active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
FROM(mark(X)) → FROM(X)
QUOT(X1, active(X2)) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MINUS(X, Y)
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
FROM(active(X)) → FROM(X)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
MARK(zWquot(X1, X2)) → MARK(X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → QUOT(minus(X, Y), s(Y))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
ACTIVE(zWquot(nil, XS)) → MARK(nil)
ACTIVE(zWquot(XS, nil)) → MARK(nil)
MINUS(X1, active(X2)) → MINUS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → CONS(quot(X, Y), zWquot(XS, YS))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MINUS(active(X1), X2) → MINUS(X1, X2)
MARK(minus(X1, X2)) → MINUS(mark(X1), mark(X2))
MARK(quot(X1, X2)) → QUOT(mark(X1), mark(X2))
S(active(X)) → S(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MARK(quot(X1, X2)) → MARK(X1)
ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → QUOT(X, Y)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(from(X)) → FROM(s(X))
SEL(mark(X1), X2) → SEL(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
SEL(X1, active(X2)) → SEL(X1, X2)
MARK(s(X)) → MARK(X)
MARK(from(X)) → FROM(mark(X))
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → ZWQUOT(XS, YS)
MARK(quot(X1, X2)) → MARK(X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(quot(s(X), s(Y))) → S(quot(minus(X, Y), s(Y)))
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
QUOT(active(X1), X2) → QUOT(X1, X2)
ACTIVE(quot(0, s(Y))) → MARK(0)
MARK(minus(X1, X2)) → MARK(X2)
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
S(mark(X)) → S(X)
MARK(from(X)) → MARK(X)
MARK(s(X)) → S(mark(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(minus(X, 0)) → MARK(0)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
ACTIVE(from(X)) → S(X)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
QUOT(mark(X1), X2) → QUOT(X1, X2)
MARK(0) → ACTIVE(0)
MARK(zWquot(X1, X2)) → ZWQUOT(mark(X1), mark(X2))
ACTIVE(sel(s(N), cons(X, XS))) → SEL(N, XS)
MARK(minus(X1, X2)) → MARK(X1)
MARK(nil) → ACTIVE(nil)
ACTIVE(from(X)) → CONS(X, from(s(X)))
MARK(sel(X1, X2)) → MARK(X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
FROM(mark(X)) → FROM(X)
QUOT(X1, active(X2)) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MINUS(X, Y)
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
FROM(active(X)) → FROM(X)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
MARK(zWquot(X1, X2)) → MARK(X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → QUOT(minus(X, Y), s(Y))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
ACTIVE(zWquot(nil, XS)) → MARK(nil)
ACTIVE(zWquot(XS, nil)) → MARK(nil)
MINUS(X1, active(X2)) → MINUS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → CONS(quot(X, Y), zWquot(XS, YS))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MINUS(active(X1), X2) → MINUS(X1, X2)
MARK(minus(X1, X2)) → MINUS(mark(X1), mark(X2))
MARK(quot(X1, X2)) → QUOT(mark(X1), mark(X2))
S(active(X)) → S(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MARK(quot(X1, X2)) → MARK(X1)
ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → QUOT(X, Y)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(quot(s(X), s(Y))) → MINUS(X, Y)
ACTIVE(from(X)) → FROM(s(X))
SEL(mark(X1), X2) → SEL(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
SEL(X1, active(X2)) → SEL(X1, X2)
MARK(s(X)) → MARK(X)
MARK(from(X)) → FROM(mark(X))
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
CONS(X1, active(X2)) → CONS(X1, X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → ZWQUOT(XS, YS)
MARK(quot(X1, X2)) → MARK(X2)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(quot(s(X), s(Y))) → S(quot(minus(X, Y), s(Y)))
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
QUOT(active(X1), X2) → QUOT(X1, X2)
ACTIVE(quot(0, s(Y))) → MARK(0)
MARK(minus(X1, X2)) → MARK(X2)
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
S(mark(X)) → S(X)
MARK(from(X)) → MARK(X)
MARK(s(X)) → S(mark(X))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(minus(X, 0)) → MARK(0)
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
ACTIVE(from(X)) → S(X)
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
QUOT(mark(X1), X2) → QUOT(X1, X2)
MARK(0) → ACTIVE(0)
MARK(zWquot(X1, X2)) → ZWQUOT(mark(X1), mark(X2))
ACTIVE(sel(s(N), cons(X, XS))) → SEL(N, XS)
MARK(minus(X1, X2)) → MARK(X1)
MARK(nil) → ACTIVE(nil)
ACTIVE(from(X)) → CONS(X, from(s(X)))
MARK(sel(X1, X2)) → MARK(X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
ZWQUOT(X1, mark(X2)) → ZWQUOT(X1, X2)
ZWQUOT(X1, active(X2)) → ZWQUOT(X1, X2)
ZWQUOT(active(X1), X2) → ZWQUOT(X1, X2)
ZWQUOT(mark(X1), X2) → ZWQUOT(X1, X2)
The value of delta used in the strict ordering is 16.
POL(active(x1)) = 4 + (2)x_1
POL(mark(x1)) = 4 + (2)x_1
POL(ZWQUOT(x1, x2)) = (4)x_1 + (4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
QUOT(X1, active(X2)) → QUOT(X1, X2)
QUOT(mark(X1), X2) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
QUOT(active(X1), X2) → QUOT(X1, X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
QUOT(X1, active(X2)) → QUOT(X1, X2)
QUOT(mark(X1), X2) → QUOT(X1, X2)
QUOT(X1, mark(X2)) → QUOT(X1, X2)
QUOT(active(X1), X2) → QUOT(X1, X2)
The value of delta used in the strict ordering is 1.
POL(active(x1)) = 3 + (3)x_1
POL(QUOT(x1, x2)) = x_1 + (4)x_2
POL(mark(x1)) = 1 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
MINUS(X1, active(X2)) → MINUS(X1, X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(active(X1), X2) → MINUS(X1, X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MINUS(X1, active(X2)) → MINUS(X1, X2)
MINUS(mark(X1), X2) → MINUS(X1, X2)
MINUS(X1, mark(X2)) → MINUS(X1, X2)
MINUS(active(X1), X2) → MINUS(X1, X2)
The value of delta used in the strict ordering is 16.
POL(active(x1)) = 4 + (2)x_1
POL(MINUS(x1, x2)) = (4)x_1 + (4)x_2
POL(mark(x1)) = 4 + (2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
The value of delta used in the strict ordering is 16.
POL(active(x1)) = 4 + (2)x_1
POL(mark(x1)) = 4 + (2)x_1
POL(SEL(x1, x2)) = (4)x_1 + (4)x_2
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
S(mark(X)) → S(X)
S(active(X)) → S(X)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
S(mark(X)) → S(X)
S(active(X)) → S(X)
The value of delta used in the strict ordering is 16.
POL(active(x1)) = 4 + x_1
POL(mark(x1)) = 4 + (4)x_1
POL(S(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
CONS(X1, active(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
CONS(X1, active(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
The value of delta used in the strict ordering is 1.
POL(active(x1)) = 1 + (4)x_1
POL(CONS(x1, x2)) = (2)x_1 + x_2
POL(mark(x1)) = 1 + (2)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
FROM(mark(X)) → FROM(X)
FROM(active(X)) → FROM(X)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
FROM(mark(X)) → FROM(X)
FROM(active(X)) → FROM(X)
The value of delta used in the strict ordering is 16.
POL(active(x1)) = 4 + (4)x_1
POL(mark(x1)) = 4 + x_1
POL(FROM(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(from(X)) → ACTIVE(from(mark(X)))
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel(X1, X2)) → MARK(X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))
Used ordering: Polynomial interpretation [25,35]:
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(minus(X1, X2)) → MARK(X1)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(sel(X1, X2)) → MARK(X2)
The value of delta used in the strict ordering is 2.
POL(sel(x1, x2)) = 2
POL(from(x1)) = 2
POL(minus(x1, x2)) = 2
POL(mark(x1)) = 0
POL(0) = 0
POL(ACTIVE(x1)) = x_1
POL(cons(x1, x2)) = 0
POL(active(x1)) = 2 + (4)x_1
POL(MARK(x1)) = 2
POL(quot(x1, x2)) = 2
POL(s(x1)) = 0
POL(zWquot(x1, x2)) = 2
POL(nil) = 2
from(active(X)) → from(X)
from(mark(X)) → from(X)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(minus(s(X), s(Y))) → MARK(minus(X, Y))
MARK(s(X)) → MARK(X)
MARK(zWquot(X1, X2)) → ACTIVE(zWquot(mark(X1), mark(X2)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(quot(X1, X2)) → MARK(X1)
MARK(minus(X1, X2)) → ACTIVE(minus(mark(X1), mark(X2)))
MARK(zWquot(X1, X2)) → MARK(X2)
MARK(quot(X1, X2)) → ACTIVE(quot(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(quot(X1, X2)) → MARK(X2)
MARK(from(X)) → ACTIVE(from(mark(X)))
ACTIVE(zWquot(cons(X, XS), cons(Y, YS))) → MARK(cons(quot(X, Y), zWquot(XS, YS)))
MARK(zWquot(X1, X2)) → MARK(X1)
ACTIVE(quot(s(X), s(Y))) → MARK(s(quot(minus(X, Y), s(Y))))
MARK(minus(X1, X2)) → MARK(X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(minus(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → MARK(X2)
active(from(X)) → mark(cons(X, from(s(X))))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
active(minus(X, 0)) → mark(0)
active(minus(s(X), s(Y))) → mark(minus(X, Y))
active(quot(0, s(Y))) → mark(0)
active(quot(s(X), s(Y))) → mark(s(quot(minus(X, Y), s(Y))))
active(zWquot(XS, nil)) → mark(nil)
active(zWquot(nil, XS)) → mark(nil)
active(zWquot(cons(X, XS), cons(Y, YS))) → mark(cons(quot(X, Y), zWquot(XS, YS)))
mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(s(X)) → active(s(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(0) → active(0)
mark(minus(X1, X2)) → active(minus(mark(X1), mark(X2)))
mark(quot(X1, X2)) → active(quot(mark(X1), mark(X2)))
mark(zWquot(X1, X2)) → active(zWquot(mark(X1), mark(X2)))
mark(nil) → active(nil)
from(mark(X)) → from(X)
from(active(X)) → from(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
minus(mark(X1), X2) → minus(X1, X2)
minus(X1, mark(X2)) → minus(X1, X2)
minus(active(X1), X2) → minus(X1, X2)
minus(X1, active(X2)) → minus(X1, X2)
quot(mark(X1), X2) → quot(X1, X2)
quot(X1, mark(X2)) → quot(X1, X2)
quot(active(X1), X2) → quot(X1, X2)
quot(X1, active(X2)) → quot(X1, X2)
zWquot(mark(X1), X2) → zWquot(X1, X2)
zWquot(X1, mark(X2)) → zWquot(X1, X2)
zWquot(active(X1), X2) → zWquot(X1, X2)
zWquot(X1, active(X2)) → zWquot(X1, X2)