(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:
Polynomial interpretation [POLO]:
POL(0) = 0
POL(a__tail(x1)) = 3 + x1
POL(a__zeros) = 1
POL(cons(x1, x2)) = x1 + x2
POL(mark(x1)) = 2 + x1
POL(tail(x1)) = 3 + x1
POL(zeros) = 0
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(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))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__tail(X) → tail(X)
Q is empty.
(3) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a__tail(x1)) = 2 + x1
POL(cons(x1, x2)) = 1 + x1 + x2
POL(mark(x1)) = 2 + 2·x1
POL(tail(x1)) = 1 + x1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__tail(X) → tail(X)
(4) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
mark(tail(X)) → a__tail(mark(X))
Q is empty.
(5) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(a__tail(x1)) = x1
POL(mark(x1)) = x1
POL(tail(x1)) = 1 + x1
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))
(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