Termination Proof using AProVE

Term Rewriting System R:
[X, Y, Z, X1, X2]
dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

DBL(s(X)) -> S(n_{_s}(n_{_dbl}(activate(X))))
DBL(s(X)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(Y)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
SEL(s(X), cons(Y, Z)) -> SEL(activate(X), activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)
FROM(X) -> ACTIVATE(X)
DBL1(s(X)) -> DBL1(activate(X))
DBL1(s(X)) -> ACTIVATE(X)
SEL1(0, cons(X, Y)) -> ACTIVATE(X)
SEL1(s(X), cons(Y, Z)) -> SEL1(activate(X), activate(Z))
SEL1(s(X), cons(Y, Z)) -> ACTIVATE(X)
SEL1(s(X), cons(Y, Z)) -> ACTIVATE(Z)
QUOTE(s(X)) -> QUOTE(activate(X))
QUOTE(s(X)) -> ACTIVATE(X)
QUOTE(dbl(X)) -> DBL1(X)
QUOTE(sel(X, Y)) -> SEL1(X, Y)
ACTIVATE(n_{_s}(X)) -> S(X)
ACTIVATE(n_{_dbl}(X)) -> DBL(activate(X))
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(X)) -> DBLS(activate(X))
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(X1, X2)) -> SEL(activate(X1), activate(X2))
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_indx}(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_from}(X)) -> FROM(X)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

DBLS(cons(X, Y)) -> ACTIVATE(Y)
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
SEL(s(X), cons(Y, Z)) -> SEL(activate(X), activate(Z))
INDX(cons(X, Y), Z) -> ACTIVATE(Y)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(X1, X2)) -> SEL(activate(X1), activate(X2))
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(X)) -> DBLS(activate(X))
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> DBL(activate(X))
DBL(s(X)) -> ACTIVATE(X)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

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

SEL(s(X), cons(Y, Z)) -> SEL(activate(X), activate(Z))

14 new Dependency Pairs are created:

SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 5

             ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

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

ACTIVATE(n_{_dbl}(X)) -> DBL(activate(X))

seven new Dependency Pairs are created:

ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 5

             ↳Nar

...

               →DP Problem 6

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

DBLS(cons(X, Y)) -> ACTIVATE(Y)
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(X1, X2)) -> SEL(activate(X1), activate(X2))
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(X)) -> DBLS(activate(X))
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

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

ACTIVATE(n_{_dbls}(X)) -> DBLS(activate(X))

seven new Dependency Pairs are created:

ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 5

             ↳Nar

...

               →DP Problem 7

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(X1, X2)) -> SEL(activate(X1), activate(X2))
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
DBLS(cons(X, Y)) -> ACTIVATE(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

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

ACTIVATE(n_{_sel}(X1, X2)) -> SEL(activate(X1), activate(X2))

14 new Dependency Pairs are created:

ACTIVATE(n_{_sel}(n_{_s}(X'), X2)) -> SEL(s(X'), activate(X2))
ACTIVATE(n_{_sel}(n_{_dbl}(X'), X2)) -> SEL(dbl(activate(X')), activate(X2))
ACTIVATE(n_{_sel}(n_{_dbls}(X'), X2)) -> SEL(dbls(activate(X')), activate(X2))
ACTIVATE(n_{_sel}(n_{_sel}(X1'', X2''), X2)) -> SEL(sel(activate(X1''), activate(X2'')), activate(X2))
ACTIVATE(n_{_sel}(n_{_indx}(X1'', X2''), X2)) -> SEL(indx(activate(X1''), X2''), activate(X2))
ACTIVATE(n_{_sel}(n_{_from}(X'), X2)) -> SEL(from(X'), activate(X2))
ACTIVATE(n_{_sel}(X1', X2)) -> SEL(X1', activate(X2))
ACTIVATE(n_{_sel}(X1, n_{_s}(X'))) -> SEL(activate(X1), s(X'))
ACTIVATE(n_{_sel}(X1, n_{_dbl}(X'))) -> SEL(activate(X1), dbl(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_dbls}(X'))) -> SEL(activate(X1), dbls(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_sel}(X1'', X2''))) -> SEL(activate(X1), sel(activate(X1''), activate(X2'')))
ACTIVATE(n_{_sel}(X1, n_{_indx}(X1'', X2''))) -> SEL(activate(X1), indx(activate(X1''), X2''))
ACTIVATE(n_{_sel}(X1, n_{_from}(X'))) -> SEL(activate(X1), from(X'))
ACTIVATE(n_{_sel}(X1, X2')) -> SEL(activate(X1), X2')

