Termination w.r.t. Q proof of /tmp/tmpERV9gs/Ex1_2_AEL03

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

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 Narrowing (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 QDP

↳7 QDPOrderProof (⇔)

↳8 QDP

↳9 QDPOrderProof (⇔)

↳10 QDP

↳11 QDPOrderProof (⇔)

↳12 QDP

↳13 DependencyGraphProof (⇔)

↳14 QDP

↳15 QDPOrderProof (⇔)

↳16 QDP

↳17 QDPOrderProof (⇔)

↳18 QDP

↳19 DependencyGraphProof (⇔)

↳20 QDP

↳21 QDPOrderProof (⇔)

↳22 QDP

↳23 QDPOrderProof (⇔)

↳24 QDP

↳25 QDPOrderProof (⇔)

↳26 QDP

↳27 QDPOrderProof (⇔)

↳28 QDP

↳29 QDPOrderProof (⇔)

↳30 QDP

↳31 DependencyGraphProof (⇔)

↳32 QDP

↳33 QDPOrderProof (⇔)

↳34 QDP

↳35 QDPOrderProof (⇔)

↳36 QDP

↳37 QDPOrderProof (⇔)

↳38 QDP

↳39 QDPOrderProof (⇔)

↳40 QDP

↳41 QDPOrderProof (⇔)

↳42 QDP

↳43 QDPOrderProof (⇔)

↳44 QDP

↳45 QDPOrderProof (⇔)

↳46 QDP

↳47 DependencyGraphProof (⇔)

↳48 AND

  ↳49 QDP

    ↳50 QDPOrderProof (⇔)

    ↳51 QDP

    ↳52 QDPOrderProof (⇔)

    ↳53 QDP

    ↳54 QDPOrderProof (⇔)

    ↳55 QDP

    ↳56 QDPOrderProof (⇔)

    ↳57 QDP

    ↳58 QDPOrderProof (⇔)

    ↳59 QDP

    ↳60 DependencyGraphProof (⇔)

    ↳61 QDP

    ↳62 QDPOrderProof (⇔)

    ↳63 QDP

    ↳64 PisEmptyProof (⇔)

    ↳65 TRUE

  ↳66 QDP

    ↳67 QDPOrderProof (⇔)

    ↳68 QDP

    ↳69 PisEmptyProof (⇔)

    ↳70 TRUE

(0) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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:

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → A__2NDSPOS(mark(X), a__from(0))
A__PI(X) → MARK(X)
A__PI(X) → A__FROM(0)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(3) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule A__PI(X) → A__2NDSPOS(mark(X), a__from(0)) at position [1] we obtained the following new rules [LPAR04]:

A__PI(y0) → A__2NDSPOS(mark(y0), cons(mark(0), from(s(0))))
A__PI(y0) → A__2NDSPOS(mark(y0), from(0))

(4) Obligation:

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

A__FROM(X) → MARK(X)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__PI(X) → MARK(X)
A__PI(X) → A__FROM(0)
A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__SQUARE(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PI(y0) → A__2NDSPOS(mark(y0), cons(mark(0), from(s(0))))
A__PI(y0) → A__2NDSPOS(mark(y0), from(0))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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 2 less nodes.

(6) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
A__PI(X) → MARK(X)
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(square(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__PI(X) → A__FROM(0)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

MARK(square(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(from(x₁)) = 0A + 0A · x₁

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

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

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = -I + 0A · x₁

POL(A__PI(x₁)) = 0A + 0A · x₁

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

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

POL(0) = 0A

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

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

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

POL(square(x₁)) = 2A + 2A · x₁

POL(A__SQUARE(x₁)) = 2A + 0A · x₁

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 2A + 2A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 0A + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(8) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
A__PI(X) → MARK(X)
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__PI(X) → A__FROM(0)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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.

