Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(negrecip1(X)) -> NEGRECIP1(proper1(X))
2NDSPOS2(ok1(X1), ok1(X2)) -> 2NDSPOS2(X1, X2)
ACTIVE1(2ndsneg2(X1, X2)) -> 2NDSNEG2(X1, active1(X2))
S1(mark1(X)) -> S1(X)
ACTIVE1(times2(X1, X2)) -> TIMES2(X1, active1(X2))
ACTIVE1(square1(X)) -> SQUARE1(active1(X))
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
PROPER1(pi1(X)) -> PI1(proper1(X))
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
TOP1(mark1(X)) -> TOP1(proper1(X))
2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)
ACTIVE1(pi1(X)) -> PI1(active1(X))
ACTIVE1(pi1(X)) -> 2NDSPOS2(X, from1(0))
ACTIVE1(times2(s1(X), Y)) -> TIMES2(X, Y)
ACTIVE1(2ndspos2(X1, X2)) -> 2NDSPOS2(active1(X1), X2)
ACTIVE1(2ndspos2(X1, X2)) -> 2NDSPOS2(X1, active1(X2))
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(square1(X)) -> SQUARE1(proper1(X))
PLUS2(mark1(X1), X2) -> PLUS2(X1, X2)
RCONS2(ok1(X1), ok1(X2)) -> RCONS2(X1, X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
ACTIVE1(times2(X1, X2)) -> TIMES2(active1(X1), X2)
ACTIVE1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> RCONS2(negrecip1(Y), 2ndspos2(N, Z))
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(square1(X)) -> PROPER1(X)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> NEGRECIP1(active1(X))
TIMES2(X1, mark1(X2)) -> TIMES2(X1, X2)
ACTIVE1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> POSRECIP1(Y)
SQUARE1(mark1(X)) -> SQUARE1(X)
ACTIVE1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> RCONS2(posrecip1(Y), 2ndsneg2(N, Z))
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
PROPER1(2ndsneg2(X1, X2)) -> 2NDSNEG2(proper1(X1), proper1(X2))
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> 2NDSNEG2(N, Z)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
POSRECIP1(ok1(X)) -> POSRECIP1(X)
ACTIVE1(times2(s1(X), Y)) -> PLUS2(Y, times2(X, Y))
ACTIVE1(plus2(X1, X2)) -> PLUS2(X1, active1(X2))
ACTIVE1(s1(X)) -> ACTIVE1(X)
2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)
ACTIVE1(2ndsneg2(X1, X2)) -> 2NDSNEG2(active1(X1), X2)
PI1(ok1(X)) -> PI1(X)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
RCONS2(X1, mark1(X2)) -> RCONS2(X1, X2)
PROPER1(pi1(X)) -> PROPER1(X)
ACTIVE1(pi1(X)) -> FROM1(0)
ACTIVE1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> NEGRECIP1(Y)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
ACTIVE1(square1(X)) -> ACTIVE1(X)
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
PROPER1(times2(X1, X2)) -> TIMES2(proper1(X1), proper1(X2))
2NDSNEG2(ok1(X1), ok1(X2)) -> 2NDSNEG2(X1, X2)
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(posrecip1(X)) -> POSRECIP1(proper1(X))
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> PLUS2(active1(X1), X2)
NEGRECIP1(mark1(X)) -> NEGRECIP1(X)
ACTIVE1(rcons2(X1, X2)) -> RCONS2(active1(X1), X2)
2NDSPOS2(mark1(X1), X2) -> 2NDSPOS2(X1, X2)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X2)
ACTIVE1(rcons2(X1, X2)) -> RCONS2(X1, active1(X2))
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)
TOP1(ok1(X)) -> ACTIVE1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(2ndspos2(X1, X2)) -> 2NDSPOS2(proper1(X1), proper1(X2))
ACTIVE1(pi1(X)) -> ACTIVE1(X)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(square1(X)) -> TIMES2(X, X)
PI1(mark1(X)) -> PI1(X)
PROPER1(rcons2(X1, X2)) -> RCONS2(proper1(X1), proper1(X2))
PLUS2(ok1(X1), ok1(X2)) -> PLUS2(X1, X2)
RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
POSRECIP1(mark1(X)) -> POSRECIP1(X)
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
PROPER1(negrecip1(X)) -> PROPER1(X)
SQUARE1(ok1(X)) -> SQUARE1(X)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> 2NDSPOS2(N, Z)
ACTIVE1(plus2(s1(X), Y)) -> PLUS2(X, Y)
PROPER1(plus2(X1, X2)) -> PLUS2(proper1(X1), proper1(X2))
TIMES2(mark1(X1), X2) -> TIMES2(X1, X2)
ACTIVE1(posrecip1(X)) -> POSRECIP1(active1(X))
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
2NDSNEG2(X1, mark1(X2)) -> 2NDSNEG2(X1, X2)
ACTIVE1(from1(X)) -> S1(X)
NEGRECIP1(ok1(X)) -> NEGRECIP1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(plus2(s1(X), Y)) -> S1(plus2(X, Y))
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(negrecip1(X)) -> NEGRECIP1(proper1(X))
2NDSPOS2(ok1(X1), ok1(X2)) -> 2NDSPOS2(X1, X2)
ACTIVE1(2ndsneg2(X1, X2)) -> 2NDSNEG2(X1, active1(X2))
