Termination w.r.t. Q proof of /tmp/tmpv33QA9/Ex4_7_37_Bor03

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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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:

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 10 SCCs with 31 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ 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(mark(X1), X2) → ZWQUOT(X1, X2)
ZWQUOT(ok(X1), ok(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

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

          ↳ QDP

          ↳ QDP

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

QUOT(mark(X1), X2) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
QUOT(ok(X1), ok(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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(ok(X1), ok(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

          ↳ QDP

          ↳ QDP

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

MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(ok(X1), ok(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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(ok(X1), ok(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

          ↳ QDP

          ↳ QDP

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

SEL(mark(X1), X2) → SEL(X1, X2)
SEL(ok(X1), ok(X2)) → SEL(X1, X2)
SEL(X1, mark(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

active(from(X)) → mark(cons(X, from(s(X))))
active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

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

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

          ↳ QDP

          ↳ QDP

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(ok(X1), ok(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

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

          ↳ QDP

          ↳ QDP

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

FROM(mark(X)) → FROM(X)
FROM(ok(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

PROPER(minus(X1, X2)) → PROPER(X2)
PROPER(minus(X1, X2)) → PROPER(X1)
PROPER(s(X)) → PROPER(X)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(zWquot(X1, X2)) → PROPER(X1)
PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(from(X)) → PROPER(X)
PROPER(zWquot(X1, X2)) → PROPER(X2)
PROPER(quot(X1, X2)) → PROPER(X2)
PROPER(quot(X1, X2)) → PROPER(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

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

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

PROPER(minus(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(minus(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(zWquot(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(sel(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(sel(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(from(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(zWquot(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(quot(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(quot(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

ACTIVE(quot(X1, X2)) → ACTIVE(X2)
ACTIVE(zWquot(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(minus(X1, X2)) → ACTIVE(X1)
ACTIVE(zWquot(X1, X2)) → ACTIVE(X1)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(quot(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(minus(X1, X2)) → ACTIVE(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

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

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

ACTIVE(quot(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(zWquot(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(minus(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(zWquot(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(quot(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(from(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(minus(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

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

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
from(mark(X)) → mark(from(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
s(mark(X)) → mark(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)
from(ok(X)) → ok(from(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(ok(X)) → ok(s(X))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = 2·x₁
POL(active(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(from(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(minus(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(ok(x₁)) = 2·x₁
POL(proper(x₁)) = x₁
POL(quot(x₁, x₂)) = x₁ + 2·x₂
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(zWquot(x₁, x₂)) = 2·x₁ + 2·x₂

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

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

TOP(mark(X)) → TOP(proper(X))
TOP(ok(X)) → TOP(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

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

TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(0)) → TOP(ok(0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(0)) → TOP(ok(0))
TOP(ok(X)) → TOP(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(quot(0, s(x0)))) → TOP(mark(0))
TOP(ok(quot(s(x0), s(x1)))) → TOP(mark(s(quot(minus(x0, x1), s(x1)))))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(minus(x0, 0))) → TOP(mark(0))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(ok(zWquot(x0, nil))) → TOP(mark(nil))
TOP(ok(zWquot(nil, x0))) → TOP(mark(nil))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(minus(x0, x1))) → TOP(minus(active(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

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

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quot(0, s(x0)))) → TOP(mark(0))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(minus(x0, 0))) → TOP(mark(0))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(ok(zWquot(nil, x0))) → TOP(mark(nil))
TOP(ok(zWquot(x0, nil))) → TOP(mark(nil))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(quot(s(x0), s(x1)))) → TOP(mark(s(quot(minus(x0, x1), s(x1)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 6 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(quot(s(x0), s(x1)))) → TOP(mark(s(quot(minus(x0, x1), s(x1)))))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

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.

TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))

The remaining pairs can at least be oriented weakly.

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(quot(s(x0), s(x1)))) → TOP(mark(s(quot(minus(x0, x1), s(x1)))))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(x0), x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂
POL(from(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(minus(x₁, x₂)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = x₁
POL(quot(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 1 + x₂
POL(zWquot(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

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

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(quot(s(x0), s(x1)))) → TOP(mark(s(quot(minus(x0, x1), s(x1)))))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

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.

TOP(ok(quot(s(x0), s(x1)))) → TOP(mark(s(quot(minus(x0, x1), s(x1)))))

The remaining pairs can at least be oriented weakly.

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(x0), x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(minus(x₁, x₂)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quot(x₁, x₂)) = 1
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(zWquot(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

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.

TOP(ok(zWquot(cons(x0, x1), cons(x2, x3)))) → TOP(mark(cons(quot(x0, x2), zWquot(x1, x3))))

The remaining pairs can at least be oriented weakly.

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(x0), x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = x₁
POL(minus(x₁, x₂)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quot(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(zWquot(x₁, x₂)) = 1

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

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.

TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))

The remaining pairs can at least be oriented weakly.

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(x0), x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 1
POL(mark(x₁)) = x₁
POL(minus(x₁, x₂)) = 0
POL(nil) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quot(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(zWquot(x₁, x₂)) = 0

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

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

TOP(ok(zWquot(x0, x1))) → TOP(zWquot(x0, active(x1)))
TOP(ok(zWquot(x0, x1))) → TOP(zWquot(active(x0), x1))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(mark(zWquot(x0, x1))) → TOP(zWquot(proper(x0), proper(x1)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(minus(s(x0), s(x1)))) → TOP(mark(minus(x0, x1)))
TOP(ok(quot(x0, x1))) → TOP(quot(active(x0), x1))
TOP(mark(quot(x0, x1))) → TOP(quot(proper(x0), proper(x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(minus(x0, x1))) → TOP(minus(x0, active(x1)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(minus(x0, x1))) → TOP(minus(proper(x0), proper(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(quot(x0, x1))) → TOP(quot(x0, active(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(minus(x0, x1))) → TOP(minus(active(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)))
active(from(X)) → from(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(s(X)) → s(active(X))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(minus(X1, X2)) → minus(active(X1), X2)
active(minus(X1, X2)) → minus(X1, active(X2))
active(quot(X1, X2)) → quot(active(X1), X2)
active(quot(X1, X2)) → quot(X1, active(X2))
active(zWquot(X1, X2)) → zWquot(active(X1), X2)
active(zWquot(X1, X2)) → zWquot(X1, active(X2))
zWquot(mark(X1), X2) → mark(zWquot(X1, X2))
zWquot(X1, mark(X2)) → mark(zWquot(X1, X2))
zWquot(ok(X1), ok(X2)) → ok(zWquot(X1, X2))
quot(mark(X1), X2) → mark(quot(X1, X2))
quot(X1, mark(X2)) → mark(quot(X1, X2))
quot(ok(X1), ok(X2)) → ok(quot(X1, X2))
minus(mark(X1), X2) → mark(minus(X1, X2))
minus(X1, mark(X2)) → mark(minus(X1, X2))
minus(ok(X1), ok(X2)) → ok(minus(X1, X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
proper(from(X)) → from(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(0) → ok(0)
proper(minus(X1, X2)) → minus(proper(X1), proper(X2))
proper(quot(X1, X2)) → quot(proper(X1), proper(X2))
proper(zWquot(X1, X2)) → zWquot(proper(X1), proper(X2))
proper(nil) → ok(nil)

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