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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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:

MINUS2(ok1(X1), ok1(X2)) -> MINUS2(X1, X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> ZWQUOT2(XS, YS)
PROPER1(minus2(X1, X2)) -> MINUS2(proper1(X1), proper1(X2))
ACTIVE1(from1(X)) -> FROM1(active1(X))
ACTIVE1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> QUOT2(X, Y)
MINUS2(mark1(X1), X2) -> MINUS2(X1, X2)
S1(mark1(X)) -> S1(X)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X2)
PROPER1(quot2(X1, X2)) -> PROPER1(X1)
PROPER1(quot2(X1, X2)) -> QUOT2(proper1(X1), proper1(X2))
PROPER1(zWquot2(X1, X2)) -> ZWQUOT2(proper1(X1), proper1(X2))
PROPER1(minus2(X1, X2)) -> PROPER1(X1)
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(minus2(X1, X2)) -> MINUS2(X1, active1(X2))
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
ACTIVE1(quot2(s1(X), s1(Y))) -> S1(quot2(minus2(X, Y), s1(Y)))
ACTIVE1(quot2(s1(X), s1(Y))) -> QUOT2(minus2(X, Y), s1(Y))
PROPER1(zWquot2(X1, X2)) -> PROPER1(X2)
ACTIVE1(s1(X)) -> S1(active1(X))
ACTIVE1(quot2(X1, X2)) -> QUOT2(X1, active1(X2))
ZWQUOT2(X1, mark1(X2)) -> ZWQUOT2(X1, X2)
PROPER1(s1(X)) -> PROPER1(X)
ZWQUOT2(ok1(X1), ok1(X2)) -> ZWQUOT2(X1, X2)
QUOT2(ok1(X1), ok1(X2)) -> QUOT2(X1, X2)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X2)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X1)
QUOT2(mark1(X1), X2) -> QUOT2(X1, X2)
QUOT2(X1, mark1(X2)) -> QUOT2(X1, X2)
PROPER1(zWquot2(X1, X2)) -> PROPER1(X1)
ACTIVE1(quot2(X1, X2)) -> QUOT2(active1(X1), X2)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
PROPER1(minus2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
ACTIVE1(quot2(s1(X), s1(Y))) -> MINUS2(X, Y)
ACTIVE1(minus2(X1, X2)) -> MINUS2(active1(X1), X2)
ACTIVE1(sel2(s1(N), cons2(X, XS))) -> SEL2(N, XS)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
PROPER1(s1(X)) -> S1(proper1(X))
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X2)
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
MINUS2(X1, mark1(X2)) -> MINUS2(X1, X2)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
ACTIVE1(zWquot2(X1, X2)) -> ZWQUOT2(active1(X1), X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> CONS2(quot2(X, Y), zWquot2(XS, YS))
ACTIVE1(zWquot2(X1, X2)) -> ZWQUOT2(X1, active1(X2))
ACTIVE1(from1(X)) -> S1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
ACTIVE1(minus2(s1(X), s1(Y))) -> MINUS2(X, Y)
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(quot2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
FROM1(ok1(X)) -> FROM1(X)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X1)
ZWQUOT2(mark1(X1), X2) -> ZWQUOT2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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:

MINUS2(ok1(X1), ok1(X2)) -> MINUS2(X1, X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> FROM1(s1(X))
PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> ZWQUOT2(XS, YS)
PROPER1(minus2(X1, X2)) -> MINUS2(proper1(X1), proper1(X2))
ACTIVE1(from1(X)) -> FROM1(active1(X))
ACTIVE1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> QUOT2(X, Y)
MINUS2(mark1(X1), X2) -> MINUS2(X1, X2)
S1(mark1(X)) -> S1(X)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X2)
PROPER1(quot2(X1, X2)) -> PROPER1(X1)
PROPER1(quot2(X1, X2)) -> QUOT2(proper1(X1), proper1(X2))
PROPER1(zWquot2(X1, X2)) -> ZWQUOT2(proper1(X1), proper1(X2))
PROPER1(minus2(X1, X2)) -> PROPER1(X1)
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(minus2(X1, X2)) -> MINUS2(X1, active1(X2))
FROM1(mark1(X)) -> FROM1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
ACTIVE1(quot2(s1(X), s1(Y))) -> S1(quot2(minus2(X, Y), s1(Y)))
ACTIVE1(quot2(s1(X), s1(Y))) -> QUOT2(minus2(X, Y), s1(Y))
PROPER1(zWquot2(X1, X2)) -> PROPER1(X2)
ACTIVE1(s1(X)) -> S1(active1(X))
ACTIVE1(quot2(X1, X2)) -> QUOT2(X1, active1(X2))
ZWQUOT2(X1, mark1(X2)) -> ZWQUOT2(X1, X2)
PROPER1(s1(X)) -> PROPER1(X)
ZWQUOT2(ok1(X1), ok1(X2)) -> ZWQUOT2(X1, X2)
QUOT2(ok1(X1), ok1(X2)) -> QUOT2(X1, X2)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X2)
TOP1(mark1(X)) -> PROPER1(X)
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X1)
QUOT2(mark1(X1), X2) -> QUOT2(X1, X2)
QUOT2(X1, mark1(X2)) -> QUOT2(X1, X2)
PROPER1(zWquot2(X1, X2)) -> PROPER1(X1)
ACTIVE1(quot2(X1, X2)) -> QUOT2(active1(X1), X2)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
PROPER1(minus2(X1, X2)) -> PROPER1(X2)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
ACTIVE1(quot2(s1(X), s1(Y))) -> MINUS2(X, Y)
ACTIVE1(minus2(X1, X2)) -> MINUS2(active1(X1), X2)
ACTIVE1(sel2(s1(N), cons2(X, XS))) -> SEL2(N, XS)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
PROPER1(s1(X)) -> S1(proper1(X))
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X2)
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
MINUS2(X1, mark1(X2)) -> MINUS2(X1, X2)
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
ACTIVE1(zWquot2(X1, X2)) -> ZWQUOT2(active1(X1), X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> CONS2(quot2(X, Y), zWquot2(XS, YS))
ACTIVE1(zWquot2(X1, X2)) -> ZWQUOT2(X1, active1(X2))
ACTIVE1(from1(X)) -> S1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
ACTIVE1(minus2(s1(X), s1(Y))) -> MINUS2(X, Y)
TOP1(ok1(X)) -> TOP1(active1(X))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(quot2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
FROM1(ok1(X)) -> FROM1(X)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X1)
ZWQUOT2(mark1(X1), X2) -> ZWQUOT2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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 10 SCCs with 31 less nodes.

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

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

ZWQUOT2(X1, mark1(X2)) -> ZWQUOT2(X1, X2)
ZWQUOT2(ok1(X1), ok1(X2)) -> ZWQUOT2(X1, X2)
ZWQUOT2(mark1(X1), X2) -> ZWQUOT2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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.


ZWQUOT2(X1, mark1(X2)) -> ZWQUOT2(X1, X2)
ZWQUOT2(ok1(X1), ok1(X2)) -> ZWQUOT2(X1, X2)
ZWQUOT2(mark1(X1), X2) -> ZWQUOT2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ZWQUOT2(x1, x2)) = x1 + 2·x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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

QUOT2(mark1(X1), X2) -> QUOT2(X1, X2)
QUOT2(X1, mark1(X2)) -> QUOT2(X1, X2)
QUOT2(ok1(X1), ok1(X2)) -> QUOT2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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.


QUOT2(mark1(X1), X2) -> QUOT2(X1, X2)
QUOT2(X1, mark1(X2)) -> QUOT2(X1, X2)
QUOT2(ok1(X1), ok1(X2)) -> QUOT2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(QUOT2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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

MINUS2(ok1(X1), ok1(X2)) -> MINUS2(X1, X2)
MINUS2(mark1(X1), X2) -> MINUS2(X1, X2)
MINUS2(X1, mark1(X2)) -> MINUS2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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.


MINUS2(ok1(X1), ok1(X2)) -> MINUS2(X1, X2)
MINUS2(mark1(X1), X2) -> MINUS2(X1, X2)
MINUS2(X1, mark1(X2)) -> MINUS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(MINUS2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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

SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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.


SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SEL2(x1, x2)) = x1 + 2·x2   
POL(mark1(x1)) = 1 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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)
S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = 2·x1 + x2   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 1 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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)
FROM1(ok1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FROM1(x1)) = 2·x1   
POL(mark1(x1)) = 2 + 2·x1   
POL(ok1(x1)) = 2 + 2·x1   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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

PROPER1(quot2(X1, X2)) -> PROPER1(X1)
PROPER1(minus2(X1, X2)) -> PROPER1(X2)
PROPER1(zWquot2(X1, X2)) -> PROPER1(X2)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(zWquot2(X1, X2)) -> PROPER1(X1)
PROPER1(minus2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(quot2(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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(quot2(X1, X2)) -> PROPER1(X1)
PROPER1(minus2(X1, X2)) -> PROPER1(X2)
PROPER1(zWquot2(X1, X2)) -> PROPER1(X2)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(zWquot2(X1, X2)) -> PROPER1(X1)
PROPER1(minus2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(quot2(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = 2·x1   
POL(cons2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(from1(x1)) = 2 + 2·x1   
POL(minus2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(quot2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(s1(x1)) = 2 + 2·x1   
POL(sel2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(zWquot2(x1, x2)) = 2 + 2·x1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X2)

The TRS R consists of the following rules:

active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(quot2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(zWquot2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(minus2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = 2·x1   
POL(cons2(x1, x2)) = 2 + 2·x1   
POL(from1(x1)) = 2 + 2·x1   
POL(minus2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(quot2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(s1(x1)) = 2 + 2·x1   
POL(sel2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(zWquot2(x1, x2)) = 2 + 2·x1 + 2·x2   

The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ 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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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

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(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(minus2(X, 0)) -> mark1(0)
active1(minus2(s1(X), s1(Y))) -> mark1(minus2(X, Y))
active1(quot2(0, s1(Y))) -> mark1(0)
active1(quot2(s1(X), s1(Y))) -> mark1(s1(quot2(minus2(X, Y), s1(Y))))
active1(zWquot2(XS, nil)) -> mark1(nil)
active1(zWquot2(nil, XS)) -> mark1(nil)
active1(zWquot2(cons2(X, XS), cons2(Y, YS))) -> mark1(cons2(quot2(X, Y), zWquot2(XS, YS)))
active1(from1(X)) -> from1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(s1(X)) -> s1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(minus2(X1, X2)) -> minus2(active1(X1), X2)
active1(minus2(X1, X2)) -> minus2(X1, active1(X2))
active1(quot2(X1, X2)) -> quot2(active1(X1), X2)
active1(quot2(X1, X2)) -> quot2(X1, active1(X2))
active1(zWquot2(X1, X2)) -> zWquot2(active1(X1), X2)
active1(zWquot2(X1, X2)) -> zWquot2(X1, active1(X2))
from1(mark1(X)) -> mark1(from1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
minus2(mark1(X1), X2) -> mark1(minus2(X1, X2))
minus2(X1, mark1(X2)) -> mark1(minus2(X1, X2))
quot2(mark1(X1), X2) -> mark1(quot2(X1, X2))
quot2(X1, mark1(X2)) -> mark1(quot2(X1, X2))
zWquot2(mark1(X1), X2) -> mark1(zWquot2(X1, X2))
zWquot2(X1, mark1(X2)) -> mark1(zWquot2(X1, X2))
proper1(from1(X)) -> from1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(minus2(X1, X2)) -> minus2(proper1(X1), proper1(X2))
proper1(quot2(X1, X2)) -> quot2(proper1(X1), proper1(X2))
proper1(zWquot2(X1, X2)) -> zWquot2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
from1(ok1(X)) -> ok1(from1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
minus2(ok1(X1), ok1(X2)) -> ok1(minus2(X1, X2))
quot2(ok1(X1), ok1(X2)) -> ok1(quot2(X1, X2))
zWquot2(ok1(X1), ok1(X2)) -> ok1(zWquot2(X1, X2))
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.