Termination w.r.t. Q proof of /tmp/aprove_benchmark_example

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:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

FROM(mark(X)) → FROM(X)
QUOT(X1, active(X2)) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MINUS(X, Y)
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
FROM(active(X)) → FROM(X)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
MARK(zWquot(X1, X2)) → MARK(X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → QUOT(minus(X, Y), s(Y))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
ACTIVE(zWquot(nil, XS)) → MARK(nil)
ACTIVE(zWquot(XS, nil)) → MARK(nil)
MINUS(X1, active(X2)) → MINUS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → CONS(quot(X, Y), zWquot(XS, YS))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MINUS(active(X1), X2) → MINUS(X1, X2)
MARK(minus(X1, X2)) → MINUS(mark(X1), mark(X2))
MARK(quot(X1, X2)) → QUOT(mark(X1), mark(X2))
S(active(X)) → S(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MARK(quot(X1, X2)) → MARK(X1)
ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → QUOT(X, Y)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(from(X)) → FROM(s(X))
SEL(mark(X1), X2) → SEL(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
SEL(X1, active(X2)) → SEL(X1, X2)
MARK(s(X)) → MARK(X)
MARK(from(X)) → FROM(mark(X))
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → ZWQUOT(XS, YS)
MARK(quot(X1, X2)) → MARK(X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(quot(s(X), s(Y))) → S(quot(minus(X, Y), s(Y)))
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
QUOT(active(X1), X2) → QUOT(X1, X2)
ACTIVE(quot(0, s(Y))) → MARK(0)
MARK(minus(X1, X2)) → MARK(X2)
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
S(mark(X)) → S(X)
MARK(from(X)) → MARK(X)
MARK(s(X)) → S(mark(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(minus(X, 0)) → MARK(0)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
ACTIVE(from(X)) → S(X)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
QUOT(mark(X1), X2) → QUOT(X1, X2)
MARK(0) → ACTIVE(0)
MARK(zWquot(X1, X2)) → ZWQUOT(mark(X1), mark(X2))
ACTIVE(sel(s(N), cons(X, XS))) → SEL(N, XS)
MARK(minus(X1, X2)) → MARK(X1)
MARK(nil) → ACTIVE(nil)
ACTIVE(from(X)) → CONS(X, from(s(X)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

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:

FROM(mark(X)) → FROM(X)
QUOT(X1, active(X2)) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MINUS(X, Y)
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
FROM(active(X)) → FROM(X)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
MARK(zWquot(X1, X2)) → MARK(X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → QUOT(minus(X, Y), s(Y))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
ACTIVE(zWquot(nil, XS)) → MARK(nil)
ACTIVE(zWquot(XS, nil)) → MARK(nil)
MINUS(X1, active(X2)) → MINUS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → CONS(quot(X, Y), zWquot(XS, YS))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MINUS(active(X1), X2) → MINUS(X1, X2)
MARK(minus(X1, X2)) → MINUS(mark(X1), mark(X2))
MARK(quot(X1, X2)) → QUOT(mark(X1), mark(X2))
S(active(X)) → S(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MARK(quot(X1, X2)) → MARK(X1)
ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → QUOT(X, Y)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(from(X)) → FROM(s(X))
SEL(mark(X1), X2) → SEL(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
SEL(X1, active(X2)) → SEL(X1, X2)
MARK(s(X)) → MARK(X)
MARK(from(X)) → FROM(mark(X))
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → ZWQUOT(XS, YS)
MARK(quot(X1, X2)) → MARK(X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(quot(s(X), s(Y))) → S(quot(minus(X, Y), s(Y)))
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
QUOT(active(X1), X2) → QUOT(X1, X2)
ACTIVE(quot(0, s(Y))) → MARK(0)
MARK(minus(X1, X2)) → MARK(X2)
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
S(mark(X)) → S(X)
MARK(from(X)) → MARK(X)
MARK(s(X)) → S(mark(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(minus(X, 0)) → MARK(0)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
ACTIVE(from(X)) → S(X)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
QUOT(mark(X1), X2) → QUOT(X1, X2)
MARK(0) → ACTIVE(0)
MARK(zWquot(X1, X2)) → ZWQUOT(mark(X1), mark(X2))
ACTIVE(sel(s(N), cons(X, XS))) → SEL(N, XS)
MARK(minus(X1, X2)) → MARK(X1)
MARK(nil) → ACTIVE(nil)
ACTIVE(from(X)) → CONS(X, from(s(X)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 8 SCCs with 24 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