A__PI(X) → A__FROM(0)

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

POL(MARK(x₁)) = -I + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(A__FROM(x₁)) = -I + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

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

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

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

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 1A + 0A · x₁

POL(A__PI(x₁)) = 1A + 0A · x₁

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

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

POL(0) = 0A

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

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

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

POL(square(x₁)) = 0A + 0A · x₁

POL(A__SQUARE(x₁)) = 0A + 0A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(a__from(x₁)) = -I + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(10) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(pi(X)) → A__PI(mark(X))
A__PI(X) → MARK(X)
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

MARK(pi(X)) → A__PI(mark(X))
MARK(pi(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)) = -I + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

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

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(pi(x₁)) = 0A + 1A · x₁

POL(A__PI(x₁)) = -I + 0A · x₁

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

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

POL(0) = 0A

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

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

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

POL(square(x₁)) = 0A + 4A · x₁

POL(A__SQUARE(x₁)) = 0A + 0A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

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

POL(a__square(x₁)) = 0A + 4A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 0A + 1A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(12) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
A__PI(X) → MARK(X)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(13) DependencyGraphProof (EQUIVALENT transformation)

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

(14) Obligation:

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

A__FROM(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__SQUARE(X) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

A__SQUARE(X) → MARK(X)

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

POL(A__FROM(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = -I + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

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

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

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

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

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(square(x₁)) = 1A + 1A · x₁

POL(A__SQUARE(x₁)) = 0A + 1A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 1A + 1A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 0A + 0A · x₁

POL(pi(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(16) Obligation:

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

A__FROM(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__SQUARE(X) → A__TIMES(mark(X), mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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.

A__SQUARE(X) → A__TIMES(mark(X), mark(X))

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

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(square(x₁)) = 2A + 2A · x₁

POL(A__SQUARE(x₁)) = 1A + 2A · x₁

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

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

POL(a__square(x₁)) = 2A + 2A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 0A + 0A · x₁

POL(pi(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(18) Obligation:

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

A__FROM(X) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → A__SQUARE(mark(X))
A__TIMES(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__PLUS(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(19) DependencyGraphProof (EQUIVALENT transformation)

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

(20) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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.

A__TIMES(s(X), Y) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(N)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(N)

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(from(x₁)) = 5A + 0A · x₁

POL(A__FROM(x₁)) = 5A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(s(x₁)) = 5A + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + 1A · x₁ + 0A · x₂

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

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

POL(0) = 3A

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

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

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

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

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

POL(a__square(x₁)) = 1A + 1A · x₁

POL(square(x₁)) = 1A + 1A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 1A · x₁

POL(pi(x₁)) = 5A + 1A · x₁

POL(a__from(x₁)) = 5A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(22) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

MARK(times(X1, X2)) → MARK(X2)

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(from(x₁)) = -I + 0A · x₁

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

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

POL(a__square(x₁)) = 1A + 1A · x₁

POL(square(x₁)) = 1A + 1A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 0A + 0A · x₁

POL(pi(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(24) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

MARK(2ndspos(X1, X2)) → MARK(X1)

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(from(x₁)) = 4A + 0A · x₁

POL(A__FROM(x₁)) = 4A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = 5A + 1A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(s(x₁)) = 4A + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = 5A + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 1A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 1A · x₁

POL(pi(x₁)) = 5A + 1A · x₁

POL(a__from(x₁)) = 4A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 5A + 1A · x₁ + 0A · x₂

POL(rnil) = 1A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(26) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(from(X)) → MARK(X)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Y)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Y)
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(MARK(x₁)) = -I + 0A · x₁

POL(from(x₁)) = 0A + 5A · x₁

POL(A__FROM(x₁)) = 0A + 0A · x₁

POL(mark(x₁)) = -I + 0A · x₁

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

POL(A__2NDSPOS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSNEG(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 1A + 0A · x₁

POL(square(x₁)) = 1A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 1A · x₁

POL(pi(x₁)) = 5A + 1A · x₁

POL(a__from(x₁)) = 0A + 5A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(28) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
A__FROM(X) → MARK(X)
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

A__FROM(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(from(x₁)) = 1A + 1A · x₁

POL(A__FROM(x₁)) = 1A + 1A · x₁

POL(mark(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

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

POL(A__2NDSNEG(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

POL(s(x₁)) = 0A + 0A · x₁

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

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

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

POL(a__square(x₁)) = -I + 0A · x₁

POL(square(x₁)) = -I + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 2A · x₁

POL(pi(x₁)) = 1A + 2A · x₁

POL(a__from(x₁)) = 1A + 1A · x₁

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(30) Obligation:

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

MARK(from(X)) → A__FROM(mark(X))
MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(31) DependencyGraphProof (EQUIVALENT transformation)

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

(32) Obligation:

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

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

MARK(2ndsneg(X1, X2)) → MARK(X1)

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(2ndspos(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(pi(x₁)) = 1A + 1A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 1A + 1A · x₁ + 0A · x₂

POL(rnil) = 1A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(34) Obligation:

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

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))
A__TIMES(s(X), Y) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(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.