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

           →DP Problem 5

             ↳Nar

...

               →DP Problem 8

                 ↳Narrowing Transformation

       →DP Problem 2

         ↳Remaining

       →DP Problem 3

         ↳Remaining

       →DP Problem 4

         ↳Remaining

Dependency Pairs:

INDX(cons(X, Y), Z) -> ACTIVATE(Y)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_sel}(X1, X2')) -> SEL(activate(X1), X2')
ACTIVATE(n_{_sel}(X1, n_{_from}(X'))) -> SEL(activate(X1), from(X'))
ACTIVATE(n_{_sel}(X1, n_{_indx}(X1'', X2''))) -> SEL(activate(X1), indx(activate(X1''), X2''))
ACTIVATE(n_{_sel}(X1, n_{_sel}(X1'', X2''))) -> SEL(activate(X1), sel(activate(X1''), activate(X2'')))
ACTIVATE(n_{_sel}(X1, n_{_dbls}(X'))) -> SEL(activate(X1), dbls(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_dbl}(X'))) -> SEL(activate(X1), dbl(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_s}(X'))) -> SEL(activate(X1), s(X'))
ACTIVATE(n_{_sel}(X1', X2)) -> SEL(X1', activate(X2))
ACTIVATE(n_{_sel}(n_{_from}(X'), X2)) -> SEL(from(X'), activate(X2))
ACTIVATE(n_{_sel}(n_{_indx}(X1'', X2''), X2)) -> SEL(indx(activate(X1''), X2''), activate(X2))
ACTIVATE(n_{_sel}(n_{_sel}(X1'', X2''), X2)) -> SEL(sel(activate(X1''), activate(X2'')), activate(X2))
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
ACTIVATE(n_{_sel}(n_{_dbls}(X'), X2)) -> SEL(dbls(activate(X')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_sel}(n_{_dbl}(X'), X2)) -> SEL(dbl(activate(X')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_s}(X'), X2)) -> SEL(s(X'), activate(X2))
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(X1, X2)) -> INDX(activate(X1), X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
SEL(0, cons(X, Y)) -> ACTIVATE(X)
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

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

ACTIVATE(n_{_indx}(X1, X2)) -> INDX(activate(X1), X2)

seven new Dependency Pairs are created:

ACTIVATE(n_{_indx}(n_{_s}(X'), X2)) -> INDX(s(X'), X2)
ACTIVATE(n_{_indx}(n_{_dbl}(X'), X2)) -> INDX(dbl(activate(X')), X2)
ACTIVATE(n_{_indx}(n_{_dbls}(X'), X2)) -> INDX(dbls(activate(X')), X2)
ACTIVATE(n_{_indx}(n_{_sel}(X1'', X2''), X2)) -> INDX(sel(activate(X1''), activate(X2'')), X2)
ACTIVATE(n_{_indx}(n_{_indx}(X1'', X2''), X2)) -> INDX(indx(activate(X1''), X2''), X2)
ACTIVATE(n_{_indx}(n_{_from}(X'), X2)) -> INDX(from(X'), X2)
ACTIVATE(n_{_indx}(X1', X2)) -> INDX(X1', X2)

The transformation is resulting in one new DP problem:

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_indx}(X1', X2)) -> INDX(X1', X2)
ACTIVATE(n_{_indx}(n_{_from}(X'), X2)) -> INDX(from(X'), X2)
ACTIVATE(n_{_indx}(n_{_indx}(X1'', X2''), X2)) -> INDX(indx(activate(X1''), X2''), X2)
ACTIVATE(n_{_indx}(n_{_sel}(X1'', X2''), X2)) -> INDX(sel(activate(X1''), activate(X2'')), X2)
ACTIVATE(n_{_indx}(n_{_dbls}(X'), X2)) -> INDX(dbls(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_indx}(n_{_dbl}(X'), X2)) -> INDX(dbl(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(n_{_s}(X'), X2)) -> INDX(s(X'), X2)
ACTIVATE(n_{_sel}(X1, X2')) -> SEL(activate(X1), X2')
ACTIVATE(n_{_sel}(X1, n_{_from}(X'))) -> SEL(activate(X1), from(X'))
ACTIVATE(n_{_sel}(X1, n_{_indx}(X1'', X2''))) -> SEL(activate(X1), indx(activate(X1''), X2''))
ACTIVATE(n_{_sel}(X1, n_{_sel}(X1'', X2''))) -> SEL(activate(X1), sel(activate(X1''), activate(X2'')))
ACTIVATE(n_{_sel}(X1, n_{_dbls}(X'))) -> SEL(activate(X1), dbls(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_dbl}(X'))) -> SEL(activate(X1), dbl(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_s}(X'))) -> SEL(activate(X1), s(X'))
ACTIVATE(n_{_sel}(X1', X2)) -> SEL(X1', activate(X2))
ACTIVATE(n_{_sel}(n_{_from}(X'), X2)) -> SEL(from(X'), activate(X2))
ACTIVATE(n_{_sel}(n_{_indx}(X1'', X2''), X2)) -> SEL(indx(activate(X1''), X2''), activate(X2))
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
ACTIVATE(n_{_sel}(n_{_sel}(X1'', X2''), X2)) -> SEL(sel(activate(X1''), activate(X2'')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_sel}(n_{_dbls}(X'), X2)) -> SEL(dbls(activate(X')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_dbl}(X'), X2)) -> SEL(dbl(activate(X')), activate(X2))
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_s}(X'), X2)) -> SEL(s(X'), activate(X2))
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
DBL1(s(X)) -> DBL1(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(activate(X), activate(Z))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
QUOTE(s(X)) -> QUOTE(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_indx}(X1', X2)) -> INDX(X1', X2)
ACTIVATE(n_{_indx}(n_{_from}(X'), X2)) -> INDX(from(X'), X2)
ACTIVATE(n_{_indx}(n_{_indx}(X1'', X2''), X2)) -> INDX(indx(activate(X1''), X2''), X2)
ACTIVATE(n_{_indx}(n_{_sel}(X1'', X2''), X2)) -> INDX(sel(activate(X1''), activate(X2'')), X2)
ACTIVATE(n_{_indx}(n_{_dbls}(X'), X2)) -> INDX(dbls(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_indx}(n_{_dbl}(X'), X2)) -> INDX(dbl(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(n_{_s}(X'), X2)) -> INDX(s(X'), X2)
ACTIVATE(n_{_sel}(X1, X2')) -> SEL(activate(X1), X2')
ACTIVATE(n_{_sel}(X1, n_{_from}(X'))) -> SEL(activate(X1), from(X'))
ACTIVATE(n_{_sel}(X1, n_{_indx}(X1'', X2''))) -> SEL(activate(X1), indx(activate(X1''), X2''))
ACTIVATE(n_{_sel}(X1, n_{_sel}(X1'', X2''))) -> SEL(activate(X1), sel(activate(X1''), activate(X2'')))
ACTIVATE(n_{_sel}(X1, n_{_dbls}(X'))) -> SEL(activate(X1), dbls(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_dbl}(X'))) -> SEL(activate(X1), dbl(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_s}(X'))) -> SEL(activate(X1), s(X'))
