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

(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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, Z))) → MARK(2ndspos(s(N), cons2(X, Z)))
ACTIVE(2ndspos(s(N), cons(X, Z))) → 2NDSPOS(s(N), cons2(X, Z))
ACTIVE(2ndspos(s(N), cons(X, Z))) → CONS2(X, Z)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → RCONS(posrecip(Y), 2ndsneg(N, Z))
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → POSRECIP(Y)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → 2NDSNEG(N, Z)
ACTIVE(2ndsneg(0, Z)) → MARK(rnil)
ACTIVE(2ndsneg(s(N), cons(X, Z))) → MARK(2ndsneg(s(N), cons2(X, Z)))
ACTIVE(2ndsneg(s(N), cons(X, Z))) → 2NDSNEG(s(N), cons2(X, Z))
ACTIVE(2ndsneg(s(N), cons(X, Z))) → CONS2(X, Z)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → RCONS(negrecip(Y), 2ndspos(N, Z))
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → NEGRECIP(Y)
ACTIVE(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → ACTIVE(cons2(X1, mark(X2)))
MARK(cons2(X1, X2)) → CONS2(X1, mark(X2))
MARK(cons2(X1, X2)) → MARK(X2)
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)
CONS2(mark(X1), X2) → CONS2(X1, X2)
CONS2(X1, mark(X2)) → CONS2(X1, X2)
CONS2(active(X1), X2) → CONS2(X1, X2)
CONS2(X1, active(X2)) → CONS2(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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 14 SCCs with 38 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SQUARE(active(X)) → SQUARE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SQUARE(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SQUARE(mark(X)) → SQUARE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SQUARE(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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) PisEmptyProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TIMES(X1, active(X2)) → TIMES(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TIMES(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(14) 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)

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TIMES(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

TIMES(X1, mark(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

TIMES(X1, mark(X2)) → TIMES(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TIMES(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(19) PisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE

(21) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(22) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, active(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(23) 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)

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(24) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

PLUS(X1, mark(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PLUS(X1, mark(X2)) → PLUS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PLUS(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(27) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(28) PisEmptyProof (EQUIVALENT transformation)

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

(29) TRUE

(30) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(31) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PI(active(X)) → PI(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PI(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(32) Obligation:

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

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PI(mark(X)) → PI(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PI(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(34) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(35) PisEmptyProof (EQUIVALENT transformation)

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

(36) TRUE

(37) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(38) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NEGRECIP(active(X)) → NEGRECIP(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NEGRECIP(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(39) Obligation:

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

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(40) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

NEGRECIP(mark(X)) → NEGRECIP(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
NEGRECIP(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(41) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(42) PisEmptyProof (EQUIVALENT transformation)

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

(43) TRUE

(44) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSNEG(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(46) 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)

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(47) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSNEG(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

2NDSNEG(X1, mark(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSNEG(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(50) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(51) PisEmptyProof (EQUIVALENT transformation)

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

(52) TRUE

(53) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(54) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

POSRECIP(active(X)) → POSRECIP(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
POSRECIP(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(55) Obligation:

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

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(56) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

POSRECIP(mark(X)) → POSRECIP(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
POSRECIP(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(57) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(58) PisEmptyProof (EQUIVALENT transformation)

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

(59) TRUE

(60) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(61) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

RCONS(X1, active(X2)) → RCONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
RCONS(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(62) 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)

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(63) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
RCONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

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

RCONS(X1, mark(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

RCONS(X1, mark(X2)) → RCONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
RCONS(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(67) PisEmptyProof (EQUIVALENT transformation)

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

(68) TRUE

(69) Obligation:

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

CONS2(X1, mark(X2)) → CONS2(X1, X2)
CONS2(mark(X1), X2) → CONS2(X1, X2)
CONS2(active(X1), X2) → CONS2(X1, X2)
CONS2(X1, active(X2)) → CONS2(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(70) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS2(X1, active(X2)) → CONS2(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS2(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(71) Obligation:

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

CONS2(X1, mark(X2)) → CONS2(X1, X2)
CONS2(mark(X1), X2) → CONS2(X1, X2)
CONS2(active(X1), X2) → CONS2(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS2(mark(X1), X2) → CONS2(X1, X2)
CONS2(active(X1), X2) → CONS2(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS2(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

CONS2(X1, mark(X2)) → CONS2(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS2(X1, mark(X2)) → CONS2(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS2(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSPOS(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(80) 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)

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSPOS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

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

2NDSPOS(X1, mark(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSPOS(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(84) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(85) PisEmptyProof (EQUIVALENT transformation)

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

(86) TRUE

(87) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(active(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
S(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(92) PisEmptyProof (EQUIVALENT transformation)

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

(93) TRUE

(94) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, active(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
mark(x1)  =  x1
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(96) 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)

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(97) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x1
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(98) Obligation:

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

CONS(X1, mark(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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(99) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(X1, mark(X2)) → CONS(X1, X2)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  x2
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(100) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(101) PisEmptyProof (EQUIVALENT transformation)

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

(102) TRUE

(103) 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(active(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1)  =  x1
active(x1)  =  active(x1)
mark(x1)  =  x1

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

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

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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(mark(X)) → FROM(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FROM(x1)  =  x1
mark(x1)  =  mark(x1)

Lexicographic Path Order [LPO].
Precedence:

trivial

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

Q DP problem:
P is empty.
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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(108) PisEmptyProof (EQUIVALENT transformation)