A__TIMES(s(X), Y) → MARK(X)

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

POL(MARK(x₁)) = 1A + 0A · x₁

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(A__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 1A + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = 0A + 0A · x₁

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

POL(a__square(x₁)) = 2A + 1A · x₁

POL(square(x₁)) = 2A + 1A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(pi(x₁)) = 1A + 0A · x₁

POL(a__from(x₁)) = 1A + 0A · x₁

POL(from(x₁)) = 1A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(36) Obligation:

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

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → A__2NDSPOS(mark(X1), mark(X2))

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(2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(A__2NDSPOS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 3A + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(a__pi(x₁)) = 4A + 0A · x₁

POL(pi(x₁)) = 4A + 0A · x₁

POL(a__from(x₁)) = 3A + 0A · x₁

POL(from(x₁)) = 3A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + 0A · x₁ + 1A · x₂

POL(rnil) = 0A

POL(nil) = 1A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(38) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(rcons(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__2NDSPOS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(mark(x₁)) = -I + 0A · x₁

POL(MARK(x₁)) = -I + 0A · x₁

POL(2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = 0A + 0A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(pi(x₁)) = 1A + 0A · x₁

POL(a__from(x₁)) = 0A + 1A · x₁

POL(from(x₁)) = 0A + 1A · x₁

POL(a__2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(rnil) = 1A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(40) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(posrecip(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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.

MARK(posrecip(X)) → MARK(X)

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

POL(A__2NDSPOS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(s(x₁)) = 0A + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 5A + 5A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 5A + 5A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(posrecip(x₁)) = 1A + 1A · x₁

POL(negrecip(x₁)) = -I + 0A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = -I + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 5A + 5A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 5A + 5A · x₁

POL(pi(x₁)) = 5A + 5A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 5A + 5A · x₁ + 0A · x₂

POL(rnil) = 0A

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(42) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(negrecip(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(negrecip(X)) → MARK(X)

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

POL(A__2NDSPOS(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = -I + -I · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

POL(2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

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

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

POL(0) = 0A

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

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

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

POL(negrecip(x₁)) = 1A + 1A · x₁

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(pi(x₁)) = 1A + 1A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = 0A + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 1A + 0A · x₁ + 0A · x₂

POL(rnil) = 1A

POL(posrecip(x₁)) = 4A + 0A · x₁

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(44) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(2ndsneg(X1, X2)) → A__2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(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__2NDSPOS(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(s(x₁)) = -I + 0A · x₁

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

POL(A__2NDSNEG(x₁, x₂)) = 0A + -I · x₁ + 0A · x₂

POL(mark(x₁)) = 0A + 0A · x₁

POL(MARK(x₁)) = 0A + 0A · x₁

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

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

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

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

POL(0) = 0A

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

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

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

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

POL(a__square(x₁)) = 0A + 0A · x₁

POL(square(x₁)) = 0A + 0A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = 1A + -I · x₁ + 1A · x₂

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(pi(x₁)) = 1A + 1A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 1A + -I · x₁ + 1A · x₂

POL(rnil) = 1A

POL(posrecip(x₁)) = 0A + -I · x₁

POL(negrecip(x₁)) = 0A + -I · x₁

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(46) Obligation:

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

A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → MARK(Z)
A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → MARK(Z)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__PLUS(s(X), Y) → MARK(Y)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes.