QUOT(X1, active(X2)) → QUOT(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
QUOT(mark(X1), X2) → QUOT(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
QUOT(X1, mark(X2)) → QUOT(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
QUOT(active(X1), X2) → QUOT(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

MINUS(X1, active(X2)) → MINUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
MINUS(mark(X1), X2) → MINUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
MINUS(X1, mark(X2)) → MINUS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
MINUS(active(X1), X2) → MINUS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

SEL(mark(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
SEL(X1, active(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(X1, mark(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(active(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

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

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

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

From the DPs we obtained the following set of size-change graphs:

FROM(mark(X)) → FROM(X)
The graph contains the following edges 1 > 1
FROM(active(X)) → FROM(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.

MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 1
POL(mark(x₁)) = 0
POL(minus(x₁, x₂)) = 1
POL(nil) = 0
POL(quot(x₁, x₂)) = 1
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 1
POL(zWquot(x₁, x₂)) = 1

The following usable rules [17] were oriented:

zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
from(active(X)) → from(X)
from(mark(X)) → from(X)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

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

MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(zWquot(zWquot(x0, x1), y1)) → ACTIVE(zWquot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(minus(x0, x1), y1)) → ACTIVE(zWquot(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(y0, x1)) → ACTIVE(zWquot(mark(y0), x1))
MARK(zWquot(y0, quot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(zWquot(cons(x0, x1), y1)) → ACTIVE(zWquot(active(cons(mark(x0), x1)), mark(y1)))
MARK(zWquot(y0, minus(x0, x1))) → ACTIVE(zWquot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(zWquot(quot(x0, x1), y1)) → ACTIVE(zWquot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(y0, s(x0))) → ACTIVE(zWquot(mark(y0), active(s(mark(x0)))))
MARK(zWquot(y0, zWquot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(zWquot(s(x0), y1)) → ACTIVE(zWquot(active(s(mark(x0))), mark(y1)))
MARK(zWquot(sel(x0, x1), y1)) → ACTIVE(zWquot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(y0, from(x0))) → ACTIVE(zWquot(mark(y0), active(from(mark(x0)))))
MARK(zWquot(y0, nil)) → ACTIVE(zWquot(mark(y0), active(nil)))
MARK(zWquot(from(x0), y1)) → ACTIVE(zWquot(active(from(mark(x0))), mark(y1)))
MARK(zWquot(nil, y1)) → ACTIVE(zWquot(active(nil), mark(y1)))
MARK(zWquot(x0, y1)) → ACTIVE(zWquot(x0, mark(y1)))
MARK(zWquot(y0, sel(x0, x1))) → ACTIVE(zWquot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(zWquot(y0, cons(x0, x1))) → ACTIVE(zWquot(mark(y0), active(cons(mark(x0), x1))))
MARK(zWquot(y0, 0)) → ACTIVE(zWquot(mark(y0), active(0)))
MARK(zWquot(0, y1)) → ACTIVE(zWquot(active(0), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

