TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60012 ms.
Problem 1 was processed with processor DependencyGraph (2819ms). | Problem 2 was processed with processor SubtermCriterion (2ms). | Problem 3 was processed with processor SubtermCriterion (0ms). | | Problem 10 was processed with processor ReductionPairSAT (84ms). | | | Problem 14 was processed with processor ReductionPairSAT (65ms). | Problem 4 was processed with processor SubtermCriterion (0ms). | | Problem 11 was processed with processor ReductionPairSAT (37ms). | | | Problem 15 was processed with processor ReductionPairSAT (45ms). | Problem 5 was processed with processor SubtermCriterion (2ms). | | Problem 12 was processed with processor ReductionPairSAT (49ms). | | | Problem 16 was processed with processor ReductionPairSAT (48ms). | Problem 6 was processed with processor SubtermCriterion (0ms). | Problem 7 was processed with processor ReductionPairSAT (13861ms). | | Problem 13 was processed with processor ReductionPairSAT (9220ms). | | | Problem 17 was processed with processor ReductionPairSAT (8276ms). | | | | Problem 18 remains open; application of the following processors failed [DependencyGraph (220ms), ReductionPairSAT (timeout)]. | Problem 8 was processed with processor SubtermCriterion (2ms). | Problem 9 was processed with processor SubtermCriterion (1ms).
active#(sel(0, cons(X, Y))) | → | mark#(X) | mark#(dbls(X)) | → | mark#(X) | |
mark#(dbls(X)) | → | active#(dbls(mark(X))) | mark#(dbl(X)) | → | active#(dbl(mark(X))) | |
mark#(from(X)) | → | active#(from(X)) | mark#(indx(X1, X2)) | → | mark#(X1) | |
mark#(sel(X1, X2)) | → | active#(sel(mark(X1), mark(X2))) | active#(from(X)) | → | mark#(cons(X, from(s(X)))) | |
active#(dbl(s(X))) | → | mark#(s(s(dbl(X)))) | mark#(dbl(X)) | → | mark#(X) | |
mark#(s(X)) | → | active#(s(X)) | active#(sel(s(X), cons(Y, Z))) | → | mark#(sel(X, Z)) | |
active#(dbls(cons(X, Y))) | → | mark#(cons(dbl(X), dbls(Y))) | active#(indx(nil, X)) | → | mark#(nil) | |
active#(dbl(0)) | → | mark#(0) | mark#(indx(X1, X2)) | → | active#(indx(mark(X1), X2)) | |
mark#(sel(X1, X2)) | → | mark#(X2) | active#(indx(cons(X, Y), Z)) | → | mark#(cons(sel(X, Z), indx(Y, Z))) | |
mark#(sel(X1, X2)) | → | mark#(X1) | active#(dbls(nil)) | → | mark#(nil) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
mark#(dbls(X)) | → | active#(dbls(mark(X))) | dbls#(active(X)) | → | dbls#(X) | |
mark#(dbls(X)) | → | dbls#(mark(X)) | active#(dbl(s(X))) | → | s#(dbl(X)) | |
active#(dbls(cons(X, Y))) | → | cons#(dbl(X), dbls(Y)) | dbls#(mark(X)) | → | dbls#(X) | |
active#(indx(nil, X)) | → | mark#(nil) | mark#(indx(X1, X2)) | → | active#(indx(mark(X1), X2)) | |
mark#(dbl(X)) | → | dbl#(mark(X)) | active#(dbl(s(X))) | → | dbl#(X) | |
mark#(sel(X1, X2)) | → | mark#(X1) | indx#(mark(X1), X2) | → | indx#(X1, X2) | |
mark#(sel(X1, X2)) | → | sel#(mark(X1), mark(X2)) | active#(sel(s(X), cons(Y, Z))) | → | sel#(X, Z) | |
active#(dbl(s(X))) | → | s#(s(dbl(X))) | mark#(dbl(X)) | → | active#(dbl(mark(X))) | |
mark#(indx(X1, X2)) | → | indx#(mark(X1), X2) | mark#(sel(X1, X2)) | → | active#(sel(mark(X1), mark(X2))) | |
active#(from(X)) | → | mark#(cons(X, from(s(X)))) | sel#(X1, mark(X2)) | → | sel#(X1, X2) | |
mark#(s(X)) | → | active#(s(X)) | active#(indx(cons(X, Y), Z)) | → | cons#(sel(X, Z), indx(Y, Z)) | |
active#(from(X)) | → | s#(X) | sel#(active(X1), X2) | → | sel#(X1, X2) | |
cons#(X1, active(X2)) | → | cons#(X1, X2) | sel#(X1, active(X2)) | → | sel#(X1, X2) | |
