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 PisEmptyProof (⇔)
    ↳18 TRUE
  ↳19 QDP
    ↳20 QDPOrderProof (⇔)
    ↳21 QDP
    ↳22 QDPOrderProof (⇔)
    ↳23 QDP
    ↳24 PisEmptyProof (⇔)
    ↳25 TRUE
  ↳26 QDP
    ↳27 QDPOrderProof (⇔)
    ↳28 QDP
    ↳29 QDPOrderProof (⇔)
    ↳30 QDP
    ↳31 PisEmptyProof (⇔)
    ↳32 TRUE
  ↳33 QDP
    ↳34 QDPOrderProof (⇔)
    ↳35 QDP
    ↳36 QDPOrderProof (⇔)
    ↳37 QDP
    ↳38 PisEmptyProof (⇔)
    ↳39 TRUE
  ↳40 QDP
    ↳41 QDPOrderProof (⇔)
    ↳42 QDP
    ↳43 QDPOrderProof (⇔)
    ↳44 QDP
    ↳45 PisEmptyProof (⇔)
    ↳46 TRUE
  ↳47 QDP
    ↳48 QDPOrderProof (⇔)
    ↳49 QDP
    ↳50 QDPOrderProof (⇔)
    ↳51 QDP
    ↳52 PisEmptyProof (⇔)
    ↳53 TRUE
  ↳54 QDP
    ↳55 QDPOrderProof (⇔)
    ↳56 QDP
    ↳57 QDPOrderProof (⇔)
    ↳58 QDP
    ↳59 PisEmptyProof (⇔)
    ↳60 TRUE
  ↳61 QDP
    ↳62 QDPOrderProof (⇔)
    ↳63 QDP
    ↳64 QDPOrderProof (⇔)
    ↳65 QDP
    ↳66 PisEmptyProof (⇔)
    ↳67 TRUE
  ↳68 QDP
    ↳69 QDPOrderProof (⇔)
    ↳70 QDP
    ↳71 QDPOrderProof (⇔)
    ↳72 QDP
    ↳73 PisEmptyProof (⇔)
    ↳74 TRUE
  ↳75 QDP
    ↳76 QDPOrderProof (⇔)
    ↳77 QDP
    ↳78 QDPOrderProof (⇔)
    ↳79 QDP
    ↳80 PisEmptyProof (⇔)
    ↳81 TRUE
  ↳82 QDP
    ↳83 QDPOrderProof (⇔)
    ↳84 QDP
    ↳85 QDPOrderProof (⇔)
    ↳86 QDP
    ↳87 PisEmptyProof (⇔)
    ↳88 TRUE
  ↳89 QDP
    ↳90 QDPOrderProof (⇔)
    ↳91 QDP
    ↳92 QDPOrderProof (⇔)
    ↳93 QDP
    ↳94 QDPOrderProof (⇔)
    ↳95 QDP
    ↳96 QDPOrderProof (⇔)
    ↳97 QDP
    ↳98 QDPOrderProof (⇔)
    ↳99 QDP
    ↳100 QDPOrderProof (⇔)
    ↳101 QDP
    ↳102 PisEmptyProof (⇔)
    ↳103 TRUE
  ↳104 QDP
    ↳105 QDPOrderProof (⇔)
    ↳106 QDP
    ↳107 QDPOrderProof (⇔)
    ↳108 QDP
    ↳109 QDPOrderProof (⇔)
    ↳110 QDP
    ↳111 QDPOrderProof (⇔)
    ↳112 QDP
    ↳113 QDPOrderProof (⇔)
    ↳114 QDP
    ↳115 PisEmptyProof (⇔)
    ↳116 TRUE
  ↳117 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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)) → CONS(X, from(s(X)))
ACTIVE(from(X)) → FROM(s(X))
ACTIVE(from(X)) → S(X)
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(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)) → 2NDSPOS(X, from(0))
ACTIVE(pi(X)) → FROM(0)
ACTIVE(plus(s(X), Y)) → S(plus(X, Y))
ACTIVE(plus(s(X), Y)) → PLUS(X, Y)
ACTIVE(times(s(X), Y)) → PLUS(Y, times(X, Y))
ACTIVE(times(s(X), Y)) → TIMES(X, Y)
ACTIVE(square(X)) → TIMES(X, X)
ACTIVE(s(X)) → S(active(X))
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(posrecip(X)) → POSRECIP(active(X))
ACTIVE(posrecip(X)) → ACTIVE(X)
ACTIVE(negrecip(X)) → NEGRECIP(active(X))
ACTIVE(negrecip(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → CONS(active(X1), X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → RCONS(active(X1), X2)
ACTIVE(rcons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → RCONS(X1, active(X2))
ACTIVE(rcons(X1, X2)) → ACTIVE(X2)
ACTIVE(from(X)) → FROM(active(X))
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(2ndspos(X1, X2)) → 2NDSPOS(active(X1), X2)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → 2NDSPOS(X1, active(X2))
