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}(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)
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

Innermost 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) -> ACTIVATE(X)
LEN(cons(X, Z)) -> ACTIVATE(Z)
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)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

FST(s(X), cons(Y, Z)) -> ACTIVATE(X)

three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

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

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}(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)
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

Strategy:

innermost

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

FST(s(X), cons(Y, Z)) -> ACTIVATE(Z)

three new Dependency Pairs are created:

FST(s(X), cons(Y, n_{_fst}(X1'', X2''))) -> ACTIVATE(n_{_fst}(X1'', X2''))
FST(s(X), cons(Y, n_{_add}(X1'', X2''))) -> ACTIVATE(n_{_add}(X1'', X2''))
FST(s(X), cons(Y, n_{_len}(X''))) -> ACTIVATE(n_{_len}(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 3

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

ADD(s(X), Y) -> ACTIVATE(X)

three new Dependency Pairs are created:

ADD(s(n_{_fst}(X1'', X2'')), Y) -> ACTIVATE(n_{_fst}(X1'', X2''))
ADD(s(n_{_add}(X1'', X2'')), Y) -> ACTIVATE(n_{_add}(X1'', X2''))
ADD(s(n_{_len}(X'')), Y) -> ACTIVATE(n_{_len}(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 4

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

LEN(cons(X, Z)) -> ACTIVATE(Z)

three new Dependency Pairs are created:

LEN(cons(X, n_{_fst}(X1'', X2''))) -> ACTIVATE(n_{_fst}(X1'', X2''))
LEN(cons(X, n_{_add}(X1'', X2''))) -> ACTIVATE(n_{_add}(X1'', X2''))
LEN(cons(X, n_{_len}(X''))) -> ACTIVATE(n_{_len}(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 5

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

six new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 6

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

ACTIVATE(n_{_add}(s(n_{_fst}(X1'''', X2'''')), X2')) -> ADD(s(n_{_fst}(X1'''', X2'''')), X2')
ACTIVATE(n_{_add}(s(n_{_add}(X1'''', X2'''')), X2')) -> ADD(s(n_{_add}(X1'''', X2'''')), X2')
ACTIVATE(n_{_add}(s(n_{_len}(X'''')), X2')) -> ADD(s(n_{_len}(X'''')), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 7

                 ↳Forward Instantiation Transformation

Dependency Pairs:

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

Strategy:

innermost

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

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

three new Dependency Pairs are created:

ACTIVATE(n_{_len}(cons(X'', n_{_fst}(X1'''', X2'''')))) -> LEN(cons(X'', n_{_fst}(X1'''', X2'''')))
ACTIVATE(n_{_len}(cons(X'', n_{_add}(X1'''', X2'''')))) -> LEN(cons(X'', n_{_add}(X1'''', X2'''')))
ACTIVATE(n_{_len}(cons(X'', n_{_len}(X'''')))) -> LEN(cons(X'', n_{_len}(X'''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:08 minutes