(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a__zeros → cons(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__zeros → zeros
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__zeros → cons(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__zeros → zeros
(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