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

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 DependencyPairsProof (⇔)

↳2 QDP

↳3 DependencyGraphProof (⇔)

↳4 AND

  ↳5 QDP

  ↳6 QDP

  ↳7 QDP

  ↳8 QDP

  ↳9 QDP

  ↳10 QDP

  ↳11 QDP

  ↳12 QDP

  ↳13 QDP

  ↳14 QDP

  ↳15 QDP

  ↳16 QDP

  ↳17 QDP

    ↳18 QDPOrderProof (⇔)

    ↳19 QDP

    ↳20 QDPOrderProof (⇔)

    ↳21 QDP

(0) Obligation:

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

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(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:

ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
ACTIVE(from(X)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
ACTIVE(2ndspos(0, Z)) → MARK(rnil)
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → RCONS(posrecip(Y), 2ndsneg(N, Z))
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → POSRECIP(Y)
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → 2NDSNEG(N, Z)
ACTIVE(2ndsneg(0, Z)) → MARK(rnil)
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → RCONS(negrecip(Y), 2ndspos(N, Z))
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → NEGRECIP(Y)
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → 2NDSPOS(N, Z)
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
ACTIVE(pi(X)) → 2NDSPOS(X, from(0))
ACTIVE(pi(X)) → FROM(0)
ACTIVE(plus(0, Y)) → MARK(Y)
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
ACTIVE(plus(s(X), Y)) → S(plus(X, Y))
ACTIVE(plus(s(X), Y)) → PLUS(X, Y)
ACTIVE(times(0, Y)) → MARK(0)
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
ACTIVE(times(s(X), Y)) → PLUS(Y, times(X, Y))
ACTIVE(times(s(X), Y)) → TIMES(X, Y)
ACTIVE(square(X)) → MARK(times(X, X))
ACTIVE(square(X)) → TIMES(X, X)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(from(X)) → FROM(mark(X))
MARK(from(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(s(X)) → S(mark(X))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
MARK(2ndspos(X1, X2)) → 2NDSPOS(mark(X1), mark(X2))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(0) → ACTIVE(0)
MARK(rnil) → ACTIVE(rnil)
MARK(rcons(X1, X2)) → ACTIVE(rcons(mark(X1), mark(X2)))
MARK(rcons(X1, X2)) → RCONS(mark(X1), mark(X2))
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
MARK(posrecip(X)) → ACTIVE(posrecip(mark(X)))
MARK(posrecip(X)) → POSRECIP(mark(X))
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
MARK(2ndsneg(X1, X2)) → 2NDSNEG(mark(X1), mark(X2))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(negrecip(X)) → ACTIVE(negrecip(mark(X)))
MARK(negrecip(X)) → NEGRECIP(mark(X))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
MARK(pi(X)) → PI(mark(X))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → PLUS(mark(X1), mark(X2))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → ACTIVE(times(mark(X1), mark(X2)))
MARK(times(X1, X2)) → TIMES(mark(X1), mark(X2))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → ACTIVE(square(mark(X)))
MARK(square(X)) → SQUARE(mark(X))
MARK(square(X)) → MARK(X)
FROM(mark(X)) → FROM(X)
FROM(active(X)) → FROM(X)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
S(mark(X)) → S(X)
S(active(X)) → S(X)
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
2NDSPOS(active(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(X1, active(X2)) → 2NDSPOS(X1, X2)
RCONS(mark(X1), X2) → RCONS(X1, X2)
RCONS(X1, mark(X2)) → RCONS(X1, X2)
RCONS(active(X1), X2) → RCONS(X1, X2)
RCONS(X1, active(X2)) → RCONS(X1, X2)
POSRECIP(mark(X)) → POSRECIP(X)
POSRECIP(active(X)) → POSRECIP(X)
2NDSNEG(mark(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)
2NDSNEG(active(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(X1, active(X2)) → 2NDSNEG(X1, X2)
NEGRECIP(mark(X)) → NEGRECIP(X)
NEGRECIP(active(X)) → NEGRECIP(X)
PI(mark(X)) → PI(X)
PI(active(X)) → PI(X)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)
TIMES(mark(X1), X2) → TIMES(X1, X2)
TIMES(X1, mark(X2)) → TIMES(X1, X2)
TIMES(active(X1), X2) → TIMES(X1, X2)
TIMES(X1, active(X2)) → TIMES(X1, X2)
SQUARE(mark(X)) → SQUARE(X)
SQUARE(active(X)) → SQUARE(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 13 SCCs with 33 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

SQUARE(active(X)) → SQUARE(X)
SQUARE(mark(X)) → SQUARE(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(6) Obligation:

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

TIMES(X1, mark(X2)) → TIMES(X1, X2)
TIMES(mark(X1), X2) → TIMES(X1, X2)
TIMES(active(X1), X2) → TIMES(X1, X2)
TIMES(X1, active(X2)) → TIMES(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(7) Obligation:

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

PLUS(X1, mark(X2)) → PLUS(X1, X2)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(active(X1), X2) → PLUS(X1, X2)
PLUS(X1, active(X2)) → PLUS(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(8) Obligation:

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

PI(active(X)) → PI(X)
PI(mark(X)) → PI(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(9) Obligation:

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

NEGRECIP(active(X)) → NEGRECIP(X)
NEGRECIP(mark(X)) → NEGRECIP(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(10) Obligation:

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

2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)
2NDSNEG(mark(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(active(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(X1, active(X2)) → 2NDSNEG(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(11) Obligation:

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

POSRECIP(active(X)) → POSRECIP(X)
POSRECIP(mark(X)) → POSRECIP(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(12) Obligation:

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

RCONS(X1, mark(X2)) → RCONS(X1, X2)
RCONS(mark(X1), X2) → RCONS(X1, X2)
RCONS(active(X1), X2) → RCONS(X1, X2)
RCONS(X1, active(X2)) → RCONS(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(13) Obligation:

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

2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(active(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(X1, active(X2)) → 2NDSPOS(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(14) Obligation:

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(15) Obligation:

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

CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(16) Obligation:

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

FROM(active(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(17) Obligation:

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

MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → ACTIVE(rcons(mark(X1), mark(X2)))
ACTIVE(plus(0, Y)) → MARK(Y)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
MARK(posrecip(X)) → ACTIVE(posrecip(mark(X)))
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(negrecip(X)) → ACTIVE(negrecip(mark(X)))
ACTIVE(square(X)) → MARK(times(X, X))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → ACTIVE(times(mark(X1), mark(X2)))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → ACTIVE(square(mark(X)))
MARK(square(X)) → MARK(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(negrecip(X)) → ACTIVE(negrecip(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK
from(x1) = from
ACTIVE(x1) = x1
mark(x1) = mark
cons(x1, x2) = cons
s(x1) = s
2ndspos(x1, x2) = 2ndspos
rcons(x1, x2) = rcons
posrecip(x1) = posrecip
2ndsneg(x1, x2) = 2ndsneg
negrecip(x1) = negrecip
pi(x1) = pi
0 = 0
plus(x1, x2) = plus
times(x1, x2) = times
square(x1) = square
active(x1) = x1
rnil = rnil

Recursive Path Order [RPO].
Precedence:

[MARK, from, mark, 2ndspos, rcons, posrecip, 2ndsneg, pi, 0, plus, times, square] > [cons, negrecip]
[MARK, from, mark, 2ndspos, rcons, posrecip, 2ndsneg, pi, 0, plus, times, square] > s
[MARK, from, mark, 2ndspos, rcons, posrecip, 2ndsneg, pi, 0, plus, times, square] > rnil

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

(19) Obligation:

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

MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → ACTIVE(rcons(mark(X1), mark(X2)))
ACTIVE(plus(0, Y)) → MARK(Y)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
MARK(posrecip(X)) → ACTIVE(posrecip(mark(X)))
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
ACTIVE(square(X)) → MARK(times(X, X))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → ACTIVE(times(mark(X1), mark(X2)))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → ACTIVE(square(mark(X)))
MARK(square(X)) → MARK(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MARK(rcons(X1, X2)) → ACTIVE(rcons(mark(X1), mark(X2)))
MARK(posrecip(X)) → ACTIVE(posrecip(mark(X)))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MARK(x1) = MARK
from(x1) = from
ACTIVE(x1) = x1
mark(x1) = mark
cons(x1, x2) = cons
s(x1) = s
2ndspos(x1, x2) = 2ndspos
rcons(x1, x2) = rcons
posrecip(x1) = posrecip
2ndsneg(x1, x2) = 2ndsneg
negrecip(x1) = negrecip
pi(x1) = pi
0 = 0
plus(x1, x2) = plus
times(x1, x2) = times
square(x1) = square
active(x1) = active
rnil = rnil