S1(mark1(X)) -> S1(X)
ACTIVE1(times2(X1, X2)) -> TIMES2(X1, active1(X2))
ACTIVE1(square1(X)) -> SQUARE1(active1(X))
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
PROPER1(pi1(X)) -> PI1(proper1(X))
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
TOP1(mark1(X)) -> TOP1(proper1(X))
2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)
ACTIVE1(pi1(X)) -> PI1(active1(X))
ACTIVE1(pi1(X)) -> 2NDSPOS2(X, from1(0))
ACTIVE1(times2(s1(X), Y)) -> TIMES2(X, Y)
ACTIVE1(2ndspos2(X1, X2)) -> 2NDSPOS2(active1(X1), X2)
ACTIVE1(2ndspos2(X1, X2)) -> 2NDSPOS2(X1, active1(X2))
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(square1(X)) -> SQUARE1(proper1(X))
PLUS2(mark1(X1), X2) -> PLUS2(X1, X2)
RCONS2(ok1(X1), ok1(X2)) -> RCONS2(X1, X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
ACTIVE1(times2(X1, X2)) -> TIMES2(active1(X1), X2)
ACTIVE1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> RCONS2(negrecip1(Y), 2ndspos2(N, Z))
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(square1(X)) -> PROPER1(X)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> NEGRECIP1(active1(X))
TIMES2(X1, mark1(X2)) -> TIMES2(X1, X2)
ACTIVE1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> POSRECIP1(Y)
SQUARE1(mark1(X)) -> SQUARE1(X)
ACTIVE1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> RCONS2(posrecip1(Y), 2ndsneg2(N, Z))
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
PROPER1(2ndsneg2(X1, X2)) -> 2NDSNEG2(proper1(X1), proper1(X2))
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> 2NDSNEG2(N, Z)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
POSRECIP1(ok1(X)) -> POSRECIP1(X)
ACTIVE1(times2(s1(X), Y)) -> PLUS2(Y, times2(X, Y))
ACTIVE1(plus2(X1, X2)) -> PLUS2(X1, active1(X2))
ACTIVE1(s1(X)) -> ACTIVE1(X)
2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)
ACTIVE1(2ndsneg2(X1, X2)) -> 2NDSNEG2(active1(X1), X2)
PI1(ok1(X)) -> PI1(X)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
RCONS2(X1, mark1(X2)) -> RCONS2(X1, X2)
PROPER1(pi1(X)) -> PROPER1(X)
ACTIVE1(pi1(X)) -> FROM1(0)
ACTIVE1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> NEGRECIP1(Y)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
ACTIVE1(square1(X)) -> ACTIVE1(X)
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
PROPER1(times2(X1, X2)) -> TIMES2(proper1(X1), proper1(X2))
2NDSNEG2(ok1(X1), ok1(X2)) -> 2NDSNEG2(X1, X2)
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(posrecip1(X)) -> POSRECIP1(proper1(X))
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> PLUS2(active1(X1), X2)
NEGRECIP1(mark1(X)) -> NEGRECIP1(X)
ACTIVE1(rcons2(X1, X2)) -> RCONS2(active1(X1), X2)
2NDSPOS2(mark1(X1), X2) -> 2NDSPOS2(X1, X2)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X2)
ACTIVE1(rcons2(X1, X2)) -> RCONS2(X1, active1(X2))
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)
TOP1(ok1(X)) -> ACTIVE1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(s1(X)) -> S1(active1(X))
PROPER1(2ndspos2(X1, X2)) -> 2NDSPOS2(proper1(X1), proper1(X2))
ACTIVE1(pi1(X)) -> ACTIVE1(X)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(square1(X)) -> TIMES2(X, X)
PI1(mark1(X)) -> PI1(X)
PROPER1(rcons2(X1, X2)) -> RCONS2(proper1(X1), proper1(X2))
PLUS2(ok1(X1), ok1(X2)) -> PLUS2(X1, X2)
RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
POSRECIP1(mark1(X)) -> POSRECIP1(X)
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
PROPER1(negrecip1(X)) -> PROPER1(X)
SQUARE1(ok1(X)) -> SQUARE1(X)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> 2NDSPOS2(N, Z)
ACTIVE1(plus2(s1(X), Y)) -> PLUS2(X, Y)
PROPER1(plus2(X1, X2)) -> PLUS2(proper1(X1), proper1(X2))
TIMES2(mark1(X1), X2) -> TIMES2(X1, X2)
ACTIVE1(posrecip1(X)) -> POSRECIP1(active1(X))
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
2NDSNEG2(X1, mark1(X2)) -> 2NDSNEG2(X1, X2)
ACTIVE1(from1(X)) -> S1(X)
NEGRECIP1(ok1(X)) -> NEGRECIP1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(plus2(s1(X), Y)) -> S1(plus2(X, Y))
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 15 SCCs with 47 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SQUARE1(ok1(X)) -> SQUARE1(X)
SQUARE1(mark1(X)) -> SQUARE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SQUARE1(ok1(X)) -> SQUARE1(X)
The remaining pairs can at least be oriented weakly.