MARK(zWquot(minus(x0, x1), y1)) → ACTIVE(zWquot(active(minus(mark(x0), mark(x1))), mark(y1)))
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(cons(x0, x1), y1)) → ACTIVE(zWquot(active(cons(mark(x0), x1)), mark(y1)))
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(y0, minus(x0, x1))) → ACTIVE(zWquot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(zWquot(quot(x0, x1), y1)) → ACTIVE(zWquot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(y0, s(x0))) → ACTIVE(zWquot(mark(y0), active(s(mark(x0)))))
MARK(zWquot(y0, zWquot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(zWquot(s(x0), y1)) → ACTIVE(zWquot(active(s(mark(x0))), mark(y1)))
MARK(zWquot(sel(x0, x1), y1)) → ACTIVE(zWquot(active(sel(mark(x0), mark(x1))), mark(y1)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(zWquot(y0, from(x0))) → ACTIVE(zWquot(mark(y0), active(from(mark(x0)))))
MARK(zWquot(nil, y1)) → ACTIVE(zWquot(active(nil), mark(y1)))
MARK(zWquot(from(x0), y1)) → ACTIVE(zWquot(active(from(mark(x0))), mark(y1)))
MARK(zWquot(y0, nil)) → ACTIVE(zWquot(mark(y0), active(nil)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(zWquot(0, y1)) → ACTIVE(zWquot(active(0), mark(y1)))
MARK(zWquot(y0, 0)) → ACTIVE(zWquot(mark(y0), active(0)))
MARK(zWquot(zWquot(x0, x1), y1)) → ACTIVE(zWquot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(zWquot(y0, x1)) → ACTIVE(zWquot(mark(y0), x1))
MARK(zWquot(y0, quot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(quot(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(zWquot(x0, y1)) → ACTIVE(zWquot(x0, mark(y1)))
MARK(minus(X1, X2)) → MARK(X1)
MARK(zWquot(y0, cons(x0, x1))) → ACTIVE(zWquot(mark(y0), active(cons(mark(x0), x1))))
MARK(zWquot(y0, sel(x0, x1))) → ACTIVE(zWquot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(minus(y0, s(x0))) → ACTIVE(minus(mark(y0), active(s(mark(x0)))))
MARK(minus(y0, from(x0))) → ACTIVE(minus(mark(y0), active(from(mark(x0)))))
MARK(minus(y0, 0)) → ACTIVE(minus(mark(y0), active(0)))
MARK(minus(0, y1)) → ACTIVE(minus(active(0), mark(y1)))
MARK(minus(cons(x0, x1), y1)) → ACTIVE(minus(active(cons(mark(x0), x1)), mark(y1)))
MARK(minus(from(x0), y1)) → ACTIVE(minus(active(from(mark(x0))), mark(y1)))
MARK(minus(x0, y1)) → ACTIVE(minus(x0, mark(y1)))
MARK(minus(y0, nil)) → ACTIVE(minus(mark(y0), active(nil)))
MARK(minus(y0, minus(x0, x1))) → ACTIVE(minus(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(minus(nil, y1)) → ACTIVE(minus(active(nil), mark(y1)))
MARK(minus(zWquot(x0, x1), y1)) → ACTIVE(minus(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, cons(x0, x1))) → ACTIVE(minus(mark(y0), active(cons(mark(x0), x1))))
MARK(minus(y0, x1)) → ACTIVE(minus(mark(y0), x1))
MARK(minus(y0, sel(x0, x1))) → ACTIVE(minus(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(minus(s(x0), y1)) → ACTIVE(minus(active(s(mark(x0))), mark(y1)))
MARK(minus(minus(x0, x1), y1)) → ACTIVE(minus(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(sel(x0, x1), y1)) → ACTIVE(minus(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, quot(x0, x1))) → ACTIVE(minus(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(minus(y0, zWquot(x0, x1))) → ACTIVE(minus(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(minus(quot(x0, x1), y1)) → ACTIVE(minus(active(quot(mark(x0), mark(x1))), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

