Termination Proof using AProVE

Term Rewriting System R:
[X, Y, X1, X2, Z]
and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

AND(true, X) -> ACTIVATE(X)
IF(true, X, Y) -> ACTIVATE(X)
IF(false, X, Y) -> ACTIVATE(Y)
ADD(0, X) -> ACTIVATE(X)
ADD(s(X), Y) -> S(n_{_add}(activate(X), activate(Y)))
ADD(s(X), Y) -> ACTIVATE(X)
ADD(s(X), Y) -> ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Y)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(X)
FIRST(s(X), cons(Y, Z)) -> ACTIVATE(Z)
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
ACTIVATE(n_{_first}(X1, X2)) -> FIRST(X1, X2)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_s}(X)) -> S(X)

Furthermore, R contains one SCC.

     ↳DPs

       →DP Problem 1

         ↳Forward Instantiation Transformation

Dependency Pairs:

FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ADD(0, X) -> ACTIVATE(X)
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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(0, X) -> ACTIVATE(X)

two new Dependency Pairs are created:

ADD(0, n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ADD(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳Forward Instantiation Transformation

Dependency Pairs:

ACTIVATE(n_{_from}(X)) -> FROM(X)
ADD(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ADD(0, n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
FROM(X) -> ACTIVATE(X)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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

FROM(X) -> ACTIVATE(X)

two new Dependency Pairs are created:

FROM(n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(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:

FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ADD(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ADD(0, n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_add}(X1, X2)) -> ADD(X1, X2)
FROM(n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_from}(X)) -> FROM(X)

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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)

two new Dependency Pairs are created:

ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(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:

ADD(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
ADD(0, n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
FROM(n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_from}(X)) -> FROM(X)
FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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_{_from}(X)) -> FROM(X)

two new Dependency Pairs are created:

ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(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:

FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
ADD(0, n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
FROM(n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
ADD(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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(0, n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))

two new Dependency Pairs are created:

ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(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:

ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
ADD(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
FROM(n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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(0, n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))

two new Dependency Pairs are created:

ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

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:

ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
FROM(n_{_add}(X1'', X2'')) -> ACTIVATE(n_{_add}(X1'', X2''))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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

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

two new Dependency Pairs are created:

FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 8

                 ↳Forward Instantiation Transformation

Dependency Pairs:

FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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

FROM(n_{_from}(X'')) -> ACTIVATE(n_{_from}(X''))

two new Dependency Pairs are created:

FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 9

                 ↳Forward Instantiation Transformation

Dependency Pairs:

FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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}(0, n_{_add}(X1'''', X2''''))) -> ADD(0, n_{_add}(X1'''', X2''''))

two new Dependency Pairs are created:

ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_from}(X'''''''')))) -> ADD(0, n_{_add}(0, n_{_from}(X'''''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 10

                 ↳Forward Instantiation Transformation

Dependency Pairs:

FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_from}(X'''''''')))) -> ADD(0, n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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}(0, n_{_from}(X''''))) -> ADD(0, n_{_from}(X''''))

two new Dependency Pairs are created:

ACTIVATE(n_{_add}(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_from}(X'''''''')))) -> ADD(0, n_{_from}(n_{_from}(X'''''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 11

                 ↳Forward Instantiation Transformation

Dependency Pairs:

FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_from}(X'''''''')))) -> ADD(0, n_{_from}(n_{_from}(X'''''''')))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_from}(X'''''''')))) -> ADD(0, n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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_{_from}(n_{_add}(X1'''', X2''''))) -> FROM(n_{_add}(X1'''', X2''''))

two new Dependency Pairs are created:

ACTIVATE(n_{_from}(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_from}(X'''''''')))) -> FROM(n_{_add}(0, n_{_from}(X'''''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 12

                 ↳Forward Instantiation Transformation

Dependency Pairs:

ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_from}(X'''''''')))) -> ADD(0, n_{_from}(n_{_from}(X'''''''')))
FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_from}(X'''''''')))) -> FROM(n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_from}(X'''''''')))) -> ADD(0, n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))
FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(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_{_from}(n_{_from}(X''''))) -> FROM(n_{_from}(X''''))

two new Dependency Pairs are created:

ACTIVATE(n_{_from}(n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X'''''''')))) -> FROM(n_{_from}(n_{_from}(X'''''''')))

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 13

                 ↳Argument Filtering and Ordering

Dependency Pairs:

FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X'''''''')))) -> FROM(n_{_from}(n_{_from}(X'''''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_from}(X'''''''')))) -> ADD(0, n_{_from}(n_{_from}(X'''''''')))
FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_from}(X'''''''')))) -> FROM(n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_from}(X'''''''')))) -> ADD(0, n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

The following dependency pairs can be strictly oriented:

FROM(n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_from}(X'''''''')))) -> FROM(n_{_from}(n_{_from}(X'''''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_from}(X'''''''')))) -> ADD(0, n_{_from}(n_{_from}(X'''''''')))
FROM(n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_from}(X'''''''')))) -> FROM(n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ADD(0, n_{_add}(0, n_{_from}(X''''''))) -> ACTIVATE(n_{_add}(0, n_{_from}(X'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_from}(X'''''''')))) -> ADD(0, n_{_add}(0, n_{_from}(X'''''''')))
ADD(0, n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_add}(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> ADD(0, n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_add}(0, n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_add}(0, n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_add}(0, n_{_add}(X1'''''''', X2'''''''')))
FROM(n_{_from}(n_{_add}(X1'''''', X2''''''))) -> ACTIVATE(n_{_from}(n_{_add}(X1'''''', X2'''''')))
ACTIVATE(n_{_from}(n_{_from}(n_{_add}(X1'''''''', X2'''''''')))) -> FROM(n_{_from}(n_{_add}(X1'''''''', X2'''''''')))
ADD(0, n_{_from}(n_{_from}(X''''''))) -> ACTIVATE(n_{_from}(n_{_from}(X'''''')))

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:

n_{_from} > FROM > ACTIVATE
n_{_add} > ADD > ACTIVATE

resulting in one new DP problem.
Used Argument Filtering System:

ACTIVATE(x₁) -> ACTIVATE(x₁)
FROM(x₁) -> FROM(x₁)
n_{_from}(x₁) -> n_{_from}(x₁)
n_{_add}(x₁, x₂) -> n_{_add}(x₁, x₂)
ADD(x₁, x₂) -> ADD(x₁, x₂)

     ↳DPs

       →DP Problem 1

         ↳FwdInst

           →DP Problem 2

             ↳FwdInst

...

               →DP Problem 14

                 ↳Dependency Graph

Dependency Pair:

Rules:

and(true, X) -> activate(X)
and(false, Y) -> false
if(true, X, Y) -> activate(X)
if(false, X, Y) -> activate(Y)
add(0, X) -> activate(X)
add(s(X), Y) -> s(n_{_add}(activate(X), activate(Y)))
add(X1, X2) -> n_{_add}(X1, X2)
first(0, X) -> nil
first(s(X), cons(Y, Z)) -> cons(activate(Y), n_{_first}(activate(X), activate(Z)))
first(X1, X2) -> n_{_first}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
s(X) -> n_{_s}(X)
activate(n_{_add}(X1, X2)) -> add(X1, X2)
activate(n_{_first}(X1, X2)) -> first(X1, X2)
activate(n_{_from}(X)) -> from(X)
activate(n_{_s}(X)) -> s(X)
activate(X) -> X

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:03 minutes