(0) Obligation:

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

a__zeroscons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(tail(X)) → a__tail(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeroszeros
a__tail(X) → tail(X)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:
[azeros, atail1, mark1, tail1] > 0 > cons2
[azeros, atail1, mark1, tail1] > zeros > cons2

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__zeroscons(0, zeros)
a__tail(cons(X, XS)) → mark(XS)
mark(zeros) → a__zeros
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
a__zeroszeros


(2) Obligation:

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

mark(tail(X)) → a__tail(mark(X))
a__tail(X) → tail(X)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:
mark1 > [tail1, atail1]

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

mark(tail(X)) → a__tail(mark(X))


(4) Obligation:

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

a__tail(X) → tail(X)

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Lexicographic Path Order [LPO].
Precedence:
atail1 > tail1

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

a__tail(X) → tail(X)


(6) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

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

(8) TRUE