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

MARK(minus(y0, 0)) → ACTIVE(minus(mark(y0), active(0)))
MARK(minus(0, y1)) → ACTIVE(minus(active(0), mark(y1)))
MARK(minus(cons(x0, x1), y1)) → ACTIVE(minus(active(cons(mark(x0), x1)), mark(y1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(y0, minus(x0, x1))) → ACTIVE(zWquot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(zWquot(quot(x0, x1), y1)) → ACTIVE(zWquot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(y0, zWquot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(minus(y0, sel(x0, x1))) → ACTIVE(minus(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(zWquot(y0, from(x0))) → ACTIVE(zWquot(mark(y0), active(from(mark(x0)))))
MARK(zWquot(from(x0), y1)) → ACTIVE(zWquot(active(from(mark(x0))), mark(y1)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(zWquot(0, y1)) → ACTIVE(zWquot(active(0), mark(y1)))
MARK(zWquot(y0, 0)) → ACTIVE(zWquot(mark(y0), active(0)))
MARK(zWquot(zWquot(x0, x1), y1)) → ACTIVE(zWquot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(zWquot(y0, x1)) → ACTIVE(zWquot(mark(y0), x1))
MARK(zWquot(y0, quot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(minus(nil, y1)) → ACTIVE(minus(active(nil), mark(y1)))
MARK(minus(y0, minus(x0, x1))) → ACTIVE(minus(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(y0, nil)) → ACTIVE(minus(mark(y0), active(nil)))
MARK(minus(sel(x0, x1), y1)) → ACTIVE(minus(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(x0, y1)) → ACTIVE(zWquot(x0, mark(y1)))
MARK(minus(y0, zWquot(x0, x1))) → ACTIVE(minus(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(minus(y0, s(x0))) → ACTIVE(minus(mark(y0), active(s(mark(x0)))))
MARK(zWquot(minus(x0, x1), y1)) → ACTIVE(zWquot(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, from(x0))) → ACTIVE(minus(mark(y0), active(from(mark(x0)))))
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(minus(from(x0), y1)) → ACTIVE(minus(active(from(mark(x0))), mark(y1)))
MARK(zWquot(cons(x0, x1), y1)) → ACTIVE(zWquot(active(cons(mark(x0), x1)), mark(y1)))
MARK(minus(y0, cons(x0, x1))) → ACTIVE(minus(mark(y0), active(cons(mark(x0), x1))))
MARK(zWquot(y0, s(x0))) → ACTIVE(zWquot(mark(y0), active(s(mark(x0)))))
MARK(zWquot(s(x0), y1)) → ACTIVE(zWquot(active(s(mark(x0))), mark(y1)))
MARK(zWquot(sel(x0, x1), y1)) → ACTIVE(zWquot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(quot(X1, X2)) → MARK(X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(minus(s(x0), y1)) → ACTIVE(minus(active(s(mark(x0))), mark(y1)))
MARK(zWquot(nil, y1)) → ACTIVE(zWquot(active(nil), mark(y1)))
MARK(zWquot(y0, nil)) → ACTIVE(zWquot(mark(y0), active(nil)))
MARK(minus(X1, X2)) → MARK(X2)
MARK(minus(y0, quot(x0, x1))) → ACTIVE(minus(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(from(X)) → MARK(X)
MARK(minus(x0, y1)) → ACTIVE(minus(x0, mark(y1)))
MARK(minus(zWquot(x0, x1), y1)) → ACTIVE(minus(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(minus(y0, x1)) → ACTIVE(minus(mark(y0), x1))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(minus(minus(x0, x1), y1)) → ACTIVE(minus(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(X1, X2)) → MARK(X1)
MARK(minus(quot(x0, x1), y1)) → ACTIVE(minus(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(y0, cons(x0, x1))) → ACTIVE(zWquot(mark(y0), active(cons(mark(x0), x1))))
MARK(zWquot(y0, sel(x0, x1))) → ACTIVE(zWquot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(quot(y0, from(x0))) → ACTIVE(quot(mark(y0), active(from(mark(x0)))))
MARK(quot(y0, 0)) → ACTIVE(quot(mark(y0), active(0)))
MARK(quot(y0, s(x0))) → ACTIVE(quot(mark(y0), active(s(mark(x0)))))
MARK(quot(0, y1)) → ACTIVE(quot(active(0), mark(y1)))
MARK(quot(y0, nil)) → ACTIVE(quot(mark(y0), active(nil)))
MARK(quot(nil, y1)) → ACTIVE(quot(active(nil), mark(y1)))
MARK(quot(y0, x1)) → ACTIVE(quot(mark(y0), x1))
MARK(quot(sel(x0, x1), y1)) → ACTIVE(quot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(quot(from(x0), y1)) → ACTIVE(quot(active(from(mark(x0))), mark(y1)))
MARK(quot(x0, y1)) → ACTIVE(quot(x0, mark(y1)))
MARK(quot(cons(x0, x1), y1)) → ACTIVE(quot(active(cons(mark(x0), x1)), mark(y1)))
MARK(quot(y0, minus(x0, x1))) → ACTIVE(quot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(quot(y0, sel(x0, x1))) → ACTIVE(quot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(quot(s(x0), y1)) → ACTIVE(quot(active(s(mark(x0))), mark(y1)))
MARK(quot(y0, cons(x0, x1))) → ACTIVE(quot(mark(y0), active(cons(mark(x0), x1))))
MARK(quot(y0, zWquot(x0, x1))) → ACTIVE(quot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(quot(quot(x0, x1), y1)) → ACTIVE(quot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(minus(x0, x1), y1)) → ACTIVE(quot(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(quot(y0, quot(x0, x1))) → ACTIVE(quot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(quot(zWquot(x0, x1), y1)) → ACTIVE(quot(active(zWquot(mark(x0), mark(x1))), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

