YES Termination proof of ../tpdb/TRS/CSR/Ex1_2_Luc02c.trs
Termination of the following Term Rewriting System could be proven:

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

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons: {1}
from: {1}
s: {1}


↳ CSR
  ↳ Zantema-Transformation

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

2nd(cons(X, cons(Y, Z))) → Y
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

2nd: {1}
cons: {1}
from: {1}
s: {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:

2nd(cons(X, consInact(Y, Z))) → a(Y)
from(X) → cons(X, fromInact(s(X)))
a(x) → x
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)

Q is empty.

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

2nd(cons(X, consInact(Y, Z))) → a(Y)
from(X) → cons(X, fromInact(s(X)))
a(x) → x
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
cons(x1, x2) → consInact(x1, x2)
a(consInact(x1, x2)) → cons(x1, x2)

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

2nd(cons(X, consInact(Y, Z))) → a(Y)
a(x) → x
from(x1) → fromInact(x1)
a(consInact(x1, x2)) → cons(x1, x2)
Used ordering:
Polynomial interpretation [25]:

POL(2nd(x1)) = 2 + 2·x1   
POL(a(x1)) = 1 + 2·x1   
POL(cons(x1, x2)) = x1 + x2   
POL(consInact(x1, x2)) = x1 + x2   
POL(from(x1)) = 1 + 2·x1   
POL(fromInact(x1)) = x1   
POL(s(x1)) = 1 + x1   




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

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

from(X) → cons(X, fromInact(s(X)))
a(fromInact(x1)) → from(x1)
cons(x1, x2) → consInact(x1, x2)

Q is empty.

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

from(X) → cons(X, fromInact(s(X)))
a(fromInact(x1)) → from(x1)
cons(x1, x2) → consInact(x1, x2)

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

a(fromInact(x1)) → from(x1)
Used ordering:
Polynomial interpretation [25]:

POL(a(x1)) = 1 + 2·x1   
POL(cons(x1, x2)) = x1 + x2   
POL(consInact(x1, x2)) = x1 + x2   
POL(from(x1)) = 2·x1   
POL(fromInact(x1)) = x1   
POL(s(x1)) = x1   




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

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

from(X) → cons(X, fromInact(s(X)))
cons(x1, x2) → consInact(x1, x2)

Q is empty.

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

from(X) → cons(X, fromInact(s(X)))
cons(x1, x2) → consInact(x1, x2)

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

cons(x1, x2) → consInact(x1, x2)
Used ordering:
Polynomial interpretation [25]:

POL(cons(x1, x2)) = 1 + x1 + x2   
POL(consInact(x1, x2)) = x1 + x2   
POL(from(x1)) = 1 + 2·x1   
POL(fromInact(x1)) = x1   
POL(s(x1)) = x1   




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

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

from(X) → cons(X, fromInact(s(X)))

Q is empty.

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

from(X) → cons(X, fromInact(s(X)))

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

from(X) → cons(X, fromInact(s(X)))
Used ordering:
Polynomial interpretation [25]:

POL(cons(x1, x2)) = x1 + x2   
POL(from(x1)) = 2 + 2·x1   
POL(fromInact(x1)) = 1 + x1   
POL(s(x1)) = x1   




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