Termination Proof using AProVE

Term Rewriting System R:
[Z, X, Y, X1, X2]
fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, n_{_fst}(activate(X), activate(Z)))
fst(X1, X2) -> n_{_fst}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(n_{_len}(activate(Z)))
len(X) -> n_{_len}(X)
s(X) -> n_{_s}(X)
activate(n_{_fst}(X1, X2)) -> fst(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(n_{_len}(X)) -> len(activate(X))
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ADD(s(X), Y) -> S(n_{_add}(activate(X), Y))
ADD(s(X), Y) -> ACTIVATE(X)
LEN(cons(X, Z)) -> S(n_{_len}(activate(Z)))
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_fst}(X1, X2)) -> FST(activate(X1), activate(X2))
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_from}(X)) -> FROM(activate(X))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_s}(X)) -> S(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_len}(X)) -> LEN(activate(X))
ACTIVATE(n_{_len}(X)) -> ACTIVATE(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_len}(X)) -> ACTIVATE(X)
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_len}(X)) -> LEN(activate(X))
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X1)
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_fst}(X1, X2)) -> FST(activate(X1), activate(X2))
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, n_{_fst}(activate(X), activate(Z)))
fst(X1, X2) -> n_{_fst}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(n_{_len}(activate(Z)))
len(X) -> n_{_len}(X)
s(X) -> n_{_s}(X)
activate(n_{_fst}(X1, X2)) -> fst(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(n_{_len}(X)) -> len(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(n_{_fst}(X1, X2)) -> FST(activate(X1), activate(X2))

12 new Dependency Pairs are created:

ACTIVATE(n_{_fst}(n_{_fst}(X1'', X2''), X2)) -> FST(fst(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(n_{_from}(X'), X2)) -> FST(from(activate(X')), activate(X2))
ACTIVATE(n_{_fst}(n_{_s}(X'), X2)) -> FST(s(X'), activate(X2))
ACTIVATE(n_{_fst}(n_{_add}(X1'', X2''), X2)) -> FST(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(n_{_len}(X'), X2)) -> FST(len(activate(X')), activate(X2))
ACTIVATE(n_{_fst}(X1', X2)) -> FST(X1', activate(X2))
ACTIVATE(n_{_fst}(X1, n_{_fst}(X1'', X2''))) -> FST(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1, n_{_from}(X'))) -> FST(activate(X1), from(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_s}(X'))) -> FST(activate(X1), s(X'))
ACTIVATE(n_{_fst}(X1, n_{_add}(X1'', X2''))) -> FST(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1, n_{_len}(X'))) -> FST(activate(X1), len(activate(X')))
ACTIVATE(n_{_fst}(X1, X2')) -> FST(activate(X1), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_fst}(X1, X2')) -> FST(activate(X1), X2')
ACTIVATE(n_{_fst}(X1, n_{_len}(X'))) -> FST(activate(X1), len(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_add}(X1'', X2''))) -> FST(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1, n_{_s}(X'))) -> FST(activate(X1), s(X'))
ACTIVATE(n_{_fst}(X1, n_{_from}(X'))) -> FST(activate(X1), from(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_fst}(X1'', X2''))) -> FST(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1', X2)) -> FST(X1', activate(X2))
ACTIVATE(n_{_fst}(n_{_len}(X'), X2)) -> FST(len(activate(X')), activate(X2))
ACTIVATE(n_{_fst}(n_{_add}(X1'', X2''), X2)) -> FST(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(n_{_s}(X'), X2)) -> FST(s(X'), activate(X2))
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_fst}(n_{_from}(X'), X2)) -> FST(from(activate(X')), activate(X2))
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(n_{_fst}(X1'', X2''), X2)) -> FST(fst(activate(X1''), activate(X2'')), activate(X2))
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_len}(X)) -> LEN(activate(X))
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), activate(X2))
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_len}(X)) -> ACTIVATE(X)

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, n_{_fst}(activate(X), activate(Z)))
fst(X1, X2) -> n_{_fst}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(n_{_len}(activate(Z)))
len(X) -> n_{_len}(X)
s(X) -> n_{_s}(X)
activate(n_{_fst}(X1, X2)) -> fst(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(n_{_len}(X)) -> len(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(n_{_add}(X1, X2)) -> ADD(activate(X1), activate(X2))

12 new Dependency Pairs are created:

ACTIVATE(n_{_add}(n_{_fst}(X1'', X2''), X2)) -> ADD(fst(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_add}(n_{_from}(X'), X2)) -> ADD(from(activate(X')), activate(X2))
ACTIVATE(n_{_add}(n_{_s}(X'), X2)) -> ADD(s(X'), activate(X2))
ACTIVATE(n_{_add}(n_{_add}(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_add}(n_{_len}(X'), X2)) -> ADD(len(activate(X')), activate(X2))
ACTIVATE(n_{_add}(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(n_{_add}(X1, n_{_fst}(X1'', X2''))) -> ADD(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_add}(X1, n_{_from}(X'))) -> ADD(activate(X1), from(activate(X')))
ACTIVATE(n_{_add}(X1, n_{_s}(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(n_{_add}(X1, n_{_add}(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_add}(X1, n_{_len}(X'))) -> ADD(activate(X1), len(activate(X')))
ACTIVATE(n_{_add}(X1, X2')) -> ADD(activate(X1), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 3

                 ↳Narrowing Transformation

Dependency Pairs:

ACTIVATE(n_{_add}(X1, X2')) -> ADD(activate(X1), X2')
ACTIVATE(n_{_add}(X1, n_{_len}(X'))) -> ADD(activate(X1), len(activate(X')))
ACTIVATE(n_{_add}(X1, n_{_add}(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_add}(X1, n_{_s}(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(n_{_add}(X1, n_{_from}(X'))) -> ADD(activate(X1), from(activate(X')))
ACTIVATE(n_{_add}(X1, n_{_fst}(X1'', X2''))) -> ADD(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_add}(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(n_{_add}(n_{_len}(X'), X2)) -> ADD(len(activate(X')), activate(X2))
ACTIVATE(n_{_add}(n_{_add}(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_add}(n_{_s}(X'), X2)) -> ADD(s(X'), activate(X2))
ACTIVATE(n_{_add}(n_{_from}(X'), X2)) -> ADD(from(activate(X')), activate(X2))
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(n_{_fst}(X1'', X2''), X2)) -> ADD(fst(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(X1, n_{_len}(X'))) -> FST(activate(X1), len(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_add}(X1'', X2''))) -> FST(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1, n_{_s}(X'))) -> FST(activate(X1), s(X'))
ACTIVATE(n_{_fst}(X1, n_{_from}(X'))) -> FST(activate(X1), from(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_fst}(X1'', X2''))) -> FST(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1', X2)) -> FST(X1', activate(X2))
ACTIVATE(n_{_fst}(n_{_len}(X'), X2)) -> FST(len(activate(X')), activate(X2))
ACTIVATE(n_{_fst}(n_{_add}(X1'', X2''), X2)) -> FST(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(n_{_s}(X'), X2)) -> FST(s(X'), activate(X2))
ACTIVATE(n_{_fst}(n_{_from}(X'), X2)) -> FST(from(activate(X')), activate(X2))
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_fst}(n_{_fst}(X1'', X2''), X2)) -> FST(fst(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_len}(X)) -> ACTIVATE(X)
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_len}(X)) -> LEN(activate(X))
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X1)
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(X1, X2')) -> FST(activate(X1), X2')

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, n_{_fst}(activate(X), activate(Z)))
fst(X1, X2) -> n_{_fst}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(n_{_len}(activate(Z)))
len(X) -> n_{_len}(X)
s(X) -> n_{_s}(X)
activate(n_{_fst}(X1, X2)) -> fst(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(n_{_len}(X)) -> len(activate(X))
activate(X) -> X

On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

ACTIVATE(n_{_len}(X)) -> LEN(activate(X))

six new Dependency Pairs are created:

ACTIVATE(n_{_len}(n_{_fst}(X1', X2'))) -> LEN(fst(activate(X1'), activate(X2')))
ACTIVATE(n_{_len}(n_{_from}(X''))) -> LEN(from(activate(X'')))
ACTIVATE(n_{_len}(n_{_s}(X''))) -> LEN(s(X''))
ACTIVATE(n_{_len}(n_{_add}(X1', X2'))) -> LEN(add(activate(X1'), activate(X2')))
ACTIVATE(n_{_len}(n_{_len}(X''))) -> LEN(len(activate(X'')))
ACTIVATE(n_{_len}(X'')) -> LEN(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 2

             ↳Nar

...