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

MARK(quot(y0, s(x0))) → ACTIVE(quot(mark(y0), active(s(mark(x0)))))
MARK(minus(0, y1)) → ACTIVE(minus(active(0), mark(y1)))
MARK(minus(y0, 0)) → ACTIVE(minus(mark(y0), active(0)))
MARK(minus(cons(x0, x1), y1)) → ACTIVE(minus(active(cons(mark(x0), x1)), mark(y1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(zWquot(y0, minus(x0, x1))) → ACTIVE(zWquot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(zWquot(quot(x0, x1), y1)) → ACTIVE(zWquot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(cons(x0, x1), y1)) → ACTIVE(quot(active(cons(mark(x0), x1)), mark(y1)))
MARK(zWquot(y0, zWquot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(zWquot(y0, from(x0))) → ACTIVE(zWquot(mark(y0), active(from(mark(x0)))))
MARK(minus(y0, sel(x0, x1))) → ACTIVE(minus(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(zWquot(from(x0), y1)) → ACTIVE(zWquot(active(from(mark(x0))), mark(y1)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(quot(y0, quot(x0, x1))) → ACTIVE(quot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(zWquot(y0, 0)) → ACTIVE(zWquot(mark(y0), active(0)))
MARK(zWquot(0, y1)) → ACTIVE(zWquot(active(0), mark(y1)))
MARK(zWquot(zWquot(x0, x1), y1)) → ACTIVE(zWquot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(0, y1)) → ACTIVE(quot(active(0), mark(y1)))
MARK(quot(y0, 0)) → ACTIVE(quot(mark(y0), active(0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(quot(nil, y1)) → ACTIVE(quot(active(nil), mark(y1)))
MARK(quot(y0, nil)) → ACTIVE(quot(mark(y0), active(nil)))
MARK(zWquot(y0, x1)) → ACTIVE(zWquot(mark(y0), x1))
MARK(zWquot(y0, quot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(minus(y0, nil)) → ACTIVE(minus(mark(y0), active(nil)))
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(y0, minus(x0, x1))) → ACTIVE(minus(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(minus(nil, y1)) → ACTIVE(minus(active(nil), mark(y1)))
MARK(quot(from(x0), y1)) → ACTIVE(quot(active(from(mark(x0))), mark(y1)))
MARK(quot(s(x0), y1)) → ACTIVE(quot(active(s(mark(x0))), mark(y1)))
MARK(minus(sel(x0, x1), y1)) → ACTIVE(minus(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(x0, y1)) → ACTIVE(zWquot(x0, mark(y1)))
MARK(quot(minus(x0, x1), y1)) → ACTIVE(quot(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, zWquot(x0, x1))) → ACTIVE(minus(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(quot(y0, from(x0))) → ACTIVE(quot(mark(y0), active(from(mark(x0)))))
MARK(minus(y0, s(x0))) → ACTIVE(minus(mark(y0), active(s(mark(x0)))))
MARK(zWquot(minus(x0, x1), y1)) → ACTIVE(zWquot(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, from(x0))) → ACTIVE(minus(mark(y0), active(from(mark(x0)))))
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(minus(from(x0), y1)) → ACTIVE(minus(active(from(mark(x0))), mark(y1)))
MARK(quot(sel(x0, x1), y1)) → ACTIVE(quot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(cons(x0, x1), y1)) → ACTIVE(zWquot(active(cons(mark(x0), x1)), mark(y1)))
MARK(minus(y0, cons(x0, x1))) → ACTIVE(minus(mark(y0), active(cons(mark(x0), x1))))
MARK(quot(x0, y1)) → ACTIVE(quot(x0, mark(y1)))
MARK(zWquot(y0, s(x0))) → ACTIVE(zWquot(mark(y0), active(s(mark(x0)))))
MARK(zWquot(s(x0), y1)) → ACTIVE(zWquot(active(s(mark(x0))), mark(y1)))
MARK(zWquot(sel(x0, x1), y1)) → ACTIVE(zWquot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(quot(X1, X2)) → MARK(X2)
MARK(quot(y0, zWquot(x0, x1))) → ACTIVE(quot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(y0, nil)) → ACTIVE(zWquot(mark(y0), active(nil)))
MARK(zWquot(nil, y1)) → ACTIVE(zWquot(active(nil), mark(y1)))
MARK(minus(s(x0), y1)) → ACTIVE(minus(active(s(mark(x0))), mark(y1)))
MARK(minus(X1, X2)) → MARK(X2)
MARK(minus(y0, quot(x0, x1))) → ACTIVE(minus(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(quot(zWquot(x0, x1), y1)) → ACTIVE(quot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(from(X)) → MARK(X)
MARK(quot(y0, x1)) → ACTIVE(quot(mark(y0), x1))
MARK(minus(x0, y1)) → ACTIVE(minus(x0, mark(y1)))
MARK(minus(zWquot(x0, x1), y1)) → ACTIVE(minus(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(quot(y0, minus(x0, x1))) → ACTIVE(quot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(quot(y0, sel(x0, x1))) → ACTIVE(quot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(minus(y0, x1)) → ACTIVE(minus(mark(y0), x1))
MARK(quot(y0, cons(x0, x1))) → ACTIVE(quot(mark(y0), active(cons(mark(x0), x1))))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(quot(quot(x0, x1), y1)) → ACTIVE(quot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(minus(minus(x0, x1), y1)) → ACTIVE(minus(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
MARK(zWquot(y0, sel(x0, x1))) → ACTIVE(zWquot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(zWquot(y0, cons(x0, x1))) → ACTIVE(zWquot(mark(y0), active(cons(mark(x0), x1))))
MARK(minus(quot(x0, x1), y1)) → ACTIVE(minus(active(quot(mark(x0), mark(x1))), mark(y1)))

