

POL(A__DBL(x₁)) = 3A + 0A · x₁

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

POL(dbl(x₁)) = 3A + 0A · x₁

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

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

POL(recip(x₁)) = -I + 0A · x₁

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

POL(a__terms(x₁)) = 3A + 1A · x₁

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

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__first(X1, X2) → first(X1, X2)
a__dbl(X) → dbl(X)
a__add(X1, X2) → add(X1, X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(8) Obligation:

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

A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(0, X) → MARK(X)
MARK(terms(X)) → A__TERMS(mark(X))
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(recip(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(terms(X)) → A__TERMS(mark(X))

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

POL(A__SQR(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

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

POL(a__sqr(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

POL(0) = 0A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(terms(x₁)) = 4A + 4A · x₁

POL(A__TERMS(x₁)) = 0A + 3A · x₁

POL(sqr(x₁)) = -I + 0A · x₁

POL(A__DBL(x₁)) = 0A + 0A · x₁

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

POL(dbl(x₁)) = -I + 0A · x₁

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

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

POL(recip(x₁)) = -I + 0A · x₁

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

POL(a__terms(x₁)) = 4A + 4A · x₁

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

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__first(X1, X2) → first(X1, X2)
a__dbl(X) → dbl(X)
a__add(X1, X2) → add(X1, X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(10) Obligation:

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

A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(0, X) → MARK(X)
A__TERMS(N) → A__SQR(mark(N))
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(recip(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(11) DependencyGraphProof (EQUIVALENT transformation)

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

(12) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(recip(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
A__SQR(s(X)) → A__DBL(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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.

MARK(recip(X)) → MARK(X)

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

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

POL(0) = 0A

POL(MARK(x₁)) = -I + 0A · x₁

POL(sqr(x₁)) = 0A + 0A · x₁

POL(A__SQR(x₁)) = -I + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(a__sqr(x₁)) = 0A + 0A · x₁

POL(a__dbl(x₁)) = -I + 0A · x₁

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

POL(dbl(x₁)) = -I + 0A · x₁

POL(A__DBL(x₁)) = -I + 0A · x₁

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

POL(cons(x₁, x₂)) = 5A + 0A · x₁ + -I · x₂

POL(recip(x₁)) = -I + 4A · x₁

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

POL(terms(x₁)) = 5A + 5A · x₁

POL(a__terms(x₁)) = 5A + 5A · x₁

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

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(14) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(sqr(X)) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))
A__SQR(s(X)) → MARK(X)
A__SQR(s(X)) → A__DBL(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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.

MARK(sqr(X)) → MARK(X)
A__SQR(s(X)) → MARK(X)
A__SQR(s(X)) → A__DBL(mark(X))

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

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

POL(0) = 0A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = 5A + 5A · x₁

POL(A__SQR(x₁)) = 5A + 5A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

POL(a__sqr(x₁)) = 5A + 5A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

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

POL(dbl(x₁)) = -I + 0A · x₁

POL(A__DBL(x₁)) = 0A + 0A · x₁

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

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

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

POL(terms(x₁)) = -I + 0A · x₁

POL(a__terms(x₁)) = 0A + 0A · x₁

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

POL(nil) = 0A

POL(recip(x₁)) = 0A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(16) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(X1, X2)) → MARK(X2)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(add(X1, X2)) → MARK(X2)

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

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

POL(0) = 0A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = -I + 1A · x₁

POL(A__SQR(x₁)) = 0A + 1A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(a__sqr(x₁)) = -I + 1A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

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

POL(dbl(x₁)) = 0A + 0A · x₁

POL(A__DBL(x₁)) = 0A + 0A · x₁

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

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

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

POL(terms(x₁)) = -I + 1A · x₁

POL(a__terms(x₁)) = -I + 1A · x₁

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

POL(nil) = 0A

POL(recip(x₁)) = -I + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(18) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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.

MARK(cons(X1, X2)) → MARK(X1)

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

POL(A__ADD(x₁, x₂)) = 4A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(MARK(x₁)) = 4A + 0A · x₁

POL(sqr(x₁)) = -I + 0A · x₁

POL(A__SQR(x₁)) = 4A + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(a__sqr(x₁)) = 0A + 0A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

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

POL(dbl(x₁)) = -I + 0A · x₁

POL(A__DBL(x₁)) = 4A + 0A · x₁

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

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

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

POL(terms(x₁)) = 5A + 0A · x₁

POL(a__terms(x₁)) = 5A + 0A · x₁

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

POL(nil) = 0A

POL(recip(x₁)) = 0A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(20) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(first(X1, X2)) → MARK(X2)

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

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

POL(0) = 1A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = -I + 0A · x₁

POL(A__SQR(x₁)) = -I + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(a__sqr(x₁)) = 0A + 0A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

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

POL(dbl(x₁)) = -I + 0A · x₁

POL(A__DBL(x₁)) = 0A + 0A · x₁

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

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

POL(terms(x₁)) = 5A + -I · x₁

POL(a__terms(x₁)) = 5A + -I · x₁

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

POL(nil) = 0A

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

POL(recip(x₁)) = 0A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(22) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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.