ACTIVATE(n_{_sel}(X1', X2)) -> SEL(X1', activate(X2))
ACTIVATE(n_{_sel}(n_{_from}(X'), X2)) -> SEL(from(X'), activate(X2))
ACTIVATE(n_{_sel}(n_{_indx}(X1'', X2''), X2)) -> SEL(indx(activate(X1''), X2''), activate(X2))
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
ACTIVATE(n_{_sel}(n_{_sel}(X1'', X2''), X2)) -> SEL(sel(activate(X1''), activate(X2'')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_sel}(n_{_dbls}(X'), X2)) -> SEL(dbls(activate(X')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_dbl}(X'), X2)) -> SEL(dbl(activate(X')), activate(X2))
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_s}(X'), X2)) -> SEL(s(X'), activate(X2))
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
DBL1(s(X)) -> DBL1(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(activate(X), activate(Z))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
QUOTE(s(X)) -> QUOTE(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_indx}(X1', X2)) -> INDX(X1', X2)
ACTIVATE(n_{_indx}(n_{_from}(X'), X2)) -> INDX(from(X'), X2)
ACTIVATE(n_{_indx}(n_{_indx}(X1'', X2''), X2)) -> INDX(indx(activate(X1''), X2''), X2)
ACTIVATE(n_{_indx}(n_{_sel}(X1'', X2''), X2)) -> INDX(sel(activate(X1''), activate(X2'')), X2)
ACTIVATE(n_{_indx}(n_{_dbls}(X'), X2)) -> INDX(dbls(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_indx}(n_{_dbl}(X'), X2)) -> INDX(dbl(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(n_{_s}(X'), X2)) -> INDX(s(X'), X2)
ACTIVATE(n_{_sel}(X1, X2')) -> SEL(activate(X1), X2')
ACTIVATE(n_{_sel}(X1, n_{_from}(X'))) -> SEL(activate(X1), from(X'))
ACTIVATE(n_{_sel}(X1, n_{_indx}(X1'', X2''))) -> SEL(activate(X1), indx(activate(X1''), X2''))
ACTIVATE(n_{_sel}(X1, n_{_sel}(X1'', X2''))) -> SEL(activate(X1), sel(activate(X1''), activate(X2'')))
ACTIVATE(n_{_sel}(X1, n_{_dbls}(X'))) -> SEL(activate(X1), dbls(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_dbl}(X'))) -> SEL(activate(X1), dbl(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_s}(X'))) -> SEL(activate(X1), s(X'))
ACTIVATE(n_{_sel}(X1', X2)) -> SEL(X1', activate(X2))
ACTIVATE(n_{_sel}(n_{_from}(X'), X2)) -> SEL(from(X'), activate(X2))
ACTIVATE(n_{_sel}(n_{_indx}(X1'', X2''), X2)) -> SEL(indx(activate(X1''), X2''), activate(X2))
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
ACTIVATE(n_{_sel}(n_{_sel}(X1'', X2''), X2)) -> SEL(sel(activate(X1''), activate(X2'')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_sel}(n_{_dbls}(X'), X2)) -> SEL(dbls(activate(X')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_dbl}(X'), X2)) -> SEL(dbl(activate(X')), activate(X2))
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_s}(X'), X2)) -> SEL(s(X'), activate(X2))
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
DBL1(s(X)) -> DBL1(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(activate(X), activate(Z))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
QUOTE(s(X)) -> QUOTE(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