(48) Complex Obligation (AND)

(49) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → MARK(Y)
MARK(times(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(times(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__PLUS(x₁, x₂)) = 3A + 0A · x₁ + 0A · x₂

POL(0) = 0A

POL(MARK(x₁)) = 3A + 0A · x₁

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

POL(mark(x₁)) = -I + 0A · x₁

POL(s(x₁)) = 3A + 0A · x₁

POL(times(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(A__TIMES(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(a__times(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

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

POL(a__square(x₁)) = 5A + 2A · x₁

POL(square(x₁)) = 5A + 2A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 1A · x₁ + -I · x₂

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

POL(a__pi(x₁)) = 1A + 1A · x₁

POL(pi(x₁)) = 1A + 1A · x₁

POL(a__from(x₁)) = -I + 4A · x₁

POL(from(x₁)) = -I + 4A · x₁

POL(a__2ndspos(x₁, x₂)) = -I + 1A · x₁ + -I · x₂

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

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

POL(rnil) = 0A

POL(posrecip(x₁)) = 3A + -I · x₁

POL(negrecip(x₁)) = 0A + -I · x₁

POL(nil) = 0A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(51) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(52) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(plus(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__PLUS(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(0) = 1A

POL(MARK(x₁)) = -I + 0A · x₁

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

POL(mark(x₁)) = 0A + 0A · x₁

POL(s(x₁)) = 0A + 0A · x₁

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

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

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

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

POL(a__square(x₁)) = 5A + 5A · x₁

POL(square(x₁)) = 5A + 5A · x₁

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

POL(a__2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndsneg(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(a__pi(x₁)) = 3A + 0A · x₁

POL(pi(x₁)) = 3A + 0A · x₁

POL(a__from(x₁)) = 3A + -I · x₁

POL(from(x₁)) = 3A + -I · x₁

POL(a__2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

POL(2ndspos(x₁, x₂)) = -I + 0A · x₁ + 0A · x₂

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

POL(rnil) = 0A

POL(posrecip(x₁)) = 2A + -I · x₁

POL(negrecip(x₁)) = 0A + -I · x₁

POL(nil) = 2A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(53) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(X)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__PLUS(s(X), Y) → MARK(X)

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

POL(A__PLUS(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(0) = 0A

POL(MARK(x₁)) = 0A + 0A · x₁

POL(plus(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(mark(x₁)) = 1A + 0A · x₁

POL(s(x₁)) = -I + 0A · x₁

POL(times(x₁, x₂)) = 4A + -I · x₁ + 1A · x₂

POL(A__TIMES(x₁, x₂)) = 4A + -I · x₁ + 1A · x₂

POL(a__times(x₁, x₂)) = 4A + -I · x₁ + 1A · x₂

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

POL(a__square(x₁)) = 4A + 2A · x₁

POL(square(x₁)) = 4A + 2A · x₁

POL(a__plus(x₁, x₂)) = 4A + 1A · x₁ + 0A · x₂

POL(a__2ndsneg(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(2ndsneg(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(a__pi(x₁)) = 1A + 0A · x₁

POL(pi(x₁)) = -I + 0A · x₁

POL(a__from(x₁)) = 0A + 0A · x₁

POL(from(x₁)) = -I + 0A · x₁

POL(a__2ndspos(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

POL(2ndspos(x₁, x₂)) = 0A + -I · x₁ + -I · x₂

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

POL(rnil) = 0A

POL(posrecip(x₁)) = -I + 0A · x₁

POL(negrecip(x₁)) = 1A + -I · x₁

POL(nil) = 1A

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(55) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
MARK(rcons(X1, X2)) → MARK(X2)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(rcons(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__PLUS(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 0 /

· x₂

POL(0) =
/ 0 \

\ 0 /

POL(MARK(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(plus(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 1 /

· x₂

POL(mark(x₁)) =
/ 0 \

\ 0 /

+
/ 1 1 \

\ 0 1 /

· x₁

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁

POL(times(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(A__TIMES(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__times(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(rcons(x₁, x₂)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(a__square(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(square(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__plus(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 1 \

\ 0 1 /

· x₂

POL(a__2ndsneg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(2ndsneg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(a__pi(x₁)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(pi(x₁)) =
/ 1 \

\ 1 /

+
/ 0 1 \

\ 0 0 /

· x₁

POL(a__from(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(from(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__2ndspos(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(2ndspos(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 1 /