MARK(first(X1, X2)) → MARK(X1)

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

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

POL(0) = 5A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = -I + 5A · x₁

POL(A__SQR(x₁)) = 0A + 5A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(a__sqr(x₁)) = -I + 5A · x₁

POL(a__dbl(x₁)) = 5A + 0A · x₁

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

POL(dbl(x₁)) = 5A + 0A · x₁

POL(A__DBL(x₁)) = 5A + 0A · x₁

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

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

POL(terms(x₁)) = 0A + -I · x₁

POL(a__terms(x₁)) = 0A + -I · x₁

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

POL(nil) = 0A

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

POL(recip(x₁)) = 2A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(24) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → A__DBL(mark(X))
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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.

MARK(dbl(X)) → A__DBL(mark(X))

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

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

POL(0) = 0A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = -I + 5A · x₁

POL(A__SQR(x₁)) = 0A + 5A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

POL(a__sqr(x₁)) = -I + 5A · x₁

POL(a__dbl(x₁)) = 0A + 5A · x₁

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

POL(dbl(x₁)) = 0A + 5A · x₁

POL(A__DBL(x₁)) = -I + 0A · x₁

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

POL(terms(x₁)) = 0A + 0A · x₁

POL(a__terms(x₁)) = 0A + 0A · x₁

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

POL(nil) = 0A

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

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

POL(recip(x₁)) = 0A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(26) Obligation:

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

A__ADD(0, X) → MARK(X)
MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(X1, X2)) → MARK(X1)
A__DBL(s(X)) → A__DBL(mark(X))
A__DBL(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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 2 SCCs with 1 less node.

(28) Complex Obligation (AND)

(29) Obligation:

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

MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(0, X) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__ADD(s(X), Y) → MARK(Y)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ADD(s(X), Y) → MARK(Y)

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

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = 1A + 1A · x₁

POL(A__SQR(x₁)) = -I + 1A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 1A + 0A · x₁

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

POL(a__sqr(x₁)) = 1A + 1A · x₁

POL(a__dbl(x₁)) = 0A + 0A · x₁

POL(0) = 0A

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

POL(dbl(x₁)) = 0A + 0A · x₁

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

POL(terms(x₁)) = 0A + -I · x₁

POL(a__terms(x₁)) = 0A + -I · x₁

POL(a__first(x₁, x₂)) = 3A + 0A · x₁ + -I · x₂

POL(nil) = 0A

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

POL(first(x₁, x₂)) = 3A + 0A · x₁ + -I · x₂

POL(recip(x₁)) = 0A + 0A · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(31) Obligation:

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

MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
A__ADD(0, X) → MARK(X)
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__ADD(0, X) → MARK(X)

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

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = -I + 1A · x₁

POL(A__SQR(x₁)) = 0A + 1A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

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

POL(a__sqr(x₁)) = -I + 1A · x₁

POL(a__dbl(x₁)) = -I + 0A · x₁

POL(0) = 1A

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

POL(dbl(x₁)) = -I + 0A · x₁

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

POL(terms(x₁)) = -I + 0A · x₁

POL(a__terms(x₁)) = -I + 0A · x₁

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

POL(nil) = 0A

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

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

POL(recip(x₁)) = 0A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(33) Obligation:

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

MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(dbl(X)) → MARK(X)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(dbl(X)) → MARK(X)

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

POL(MARK(x₁)) = 0A + 0A · x₁

POL(sqr(x₁)) = 1A + 1A · x₁

POL(A__SQR(x₁)) = 1A + 1A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

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

POL(a__sqr(x₁)) = 1A + 1A · x₁

POL(a__dbl(x₁)) = 1A + 1A · x₁

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

POL(dbl(x₁)) = 1A + 1A · x₁

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

POL(0) = 1A

POL(terms(x₁)) = 0A + -I · x₁

POL(a__terms(x₁)) = 0A + -I · x₁

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

POL(nil) = 1A

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

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

POL(recip(x₁)) = 5A + -I · x₁

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(35) Obligation:

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

MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(sqr(X)) → A__SQR(mark(X))
A__SQR(s(X)) → A__ADD(a__sqr(mark(X)), a__dbl(mark(X)))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
A__SQR(s(X)) → A__SQR(mark(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK(x1)
sqr(x1) = sqr(x1)
A__SQR(x1) = A__SQR(x1)
mark(x1) = x1
s(x1) = s(x1)
A__ADD(x1, x2) = A__ADD(x1, x2)
a__sqr(x1) = a__sqr(x1)
a__dbl(x1) = a__dbl(x1)
add(x1, x2) = add(x1, x2)
a__add(x1, x2) = a__add(x1, x2)
0 = 0
dbl(x1) = dbl(x1)
terms(x1) = terms
a__terms(x1) = a__terms
a__first(x1, x2) = a__first(x1, x2)
nil = nil
cons(x1, x2) = x1
first(x1, x2) = first(x1, x2)
recip(x1) = recip