               →DP Problem 4

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

ACTIVATE(n_{_len}(X'')) -> LEN(X'')
ACTIVATE(n_{_len}(n_{_len}(X''))) -> LEN(len(activate(X'')))
ACTIVATE(n_{_len}(n_{_add}(X1', X2'))) -> LEN(add(activate(X1'), activate(X2')))
ACTIVATE(n_{_len}(n_{_s}(X''))) -> LEN(s(X''))
ACTIVATE(n_{_len}(n_{_from}(X''))) -> LEN(from(activate(X'')))
LEN(cons(X, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_len}(n_{_fst}(X1', X2'))) -> LEN(fst(activate(X1'), activate(X2')))
ACTIVATE(n_{_add}(X1, n_{_len}(X'))) -> ADD(activate(X1), len(activate(X')))
ACTIVATE(n_{_add}(X1, n_{_add}(X1'', X2''))) -> ADD(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_add}(X1, n_{_s}(X'))) -> ADD(activate(X1), s(X'))
ACTIVATE(n_{_add}(X1, n_{_from}(X'))) -> ADD(activate(X1), from(activate(X')))
ACTIVATE(n_{_add}(X1, n_{_fst}(X1'', X2''))) -> ADD(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_add}(X1', X2)) -> ADD(X1', activate(X2))
ACTIVATE(n_{_add}(n_{_len}(X'), X2)) -> ADD(len(activate(X')), activate(X2))
ACTIVATE(n_{_add}(n_{_add}(X1'', X2''), X2)) -> ADD(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_add}(n_{_s}(X'), X2)) -> ADD(s(X'), activate(X2))
ACTIVATE(n_{_add}(n_{_from}(X'), X2)) -> ADD(from(activate(X')), activate(X2))
ACTIVATE(n_{_add}(n_{_fst}(X1'', X2''), X2)) -> ADD(fst(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(X1, X2')) -> FST(activate(X1), X2')
ACTIVATE(n_{_fst}(X1, n_{_len}(X'))) -> FST(activate(X1), len(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_add}(X1'', X2''))) -> FST(activate(X1), add(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1, n_{_s}(X'))) -> FST(activate(X1), s(X'))
ACTIVATE(n_{_fst}(X1, n_{_from}(X'))) -> FST(activate(X1), from(activate(X')))
ACTIVATE(n_{_fst}(X1, n_{_fst}(X1'', X2''))) -> FST(activate(X1), fst(activate(X1''), activate(X2'')))
ACTIVATE(n_{_fst}(X1', X2)) -> FST(X1', activate(X2))
ACTIVATE(n_{_fst}(n_{_len}(X'), X2)) -> FST(len(activate(X')), activate(X2))
ACTIVATE(n_{_fst}(n_{_add}(X1'', X2''), X2)) -> FST(add(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_fst}(n_{_s}(X'), X2)) -> FST(s(X'), activate(X2))
FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_fst}(n_{_from}(X'), X2)) -> FST(from(activate(X')), activate(X2))
FST(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(n_{_fst}(X1'', X2''), X2)) -> FST(fst(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_len}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_add}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_from}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_fst}(X1, X2)) -> ACTIVATE(X1)
ADD(s(X), Y) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2')) -> ADD(activate(X1), X2')

Rules:

fst(0, Z) -> nil
fst(s(X), cons(Y, Z)) -> cons(Y, n_{_fst}(activate(X), activate(Z)))
fst(X1, X2) -> n_{_fst}(X1, X2)
from(X) -> cons(X, n_{_from}(n_{_s}(X)))
from(X) -> n_{_from}(X)
add(0, X) -> X
add(s(X), Y) -> s(n_{_add}(activate(X), Y))
add(X1, X2) -> n_{_add}(X1, X2)
len(nil) -> 0
len(cons(X, Z)) -> s(n_{_len}(activate(Z)))
len(X) -> n_{_len}(X)
s(X) -> n_{_s}(X)
activate(n_{_fst}(X1, X2)) -> fst(activate(X1), activate(X2))
activate(n_{_from}(X)) -> from(activate(X))
activate(n_{_s}(X)) -> s(X)
activate(n_{_add}(X1, X2)) -> add(activate(X1), activate(X2))
activate(n_{_len}(X)) -> len(activate(X))
activate(X) -> X

Termination of R could not be shown.
Duration:
0:11 minutes