/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 79 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) UsableRulesProof [EQUIVALENT, 0 ms]
        (17) QDP
        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (19) YES
    (20) QDP
        (21) UsableRulesProof [EQUIVALENT, 0 ms]
        (22) QDP
        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (24) YES
    (25) QDP
        (26) UsableRulesProof [EQUIVALENT, 0 ms]
        (27) QDP
        (28) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (29) YES
    (30) QDP
        (31) UsableRulesProof [EQUIVALENT, 0 ms]
        (32) QDP
        (33) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (34) YES
    (35) QDP
        (36) UsableRulesProof [EQUIVALENT, 0 ms]
        (37) QDP
        (38) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (39) YES
    (40) QDP
        (41) UsableRulesProof [EQUIVALENT, 0 ms]
        (42) QDP
        (43) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (44) YES
    (45) QDP
        (46) UsableRulesProof [EQUIVALENT, 0 ms]
        (47) QDP
        (48) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (49) YES
    (50) QDP
        (51) UsableRulesProof [EQUIVALENT, 0 ms]
        (52) QDP
        (53) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (54) YES
    (55) QDP
        (56) UsableRulesProof [EQUIVALENT, 0 ms]
        (57) QDP
        (58) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (59) YES
    (60) QDP
        (61) UsableRulesProof [EQUIVALENT, 0 ms]
        (62) QDP
        (63) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (64) YES
    (65) QDP
        (66) QDPOrderProof [EQUIVALENT, 422 ms]
        (67) QDP
        (68) QDPOrderProof [EQUIVALENT, 608 ms]
        (69) QDP
        (70) QDPOrderProof [EQUIVALENT, 552 ms]
        (71) QDP
        (72) QDPOrderProof [EQUIVALENT, 534 ms]
        (73) QDP
        (74) QDPOrderProof [EQUIVALENT, 538 ms]
        (75) QDP
        (76) QDPOrderProof [EQUIVALENT, 549 ms]
        (77) QDP
        (78) QDPOrderProof [EQUIVALENT, 522 ms]
        (79) QDP
        (80) QDPOrderProof [EQUIVALENT, 224 ms]
        (81) QDP
        (82) QDPOrderProof [EQUIVALENT, 529 ms]
        (83) QDP
        (84) QDPOrderProof [EQUIVALENT, 539 ms]
        (85) QDP
        (86) QDPOrderProof [EQUIVALENT, 577 ms]
        (87) QDP
        (88) QDPOrderProof [EQUIVALENT, 12 ms]
        (89) QDP
        (90) PisEmptyProof [EQUIVALENT, 0 ms]
        (91) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 13 SCCs with 33 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

----------------------------------------

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SQUARE(active(X)) -> SQUARE(X)
   SQUARE(mark(X)) -> SQUARE(X)

The TRS R consists of the following rules:

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

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

(6) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   SQUARE(active(X)) -> SQUARE(X)
   SQUARE(mark(X)) -> SQUARE(X)

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

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*SQUARE(active(X)) -> SQUARE(X)
The graph contains the following edges 1 > 1


*SQUARE(mark(X)) -> SQUARE(X)
The graph contains the following edges 1 > 1


----------------------------------------

(9)
YES

----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TIMES(X1, mark(X2)) -> TIMES(X1, X2)
   TIMES(mark(X1), X2) -> TIMES(X1, X2)
   TIMES(active(X1), X2) -> TIMES(X1, X2)
   TIMES(X1, active(X2)) -> TIMES(X1, X2)

The TRS R consists of the following rules:

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

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

(11) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   TIMES(X1, mark(X2)) -> TIMES(X1, X2)
   TIMES(mark(X1), X2) -> TIMES(X1, X2)
   TIMES(active(X1), X2) -> TIMES(X1, X2)
   TIMES(X1, active(X2)) -> TIMES(X1, X2)

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

(13) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*TIMES(X1, mark(X2)) -> TIMES(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*TIMES(mark(X1), X2) -> TIMES(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*TIMES(active(X1), X2) -> TIMES(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*TIMES(X1, active(X2)) -> TIMES(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(14)
YES

----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
   PLUS(mark(X1), X2) -> PLUS(X1, X2)
   PLUS(active(X1), X2) -> PLUS(X1, X2)
   PLUS(X1, active(X2)) -> PLUS(X1, X2)

The TRS R consists of the following rules:

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

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

(16) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(17)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUS(X1, mark(X2)) -> PLUS(X1, X2)
   PLUS(mark(X1), X2) -> PLUS(X1, X2)
   PLUS(active(X1), X2) -> PLUS(X1, X2)
   PLUS(X1, active(X2)) -> PLUS(X1, X2)

