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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
  ↳5 QDP
    ↳6 QDPOrderProof (⇔)
    ↳7 QDP
    ↳8 QDPOrderProof (⇔)
    ↳9 QDP
    ↳10 PisEmptyProof (⇔)
    ↳11 TRUE
  ↳12 QDP
    ↳13 QDPOrderProof (⇔)
    ↳14 QDP
    ↳15 QDPOrderProof (⇔)
    ↳16 QDP
    ↳17 QDPOrderProof (⇔)
    ↳18 QDP
    ↳19 PisEmptyProof (⇔)
    ↳20 TRUE
  ↳21 QDP
    ↳22 QDPOrderProof (⇔)
    ↳23 QDP
    ↳24 QDPOrderProof (⇔)
    ↳25 QDP
    ↳26 QDPOrderProof (⇔)
    ↳27 QDP
    ↳28 PisEmptyProof (⇔)
    ↳29 TRUE
  ↳30 QDP
    ↳31 QDPOrderProof (⇔)
    ↳32 QDP
    ↳33 QDPOrderProof (⇔)
    ↳34 QDP
    ↳35 PisEmptyProof (⇔)
    ↳36 TRUE
  ↳37 QDP
    ↳38 QDPOrderProof (⇔)
    ↳39 QDP
    ↳40 QDPOrderProof (⇔)
    ↳41 QDP
    ↳42 PisEmptyProof (⇔)
    ↳43 TRUE
  ↳44 QDP
    ↳45 QDPOrderProof (⇔)
    ↳46 QDP
    ↳47 QDPOrderProof (⇔)
    ↳48 QDP
    ↳49 QDPOrderProof (⇔)
    ↳50 QDP
    ↳51 PisEmptyProof (⇔)
    ↳52 TRUE
  ↳53 QDP
    ↳54 QDPOrderProof (⇔)
    ↳55 QDP
    ↳56 QDPOrderProof (⇔)
    ↳57 QDP
    ↳58 PisEmptyProof (⇔)
    ↳59 TRUE
  ↳60 QDP
    ↳61 QDPOrderProof (⇔)
    ↳62 QDP
    ↳63 QDPOrderProof (⇔)
    ↳64 QDP
    ↳65 QDPOrderProof (⇔)
    ↳66 QDP
    ↳67 PisEmptyProof (⇔)
    ↳68 TRUE
  ↳69 QDP
    ↳70 QDPOrderProof (⇔)
    ↳71 QDP
    ↳72 QDPOrderProof (⇔)
    ↳73 QDP
    ↳74 QDPOrderProof (⇔)
    ↳75 QDP
    ↳76 PisEmptyProof (⇔)
    ↳77 TRUE
  ↳78 QDP
    ↳79 QDPOrderProof (⇔)
    ↳80 QDP
    ↳81 QDPOrderProof (⇔)
    ↳82 QDP
    ↳83 PisEmptyProof (⇔)
    ↳84 TRUE
  ↳85 QDP
    ↳86 QDPOrderProof (⇔)
    ↳87 QDP
    ↳88 QDPOrderProof (⇔)
    ↳89 QDP
    ↳90 QDPOrderProof (⇔)
    ↳91 QDP
    ↳92 PisEmptyProof (⇔)
    ↳93 TRUE
  ↳94 QDP
    ↳95 QDPOrderProof (⇔)
    ↳96 QDP
    ↳97 QDPOrderProof (⇔)
    ↳98 QDP
    ↳99 PisEmptyProof (⇔)
    ↳100 TRUE
  ↳101 QDP

(0) Obligation:

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

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

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

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

(4) Complex Obligation (AND)

(5) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(6) 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

Recursive path order with status [RPO].
Precedence:
trivial

Status:
active1: multiset

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

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

(8) 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: Recursive path order with status [RPO].
Precedence:
mark1 > SQUARE1

Status:
SQUARE1: [1]
mark1: multiset

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

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

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

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

(13) 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)
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)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Precedence:
trivial

Status:
mark1: multiset
active1: multiset

The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

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

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

(15) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
TIMES(x1, x2)  =  TIMES(x1)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Precedence:
mark1 > TIMES1

Status:
TIMES1: multiset
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

(16) Obligation:

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

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

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


TIMES(active(X1), X2) → TIMES(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
active1 > TIMES2

Status:
TIMES2: [1,2]
active1: multiset

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

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

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

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

(22) 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)
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)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Precedence:
trivial