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

(109) TRUE

(110) 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, Z))) → MARK(2ndspos(s(N), cons2(X, Z)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
ACTIVE(2ndsneg(s(N), cons(X, Z))) → MARK(2ndsneg(s(N), cons2(X, Z)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
MARK(cons2(X1, X2)) → ACTIVE(cons2(X1, mark(X2)))
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → ACTIVE(rcons(mark(X1), mark(X2)))
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
MARK(posrecip(X)) → ACTIVE(posrecip(mark(X)))
ACTIVE(plus(0, Y)) → MARK(Y)
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
MARK(negrecip(X)) → ACTIVE(negrecip(mark(X)))
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
ACTIVE(square(X)) → MARK(times(X, 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(111) 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(cons2(X1, X2)) → ACTIVE(cons2(X1, mark(X2)))
MARK(posrecip(X)) → ACTIVE(posrecip(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(x1)
cons(x1, x2)  =  cons
s(x1)  =  s
2ndspos(x1, x2)  =  2ndspos
cons2(x1, x2)  =  cons2
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(x1)
rnil  =  rnil

Lexicographic Path Order [LPO].
Precedence:

active₁ > mark₁ > [MARK, from, 2ndspos, rcons, 2ndsneg, pi, plus, times, square] > [cons, s, cons2] > [posrecip, negrecip, 0, rnil]

The following usable rules [FROCOS05] were oriented:

from(active(X)) → from(X)
from(mark(X)) → from(X)
s(active(X)) → s(X)
s(mark(X)) → s(X)
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)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(X1, X2)
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)
posrecip(active(X)) → posrecip(X)
posrecip(mark(X)) → posrecip(X)
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)
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)
negrecip(active(X)) → negrecip(X)
negrecip(mark(X)) → negrecip(X)
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)
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)
pi(active(X)) → pi(X)
pi(mark(X)) → pi(X)
square(active(X)) → square(X)
square(mark(X)) → square(X)

(112) 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, Z))) → MARK(2ndspos(s(N), cons2(X, Z)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
ACTIVE(2ndsneg(s(N), cons(X, Z))) → MARK(2ndsneg(s(N), cons2(X, Z)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
MARK(cons2(X1, X2)) → MARK(X2)
MARK(rcons(X1, X2)) → ACTIVE(rcons(mark(X1), mark(X2)))
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
ACTIVE(plus(0, Y)) → MARK(Y)
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
ACTIVE(square(X)) → MARK(times(X, 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.

(113) 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)))

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(x1, x2)
s(x1)  =  s(x1)
2ndspos(x1, x2)  =  2ndspos
cons2(x1, x2)  =  cons2(x2)
rcons(x1, x2)  =  rcons
posrecip(x1)  =  posrecip(x1)
2ndsneg(x1, x2)  =  2ndsneg
negrecip(x1)  =  negrecip(x1)
pi(x1)  =  pi
0  =  0
plus(x1, x2)  =  plus
times(x1, x2)  =  times
square(x1)  =  square
active(x1)  =  x1
rnil  =  rnil

Lexicographic Path Order [LPO].
Precedence:

[MARK, from, 2ndspos, 2ndsneg, pi, plus, times, square] > [cons₂, s₁] > [mark, cons2₁, rcons, posrecip₁, negrecip₁, 0]
rnil > [mark, cons2₁, rcons, posrecip₁, negrecip₁, 0]

The following usable rules [FROCOS05] were oriented:

from(active(X)) → from(X)
from(mark(X)) → from(X)
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)
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)
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)
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)
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)
pi(active(X)) → pi(X)
pi(mark(X)) → pi(X)
square(active(X)) → square(X)
square(mark(X)) → square(X)

(114) 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, Z))) → MARK(2ndspos(s(N), cons2(X, Z)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(2ndspos(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
MARK(s(X)) → MARK(X)
MARK(2ndspos(X1, X2)) → ACTIVE(2ndspos(mark(X1), mark(X2)))
ACTIVE(2ndsneg(s(N), cons(X, Z))) → MARK(2ndsneg(s(N), cons2(X, Z)))
MARK(2ndspos(X1, X2)) → MARK(X1)
MARK(2ndspos(X1, X2)) → MARK(X2)
ACTIVE(2ndsneg(s(N), cons2(X, cons(Y, Z)))) → MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
MARK(cons2(X1, X2)) → MARK(X2)
ACTIVE(pi(X)) → MARK(2ndspos(X, from(0)))
MARK(rcons(X1, X2)) → MARK(X1)
MARK(rcons(X1, X2)) → MARK(X2)
ACTIVE(plus(0, Y)) → MARK(Y)
MARK(posrecip(X)) → MARK(X)
MARK(2ndsneg(X1, X2)) → ACTIVE(2ndsneg(mark(X1), mark(X2)))
ACTIVE(plus(s(X), Y)) → MARK(s(plus(X, Y)))
MARK(2ndsneg(X1, X2)) → MARK(X1)
MARK(2ndsneg(X1, X2)) → MARK(X2)
ACTIVE(times(s(X), Y)) → MARK(plus(Y, times(X, Y)))
MARK(negrecip(X)) → MARK(X)
MARK(pi(X)) → ACTIVE(pi(mark(X)))
ACTIVE(square(X)) → MARK(times(X, 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, Z))) → mark(2ndspos(s(N), cons2(X, Z)))
active(2ndspos(s(N), cons2(X, cons(Y, Z)))) → mark(rcons(posrecip(Y), 2ndsneg(N, Z)))
active(2ndsneg(0, Z)) → mark(rnil)
active(2ndsneg(s(N), cons(X, Z))) → mark(2ndsneg(s(N), cons2(X, Z)))
active(2ndsneg(s(N), cons2(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(cons2(X1, X2)) → active(cons2(X1, mark(X2)))
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)
cons2(mark(X1), X2) → cons2(X1, X2)
cons2(X1, mark(X2)) → cons2(X1, X2)
cons2(active(X1), X2) → cons2(X1, X2)
cons2(X1, active(X2)) → cons2(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.