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

(18) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*PLUS(X1, mark(X2)) -> PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*PLUS(mark(X1), X2) -> PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*PLUS(active(X1), X2) -> PLUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*PLUS(X1, active(X2)) -> PLUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(19)
YES

----------------------------------------

(20)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PI(active(X)) -> PI(X)
   PI(mark(X)) -> PI(X)

The TRS R consists of the following rules:

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

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

(21) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(22)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PI(active(X)) -> PI(X)
   PI(mark(X)) -> PI(X)

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

(23) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*PI(active(X)) -> PI(X)
The graph contains the following edges 1 > 1


*PI(mark(X)) -> PI(X)
The graph contains the following edges 1 > 1


----------------------------------------

(24)
YES

----------------------------------------

(25)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   NEGRECIP(active(X)) -> NEGRECIP(X)
   NEGRECIP(mark(X)) -> NEGRECIP(X)

The TRS R consists of the following rules:

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

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

(26) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(27)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   NEGRECIP(active(X)) -> NEGRECIP(X)
   NEGRECIP(mark(X)) -> NEGRECIP(X)

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

(28) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*NEGRECIP(active(X)) -> NEGRECIP(X)
The graph contains the following edges 1 > 1


*NEGRECIP(mark(X)) -> NEGRECIP(X)
The graph contains the following edges 1 > 1


----------------------------------------

(29)
YES

----------------------------------------

(30)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
   2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
   2NDSNEG(active(X1), X2) -> 2NDSNEG(X1, X2)
   2NDSNEG(X1, active(X2)) -> 2NDSNEG(X1, X2)

The TRS R consists of the following rules:

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

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

(31) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(32)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
   2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
   2NDSNEG(active(X1), X2) -> 2NDSNEG(X1, X2)
   2NDSNEG(X1, active(X2)) -> 2NDSNEG(X1, X2)

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

(33) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*2NDSNEG(X1, mark(X2)) -> 2NDSNEG(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*2NDSNEG(mark(X1), X2) -> 2NDSNEG(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*2NDSNEG(active(X1), X2) -> 2NDSNEG(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*2NDSNEG(X1, active(X2)) -> 2NDSNEG(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(34)
YES

----------------------------------------

(35)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   POSRECIP(active(X)) -> POSRECIP(X)
   POSRECIP(mark(X)) -> POSRECIP(X)

The TRS R consists of the following rules:

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

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

(36) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(37)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   POSRECIP(active(X)) -> POSRECIP(X)
   POSRECIP(mark(X)) -> POSRECIP(X)

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

(38) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*POSRECIP(active(X)) -> POSRECIP(X)
The graph contains the following edges 1 > 1


*POSRECIP(mark(X)) -> POSRECIP(X)
The graph contains the following edges 1 > 1


----------------------------------------

(39)
YES

----------------------------------------

(40)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   RCONS(X1, mark(X2)) -> RCONS(X1, X2)
   RCONS(mark(X1), X2) -> RCONS(X1, X2)
   RCONS(active(X1), X2) -> RCONS(X1, X2)
   RCONS(X1, active(X2)) -> RCONS(X1, X2)

The TRS R consists of the following rules:

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

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

(41) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(42)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   RCONS(X1, mark(X2)) -> RCONS(X1, X2)
   RCONS(mark(X1), X2) -> RCONS(X1, X2)
   RCONS(active(X1), X2) -> RCONS(X1, X2)
   RCONS(X1, active(X2)) -> RCONS(X1, X2)

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

(43) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*RCONS(X1, mark(X2)) -> RCONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*RCONS(mark(X1), X2) -> RCONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*RCONS(active(X1), X2) -> RCONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*RCONS(X1, active(X2)) -> RCONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(44)
YES

----------------------------------------

(45)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
   2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
   2NDSPOS(active(X1), X2) -> 2NDSPOS(X1, X2)
   2NDSPOS(X1, active(X2)) -> 2NDSPOS(X1, X2)

The TRS R consists of the following rules:

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

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

(46) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(47)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
   2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
   2NDSPOS(active(X1), X2) -> 2NDSPOS(X1, X2)
   2NDSPOS(X1, active(X2)) -> 2NDSPOS(X1, X2)

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

(48) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*2NDSPOS(X1, mark(X2)) -> 2NDSPOS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*2NDSPOS(mark(X1), X2) -> 2NDSPOS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*2NDSPOS(active(X1), X2) -> 2NDSPOS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*2NDSPOS(X1, active(X2)) -> 2NDSPOS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(49)
YES

----------------------------------------

(50)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

The TRS R consists of the following rules:

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

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

(51) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(52)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   S(active(X)) -> S(X)
   S(mark(X)) -> S(X)

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

(53) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*S(active(X)) -> S(X)
The graph contains the following edges 1 > 1


*S(mark(X)) -> S(X)
The graph contains the following edges 1 > 1


----------------------------------------

(54)
YES

----------------------------------------

(55)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

The TRS R consists of the following rules:

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

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

(56) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(57)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   CONS(X1, mark(X2)) -> CONS(X1, X2)
   CONS(mark(X1), X2) -> CONS(X1, X2)
   CONS(active(X1), X2) -> CONS(X1, X2)
   CONS(X1, active(X2)) -> CONS(X1, X2)

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

(58) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*CONS(X1, mark(X2)) -> CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


*CONS(mark(X1), X2) -> CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*CONS(active(X1), X2) -> CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2


*CONS(X1, active(X2)) -> CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2


----------------------------------------

(59)
YES

----------------------------------------

(60)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   FROM(active(X)) -> FROM(X)
   FROM(mark(X)) -> FROM(X)

The TRS R consists of the following rules:

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

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

(61) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(62)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   FROM(active(X)) -> FROM(X)
   FROM(mark(X)) -> FROM(X)

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

(63) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*FROM(active(X)) -> FROM(X)
The graph contains the following edges 1 > 1


*FROM(mark(X)) -> FROM(X)
The graph contains the following edges 1 > 1


----------------------------------------

(64)
YES

----------------------------------------

(65)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(66) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2))
   MARK(s(X)) -> ACTIVE(s(mark(X)))
   MARK(rcons(X1, X2)) -> ACTIVE(rcons(mark(X1), mark(X2)))
   MARK(posrecip(X)) -> ACTIVE(posrecip(mark(X)))
   MARK(negrecip(X)) -> ACTIVE(negrecip(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( 2ndsneg_2(x_1, x_2) ) = 2
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( 2ndspos_2(x_1, x_2) ) = 2
POL( cons_2(x_1, x_2) ) = 0
POL( from_1(x_1) ) = 2
POL( negrecip_1(x_1) ) = max{0, -2}
POL( pi_1(x_1) ) = 2
POL( plus_2(x_1, x_2) ) = 2
POL( posrecip_1(x_1) ) = 1
POL( rcons_2(x_1, x_2) ) = max{0, -2}
POL( s_1(x_1) ) = max{0, -2}
POL( square_1(x_1) ) = 2
POL( times_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = max{0, -2}
POL( active_1(x_1) ) = max{0, 2x_1 - 2}
POL( 0 ) = 2
POL( rnil ) = 2
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   from(active(X)) -> from(X)
   from(mark(X)) -> from(X)
   s(active(X)) -> s(X)
   s(mark(X)) -> s(X)
   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)
   posrecip(active(X)) -> posrecip(X)
   posrecip(mark(X)) -> posrecip(X)
   2ndsneg(X1, mark(X2)) -> 2ndsneg(X1, X2)
   2ndsneg(mark(X1), X2) -> 2ndsneg(X1, X2)
   2ndsneg(active(X1), X2) -> 2ndsneg(X1, X2)
   2ndsneg(X1, active(X2)) -> 2ndsneg(X1, X2)
   rcons(X1, mark(X2)) -> rcons(X1, X2)
   rcons(mark(X1), X2) -> rcons(X1, X2)
   rcons(active(X1), X2) -> rcons(X1, X2)
   rcons(X1, active(X2)) -> rcons(X1, X2)
   negrecip(active(X)) -> negrecip(X)
   negrecip(mark(X)) -> negrecip(X)
   2ndspos(X1, mark(X2)) -> 2ndspos(X1, X2)
   2ndspos(mark(X1), X2) -> 2ndspos(X1, X2)
   2ndspos(active(X1), X2) -> 2ndspos(X1, X2)
   2ndspos(X1, active(X2)) -> 2ndspos(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   times(X1, mark(X2)) -> times(X1, X2)
   times(mark(X1), X2) -> times(X1, X2)
   times(active(X1), X2) -> times(X1, X2)
   times(X1, active(X2)) -> times(X1, X2)
   pi(active(X)) -> pi(X)
   pi(mark(X)) -> pi(X)
   square(active(X)) -> square(X)
   square(mark(X)) -> square(X)


----------------------------------------

(67)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(68) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(rcons(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[1A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(square(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(69)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(70) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(posrecip(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[2A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(square(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(71)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(72) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(negrecip(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(square(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(73)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(74) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(square(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(square(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(75)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(76) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(pi(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[1A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[1A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(square(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(77)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(78) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(square(X)) -> MARK(times(X, X))
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[-I]] 	 +  	[[-I]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(square(x_1)) =  	[[4A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(79)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(80) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(square(X)) -> ACTIVE(square(mark(X)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:

POL( 2ndsneg_2(x_1, x_2) ) = 2
POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1}
POL( 2ndspos_2(x_1, x_2) ) = 2
POL( from_1(x_1) ) = 2
POL( pi_1(x_1) ) = 2
POL( plus_2(x_1, x_2) ) = 2
POL( square_1(x_1) ) = 0
POL( times_2(x_1, x_2) ) = 2
POL( mark_1(x_1) ) = 1
POL( active_1(x_1) ) = max{0, -2}
POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2}
POL( s_1(x_1) ) = max{0, 2x_1 - 2}
POL( rcons_2(x_1, x_2) ) = max{0, 2x_1 - 2}
POL( posrecip_1(x_1) ) = x_1 + 1
POL( negrecip_1(x_1) ) = x_1 + 1
POL( 0 ) = 0
POL( rnil ) = 0
POL( MARK_1(x_1) ) = 1

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   from(active(X)) -> from(X)
   from(mark(X)) -> from(X)
   2ndsneg(X1, mark(X2)) -> 2ndsneg(X1, X2)
   2ndsneg(mark(X1), X2) -> 2ndsneg(X1, X2)
   2ndsneg(active(X1), X2) -> 2ndsneg(X1, X2)
   2ndsneg(X1, active(X2)) -> 2ndsneg(X1, X2)
   2ndspos(X1, mark(X2)) -> 2ndspos(X1, X2)
   2ndspos(mark(X1), X2) -> 2ndspos(X1, X2)
   2ndspos(active(X1), X2) -> 2ndspos(X1, X2)
   2ndspos(X1, active(X2)) -> 2ndspos(X1, X2)
   plus(X1, mark(X2)) -> plus(X1, X2)
   plus(mark(X1), X2) -> plus(X1, X2)
   plus(active(X1), X2) -> plus(X1, X2)
   plus(X1, active(X2)) -> plus(X1, X2)
   times(X1, mark(X2)) -> times(X1, X2)
   times(mark(X1), X2) -> times(X1, X2)
   times(active(X1), X2) -> times(X1, X2)
   times(X1, active(X2)) -> times(X1, X2)
   pi(active(X)) -> pi(X)
   pi(mark(X)) -> pi(X)
   square(active(X)) -> square(X)
   square(mark(X)) -> square(X)


----------------------------------------

(81)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(82) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(from(X)) -> MARK(X)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[-I]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[1A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[0A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(active(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(square(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(83)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(84) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(cons(X1, X2)) -> MARK(X1)
The remaining pairs can at least be oriented weakly.
Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

   <<<
 POL(MARK(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(from(x_1)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1
>>>

   <<<
 POL(ACTIVE(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(mark(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[3A]] 	 +  	[[1A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(s(x_1)) =  	[[2A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndspos(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(rcons(x_1, x_2)) =  	[[3A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(posrecip(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(2ndsneg(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(negrecip(x_1)) =  	[[0A]] 	 +  	[[-I]] 	* 	x_1
>>>

   <<<
 POL(pi(x_1)) =  	[[4A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(0) =  	[[3A]]
>>>

   <<<
 POL(plus(x_1, x_2)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(times(x_1, x_2)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1 	 +  	[[0A]] 	* 	x_2
>>>

   <<<
 POL(active(x_1)) =  	[[0A]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(square(x_1)) =  	[[-I]] 	 +  	[[0A]] 	* 	x_1
>>>

   <<<
 POL(rnil) =  	[[0A]]
>>>


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(85)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

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

(86) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   ACTIVE(2ndspos(s(N), cons(X, cons(Y, Z)))) -> MARK(rcons(posrecip(Y), 2ndsneg(N, Z)))
   ACTIVE(2ndsneg(s(N), cons(X, cons(Y, Z)))) -> MARK(rcons(negrecip(Y), 2ndspos(N, Z)))
   MARK(s(X)) -> MARK(X)
   ACTIVE(pi(X)) -> MARK(2ndspos(X, from(0)))
   MARK(2ndspos(X1, X2)) -> MARK(X1)
   MARK(2ndspos(X1, X2)) -> MARK(X2)
   ACTIVE(plus(0, Y)) -> MARK(Y)
   MARK(rcons(X1, X2)) -> MARK(X2)
   ACTIVE(plus(s(X), Y)) -> MARK(s(plus(X, Y)))
   ACTIVE(times(s(X), Y)) -> MARK(plus(Y, times(X, Y)))
   MARK(2ndsneg(X1, X2)) -> MARK(X1)
   MARK(2ndsneg(X1, X2)) -> MARK(X2)
   MARK(plus(X1, X2)) -> MARK(X1)
   MARK(plus(X1, X2)) -> MARK(X2)
   MARK(times(X1, X2)) -> MARK(X1)
   MARK(times(X1, X2)) -> MARK(X2)
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
MARK(x1)  =  x1

from(x1)  =  from

ACTIVE(x1)  =  x1

mark(x1)  =  x1

cons(x1, x2)  =  x2

s(x1)  =  s(x1)

