YES Termination proof of ../tpdb/TRS/CSR/Ex4_7_77_Bor03.trs
Termination of the following Term Rewriting System could be proven:

Q restricted 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}


↳ CSR
  ↳ Zantema-Transformation

Q restricted 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}

We applied the Zantema [34] to transform the context-sensitive TRS to an usual TRS.

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

zeros → cons(0, zerosInact)
tail(cons(X, XS)) → a(XS)
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

Q is empty.

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

zeros → cons(0, zerosInact)
tail(cons(X, XS)) → a(XS)
a(x) → x
zeros → zerosInact
a(zerosInact) → zeros

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

tail(cons(X, XS)) → a(XS)
a(x) → x
a(zerosInact) → zeros
Used ordering:
Polynomial interpretation [25]:

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




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

zeros → cons(0, zerosInact)
zeros → zerosInact

Q is empty.

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

zeros → cons(0, zerosInact)
zeros → zerosInact

Q is empty.
The following rules can be removed by the rule removal processor [15] because they are oriented strictly by a polynomial ordering:

zeros → cons(0, zerosInact)
zeros → zerosInact
Used ordering:
Polynomial interpretation [25]:

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




↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RisEmptyProof

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

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