active#(dbl(0)) | → | mark#(0) | from#(active(X)) | → | from#(X) | |
indx#(active(X1), X2) | → | indx#(X1, X2) | sel#(mark(X1), X2) | → | sel#(X1, X2) | |
active#(sel(0, cons(X, Y))) | → | mark#(X) | mark#(dbls(X)) | → | mark#(X) | |
cons#(mark(X1), X2) | → | cons#(X1, X2) | active#(dbls(cons(X, Y))) | → | dbls#(Y) | |
mark#(cons(X1, X2)) | → | cons#(X1, X2) | from#(mark(X)) | → | from#(X) | |
active#(indx(cons(X, Y), Z)) | → | indx#(Y, Z) | active#(dbl(s(X))) | → | mark#(s(s(dbl(X)))) | |
mark#(from(X)) | → | from#(X) | active#(sel(s(X), cons(Y, Z))) | → | mark#(sel(X, Z)) | |
mark#(nil) | → | active#(nil) | active#(dbls(cons(X, Y))) | → | mark#(cons(dbl(X), dbls(Y))) | |
active#(indx(cons(X, Y), Z)) | → | sel#(X, Z) | mark#(sel(X1, X2)) | → | mark#(X2) | |
active#(from(X)) | → | cons#(X, from(s(X))) | cons#(X1, mark(X2)) | → | cons#(X1, X2) | |
active#(indx(cons(X, Y), Z)) | → | mark#(cons(sel(X, Z), indx(Y, Z))) | active#(dbls(cons(X, Y))) | → | dbl#(X) | |
mark#(0) | → | active#(0) | indx#(X1, active(X2)) | → | indx#(X1, X2) | |
mark#(from(X)) | → | active#(from(X)) | mark#(indx(X1, X2)) | → | mark#(X1) | |
cons#(active(X1), X2) | → | cons#(X1, X2) | dbl#(mark(X)) | → | dbl#(X) | |
mark#(dbl(X)) | → | mark#(X) | mark#(cons(X1, X2)) | → | active#(cons(X1, X2)) | |
indx#(X1, mark(X2)) | → | indx#(X1, X2) | s#(mark(X)) | → | s#(X) | |
mark#(s(X)) | → | s#(X) | s#(active(X)) | → | s#(X) | |
dbl#(active(X)) | → | dbl#(X) | active#(dbls(nil)) | → | mark#(nil) | |
active#(from(X)) | → | from#(s(X)) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
dbls#(active(X)) → dbls#(X) | dbls#(mark(X)) → dbls#(X) |
indx#(X1, active(X2)) → indx#(X1, X2) | indx#(active(X1), X2) → indx#(X1, X2) |
indx#(mark(X1), X2) → indx#(X1, X2) | indx#(X1, mark(X2)) → indx#(X1, X2) |
from#(active(X)) → from#(X) | from#(mark(X)) → from#(X) |
dbl#(mark(X)) → dbl#(X) | dbl#(active(X)) → dbl#(X) |
s#(mark(X)) → s#(X) | s#(active(X)) → s#(X) |
cons#(X1, active(X2)) → cons#(X1, X2) | cons#(mark(X1), X2) → cons#(X1, X2) |
cons#(active(X1), X2) → cons#(X1, X2) | cons#(X1, mark(X2)) → cons#(X1, X2) |
sel#(mark(X1), X2) → sel#(X1, X2) | sel#(active(X1), X2) → sel#(X1, X2) |
sel#(X1, active(X2)) → sel#(X1, X2) | sel#(X1, mark(X2)) → sel#(X1, X2) |
mark#(0) → active#(0) | active#(sel(0, cons(X, Y))) → mark#(X) |
mark#(dbls(X)) → mark#(X) | mark#(dbls(X)) → active#(dbls(mark(X))) |
mark#(dbl(X)) → active#(dbl(mark(X))) | mark#(indx(X1, X2)) → mark#(X1) |
mark#(from(X)) → active#(from(X)) | active#(from(X)) → mark#(cons(X, from(s(X)))) |
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2))) | active#(dbl(s(X))) → mark#(s(s(dbl(X)))) |
mark#(s(X)) → active#(s(X)) | mark#(dbl(X)) → mark#(X) |
mark#(cons(X1, X2)) → active#(cons(X1, X2)) | active#(sel(s(X), cons(Y, Z))) → mark#(sel(X, Z)) |
mark#(nil) → active#(nil) | active#(dbls(cons(X, Y))) → mark#(cons(dbl(X), dbls(Y))) |
active#(indx(nil, X)) → mark#(nil) | active#(dbl(0)) → mark#(0) |
mark#(indx(X1, X2)) → active#(indx(mark(X1), X2)) | mark#(sel(X1, X2)) → mark#(X2) |
active#(indx(cons(X, Y), Z)) → mark#(cons(sel(X, Z), indx(Y, Z))) | mark#(sel(X1, X2)) → mark#(X1) |
active#(dbls(nil)) → mark#(nil) |
s#(mark(X)) | → | s#(X) | s#(active(X)) | → | s#(X) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
s#(mark(X)) | → | s#(X) | s#(active(X)) | → | s#(X) |
sel#(mark(X1), X2) | → | sel#(X1, X2) | sel#(active(X1), X2) | → | sel#(X1, X2) | |
