active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(tail(X)) → TAIL(mark(X))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
TAIL(active(X)) → TAIL(X)
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
TAIL(mark(X)) → TAIL(X)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
ACTIVE(zeros) → CONS(0, zeros)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(tail(X)) → TAIL(mark(X))
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
TAIL(active(X)) → TAIL(X)
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
TAIL(mark(X)) → TAIL(X)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(0) → ACTIVE(0)
ACTIVE(zeros) → CONS(0, zeros)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ QDP
TAIL(active(X)) → TAIL(X)
TAIL(mark(X)) → TAIL(X)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
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 + x_1
POL(mark(x1)) = 4 + (4)x_1
POL(TAIL(x1)) = (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
↳ QDP
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ 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(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
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 12.
POL(active(x1)) = 4 + (4)x_1
POL(CONS(x1, x2)) = (4)x_1 + (3)x_2
POL(mark(x1)) = 4 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
↳ QDP
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(tail(X)) → MARK(X)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(tail(cons(X, XS))) → MARK(XS)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
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)
MARK(tail(X)) → ACTIVE(tail(mark(X)))
ACTIVE(tail(cons(X, XS))) → MARK(XS)
Used ordering: Polynomial interpretation [25,35]:
MARK(zeros) → ACTIVE(zeros)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
The value of delta used in the strict ordering is 2.
POL(cons(x1, x2)) = (4)x_1 + x_2
POL(active(x1)) = x_1
POL(MARK(x1)) = (4)x_1
POL(tail(x1)) = 2 + (4)x_1
POL(zeros) = 0
POL(mark(x1)) = (4)x_1
POL(0) = 0
POL(ACTIVE(x1)) = x_1
active(zeros) → mark(cons(0, zeros))
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(tail(X)) → active(tail(mark(X)))
active(tail(cons(X, XS))) → mark(XS)
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
MARK(cons(X1, X2)) → MARK(X1)
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
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)) → MARK(X1)
Used ordering: Polynomial interpretation [25,35]:
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
The value of delta used in the strict ordering is 16.
POL(cons(x1, x2)) = 4 + (4)x_1
POL(MARK(x1)) = 4 + (4)x_1
POL(zeros) = 4
POL(0) = 0
POL(ACTIVE(x1)) = 4 + (4)x_1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ DependencyGraphProof
MARK(zeros) → ACTIVE(zeros)
ACTIVE(zeros) → MARK(cons(0, zeros))
active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
mark(zeros) → active(zeros)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(0) → active(0)
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)
tail(mark(X)) → tail(X)
tail(active(X)) → tail(X)