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(X)
activate(n_{_dbls}(X)) -> dbls(X)
activate(n_{_sel}(X1, X2)) -> sel(X1, X2)
activate(n_{_indx}(X1, X2)) -> indx(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(X)
ACTIVATE(n_{_dbls}(X)) -> DBLS(X)
ACTIVATE(n_{_sel}(X1, X2)) -> SEL(X1, X2)
ACTIVATE(n_{_indx}(X1, X2)) -> INDX(X1, X2)
ACTIVATE(n_{_from}(X)) -> FROM(X)

Furthermore, R contains four SCCs.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

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

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

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

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

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

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

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

Termination of R could not be shown.
Duration:
0:01 minutes