Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
  ↳11 RisEmptyProof (⇔)
    ↳12 TRUE
  ↳13 RisEmptyProof (⇔)
    ↳14 TRUE
  ↳15 RisEmptyProof (⇔)
    ↳16 TRUE

(0) Obligation:

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.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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)


(2) Obligation:

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.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

len(cons(X, Z)) → s(n__len(activate(Z)))
len(X) → n__len(X)


(4) Obligation:

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.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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)


(6) Obligation:

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.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

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   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

from(X) → cons(X, n__from(s(X)))
from(X) → n__from(X)
activate(n__fst(X1, X2)) → fst(X1, X2)


(8) Obligation:

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.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(activate(x1)) = 1 + x1   
POL(add(x1, x2)) = 2 + x1 + x2   
POL(n__add(x1, x2)) = x1 + x2   
POL(s(x1)) = x1   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(10) Obligation:

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

(11) RisEmptyProof (EQUIVALENT transformation)

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

(12) TRUE

(13) RisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) RisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE