active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(nats) → CONS(0, incr(nats))
MARK(s(X)) → MARK(X)
MARK(incr(X)) → INCR(mark(X))
TAIL(active(X)) → TAIL(X)
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
INCR(active(X)) → INCR(X)
HEAD(mark(X)) → HEAD(X)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(pairs) → ACTIVE(pairs)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(nats) → INCR(nats)
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(odds) → INCR(pairs)
ACTIVE(incr(cons(X, XS))) → S(X)
ACTIVE(incr(cons(X, XS))) → INCR(XS)
MARK(head(X)) → HEAD(mark(X))
S(active(X)) → S(X)
S(mark(X)) → S(X)
INCR(mark(X)) → INCR(X)
MARK(tail(X)) → TAIL(mark(X))
MARK(s(X)) → S(mark(X))
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
ACTIVE(pairs) → INCR(odds)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(pairs) → CONS(0, incr(odds))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
HEAD(active(X)) → HEAD(X)
TAIL(mark(X)) → TAIL(X)
ACTIVE(incr(cons(X, XS))) → CONS(s(X), incr(XS))
ACTIVE(head(cons(X, XS))) → MARK(X)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(0) → ACTIVE(0)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
CONS(mark(X1), X2) → CONS(X1, X2)
ACTIVE(nats) → CONS(0, incr(nats))
MARK(s(X)) → MARK(X)
MARK(incr(X)) → INCR(mark(X))
TAIL(active(X)) → TAIL(X)
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
INCR(active(X)) → INCR(X)
HEAD(mark(X)) → HEAD(X)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(pairs) → ACTIVE(pairs)
MARK(s(X)) → ACTIVE(s(mark(X)))
ACTIVE(nats) → INCR(nats)
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(odds) → INCR(pairs)
ACTIVE(incr(cons(X, XS))) → S(X)
ACTIVE(incr(cons(X, XS))) → INCR(XS)
MARK(head(X)) → HEAD(mark(X))
S(active(X)) → S(X)
S(mark(X)) → S(X)
INCR(mark(X)) → INCR(X)
MARK(tail(X)) → TAIL(mark(X))
MARK(s(X)) → S(mark(X))
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
ACTIVE(pairs) → INCR(odds)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
ACTIVE(pairs) → CONS(0, incr(odds))
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
HEAD(active(X)) → HEAD(X)
TAIL(mark(X)) → TAIL(X)
ACTIVE(incr(cons(X, XS))) → CONS(s(X), incr(XS))
ACTIVE(head(cons(X, XS))) → MARK(X)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(0) → ACTIVE(0)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(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(TAIL(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(X)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
HEAD(mark(X)) → HEAD(X)
HEAD(active(X)) → HEAD(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(HEAD(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
↳ QDP
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
↳ QDP
S(mark(X)) → S(X)
S(active(X)) → S(X)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
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 + (4)x_1
POL(mark(x1)) = 4 + x_1
POL(S(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
↳ QDP
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
INCR(mark(X)) → INCR(X)
INCR(active(X)) → INCR(X)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
INCR(mark(X)) → INCR(X)
INCR(active(X)) → INCR(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(INCR(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ 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(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
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 3.
POL(active(x1)) = 4 + (4)x_1
POL(CONS(x1, x2)) = (4)x_1 + x_2
POL(mark(x1)) = 3 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(head(cons(X, XS))) → MARK(X)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
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]:
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(head(cons(X, XS))) → MARK(X)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
The value of delta used in the strict ordering is 4.
POL(tail(x1)) = 1
POL(head(x1)) = 1
POL(pairs) = 1
POL(mark(x1)) = 4
POL(0) = 3
POL(ACTIVE(x1)) = (4)x_1
POL(cons(x1, x2)) = 0
POL(odds) = 1
POL(active(x1)) = 2
POL(MARK(x1)) = 4
POL(incr(x1)) = 1
POL(s(x1)) = 0
POL(nats) = 1
s(active(X)) → s(X)
s(mark(X)) → s(X)
head(active(X)) → head(X)
head(mark(X)) → head(X)
incr(active(X)) → incr(X)
incr(mark(X)) → incr(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(head(cons(X, XS))) → MARK(X)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(head(X)) → MARK(X)
ACTIVE(head(cons(X, XS))) → MARK(X)
Used ordering: Polynomial interpretation [25,35]:
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
The value of delta used in the strict ordering is 8.
POL(tail(x1)) = (4)x_1
POL(head(x1)) = 2 + (2)x_1
POL(pairs) = 0
POL(mark(x1)) = x_1
POL(0) = 0
POL(ACTIVE(x1)) = (4)x_1
POL(cons(x1, x2)) = x_1 + (2)x_2
POL(odds) = 0
POL(active(x1)) = x_1
POL(MARK(x1)) = (4)x_1
POL(incr(x1)) = (4)x_1
POL(s(x1)) = (4)x_1
POL(nats) = 0
s(active(X)) → s(X)
s(mark(X)) → s(X)
head(active(X)) → head(X)
head(mark(X)) → head(X)
incr(active(X)) → incr(X)
incr(mark(X)) → incr(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
mark(0) → active(0)
active(nats) → mark(cons(0, incr(nats)))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(odds) → active(odds)
active(tail(cons(X, XS))) → mark(XS)
active(pairs) → mark(cons(0, incr(odds)))
mark(head(X)) → active(head(mark(X)))
mark(incr(X)) → active(incr(mark(X)))
mark(nats) → active(nats)
mark(s(X)) → active(s(mark(X)))
active(head(cons(X, XS))) → mark(X)
mark(pairs) → active(pairs)
mark(tail(X)) → active(tail(mark(X)))
active(odds) → mark(incr(pairs))
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(head(X)) → ACTIVE(head(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(head(X)) → ACTIVE(head(mark(X)))
Used ordering: Polynomial interpretation [25,35]:
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
The value of delta used in the strict ordering is 2.
POL(tail(x1)) = 1
POL(head(x1)) = 0
POL(pairs) = 1
POL(mark(x1)) = 4
POL(0) = 0
POL(ACTIVE(x1)) = (2)x_1
POL(cons(x1, x2)) = 0
POL(odds) = 1
POL(active(x1)) = 3
POL(MARK(x1)) = 2
POL(incr(x1)) = 1
POL(s(x1)) = (2)x_1
POL(nats) = 1
head(active(X)) → head(X)
head(mark(X)) → head(X)
incr(active(X)) → incr(X)
incr(mark(X)) → incr(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(tail(X)) → MARK(X)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
Used ordering: Polynomial interpretation [25,35]:
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(s(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
The value of delta used in the strict ordering is 8.
POL(tail(x1)) = 4 + (4)x_1
POL(head(x1)) = 1 + (4)x_1
POL(pairs) = 0
POL(mark(x1)) = x_1
POL(0) = 0
POL(ACTIVE(x1)) = (2)x_1
POL(cons(x1, x2)) = (4)x_1 + (2)x_2
POL(odds) = 0
POL(active(x1)) = x_1
POL(MARK(x1)) = (2)x_1
POL(incr(x1)) = x_1
POL(s(x1)) = x_1
POL(nats) = 0
s(active(X)) → s(X)
s(mark(X)) → s(X)
head(active(X)) → head(X)
head(mark(X)) → head(X)
incr(active(X)) → incr(X)
incr(mark(X)) → incr(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
mark(0) → active(0)
active(nats) → mark(cons(0, incr(nats)))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(odds) → active(odds)
active(tail(cons(X, XS))) → mark(XS)
active(pairs) → mark(cons(0, incr(odds)))
mark(head(X)) → active(head(mark(X)))
mark(incr(X)) → active(incr(mark(X)))
mark(nats) → active(nats)
mark(s(X)) → active(s(mark(X)))
active(head(cons(X, XS))) → mark(X)
mark(pairs) → active(pairs)
mark(tail(X)) → active(tail(mark(X)))
active(odds) → mark(incr(pairs))
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)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(s(X)) → MARK(X)
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
MARK(incr(X)) → MARK(X)
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
MARK(tail(X)) → ACTIVE(tail(mark(X)))
Used ordering: Polynomial interpretation [25,35]:
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(pairs) → ACTIVE(pairs)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(s(X)) → MARK(X)
MARK(odds) → ACTIVE(odds)
ACTIVE(odds) → MARK(incr(pairs))
MARK(incr(X)) → MARK(X)
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
The value of delta used in the strict ordering is 2.
POL(tail(x1)) = 0
POL(head(x1)) = 4 + (2)x_1
POL(pairs) = 1
POL(mark(x1)) = 4
POL(0) = 3
POL(ACTIVE(x1)) = (2)x_1
POL(cons(x1, x2)) = 0
POL(odds) = 1
POL(active(x1)) = 3
POL(MARK(x1)) = 2
POL(incr(x1)) = 1
POL(s(x1)) = 1
POL(nats) = 1
incr(active(X)) → incr(X)
incr(mark(X)) → incr(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDPOrderProof
↳ QDP
ACTIVE(incr(cons(X, XS))) → MARK(cons(s(X), incr(XS)))
MARK(pairs) → ACTIVE(pairs)
MARK(s(X)) → MARK(X)
ACTIVE(pairs) → MARK(cons(0, incr(odds)))
MARK(odds) → ACTIVE(odds)
MARK(incr(X)) → ACTIVE(incr(mark(X)))
MARK(incr(X)) → MARK(X)
ACTIVE(odds) → MARK(incr(pairs))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(nats) → MARK(cons(0, incr(nats)))
MARK(nats) → ACTIVE(nats)
active(nats) → mark(cons(0, incr(nats)))
active(pairs) → mark(cons(0, incr(odds)))
active(odds) → mark(incr(pairs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(head(cons(X, XS))) → mark(X)
active(tail(cons(X, XS))) → mark(XS)
mark(nats) → active(nats)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
mark(incr(X)) → active(incr(mark(X)))
mark(pairs) → active(pairs)
mark(odds) → active(odds)
mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))
mark(tail(X)) → active(tail(mark(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)
incr(mark(X)) → incr(X)
incr(active(X)) → incr(X)
s(mark(X)) → s(X)
s(active(X)) → s(X)
head(mark(X)) → head(X)
head(active(X)) → head(X)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)