· x₂

POL(cons(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 1 1 /

· x₂

POL(rnil) =
/ 0 \

\ 0 /

POL(posrecip(x₁)) =
/ 0 \

\ 0 /

+
/ 0 1 \

\ 0 1 /

· x₁

POL(negrecip(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 1 /

· x₁

POL(nil) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(57) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
A__PLUS(s(X), Y) → MARK(Y)
MARK(s(X)) → MARK(X)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(plus(X1, X2)) → A__PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → A__TIMES(mark(X1), mark(X2))
A__TIMES(s(X), Y) → A__PLUS(mark(Y), a__times(mark(X), mark(Y)))
MARK(s(X)) → MARK(X)
A__TIMES(s(X), Y) → A__TIMES(mark(X), mark(Y))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS(x2)
0 = 0
MARK(x1) = MARK(x1)
plus(x1, x2) = plus(x1, x2)
mark(x1) = x1
s(x1) = s(x1)
times(x1, x2) = times(x1, x2)
A__TIMES(x1, x2) = A__TIMES(x1, x2)
a__times(x1, x2) = a__times(x1, x2)
a__square(x1) = a__square(x1)
square(x1) = square(x1)
a__plus(x1, x2) = a__plus(x1, x2)
a__2ndsneg(x1, x2) = x2
2ndsneg(x1, x2) = x2
a__pi(x1) = a__pi
pi(x1) = pi
a__from(x1) = a__from
from(x1) = from
a__2ndspos(x1, x2) = x2
2ndspos(x1, x2) = x2
cons(x1, x2) = x2
rnil = rnil
rcons(x1, x2) = x2
posrecip(x1) = x1
negrecip(x1) = x1
nil = nil

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

[a_{_square₁}, square₁] > [times₂, A_{_TIMES₂}, a_{_times₂}] > [A_{_PLUS₁}, MARK₁, plus₂, a_{_plus₂}] > s₁ > rnil
[a_{_pi}, pi] > 0 > [A_{_PLUS₁}, MARK₁, plus₂, a_{_plus₂}] > s₁ > rnil
[a_{_pi}, pi] > [a_{_from}, from] > rnil
nil > rnil

Status:

a_{_pi}: []
plus₂: multiset
rnil: multiset
A_{_TIMES₂}: [2,1]
square₁: [1]
0: multiset
a_{_plus₂}: multiset
from: []
MARK₁: [1]
A_{_PLUS₁}: [1]
times₂: [2,1]
pi: []
a_{_from}: []
s₁: [1]
a_{_times₂}: [2,1]
nil: multiset
a_{_square₁}: [1]

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(59) Obligation:

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

A__PLUS(0, Y) → MARK(Y)
A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))
A__PLUS(s(X), Y) → MARK(Y)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(60) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(61) Obligation:

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

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__PLUS(s(X), Y) → A__PLUS(mark(X), mark(Y))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A__PLUS(x1, x2) = A__PLUS(x1)
s(x1) = s(x1)
mark(x1) = x1
a__square(x1) = a__square(x1)
square(x1) = square(x1)
a__plus(x1, x2) = a__plus(x1, x2)
plus(x1, x2) = plus(x1, x2)
a__times(x1, x2) = a__times(x1, x2)
times(x1, x2) = times(x1, x2)
a__2ndsneg(x1, x2) = x2
2ndsneg(x1, x2) = x2
a__pi(x1) = a__pi(x1)
pi(x1) = pi(x1)
a__from(x1) = x1
from(x1) = x1
a__2ndspos(x1, x2) = a__2ndspos(x1)
2ndspos(x1, x2) = 2ndspos(x1)
cons(x1, x2) = cons
0 = 0
rnil = rnil
rcons(x1, x2) = x1
posrecip(x1) = posrecip
negrecip(x1) = negrecip
nil = nil

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

A_{_PLUS₁} > [cons, rnil, negrecip]
[a_{_square₁}, square₁] > [a_{_times₂}, times₂] > [a_{_plus₂}, plus₂] > s₁ > [cons, rnil, negrecip]
[a_{_pi₁}, pi₁, a_{_2ndspos₁}, 2ndspos₁, posrecip] > [cons, rnil, negrecip]
0 > [cons, rnil, negrecip]
nil > [cons, rnil, negrecip]