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) → 2NDSNEG(active(X1), X2)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) → 2NDSNEG(X1, active(X2))
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X2)
ACTIVE(pi(X)) → PI(active(X))
ACTIVE(pi(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → PLUS(active(X1), X2)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → PLUS(X1, active(X2))
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(times(X1, X2)) → TIMES(active(X1), X2)
ACTIVE(times(X1, X2)) → ACTIVE(X1)
ACTIVE(times(X1, X2)) → TIMES(X1, active(X2))
ACTIVE(times(X1, X2)) → ACTIVE(X2)
ACTIVE(square(X)) → SQUARE(active(X))
ACTIVE(square(X)) → ACTIVE(X)
S(mark(X)) → S(X)
POSRECIP(mark(X)) → POSRECIP(X)
NEGRECIP(mark(X)) → NEGRECIP(X)
CONS(mark(X1), X2) → CONS(X1, X2)
RCONS(mark(X1), X2) → RCONS(X1, X2)
RCONS(X1, mark(X2)) → RCONS(X1, X2)
FROM(mark(X)) → FROM(X)
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
2NDSNEG(mark(X1), X2) → 2NDSNEG(X1, X2)
2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)
PI(mark(X)) → PI(X)
PLUS(mark(X1), X2) → PLUS(X1, X2)
PLUS(X1, mark(X2)) → PLUS(X1, X2)
TIMES(mark(X1), X2) → TIMES(X1, X2)
TIMES(X1, mark(X2)) → TIMES(X1, X2)
SQUARE(mark(X)) → SQUARE(X)
PROPER(s(X)) → S(proper(X))
PROPER(s(X)) → PROPER(X)
PROPER(posrecip(X)) → POSRECIP(proper(X))
PROPER(posrecip(X)) → PROPER(X)
PROPER(negrecip(X)) → NEGRECIP(proper(X))
PROPER(negrecip(X)) → PROPER(X)
PROPER(cons(X1, X2)) → CONS(proper(X1), proper(X2))
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(rcons(X1, X2)) → RCONS(proper(X1), proper(X2))
PROPER(rcons(X1, X2)) → PROPER(X1)
PROPER(rcons(X1, X2)) → PROPER(X2)
PROPER(from(X)) → FROM(proper(X))
PROPER(from(X)) → PROPER(X)
PROPER(2ndspos(X1, X2)) → 2NDSPOS(proper(X1), proper(X2))
PROPER(2ndspos(X1, X2)) → PROPER(X1)
PROPER(2ndspos(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → 2NDSNEG(proper(X1), proper(X2))
PROPER(2ndsneg(X1, X2)) → PROPER(X1)
PROPER(2ndsneg(X1, X2)) → PROPER(X2)
PROPER(pi(X)) → PI(proper(X))
PROPER(pi(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PLUS(proper(X1), proper(X2))
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(times(X1, X2)) → TIMES(proper(X1), proper(X2))
PROPER(times(X1, X2)) → PROPER(X1)
PROPER(times(X1, X2)) → PROPER(X2)
PROPER(square(X)) → SQUARE(proper(X))
PROPER(square(X)) → PROPER(X)
S(ok(X)) → S(X)
POSRECIP(ok(X)) → POSRECIP(X)
NEGRECIP(ok(X)) → NEGRECIP(X)
CONS(ok(X1), ok(X2)) → CONS(X1, X2)
RCONS(ok(X1), ok(X2)) → RCONS(X1, X2)
FROM(ok(X)) → FROM(X)
2NDSPOS(ok(X1), ok(X2)) → 2NDSPOS(X1, X2)
2NDSNEG(ok(X1), ok(X2)) → 2NDSNEG(X1, X2)
PI(ok(X)) → PI(X)
PLUS(ok(X1), ok(X2)) → PLUS(X1, X2)
TIMES(ok(X1), ok(X2)) → TIMES(X1, X2)
SQUARE(ok(X)) → SQUARE(X)
TOP(mark(X)) → TOP(proper(X))
TOP(mark(X)) → PROPER(X)
TOP(ok(X)) → TOP(active(X))
TOP(ok(X)) → ACTIVE(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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 15 SCCs with 47 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