SQUARE1(mark1(X)) -> SQUARE1(X)
Used ordering: Polynomial interpretation [21]:

POL(SQUARE1(x1)) = x1   
POL(mark1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SQUARE1(mark1(X)) -> SQUARE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SQUARE1(mark1(X)) -> SQUARE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SQUARE1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
TIMES2(mark1(X1), X2) -> TIMES2(X1, X2)
TIMES2(X1, mark1(X2)) -> TIMES2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TIMES2(mark1(X1), X2) -> TIMES2(X1, X2)
The remaining pairs can at least be oriented weakly.

TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
TIMES2(X1, mark1(X2)) -> TIMES2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(TIMES2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
TIMES2(X1, mark1(X2)) -> TIMES2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TIMES2(X1, mark1(X2)) -> TIMES2(X1, X2)
The remaining pairs can at least be oriented weakly.

TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(TIMES2(x1, x2)) = x2   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


TIMES2(ok1(X1), ok1(X2)) -> TIMES2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(TIMES2(x1, x2)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

PLUS2(mark1(X1), X2) -> PLUS2(X1, X2)
PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)
PLUS2(ok1(X1), ok1(X2)) -> PLUS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PLUS2(ok1(X1), ok1(X2)) -> PLUS2(X1, X2)
The remaining pairs can at least be oriented weakly.

PLUS2(mark1(X1), X2) -> PLUS2(X1, X2)
PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(PLUS2(x1, x2)) = x1   
POL(mark1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

PLUS2(mark1(X1), X2) -> PLUS2(X1, X2)
PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PLUS2(mark1(X1), X2) -> PLUS2(X1, X2)
The remaining pairs can at least be oriented weakly.

PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(PLUS2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PLUS2(X1, mark1(X2)) -> PLUS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PLUS2(x1, x2)) = x2   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

PI1(mark1(X)) -> PI1(X)
PI1(ok1(X)) -> PI1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PI1(mark1(X)) -> PI1(X)
The remaining pairs can at least be oriented weakly.

