Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

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

active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

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

active(zeros) → mark(cons(0, zeros))
active(tail(cons(X, XS))) → mark(XS)
active(cons(X1, X2)) → cons(active(X1), X2)
active(tail(X)) → tail(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
tail(mark(X)) → mark(tail(X))
proper(zeros) → ok(zeros)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(tail(X)) → tail(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
tail(ok(X)) → ok(tail(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tail: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)
tail(cons(X, XS)) → XS

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tail: {1}

(3) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

tail(cons(X, XS)) → XS

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(cons(x₁, x₂)) = 2·x₁ + 2·x₂   
POL(tail(x₁)) = 1 + x₁   
POL(zeros) = 0   

(4) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

zeros → cons(0, zeros)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set

(5) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

zeros → cons(0, zeros)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0   
POL(cons(x₁, x₂)) = 1 + 2·x₁   
POL(zeros) = 2   

(6) Obligation:

Context-sensitive rewrite system:
R is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

The CSR R is empty. Hence, termination is trivially proven.

(8) TRUE