     ↳DPs

       →DP Problem 1

         ↳Nar

       →DP Problem 2

         ↳Remaining Obligation(s)

       →DP Problem 3

         ↳Remaining Obligation(s)

       →DP Problem 4

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pairs:
ACTIVATE(n_{_indx}(X1', X2)) -> INDX(X1', X2)
ACTIVATE(n_{_indx}(n_{_from}(X'), X2)) -> INDX(from(X'), X2)
ACTIVATE(n_{_indx}(n_{_indx}(X1'', X2''), X2)) -> INDX(indx(activate(X1''), X2''), X2)
ACTIVATE(n_{_indx}(n_{_sel}(X1'', X2''), X2)) -> INDX(sel(activate(X1''), activate(X2'')), X2)
ACTIVATE(n_{_indx}(n_{_dbls}(X'), X2)) -> INDX(dbls(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(Z)
ACTIVATE(n_{_indx}(n_{_dbl}(X'), X2)) -> INDX(dbl(activate(X')), X2)
INDX(cons(X, Y), Z) -> ACTIVATE(X)
ACTIVATE(n_{_indx}(n_{_s}(X'), X2)) -> INDX(s(X'), X2)
ACTIVATE(n_{_sel}(X1, X2')) -> SEL(activate(X1), X2')
ACTIVATE(n_{_sel}(X1, n_{_from}(X'))) -> SEL(activate(X1), from(X'))
ACTIVATE(n_{_sel}(X1, n_{_indx}(X1'', X2''))) -> SEL(activate(X1), indx(activate(X1''), X2''))
ACTIVATE(n_{_sel}(X1, n_{_sel}(X1'', X2''))) -> SEL(activate(X1), sel(activate(X1''), activate(X2'')))
ACTIVATE(n_{_sel}(X1, n_{_dbls}(X'))) -> SEL(activate(X1), dbls(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_dbl}(X'))) -> SEL(activate(X1), dbl(activate(X')))
ACTIVATE(n_{_sel}(X1, n_{_s}(X'))) -> SEL(activate(X1), s(X'))
ACTIVATE(n_{_sel}(X1', X2)) -> SEL(X1', activate(X2))
ACTIVATE(n_{_sel}(n_{_from}(X'), X2)) -> SEL(from(X'), activate(X2))
ACTIVATE(n_{_sel}(n_{_indx}(X1'', X2''), X2)) -> SEL(indx(activate(X1''), X2''), activate(X2))
SEL(s(X), cons(Y, Z')) -> SEL(activate(X), Z')
SEL(s(X), cons(Y, n_{_from}(X''))) -> SEL(activate(X), from(X''))
SEL(s(X), cons(Y, n_{_indx}(X1', X2'))) -> SEL(activate(X), indx(activate(X1'), X2'))
SEL(s(X), cons(Y, n_{_sel}(X1', X2'))) -> SEL(activate(X), sel(activate(X1'), activate(X2')))
SEL(s(X), cons(Y, n_{_dbls}(X''))) -> SEL(activate(X), dbls(activate(X'')))
SEL(s(X), cons(Y, n_{_dbl}(X''))) -> SEL(activate(X), dbl(activate(X'')))
SEL(s(X), cons(Y, n_{_s}(X''))) -> SEL(activate(X), s(X''))
SEL(s(X''), cons(Y, Z)) -> SEL(X'', activate(Z))
SEL(s(n_{_from}(X'')), cons(Y, Z)) -> SEL(from(X''), activate(Z))
SEL(s(n_{_indx}(X1', X2')), cons(Y, Z)) -> SEL(indx(activate(X1'), X2'), activate(Z))
SEL(s(n_{_sel}(X1', X2')), cons(Y, Z)) -> SEL(sel(activate(X1'), activate(X2')), activate(Z))
SEL(s(n_{_dbls}(X'')), cons(Y, Z)) -> SEL(dbls(activate(X'')), activate(Z))
SEL(s(n_{_dbl}(X'')), cons(Y, Z)) -> SEL(dbl(activate(X'')), activate(Z))
SEL(s(n_{_s}(X'')), cons(Y, Z)) -> SEL(s(X''), activate(Z))
ACTIVATE(n_{_sel}(n_{_sel}(X1'', X2''), X2)) -> SEL(sel(activate(X1''), activate(X2'')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(Z)
ACTIVATE(n_{_sel}(n_{_dbls}(X'), X2)) -> SEL(dbls(activate(X')), activate(X2))
SEL(s(X), cons(Y, Z)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_dbl}(X'), X2)) -> SEL(dbl(activate(X')), activate(X2))
SEL(0, cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_sel}(n_{_s}(X'), X2)) -> SEL(s(X'), activate(X2))
ACTIVATE(n_{_dbls}(X'')) -> DBLS(X'')
ACTIVATE(n_{_dbls}(n_{_from}(X''))) -> DBLS(from(X''))
ACTIVATE(n_{_dbls}(n_{_indx}(X1', X2'))) -> DBLS(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbls}(n_{_sel}(X1', X2'))) -> DBLS(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbls}(n_{_dbls}(X''))) -> DBLS(dbls(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(Y)
ACTIVATE(n_{_dbls}(n_{_dbl}(X''))) -> DBLS(dbl(activate(X'')))
DBLS(cons(X, Y)) -> ACTIVATE(X)
ACTIVATE(n_{_dbls}(n_{_s}(X''))) -> DBLS(s(X''))
ACTIVATE(n_{_dbl}(X'')) -> DBL(X'')
ACTIVATE(n_{_dbl}(n_{_from}(X''))) -> DBL(from(X''))
ACTIVATE(n_{_dbl}(n_{_indx}(X1', X2'))) -> DBL(indx(activate(X1'), X2'))
ACTIVATE(n_{_dbl}(n_{_sel}(X1', X2'))) -> DBL(sel(activate(X1'), activate(X2')))
ACTIVATE(n_{_dbl}(n_{_dbls}(X''))) -> DBL(dbls(activate(X'')))
ACTIVATE(n_{_dbl}(n_{_dbl}(X''))) -> DBL(dbl(activate(X'')))
DBL(s(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(n_{_s}(X''))) -> DBL(s(X''))
FROM(X) -> ACTIVATE(X)
ACTIVATE(n_{_from}(X)) -> FROM(X)
ACTIVATE(n_{_indx}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X2)
ACTIVATE(n_{_sel}(X1, X2)) -> ACTIVATE(X1)
ACTIVATE(n_{_dbls}(X)) -> ACTIVATE(X)
ACTIVATE(n_{_dbl}(X)) -> ACTIVATE(X)
INDX(cons(X, Y), Z) -> ACTIVATE(Y)