PI1(ok1(X)) -> PI1(X)
Used ordering: Polynomial interpretation [21]:

POL(PI1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

PI1(ok1(X)) -> PI1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PI1(ok1(X)) -> PI1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PI1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

2NDSNEG2(X1, mark1(X2)) -> 2NDSNEG2(X1, X2)
2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)
2NDSNEG2(ok1(X1), ok1(X2)) -> 2NDSNEG2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2NDSNEG2(ok1(X1), ok1(X2)) -> 2NDSNEG2(X1, X2)
The remaining pairs can at least be oriented weakly.

2NDSNEG2(X1, mark1(X2)) -> 2NDSNEG2(X1, X2)
2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(2NDSNEG2(x1, x2)) = x2   
POL(mark1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

2NDSNEG2(X1, mark1(X2)) -> 2NDSNEG2(X1, X2)
2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2NDSNEG2(X1, mark1(X2)) -> 2NDSNEG2(X1, X2)
The remaining pairs can at least be oriented weakly.

2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(2NDSNEG2(x1, x2)) = x2   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2NDSNEG2(mark1(X1), X2) -> 2NDSNEG2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(2NDSNEG2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

2NDSPOS2(mark1(X1), X2) -> 2NDSPOS2(X1, X2)
2NDSPOS2(ok1(X1), ok1(X2)) -> 2NDSPOS2(X1, X2)
2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2NDSPOS2(ok1(X1), ok1(X2)) -> 2NDSPOS2(X1, X2)
The remaining pairs can at least be oriented weakly.

2NDSPOS2(mark1(X1), X2) -> 2NDSPOS2(X1, X2)
2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(2NDSPOS2(x1, x2)) = x2   
POL(mark1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

2NDSPOS2(mark1(X1), X2) -> 2NDSPOS2(X1, X2)
2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2NDSPOS2(mark1(X1), X2) -> 2NDSPOS2(X1, X2)
The remaining pairs can at least be oriented weakly.

2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(2NDSPOS2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


2NDSPOS2(X1, mark1(X2)) -> 2NDSPOS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(2NDSPOS2(x1, x2)) = x2   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FROM1(mark1(X)) -> FROM1(X)
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(mark1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.

FROM1(ok1(X)) -> FROM1(X)
Used ordering: Polynomial interpretation [21]:

POL(FROM1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(ok1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FROM1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

RCONS2(X1, mark1(X2)) -> RCONS2(X1, X2)
RCONS2(ok1(X1), ok1(X2)) -> RCONS2(X1, X2)
RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


RCONS2(ok1(X1), ok1(X2)) -> RCONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

RCONS2(X1, mark1(X2)) -> RCONS2(X1, X2)
RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(RCONS2(x1, x2)) = x2   
POL(mark1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

RCONS2(X1, mark1(X2)) -> RCONS2(X1, X2)
RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


RCONS2(X1, mark1(X2)) -> RCONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(RCONS2(x1, x2)) = x2   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


RCONS2(mark1(X1), X2) -> RCONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(RCONS2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = x2   
POL(mark1(x1)) = 0   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS2(mark1(X1), X2) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

NEGRECIP1(mark1(X)) -> NEGRECIP1(X)
NEGRECIP1(ok1(X)) -> NEGRECIP1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


NEGRECIP1(mark1(X)) -> NEGRECIP1(X)
The remaining pairs can at least be oriented weakly.

NEGRECIP1(ok1(X)) -> NEGRECIP1(X)
Used ordering: Polynomial interpretation [21]:

POL(NEGRECIP1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

NEGRECIP1(ok1(X)) -> NEGRECIP1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


NEGRECIP1(ok1(X)) -> NEGRECIP1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(NEGRECIP1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

POSRECIP1(mark1(X)) -> POSRECIP1(X)
POSRECIP1(ok1(X)) -> POSRECIP1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


POSRECIP1(mark1(X)) -> POSRECIP1(X)
The remaining pairs can at least be oriented weakly.

POSRECIP1(ok1(X)) -> POSRECIP1(X)
Used ordering: Polynomial interpretation [21]:

POL(POSRECIP1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   
POL(ok1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

POSRECIP1(ok1(X)) -> POSRECIP1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


POSRECIP1(ok1(X)) -> POSRECIP1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(POSRECIP1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(ok1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.