Status:

plus₂: multiset
rnil: multiset
a_{_2ndspos₁}: multiset
square₁: [1]
pi₁: multiset
0: multiset
a_{_plus₂}: multiset
negrecip: []
a_{_pi₁}: multiset
A_{_PLUS₁}: multiset
cons: []
times₂: multiset
s₁: multiset
a_{_times₂}: multiset
posrecip: multiset
nil: multiset
a_{_square₁}: [1]
2ndspos₁: multiset

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(63) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) Obligation:

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

A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

A__2NDSNEG(s(N), cons(X, cons(Y, Z))) → A__2NDSPOS(mark(N), mark(Z))
A__2NDSPOS(s(N), cons(X, cons(Y, Z))) → A__2NDSNEG(mark(N), mark(Z))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(A__2NDSNEG(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(s(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(cons(x₁, x₂)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 1 0 /

· x₂

POL(A__2NDSPOS(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 0 0 /

· x₂

POL(mark(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 1 1 /

· x₁

POL(a__square(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(square(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__plus(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(plus(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 1 0 \

\ 1 1 /

· x₂

POL(a__times(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(times(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__2ndsneg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(2ndsneg(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(a__pi(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(pi(x₁)) =
/ 0 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__from(x₁)) =
/ 1 \

\ 1 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(from(x₁)) =
/ 1 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(a__2ndspos(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(2ndspos(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(0) =
/ 0 \

\ 0 /

POL(rnil) =
/ 0 \

\ 0 /

POL(rcons(x₁, x₂)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂

POL(posrecip(x₁)) =
/ 0 \

\ 0 /

+
/ 1 0 \

\ 0 0 /

· x₁

POL(negrecip(x₁)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁

POL(nil) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

a__square(X) → square(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__plus(0, Y) → mark(Y)
mark(square(X)) → a__square(mark(X))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
a__square(X) → a__times(mark(X), mark(X))
mark(pi(X)) → a__pi(mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(nil) → nil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
mark(rnil) → rnil
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(negrecip(X)) → negrecip(mark(X))
mark(posrecip(X)) → posrecip(mark(X))

(68) Obligation:

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

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

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

(69) 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) Narrowing (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) QDPOrderProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) QDPOrderProof (EQUIVALENT transformation)

(12) Obligation:

(13) DependencyGraphProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) QDPOrderProof (EQUIVALENT transformation)

(18) Obligation:

(19) DependencyGraphProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) QDPOrderProof (EQUIVALENT transformation)

(24) Obligation:

(25) QDPOrderProof (EQUIVALENT transformation)

(26) Obligation:

(27) QDPOrderProof (EQUIVALENT transformation)

(28) Obligation:

(29) QDPOrderProof (EQUIVALENT transformation)

(30) Obligation:

(31) DependencyGraphProof (EQUIVALENT transformation)

(32) Obligation:

(33) QDPOrderProof (EQUIVALENT transformation)

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) QDPOrderProof (EQUIVALENT transformation)

(38) Obligation:

(39) QDPOrderProof (EQUIVALENT transformation)

(40) Obligation:

(41) QDPOrderProof (EQUIVALENT transformation)

(42) Obligation:

(43) QDPOrderProof (EQUIVALENT transformation)

(44) Obligation:

(45) QDPOrderProof (EQUIVALENT transformation)

(46) Obligation:

(47) DependencyGraphProof (EQUIVALENT transformation)

(48) Complex Obligation (AND)

(49) Obligation:

(50) QDPOrderProof (EQUIVALENT transformation)

(51) Obligation:

(52) QDPOrderProof (EQUIVALENT transformation)

(53) Obligation:

(54) QDPOrderProof (EQUIVALENT transformation)

(55) Obligation:

(56) QDPOrderProof (EQUIVALENT transformation)

(57) Obligation:

(58) QDPOrderProof (EQUIVALENT transformation)

(59) Obligation:

(60) DependencyGraphProof (EQUIVALENT transformation)

(61) Obligation:

(62) QDPOrderProof (EQUIVALENT transformation)

(63) Obligation:

(64) PisEmptyProof (EQUIVALENT transformation)

(65) TRUE

(66) Obligation:

(67) QDPOrderProof (EQUIVALENT transformation)

(68) Obligation:

(69) PisEmptyProof (EQUIVALENT transformation)

(70) TRUE