SQUARE(ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(ok(X)) → SQUARE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SQUARE(x1)  =  SQUARE(x1)
ok(x1)  =  ok(x1)
mark(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[SQUARE1, ok1]

Status:
SQUARE1: multiset
ok1: 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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].
Quasi-Precedence:
mark1 > SQUARE1

Status:
SQUARE1: multiset
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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(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, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

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

Status:
TIMES2: [2,1]
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(14) Obligation:

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

TIMES(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[TIMES1, ok1]

Status:
TIMES1: multiset
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(16) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(17) PisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) 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(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(20) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PLUS(X1, mark(X2)) → PLUS(X1, X2)
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, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

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

Status:
PLUS2: [2,1]
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(21) Obligation:

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

PLUS(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[PLUS1, ok1]

Status:
ok1: [1]
PLUS1: multiset


The following usable rules [FROCOS05] were oriented: none

(23) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(24) PisEmptyProof (EQUIVALENT transformation)

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

(25) TRUE

(26) Obligation:

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

PI(ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(27) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[PI1, ok1]

Status:
PI1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(28) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(29) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PI(mark(X)) → PI(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > PI1

Status:
PI1: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(30) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(31) PisEmptyProof (EQUIVALENT transformation)

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

(32) TRUE

(33) 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(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(34) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


2NDSNEG(X1, mark(X2)) → 2NDSNEG(X1, X2)
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, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

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

Status:
2NDSNEG2: [2,1]
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(35) Obligation:

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

2NDSNEG(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(36) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


2NDSNEG(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[2NDSNEG1, ok1]

Status:
2NDSNEG1: multiset
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(37) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(38) PisEmptyProof (EQUIVALENT transformation)

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

(39) TRUE

(40) Obligation:

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

2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
2NDSPOS(mark(X1), X2) → 2NDSPOS(X1, X2)
2NDSPOS(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(41) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


2NDSPOS(X1, mark(X2)) → 2NDSPOS(X1, X2)
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, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

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

Status:
mark1: multiset
2NDSPOS2: [2,1]


The following usable rules [FROCOS05] were oriented: none

(42) Obligation:

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

2NDSPOS(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


2NDSPOS(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[2NDSPOS1, ok1]

Status:
ok1: [1]
2NDSPOS1: multiset


The following usable rules [FROCOS05] were oriented: none

(44) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(45) PisEmptyProof (EQUIVALENT transformation)

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

(46) TRUE

(47) Obligation:

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

FROM(ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(48) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[FROM1, ok1]

Status:
ok1: multiset
FROM1: multiset


The following usable rules [FROCOS05] were oriented: none

(49) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(50) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FROM(mark(X)) → FROM(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
mark1 > FROM1

Status:
mark1: multiset
FROM1: multiset


The following usable rules [FROCOS05] were oriented: none

(51) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(52) PisEmptyProof (EQUIVALENT transformation)

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

(53) TRUE

(54) 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(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(55) 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(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, x2)
mark(x1)  =  mark(x1)
ok(x1)  =  x1

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

Status:
RCONS2: [2,1]
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(56) Obligation:

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

RCONS(ok(X1), ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(57) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


RCONS(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[RCONS1, ok1]

Status:
RCONS1: multiset
ok1: [1]


The following usable rules [FROCOS05] were oriented: none

(58) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(59) PisEmptyProof (EQUIVALENT transformation)

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

(60) TRUE

(61) Obligation:

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

CONS(ok(X1), ok(X2)) → CONS(X1, X2)
CONS(mark(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


CONS(ok(X1), ok(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(x2)
ok(x1)  =  ok(x1)
mark(x1)  =  mark

