Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

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

first(0, X) → nil
first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → 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:

first(0, X) → nil
Used ordering:
Polynomial interpretation [25]:

POL(0) = 2   
POL(activate(x1)) = 2·x1   
POL(cons(x1, x2)) = x1 + x2   
POL(first(x1, x2)) = 2·x1 + 2·x2   
POL(from(x1)) = 2·x1   
POL(n__first(x1, x2)) = x1 + x2   
POL(n__from(x1)) = x1   
POL(nil) = 2   
POL(s(x1)) = x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

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

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
from(X) → cons(X, n__from(s(X)))
first(X1, X2) → n__first(X1, X2)
from(X) → n__from(X)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → 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, n__from(s(X)))
from(X) → n__from(X)
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 2·x1   
POL(cons(x1, x2)) = x1 + x2   
POL(first(x1, x2)) = x1 + 2·x2   
POL(from(x1)) = 2 + 2·x1   
POL(n__first(x1, x2)) = x1 + x2   
POL(n__from(x1)) = 1 + x1   
POL(s(x1)) = x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof

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

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
first(X1, X2) → n__first(X1, X2)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

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

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
first(X1, X2) → n__first(X1, X2)
activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → 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:

activate(n__first(X1, X2)) → first(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 2 + 2·x1   
POL(cons(x1, x2)) = 1 + x1 + 2·x2   
POL(first(x1, x2)) = 1 + 2·x1 + 2·x2   
POL(from(x1)) = 1 + 2·x1   
POL(n__first(x1, x2)) = 1 + 2·x1 + x2   
POL(n__from(x1)) = 2 + 2·x1   
POL(s(x1)) = 2 + 2·x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof
        ↳ QTRS
          ↳ RRRPoloQTRSProof
            ↳ QTRS
              ↳ RRRPoloQTRSProof

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

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
first(X1, X2) → n__first(X1, X2)

Q is empty.

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

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
first(X1, X2) → n__first(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:

first(s(X), cons(Y, Z)) → cons(Y, n__first(X, activate(Z)))
first(X1, X2) → n__first(X1, X2)
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = x1   
POL(cons(x1, x2)) = 1 + x1 + 2·x2   
POL(first(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(n__first(x1, x2)) = 1 + x1 + 2·x2   
POL(s(x1)) = 2 + x1   




↳ 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.