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

[sqr₁, A_{_SQR₁}, a_{_sqr₁}, 0] > [MARK₁, A_{_ADD₂}]
[sqr₁, A_{_SQR₁}, a_{_sqr₁}, 0] > [a_{_dbl₁}, dbl₁] > s₁ > [a_{_first₂}, first₂]
[sqr₁, A_{_SQR₁}, a_{_sqr₁}, 0] > [add₂, a_{_add₂}] > s₁ > [a_{_first₂}, first₂]
[sqr₁, A_{_SQR₁}, a_{_sqr₁}, 0] > nil
[terms, a_{_terms}] > recip

Status:

sqr₁: [1]
a_{_terms}: multiset
a_{_dbl₁}: multiset
a_{_add₂}: multiset
recip: []
0: multiset
first₂: multiset
a_{_first₂}: multiset
add₂: multiset
MARK₁: multiset
A_{_ADD₂}: multiset
dbl₁: multiset
s₁: [1]
A_{_SQR₁}: [1]
nil: multiset
a_{_sqr₁}: [1]
terms: multiset

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__first(X1, X2) → first(X1, X2)
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__add(X1, X2) → add(X1, X2)
a__dbl(s(X)) → s(s(a__dbl(mark(X))))

(37) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(38) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(39) TRUE

(40) Obligation:

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

A__DBL(s(X)) → A__DBL(mark(X))

The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__DBL(s(X)) → A__DBL(mark(X))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__DBL(x1) = A__DBL(x1)
s(x1) = s(x1)
mark(x1) = x1
add(x1, x2) = add(x1, x2)
a__add(x1, x2) = a__add(x1, x2)
0 = 0
dbl(x1) = dbl(x1)
a__dbl(x1) = a__dbl(x1)
terms(x1) = terms(x1)
a__terms(x1) = a__terms(x1)
sqr(x1) = sqr(x1)
a__sqr(x1) = a__sqr(x1)
a__first(x1, x2) = x2
nil = nil
cons(x1, x2) = x1
first(x1, x2) = x2
recip(x1) = recip

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

[0, dbl₁, a_{_dbl₁}, sqr₁, a_{_sqr₁}] > [add₂, a_{_add₂}] > s₁ > A_{_DBL₁} > nil
[terms₁, a_{_terms₁}] > s₁ > A_{_DBL₁} > nil
[terms₁, a_{_terms₁}] > recip > nil

Status:

sqr₁: [1]
a_{_dbl₁}: [1]
a_{_add₂}: multiset
recip: []
0: multiset
terms₁: multiset
add₂: multiset
A_{_DBL₁}: multiset
a_{_terms₁}: multiset
dbl₁: [1]
s₁: [1]
nil: multiset
a_{_sqr₁}: [1]

The following usable rules [FROCOS05] were oriented:

mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
a__add(0, X) → mark(X)
mark(dbl(X)) → a__dbl(mark(X))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
mark(0) → 0
mark(nil) → nil
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__first(X1, X2) → first(X1, X2)
a__dbl(X) → dbl(X)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(X1, X2) → add(X1, X2)

(42) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a__terms(N) → cons(recip(a__sqr(mark(N))), terms(s(N)))
a__sqr(0) → 0
a__sqr(s(X)) → s(a__add(a__sqr(mark(X)), a__dbl(mark(X))))
a__dbl(0) → 0
a__dbl(s(X)) → s(s(a__dbl(mark(X))))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__first(0, X) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
mark(terms(X)) → a__terms(mark(X))
mark(sqr(X)) → a__sqr(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(recip(X)) → recip(mark(X))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__terms(X) → terms(X)
a__sqr(X) → sqr(X)
a__add(X1, X2) → add(X1, X2)
a__dbl(X) → dbl(X)
a__first(X1, X2) → first(X1, X2)

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

(43) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) DependencyPairsProof (EQUIVALENT transformation)

(2) Obligation:

(3) QDPOrderProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) QDPOrderProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Complex Obligation (AND)

(29) Obligation:

(30) QDPOrderProof (EQUIVALENT transformation)

(31) Obligation:

(32) QDPOrderProof (EQUIVALENT transformation)

(33) Obligation:

(34) QDPOrderProof (EQUIVALENT transformation)

(35) Obligation:

(36) QDPOrderProof (EQUIVALENT transformation)

(37) Obligation:

(38) PisEmptyProof (EQUIVALENT transformation)

(39) TRUE

(40) Obligation:

(41) QDPOrderProof (EQUIVALENT transformation)

(42) Obligation:

(43) PisEmptyProof (EQUIVALENT transformation)

(44) TRUE