Recursive path order with status [RPO].
Quasi-Precedence:
ok1 > CONS1
mark > CONS1

Status:
CONS1: multiset
mark: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(63) Obligation:

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

CONS(mark(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(64) 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)

Recursive path order with status [RPO].
Quasi-Precedence:
[CONS1, mark1]

Status:
CONS1: multiset
mark1: [1]


The following usable rules [FROCOS05] were oriented: none

(65) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(66) PisEmptyProof (EQUIVALENT transformation)

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

(67) TRUE

(68) Obligation:

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

NEGRECIP(ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[NEGRECIP1, ok1]

Status:
NEGRECIP1: multiset
ok1: multiset


The following usable rules [FROCOS05] were oriented: none

(70) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(71) 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].
Quasi-Precedence:
mark1 > NEGRECIP1

Status:
NEGRECIP1: multiset
mark1: multiset


The following usable rules [FROCOS05] were oriented: none

(72) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(73) PisEmptyProof (EQUIVALENT transformation)

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

(74) TRUE

(75) Obligation:

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

POSRECIP(ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(76) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[POSRECIP1, ok1]

Status:
ok1: multiset
POSRECIP1: multiset


The following usable rules [FROCOS05] were oriented: none

(77) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(78) 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].
Quasi-Precedence:
mark1 > POSRECIP1

Status:
mark1: multiset
POSRECIP1: multiset


The following usable rules [FROCOS05] were oriented: none

(79) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(80) PisEmptyProof (EQUIVALENT transformation)

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

(81) TRUE

(82) Obligation:

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

S(ok(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(83) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[S1, ok1]

Status:
ok1: multiset
S1: multiset


The following usable rules [FROCOS05] were oriented: none

(84) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(85) 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].
Quasi-Precedence:
mark1 > S1

Status:
mark1: multiset
S1: multiset


The following usable rules [FROCOS05] were oriented: none

(86) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(87) PisEmptyProof (EQUIVALENT transformation)

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

(88) TRUE

(89) Obligation:

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

PROPER(posrecip(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(negrecip(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(rcons(X1, X2)) → PROPER(X1)
PROPER(rcons(X1, X2)) → PROPER(X2)
PROPER(from(X)) → PROPER(X)
PROPER(2ndspos(X1, X2)) → PROPER(X1)
PROPER(2ndspos(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → PROPER(X1)
PROPER(2ndsneg(X1, X2)) → PROPER(X2)
PROPER(pi(X)) → PROPER(X)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(times(X1, X2)) → PROPER(X1)
PROPER(times(X1, X2)) → PROPER(X2)
PROPER(square(X)) → PROPER(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(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.


PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(rcons(X1, X2)) → PROPER(X1)
PROPER(rcons(X1, X2)) → PROPER(X2)
PROPER(2ndspos(X1, X2)) → PROPER(X1)
PROPER(2ndspos(X1, X2)) → PROPER(X2)
PROPER(2ndsneg(X1, X2)) → PROPER(X1)
PROPER(2ndsneg(X1, X2)) → PROPER(X2)
PROPER(plus(X1, X2)) → PROPER(X1)
PROPER(plus(X1, X2)) → PROPER(X2)
PROPER(times(X1, X2)) → PROPER(X1)
PROPER(times(X1, X2)) → PROPER(X2)
PROPER(square(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
posrecip(x1)  =  x1
s(x1)  =  x1
negrecip(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
rcons(x1, x2)  =  rcons(x1, x2)
from(x1)  =  x1
2ndspos(x1, x2)  =  2ndspos(x1, x2)
2ndsneg(x1, x2)  =  2ndsneg(x1, x2)
pi(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
times(x1, x2)  =  times(x1, x2)
square(x1)  =  square(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
cons2 > [PROPER1, times2]
2ndsneg2 > [PROPER1, times2]
plus2 > [PROPER1, times2]
square1 > [PROPER1, times2]

Status:
plus2: multiset
cons2: multiset
PROPER1: multiset
2ndspos2: [2,1]
times2: multiset
rcons2: multiset
square1: [1]
2ndsneg2: [1,2]


The following usable rules [FROCOS05] were oriented: none

(91) Obligation:

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

PROPER(posrecip(X)) → PROPER(X)
PROPER(s(X)) → PROPER(X)
PROPER(negrecip(X)) → PROPER(X)
PROPER(from(X)) → PROPER(X)
PROPER(pi(X)) → PROPER(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(posrecip(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
posrecip(x1)  =  posrecip(x1)
s(x1)  =  x1
negrecip(x1)  =  x1
from(x1)  =  x1
pi(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, posrecip1]

Status:
PROPER1: multiset
posrecip1: multiset


The following usable rules [FROCOS05] were oriented: none

(93) Obligation:

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

PROPER(s(X)) → PROPER(X)
PROPER(negrecip(X)) → PROPER(X)
PROPER(from(X)) → PROPER(X)
PROPER(pi(X)) → PROPER(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(from(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
s(x1)  =  x1
negrecip(x1)  =  x1
from(x1)  =  from(x1)
pi(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, from1]

Status:
from1: multiset
PROPER1: multiset


The following usable rules [FROCOS05] were oriented: none

(95) Obligation:

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

PROPER(s(X)) → PROPER(X)
PROPER(negrecip(X)) → PROPER(X)
PROPER(pi(X)) → PROPER(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(96) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(s(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
s(x1)  =  s(x1)
negrecip(x1)  =  x1
pi(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, s1]

Status:
PROPER1: multiset
s1: multiset


The following usable rules [FROCOS05] were oriented: none

(97) Obligation:

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

PROPER(negrecip(X)) → PROPER(X)
PROPER(pi(X)) → PROPER(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(98) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(negrecip(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
negrecip(x1)  =  negrecip(x1)
pi(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[PROPER1, negrecip1]

Status:
PROPER1: multiset
negrecip1: multiset


The following usable rules [FROCOS05] were oriented: none

(99) Obligation:

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

PROPER(pi(X)) → PROPER(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


PROPER(pi(X)) → PROPER(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
pi1 > PROPER1

Status:
PROPER1: multiset
pi1: multiset


The following usable rules [FROCOS05] were oriented: none

(101) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(102) PisEmptyProof (EQUIVALENT transformation)

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

(103) TRUE

(104) Obligation:

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

ACTIVE(posrecip(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(negrecip(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → ACTIVE(X2)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X2)
ACTIVE(pi(X)) → ACTIVE(X)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(times(X1, X2)) → ACTIVE(X1)
ACTIVE(times(X1, X2)) → ACTIVE(X2)
ACTIVE(square(X)) → ACTIVE(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(105) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(rcons(X1, X2)) → ACTIVE(X1)
ACTIVE(rcons(X1, X2)) → ACTIVE(X2)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndspos(X1, X2)) → ACTIVE(X2)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X1)
ACTIVE(2ndsneg(X1, X2)) → ACTIVE(X2)
ACTIVE(plus(X1, X2)) → ACTIVE(X1)
ACTIVE(plus(X1, X2)) → ACTIVE(X2)
ACTIVE(times(X1, X2)) → ACTIVE(X1)
ACTIVE(times(X1, X2)) → ACTIVE(X2)
ACTIVE(square(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
posrecip(x1)  =  x1
s(x1)  =  x1
negrecip(x1)  =  x1
cons(x1, x2)  =  x1
rcons(x1, x2)  =  rcons(x1, x2)
from(x1)  =  x1
2ndspos(x1, x2)  =  2ndspos(x1, x2)
2ndsneg(x1, x2)  =  2ndsneg(x1, x2)
pi(x1)  =  x1
plus(x1, x2)  =  plus(x1, x2)
times(x1, x2)  =  times(x1, x2)
square(x1)  =  square(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
rcons2 > [ACTIVE1, square1]
2ndspos2 > [ACTIVE1, square1]
2ndsneg2 > [ACTIVE1, square1]
plus2 > [ACTIVE1, square1]
times2 > [ACTIVE1, square1]

Status:
plus2: multiset
2ndspos2: multiset
times2: multiset
rcons2: multiset
square1: [1]
2ndsneg2: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(106) Obligation:

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

ACTIVE(posrecip(X)) → ACTIVE(X)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(negrecip(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(pi(X)) → ACTIVE(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(107) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(posrecip(X)) → ACTIVE(X)
ACTIVE(pi(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
posrecip(x1)  =  posrecip(x1)
s(x1)  =  x1
negrecip(x1)  =  x1
cons(x1, x2)  =  x1
from(x1)  =  x1
pi(x1)  =  pi(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, pi1]