S1(mark1(X)) -> S1(X)
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = x1   
POL(mark1(x1)) = x1   
POL(ok1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = x1   
POL(mark1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X2)
PROPER1(negrecip1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X1)
PROPER1(2ndsneg2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = 1 + x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = x1 + x2   
POL(from1(x1)) = x1   
POL(negrecip1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   
POL(square1(x1)) = x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(from1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = x1 + x2   
POL(from1(x1)) = 1 + x1   
POL(negrecip1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   
POL(square1(x1)) = x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(s1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = x1 + x2   
POL(negrecip1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = 1 + x1   
POL(square1(x1)) = x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(square1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(square1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = x1 + x2   
POL(negrecip1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(square1(x1)) = 1 + x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(negrecip1(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(negrecip1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = x1 + x2   
POL(negrecip1(x1)) = 1 + x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(posrecip1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(posrecip1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = x1 + x2   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = 1 + x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(PROPER1(x1)) = x1   
POL(cons2(x1, x2)) = 1 + x1 + x2   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(rcons2(x1, x2)) = x1 + x2   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(2ndspos2(X1, X2)) -> PROPER1(X2)
PROPER1(2ndspos2(X1, X2)) -> PROPER1(X1)
The remaining pairs can at least be oriented weakly.

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = 1 + x1 + x2   
POL(PROPER1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(rcons2(x1, x2)) = x1 + x2   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(times2(X1, X2)) -> PROPER1(X1)
PROPER1(times2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(rcons2(x1, x2)) = x1 + x2   
POL(times2(x1, x2)) = 1 + x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(plus2(X1, X2)) -> PROPER1(X2)
PROPER1(plus2(X1, X2)) -> PROPER1(X1)
The remaining pairs can at least be oriented weakly.

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = x1   
POL(pi1(x1)) = x1   
POL(plus2(x1, x2)) = 1 + x1 + x2   
POL(rcons2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(pi1(X)) -> PROPER1(X)
PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(pi1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = x1   
POL(pi1(x1)) = 1 + x1   
POL(rcons2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(rcons2(X1, X2)) -> PROPER1(X2)
PROPER1(rcons2(X1, X2)) -> PROPER1(X1)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = x1   
POL(rcons2(x1, x2)) = 1 + x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(square1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(pi1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(pi1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(square1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(from1(x1)) = x1   
POL(negrecip1(x1)) = x1   
POL(pi1(x1)) = 1 + x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   
POL(square1(x1)) = x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(square1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(from1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(square1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(from1(x1)) = 1 + x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   
POL(square1(x1)) = x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(square1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(square1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   
POL(square1(x1)) = 1 + x1   
POL(times2(x1, x2)) = x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(times2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(times2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   
POL(times2(x1, x2)) = 1 + x1 + x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(posrecip1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(posrecip1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(posrecip1(x1)) = 1 + x1   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndsneg2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.

ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndsneg2(x1, x2)) = 1 + x1 + x2   
POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(rcons2(x1, x2)) = x1 + x2   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(rcons2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.

ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = x1 + x2   
POL(rcons2(x1, x2)) = 1 + x1 + x2   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(plus2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.

ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = x1   
POL(plus2(x1, x2)) = 1 + x1 + x2   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(negrecip1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(negrecip1(x1)) = 1 + x1   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(s1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = x1   
POL(s1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = x1 + x2   
POL(ACTIVE1(x1)) = x1   
POL(cons2(x1, x2)) = 1 + x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(2ndspos2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(2ndspos2(x1, x2)) = 1 + x1 + x2   
POL(ACTIVE1(x1)) = x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ PisEmptyProof
          ↳ QDP