2ndspos(x1, x2)  =  2ndspos(x1, x2)

rcons(x1, x2)  =  rcons(x1, x2)

posrecip(x1)  =  posrecip

2ndsneg(x1, x2)  =  2ndsneg(x1, x2)

negrecip(x1)  =  negrecip

pi(x1)  =  pi(x1)

0  =  0

plus(x1, x2)  =  plus(x1, x2)

times(x1, x2)  =  times(x1, x2)

active(x1)  =  x1

square(x1)  =  square(x1)

rnil  =  rnil


Recursive path order with status [RPO].
Quasi-Precedence: pi_1 > from > [s_1, negrecip] > [2ndspos_2, 2ndsneg_2] > rcons_2
pi_1 > from > [s_1, negrecip] > posrecip
pi_1 > 0 > rnil
square_1 > times_2 > 0 > rnil
square_1 > times_2 > plus_2 > [s_1, negrecip] > [2ndspos_2, 2ndsneg_2] > rcons_2
square_1 > times_2 > plus_2 > [s_1, negrecip] > posrecip

Status: from: []
s_1: [1]
2ndspos_2: multiset status
rcons_2: [1,2]
posrecip: multiset status
2ndsneg_2: multiset status
negrecip: []
pi_1: [1]
0: multiset status
plus_2: multiset status
times_2: [1,2]
square_1: [1]
rnil: multiset status


The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

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


----------------------------------------

(87)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MARK(from(X)) -> ACTIVE(from(mark(X)))
   ACTIVE(from(X)) -> MARK(cons(X, from(s(X))))
   MARK(2ndspos(X1, X2)) -> ACTIVE(2ndspos(mark(X1), mark(X2)))
   MARK(2ndsneg(X1, X2)) -> ACTIVE(2ndsneg(mark(X1), mark(X2)))
   MARK(pi(X)) -> ACTIVE(pi(mark(X)))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(times(X1, X2)) -> ACTIVE(times(mark(X1), mark(X2)))

The TRS R consists of the following rules:

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

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

(88) QDPOrderProof (EQUIVALENT)
We use the reduction pair processor [LPAR04,JAR06].


The following pairs can be oriented strictly and are deleted.

   MARK(from(X)) -> ACTIVE(from(mark(X)))
   ACTIVE(from(X)) -> MARK(cons(X, from(s(X))))
   MARK(2ndspos(X1, X2)) -> ACTIVE(2ndspos(mark(X1), mark(X2)))
   MARK(2ndsneg(X1, X2)) -> ACTIVE(2ndsneg(mark(X1), mark(X2)))
   MARK(pi(X)) -> ACTIVE(pi(mark(X)))
   MARK(plus(X1, X2)) -> ACTIVE(plus(mark(X1), mark(X2)))
   MARK(times(X1, X2)) -> ACTIVE(times(mark(X1), mark(X2)))
The remaining pairs can at least be oriented weakly.
Used ordering:  Combined order from the following AFS and order.
MARK(x1)  =  x1

from(x1)  =  from

ACTIVE(x1)  =  ACTIVE

cons(x1, x2)  =  cons

2ndspos(x1, x2)  =  2ndspos

2ndsneg(x1, x2)  =  2ndsneg

pi(x1)  =  pi

plus(x1, x2)  =  plus

times(x1, x2)  =  times


Knuth-Bendix order [KBO] with precedence:plus > ACTIVE > cons

and weight map:

   plus=1
   2ndspos=2
   pi=2
   ACTIVE=1
   cons=1
   2ndsneg=2
   from=2
   times=2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

   cons(X1, mark(X2)) -> cons(X1, X2)
   cons(mark(X1), X2) -> cons(X1, X2)
   cons(active(X1), X2) -> cons(X1, X2)
   cons(X1, active(X2)) -> cons(X1, X2)


----------------------------------------

(89)
Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:

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

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

(90) PisEmptyProof (EQUIVALENT)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
----------------------------------------

(91)
YES