Status:
posrecip1: multiset
pi1: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(108) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(negrecip(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(109) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
negrecip(x1)  =  x1
cons(x1, x2)  =  cons(x1)
from(x1)  =  from(x1)

Recursive path order with status [RPO].
Quasi-Precedence:
cons1 > ACTIVE1

Status:
from1: multiset
cons1: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(110) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(negrecip(X)) → ACTIVE(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(111) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(s(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  s(x1)
negrecip(x1)  =  x1

Recursive path order with status [RPO].
Quasi-Precedence:
[ACTIVE1, s1]

Status:
s1: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(112) Obligation:

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

ACTIVE(negrecip(X)) → ACTIVE(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(113) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


ACTIVE(negrecip(X)) → ACTIVE(X)
The remaining pairs can at least be oriented weakly.
Used ordering: Recursive path order with status [RPO].
Quasi-Precedence:
negrecip1 > ACTIVE1

Status:
negrecip1: multiset
ACTIVE1: multiset


The following usable rules [FROCOS05] were oriented: none

(114) 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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

(115) PisEmptyProof (EQUIVALENT transformation)

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

(116) TRUE

(117) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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))
active(s(X)) → s(active(X))
active(posrecip(X)) → posrecip(active(X))
active(negrecip(X)) → negrecip(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(rcons(X1, X2)) → rcons(active(X1), X2)
active(rcons(X1, X2)) → rcons(X1, active(X2))
active(from(X)) → from(active(X))
active(2ndspos(X1, X2)) → 2ndspos(active(X1), X2)
active(2ndspos(X1, X2)) → 2ndspos(X1, active(X2))
active(2ndsneg(X1, X2)) → 2ndsneg(active(X1), X2)
active(2ndsneg(X1, X2)) → 2ndsneg(X1, active(X2))
active(pi(X)) → pi(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(times(X1, X2)) → times(active(X1), X2)
active(times(X1, X2)) → times(X1, active(X2))
active(square(X)) → square(active(X))
s(mark(X)) → mark(s(X))
posrecip(mark(X)) → mark(posrecip(X))
negrecip(mark(X)) → mark(negrecip(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
rcons(mark(X1), X2) → mark(rcons(X1, X2))
rcons(X1, mark(X2)) → mark(rcons(X1, X2))
from(mark(X)) → mark(from(X))
2ndspos(mark(X1), X2) → mark(2ndspos(X1, X2))
2ndspos(X1, mark(X2)) → mark(2ndspos(X1, X2))
2ndsneg(mark(X1), X2) → mark(2ndsneg(X1, X2))
2ndsneg(X1, mark(X2)) → mark(2ndsneg(X1, X2))
pi(mark(X)) → mark(pi(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
times(mark(X1), X2) → mark(times(X1, X2))
times(X1, mark(X2)) → mark(times(X1, X2))
square(mark(X)) → mark(square(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(posrecip(X)) → posrecip(proper(X))
proper(negrecip(X)) → negrecip(proper(X))
proper(nil) → ok(nil)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(rnil) → ok(rnil)
proper(rcons(X1, X2)) → rcons(proper(X1), proper(X2))
proper(from(X)) → from(proper(X))
proper(2ndspos(X1, X2)) → 2ndspos(proper(X1), proper(X2))
proper(2ndsneg(X1, X2)) → 2ndsneg(proper(X1), proper(X2))
proper(pi(X)) → pi(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(times(X1, X2)) → times(proper(X1), proper(X2))
proper(square(X)) → square(proper(X))
s(ok(X)) → ok(s(X))
posrecip(ok(X)) → ok(posrecip(X))
negrecip(ok(X)) → ok(negrecip(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
rcons(ok(X1), ok(X2)) → ok(rcons(X1, X2))
from(ok(X)) → ok(from(X))
2ndspos(ok(X1), ok(X2)) → ok(2ndspos(X1, X2))
2ndsneg(ok(X1), ok(X2)) → ok(2ndsneg(X1, X2))
pi(ok(X)) → ok(pi(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
times(ok(X1), ok(X2)) → ok(times(X1, X2))
square(ok(X)) → ok(square(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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