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
DBL1(s(X)) -> DBL1(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
SEL1(s(X), cons(Y, Z)) -> SEL1(activate(X), activate(Z))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X
Dependency Pair:
QUOTE(s(X)) -> QUOTE(activate(X))

Rules:

dbl(0) -> 0
dbl(s(X)) -> s(n_{_s}(n_{_dbl}(activate(X))))
dbl(X) -> n_{_dbl}(X)
dbls(nil) -> nil
dbls(cons(X, Y)) -> cons(n_{_dbl}(activate(X)), n_{_dbls}(activate(Y)))
dbls(X) -> n_{_dbls}(X)
sel(0, cons(X, Y)) -> activate(X)
sel(s(X), cons(Y, Z)) -> sel(activate(X), activate(Z))
sel(X1, X2) -> n_{_sel}(X1, X2)
indx(nil, X) -> nil
indx(cons(X, Y), Z) -> cons(n_{_sel}(activate(X), activate(Z)), n_{_indx}(activate(Y), activate(Z)))
indx(X1, X2) -> n_{_indx}(X1, X2)
from(X) -> cons(activate(X), n_{_from}(n_{_s}(activate(X))))
from(X) -> n_{_from}(X)
dbl1(0) -> 01
dbl1(s(X)) -> s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) -> activate(X)
sel1(s(X), cons(Y, Z)) -> sel1(activate(X), activate(Z))
quote(0) -> 01
quote(s(X)) -> s1(quote(activate(X)))
quote(dbl(X)) -> dbl1(X)
quote(sel(X, Y)) -> sel1(X, Y)
s(X) -> n_{_s}(X)
activate(n_{_s}(X)) -> s(X)
activate(n_{_dbl}(X)) -> dbl(activate(X))
activate(n_{_dbls}(X)) -> dbls(activate(X))
activate(n_{_sel}(X1, X2)) -> sel(activate(X1), activate(X2))
activate(n_{_indx}(X1, X2)) -> indx(activate(X1), X2)
activate(n_{_from}(X)) -> from(X)
activate(X) -> X

Termination of R could not be shown.
Duration:
0:33 minutes