Status:
mark1: multiset
active1: multiset

The following usable rules [FROCOS05] were oriented: none

(23) Obligation:

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

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

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

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

Recursive path order with status [RPO].
Precedence:
mark1 > PLUS1

Status:
PLUS1: multiset
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

(25) Obligation:

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

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

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(active(X1), X2) → PLUS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
active1 > PLUS2

Status:
PLUS2: [1,2]
active1: multiset

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

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

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

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

(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

Recursive path order with status [RPO].
Precedence:
trivial

Status:
active1: multiset

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

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

(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: Recursive path order with status [RPO].
Precedence:
mark1 > PI1

Status:
PI1: [1]
mark1: multiset

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

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

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

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

(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

Recursive path order with status [RPO].
Precedence:
trivial

Status:
active1: multiset

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

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

(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: Recursive path order with status [RPO].
Precedence:
mark1 > NEGRECIP1

Status:
NEGRECIP1: [1]
mark1: multiset

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

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

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

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

(45) 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)
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)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Precedence:
trivial

Status:
mark1: multiset
active1: multiset

The following usable rules [FROCOS05] were oriented: none

(46) Obligation:

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

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

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

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

Recursive path order with status [RPO].
Precedence:
mark1 > 2NDSNEG1

Status:
2NDSNEG1: multiset
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

(48) Obligation:

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

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

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

(49) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


2NDSNEG(active(X1), X2) → 2NDSNEG(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
active1 > 2NDSNEG2

Status:
2NDSNEG2: [1,2]
active1: multiset

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

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

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

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

(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

Recursive path order with status [RPO].
Precedence:
trivial

Status:
active1: multiset

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

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

(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: Recursive path order with status [RPO].
Precedence:
mark1 > POSRECIP1

Status:
POSRECIP1: [1]
mark1: multiset

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

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

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

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

(61) 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)
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)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Precedence:
trivial

Status:
mark1: multiset
active1: multiset

The following usable rules [FROCOS05] were oriented: none

(62) Obligation:

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

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

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

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

Recursive path order with status [RPO].
Precedence:
mark1 > RCONS1

Status:
RCONS1: multiset
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

(64) Obligation:

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

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

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


RCONS(active(X1), X2) → RCONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
active1 > RCONS2

Status:
RCONS2: [1,2]
active1: multiset

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(70) 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)
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)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Precedence:
trivial

Status:
mark1: multiset
active1: multiset

The following usable rules [FROCOS05] were oriented: none

(71) Obligation:

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

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

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

(72) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
2NDSPOS(x1, x2)  =  2NDSPOS(x1)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Precedence:
mark1 > 2NDSPOS1

Status:
2NDSPOS1: multiset
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

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

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


2NDSPOS(active(X1), X2) → 2NDSPOS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
active1 > 2NDSPOS2

Status:
2NDSPOS2: [1,2]
active1: multiset

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

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

(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:

S(active(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

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

(79) 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

Recursive path order with status [RPO].
Precedence:
trivial

Status:
active1: multiset

The following usable rules [FROCOS05] were oriented: none

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

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

(81) 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: Recursive path order with status [RPO].
Precedence:
mark1 > S1

Status:
S1: [1]
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

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

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

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) TRUE

(85) Obligation:

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

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

The TRS R consists of the following rules:

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

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

(86) 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)
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)  =  mark(x1)
active(x1)  =  active(x1)

Recursive path order with status [RPO].
Precedence:
trivial

Status:
mark1: multiset
active1: multiset

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

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

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

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

(88) 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)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
CONS(x1, x2)  =  CONS(x1)
mark(x1)  =  mark(x1)
active(x1)  =  x1

Recursive path order with status [RPO].
Precedence:
mark1 > CONS1

Status:
CONS1: multiset
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

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

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

(90) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(active(X1), X2) → CONS(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Precedence:
active1 > CONS2

Status:
CONS2: [1,2]
active1: multiset

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

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

(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:

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

The TRS R consists of the following rules:

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

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

(95) 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

Recursive path order with status [RPO].
Precedence:
trivial

Status:
active1: multiset

The following usable rules [FROCOS05] were oriented: none

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

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

(97) 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: Recursive path order with status [RPO].
Precedence:
mark1 > FROM1

Status:
FROM1: [1]
mark1: multiset

The following usable rules [FROCOS05] were oriented: none

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

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

(99) PisEmptyProof (EQUIVALENT transformation)

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

(100) TRUE

(101) Obligation:

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

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

The TRS R consists of the following rules:

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

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