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(sel(y0, quot(x0, x1))) → ACTIVE(sel(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(minus(x0, x1), y1)) → ACTIVE(sel(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(quot(x0, x1), y1)) → ACTIVE(sel(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(mark(x0))), mark(y1)))
MARK(sel(y0, minus(x0, x1))) → ACTIVE(sel(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(mark(x0)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(y0, zWquot(x0, x1))) → ACTIVE(sel(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(sel(zWquot(x0, x1), y1)) → ACTIVE(sel(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

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

MARK(quot(y0, s(x0))) → ACTIVE(quot(mark(y0), active(s(mark(x0)))))
MARK(quot(cons(x0, x1), y1)) → ACTIVE(quot(active(cons(mark(x0), x1)), mark(y1)))
MARK(zWquot(y0, zWquot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(zWquot(y0, from(x0))) → ACTIVE(zWquot(mark(y0), active(from(mark(x0)))))
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(quot(y0, quot(x0, x1))) → ACTIVE(quot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(zWquot(y0, 0)) → ACTIVE(zWquot(mark(y0), active(0)))
MARK(zWquot(0, y1)) → ACTIVE(zWquot(active(0), mark(y1)))
MARK(sel(y0, quot(x0, x1))) → ACTIVE(sel(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(quot(nil, y1)) → ACTIVE(quot(active(nil), mark(y1)))
MARK(quot(y0, nil)) → ACTIVE(quot(mark(y0), active(nil)))
MARK(zWquot(y0, x1)) → ACTIVE(zWquot(mark(y0), x1))
MARK(sel(minus(x0, x1), y1)) → ACTIVE(sel(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, nil)) → ACTIVE(minus(mark(y0), active(nil)))
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(y0, minus(x0, x1))) → ACTIVE(minus(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(minus(nil, y1)) → ACTIVE(minus(active(nil), mark(y1)))
MARK(quot(from(x0), y1)) → ACTIVE(quot(active(from(mark(x0))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(mark(x0))), mark(y1)))
MARK(quot(s(x0), y1)) → ACTIVE(quot(active(s(mark(x0))), mark(y1)))
MARK(minus(sel(x0, x1), y1)) → ACTIVE(minus(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(x0, y1)) → ACTIVE(zWquot(x0, mark(y1)))
MARK(zWquot(minus(x0, x1), y1)) → ACTIVE(zWquot(active(minus(mark(x0), mark(x1))), mark(y1)))
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(minus(from(x0), y1)) → ACTIVE(minus(active(from(mark(x0))), mark(y1)))
MARK(quot(sel(x0, x1), y1)) → ACTIVE(quot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(minus(y0, cons(x0, x1))) → ACTIVE(minus(mark(y0), active(cons(mark(x0), x1))))
MARK(quot(x0, y1)) → ACTIVE(quot(x0, mark(y1)))
MARK(zWquot(y0, s(x0))) → ACTIVE(zWquot(mark(y0), active(s(mark(x0)))))
MARK(sel(y0, minus(x0, x1))) → ACTIVE(sel(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(zWquot(sel(x0, x1), y1)) → ACTIVE(zWquot(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(minus(s(x0), y1)) → ACTIVE(minus(active(s(mark(x0))), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(zWquot(x0, x1), y1)) → ACTIVE(sel(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(zWquot(x0, x1), y1)) → ACTIVE(quot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(quot(y0, x1)) → ACTIVE(quot(mark(y0), x1))
MARK(minus(zWquot(x0, x1), y1)) → ACTIVE(minus(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(y0, minus(x0, x1))) → ACTIVE(quot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(quot(y0, sel(x0, x1))) → ACTIVE(quot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(quot(quot(x0, x1), y1)) → ACTIVE(quot(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(zWquot(y0, sel(x0, x1))) → ACTIVE(zWquot(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(minus(quot(x0, x1), y1)) → ACTIVE(minus(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))
MARK(minus(0, y1)) → ACTIVE(minus(active(0), mark(y1)))
MARK(minus(y0, 0)) → ACTIVE(minus(mark(y0), active(0)))
MARK(minus(cons(x0, x1), y1)) → ACTIVE(minus(active(cons(mark(x0), x1)), mark(y1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(zWquot(y0, minus(x0, x1))) → ACTIVE(zWquot(mark(y0), active(minus(mark(x0), mark(x1)))))
MARK(sel(quot(x0, x1), y1)) → ACTIVE(sel(active(quot(mark(x0), mark(x1))), mark(y1)))
MARK(zWquot(quot(x0, x1), y1)) → ACTIVE(zWquot(active(quot(mark(x0), mark(x1))), mark(y1)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(mark(x0)))))
MARK(minus(y0, sel(x0, x1))) → ACTIVE(minus(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(zWquot(from(x0), y1)) → ACTIVE(zWquot(active(from(mark(x0))), mark(y1)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(zWquot(zWquot(x0, x1), y1)) → ACTIVE(zWquot(active(zWquot(mark(x0), mark(x1))), mark(y1)))
MARK(quot(0, y1)) → ACTIVE(quot(active(0), mark(y1)))
MARK(quot(y0, 0)) → ACTIVE(quot(mark(y0), active(0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(zWquot(y0, quot(x0, x1))) → ACTIVE(zWquot(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, zWquot(x0, x1))) → ACTIVE(sel(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(quot(minus(x0, x1), y1)) → ACTIVE(quot(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(minus(y0, zWquot(x0, x1))) → ACTIVE(minus(mark(y0), active(zWquot(mark(x0), mark(x1)))))
MARK(quot(y0, from(x0))) → ACTIVE(quot(mark(y0), active(from(mark(x0)))))
MARK(minus(y0, s(x0))) → ACTIVE(minus(mark(y0), active(s(mark(x0)))))
MARK(minus(y0, from(x0))) → ACTIVE(minus(mark(y0), active(from(mark(x0)))))
MARK(zWquot(cons(x0, x1), y1)) → ACTIVE(zWquot(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(zWquot(s(x0), y1)) → ACTIVE(zWquot(active(s(mark(x0))), mark(y1)))
MARK(quot(y0, zWquot(x0, x1))) → ACTIVE(quot(mark(y0), active(zWquot(mark(x0), mark(x1)))))
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(y0, nil)) → ACTIVE(zWquot(mark(y0), active(nil)))
MARK(zWquot(nil, y1)) → ACTIVE(zWquot(active(nil), mark(y1)))
MARK(minus(X1, X2)) → MARK(X2)
MARK(minus(y0, quot(x0, x1))) → ACTIVE(minus(mark(y0), active(quot(mark(x0), mark(x1)))))
MARK(from(X)) → MARK(X)
MARK(minus(x0, y1)) → ACTIVE(minus(x0, mark(y1)))
MARK(minus(y0, x1)) → ACTIVE(minus(mark(y0), x1))
MARK(quot(y0, cons(x0, x1))) → ACTIVE(quot(mark(y0), active(cons(mark(x0), x1))))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(minus(minus(x0, x1), y1)) → ACTIVE(minus(active(minus(mark(x0), mark(x1))), mark(y1)))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(minus(X1, X2)) → MARK(X1)
MARK(zWquot(y0, cons(x0, x1))) → ACTIVE(zWquot(mark(y0), active(cons(mark(x0), x1))))

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)

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