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

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(2ndspos2(0, Z)) -> mark1(rnil)
active1(2ndspos2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(posrecip1(Y), 2ndsneg2(N, Z)))
active1(2ndsneg2(0, Z)) -> mark1(rnil)
active1(2ndsneg2(s1(N), cons2(X, cons2(Y, Z)))) -> mark1(rcons2(negrecip1(Y), 2ndspos2(N, Z)))
active1(pi1(X)) -> mark1(2ndspos2(X, from1(0)))
active1(plus2(0, Y)) -> mark1(Y)
active1(plus2(s1(X), Y)) -> mark1(s1(plus2(X, Y)))
active1(times2(0, Y)) -> mark1(0)
active1(times2(s1(X), Y)) -> mark1(plus2(Y, times2(X, Y)))
active1(square1(X)) -> mark1(times2(X, X))
active1(s1(X)) -> s1(active1(X))
active1(posrecip1(X)) -> posrecip1(active1(X))
active1(negrecip1(X)) -> negrecip1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(active1(X1), X2)
active1(rcons2(X1, X2)) -> rcons2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(2ndspos2(X1, X2)) -> 2ndspos2(active1(X1), X2)
active1(2ndspos2(X1, X2)) -> 2ndspos2(X1, active1(X2))
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(active1(X1), X2)
active1(2ndsneg2(X1, X2)) -> 2ndsneg2(X1, active1(X2))
active1(pi1(X)) -> pi1(active1(X))
active1(plus2(X1, X2)) -> plus2(active1(X1), X2)
active1(plus2(X1, X2)) -> plus2(X1, active1(X2))
active1(times2(X1, X2)) -> times2(active1(X1), X2)
active1(times2(X1, X2)) -> times2(X1, active1(X2))
active1(square1(X)) -> square1(active1(X))
s1(mark1(X)) -> mark1(s1(X))
posrecip1(mark1(X)) -> mark1(posrecip1(X))
negrecip1(mark1(X)) -> mark1(negrecip1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
rcons2(mark1(X1), X2) -> mark1(rcons2(X1, X2))
rcons2(X1, mark1(X2)) -> mark1(rcons2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
2ndspos2(mark1(X1), X2) -> mark1(2ndspos2(X1, X2))
2ndspos2(X1, mark1(X2)) -> mark1(2ndspos2(X1, X2))
2ndsneg2(mark1(X1), X2) -> mark1(2ndsneg2(X1, X2))
2ndsneg2(X1, mark1(X2)) -> mark1(2ndsneg2(X1, X2))
pi1(mark1(X)) -> mark1(pi1(X))
plus2(mark1(X1), X2) -> mark1(plus2(X1, X2))
plus2(X1, mark1(X2)) -> mark1(plus2(X1, X2))
times2(mark1(X1), X2) -> mark1(times2(X1, X2))
times2(X1, mark1(X2)) -> mark1(times2(X1, X2))
square1(mark1(X)) -> mark1(square1(X))
proper1(0) -> ok1(0)
proper1(s1(X)) -> s1(proper1(X))
proper1(posrecip1(X)) -> posrecip1(proper1(X))
proper1(negrecip1(X)) -> negrecip1(proper1(X))
proper1(nil) -> ok1(nil)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(rnil) -> ok1(rnil)
proper1(rcons2(X1, X2)) -> rcons2(proper1(X1), proper1(X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(2ndspos2(X1, X2)) -> 2ndspos2(proper1(X1), proper1(X2))
proper1(2ndsneg2(X1, X2)) -> 2ndsneg2(proper1(X1), proper1(X2))
proper1(pi1(X)) -> pi1(proper1(X))
proper1(plus2(X1, X2)) -> plus2(proper1(X1), proper1(X2))
proper1(times2(X1, X2)) -> times2(proper1(X1), proper1(X2))
proper1(square1(X)) -> square1(proper1(X))
s1(ok1(X)) -> ok1(s1(X))
posrecip1(ok1(X)) -> ok1(posrecip1(X))
negrecip1(ok1(X)) -> ok1(negrecip1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
rcons2(ok1(X1), ok1(X2)) -> ok1(rcons2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
2ndspos2(ok1(X1), ok1(X2)) -> ok1(2ndspos2(X1, X2))
2ndsneg2(ok1(X1), ok1(X2)) -> ok1(2ndsneg2(X1, X2))
pi1(ok1(X)) -> ok1(pi1(X))
plus2(ok1(X1), ok1(X2)) -> ok1(plus2(X1, X2))
times2(ok1(X1), ok1(X2)) -> ok1(times2(X1, X2))
square1(ok1(X)) -> ok1(square1(X))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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