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:

fst(0, Z) → nil
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
len(nil) → 0
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(X1, X2)
activate(n__from(X)) → from(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__len(X)) → len(X)
activate(X) → X

Q is empty.


↳ QTRS
  ↳ RRRPoloQTRSProof

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

fst(0, Z) → nil
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
len(nil) → 0
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(X1, X2)
activate(n__from(X)) → from(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__len(X)) → len(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:

fst(0, Z) → nil
fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(0, X) → X
add(s(X), Y) → s(n__add(activate(X), Y))
len(nil) → 0
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(X1, X2)
activate(n__from(X)) → from(X)
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__len(X)) → len(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:

fst(0, Z) → nil
add(0, X) → X
len(nil) → 0
activate(n__add(X1, X2)) → add(X1, X2)
activate(n__len(X)) → len(X)
Used ordering:
Polynomial interpretation [25]:

POL(0) = 2   
POL(activate(x1)) = 2·x1   
POL(add(x1, x2)) = 1 + 2·x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(from(x1)) = 2·x1   
POL(fst(x1, x2)) = 2·x1 + 2·x2   
POL(len(x1)) = 2 + 2·x1   
POL(n__add(x1, x2)) = 1 + x1 + x2   
POL(n__from(x1)) = x1   
POL(n__fst(x1, x2)) = x1 + x2   
POL(n__len(x1)) = 2 + x1   
POL(nil) = 1   
POL(s(x1)) = x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ QTRS
      ↳ RRRPoloQTRSProof

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

fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(s(X), Y) → s(n__add(activate(X), Y))
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(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:

fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(s(X), Y) → s(n__add(activate(X), Y))
len(cons(X, Z)) → s(n__len(activate(Z)))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
len(X) → n__len(X)
activate(n__fst(X1, X2)) → fst(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:

len(cons(X, Z)) → s(n__len(activate(Z)))
len(X) → n__len(X)
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 2·x1   
POL(add(x1, x2)) = 2·x1 + 2·x2   
POL(cons(x1, x2)) = x1 + x2   
POL(from(x1)) = 2·x1   
POL(fst(x1, x2)) = 2·x1 + 2·x2   
POL(len(x1)) = 2 + 2·x1   
POL(n__add(x1, x2)) = x1 + x2   
POL(n__from(x1)) = x1   
POL(n__fst(x1, x2)) = x1 + x2   
POL(n__len(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:

fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(s(X), Y) → s(n__add(activate(X), Y))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
activate(n__fst(X1, X2)) → fst(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:

fst(s(X), cons(Y, Z)) → cons(Y, n__fst(activate(X), activate(Z)))
from(X) → cons(X, n__from(s(X)))
add(s(X), Y) → s(n__add(activate(X), Y))
fst(X1, X2) → n__fst(X1, X2)
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
activate(n__fst(X1, X2)) → fst(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:

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

POL(activate(x1)) = 2·x1   
POL(add(x1, x2)) = 2·x1 + 2·x2   
POL(cons(x1, x2)) = x1 + x2   
POL(from(x1)) = 2 + 2·x1   
POL(fst(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(n__add(x1, x2)) = x1 + x2   
POL(n__from(x1)) = 2 + x1   
POL(n__fst(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = x1   




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

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

from(X) → cons(X, n__from(s(X)))
add(s(X), Y) → s(n__add(activate(X), Y))
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
activate(n__fst(X1, X2)) → fst(X1, X2)
activate(X) → 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, n__from(s(X)))
add(s(X), Y) → s(n__add(activate(X), Y))
from(X) → n__from(X)
add(X1, X2) → n__add(X1, X2)
activate(n__fst(X1, X2)) → fst(X1, X2)
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)
activate(n__fst(X1, X2)) → fst(X1, X2)
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = x1   
POL(add(x1, x2)) = 2·x1 + 2·x2   
POL(cons(x1, x2)) = x1 + x2   
POL(from(x1)) = 1 + 2·x1   
POL(fst(x1, x2)) = 1 + x1 + x2   
POL(n__add(x1, x2)) = 2·x1 + 2·x2   
POL(n__from(x1)) = x1   
POL(n__fst(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(s(x1)) = x1   




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

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

add(s(X), Y) → s(n__add(activate(X), Y))
add(X1, X2) → n__add(X1, X2)
activate(X) → X

Q is empty.

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

add(s(X), Y) → s(n__add(activate(X), Y))
add(X1, X2) → n__add(X1, X2)
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:

add(s(X), Y) → s(n__add(activate(X), Y))
add(X1, X2) → n__add(X1, X2)
activate(X) → X
Used ordering:
Polynomial interpretation [25]:

POL(activate(x1)) = 1 + x1   
POL(add(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(n__add(x1, x2)) = 1 + x1 + x2   
POL(s(x1)) = 1 + x1   




↳ QTRS
  ↳ RRRPoloQTRSProof
    ↳ 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.