Recursive Path Order [RPO].
Precedence:

cons > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > rcons > [mark, active] > rnil
cons > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > posrecip > [mark, active] > rnil
cons > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > negrecip > [mark, active] > rnil
s > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > rcons > [mark, active] > rnil
s > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > posrecip > [mark, active] > rnil
s > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > negrecip > [mark, active] > rnil
0 > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > rcons > [mark, active] > rnil
0 > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > posrecip > [mark, active] > rnil
0 > [MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > negrecip > [mark, active] > rnil

The following usable rules [FROCOS05] were oriented:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

(21) Obligation:

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

MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(from(X)) → MARK(X)
ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
ACTIVE(plus(0, Y)) → MARK(Y)
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
ACTIVE(square(X)) → MARK(times(X, X))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
MARK(pi(X)) → MARK(X)
MARK(plus(X1, X2)) → ACTIVE(plus(mark(X1), mark(X2)))
MARK(plus(X1, X2)) → MARK(X1)
MARK(plus(X1, X2)) → MARK(X2)
MARK(times(X1, X2)) → ACTIVE(times(mark(X1), mark(X2)))
MARK(times(X1, X2)) → MARK(X1)
MARK(times(X1, X2)) → MARK(X2)
MARK(square(X)) → ACTIVE(square(mark(X)))
MARK(square(X)) → MARK(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(2ndspos(0, Z)) → mark(rnil)
active(2ndspos(s(N), cons(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, cons(Y, Z)))) → mark(rcons(negrecip(Y), 2ndspos(N, Z)))
active(pi(X)) → mark(2ndspos(X, from(0)))
active(plus(0, Y)) → mark(Y)
active(plus(s(X), Y)) → mark(s(plus(X, Y)))
active(times(0, Y)) → mark(0)
active(times(s(X), Y)) → mark(plus(Y, times(X, Y)))
active(square(X)) → mark(times(X, X))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(2ndspos(X1, X2)) → active(2ndspos(mark(X1), mark(X2)))
mark(0) → active(0)
mark(rnil) → active(rnil)
mark(rcons(X1, X2)) → active(rcons(mark(X1), mark(X2)))
mark(posrecip(X)) → active(posrecip(mark(X)))
mark(2ndsneg(X1, X2)) → active(2ndsneg(mark(X1), mark(X2)))
mark(negrecip(X)) → active(negrecip(mark(X)))
mark(pi(X)) → active(pi(mark(X)))
mark(plus(X1, X2)) → active(plus(mark(X1), mark(X2)))
mark(times(X1, X2)) → active(times(mark(X1), mark(X2)))
mark(square(X)) → active(square(mark(X)))
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
2ndspos(mark(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, mark(X2)) → 2ndspos(X1, X2)
2ndspos(active(X1), X2) → 2ndspos(X1, X2)
2ndspos(X1, active(X2)) → 2ndspos(X1, X2)
rcons(mark(X1), X2) → rcons(X1, X2)
rcons(X1, mark(X2)) → rcons(X1, X2)
rcons(active(X1), X2) → rcons(X1, X2)
rcons(X1, active(X2)) → rcons(X1, X2)
posrecip(mark(X)) → posrecip(X)
posrecip(active(X)) → posrecip(X)
2ndsneg(mark(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, mark(X2)) → 2ndsneg(X1, X2)
2ndsneg(active(X1), X2) → 2ndsneg(X1, X2)
2ndsneg(X1, active(X2)) → 2ndsneg(X1, X2)
negrecip(mark(X)) → negrecip(X)
negrecip(active(X)) → negrecip(X)
pi(mark(X)) → pi(X)
pi(active(X)) → pi(X)
plus(mark(X1), X2) → plus(X1, X2)
plus(X1, mark(X2)) → plus(X1, X2)
plus(active(X1), X2) → plus(X1, X2)
plus(X1, active(X2)) → plus(X1, X2)
times(mark(X1), X2) → times(X1, X2)
times(X1, mark(X2)) → times(X1, X2)
times(active(X1), X2) → times(X1, X2)
times(X1, active(X2)) → times(X1, X2)
square(mark(X)) → square(X)
square(active(X)) → square(X)

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