sel#(X1, active(X2)) | → | sel#(X1, X2) | sel#(X1, mark(X2)) | → | sel#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
sel#(active(X1), X2) | → | sel#(X1, X2) | sel#(mark(X1), X2) | → | sel#(X1, X2) |
sel#(X1, active(X2)) | → | sel#(X1, X2) | sel#(X1, mark(X2)) | → | sel#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil
0: all arguments are removed from 0
s: collapses to 1
indx: all arguments are removed from indx
active: collapses to 1
mark: 1
dbl: all arguments are removed from dbl
from: all arguments are removed from from
sel#: 2
sel: 1
dbls: collapses to 1
cons: collapses to 1
nil: all arguments are removed from nil
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
sel#(X1, mark(X2)) → sel#(X1, X2) |
sel#(X1, active(X2)) | → | sel#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
0: all arguments are removed from 0
s: all arguments are removed from s
indx: all arguments are removed from indx
active: 1
mark: all arguments are removed from mark
dbl: all arguments are removed from dbl
from: all arguments are removed from from
sel#: collapses to 2
sel: 1 2
dbls: all arguments are removed from dbls
cons: all arguments are removed from cons
nil: all arguments are removed from nil
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
sel#(X1, active(X2)) → sel#(X1, X2) |
indx#(X1, active(X2)) | → | indx#(X1, X2) | indx#(active(X1), X2) | → | indx#(X1, X2) | |
indx#(mark(X1), X2) | → | indx#(X1, X2) | indx#(X1, mark(X2)) | → | indx#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
indx#(active(X1), X2) | → | indx#(X1, X2) | indx#(mark(X1), X2) | → | indx#(X1, X2) |
indx#(X1, active(X2)) | → | indx#(X1, X2) | indx#(X1, mark(X2)) | → | indx#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil
0: all arguments are removed from 0
s: all arguments are removed from s
indx: all arguments are removed from indx
active: 1
mark: collapses to 1
dbl: all arguments are removed from dbl
from: all arguments are removed from from
sel: all arguments are removed from sel
dbls: all arguments are removed from dbls
indx#: 1 2
cons: all arguments are removed from cons
nil: all arguments are removed from nil
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
indx#(X1, active(X2)) → indx#(X1, X2) |
indx#(X1, mark(X2)) | → | indx#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
0: all arguments are removed from 0
s: all arguments are removed from s
indx: all arguments are removed from indx
active: collapses to 1
mark: 1
dbl: collapses to 1
from: all arguments are removed from from
sel: all arguments are removed from sel
dbls: all arguments are removed from dbls
indx#: collapses to 2
cons: 1 2
nil: all arguments are removed from nil
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
indx#(X1, mark(X2)) → indx#(X1, X2) |
cons#(X1, active(X2)) | → | cons#(X1, X2) | cons#(mark(X1), X2) | → | cons#(X1, X2) | |
cons#(active(X1), X2) | → | cons#(X1, X2) | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
cons#(mark(X1), X2) | → | cons#(X1, X2) | cons#(active(X1), X2) | → | cons#(X1, X2) |
cons#(X1, active(X2)) | → | cons#(X1, X2) | cons#(X1, mark(X2)) | → | cons#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil
cons#: 2
0: all arguments are removed from 0
s: all arguments are removed from s
indx: 1 2
active: 1
mark: collapses to 1
dbl: all arguments are removed from dbl
from: all arguments are removed from from
sel: all arguments are removed from sel
dbls: 1
cons: collapses to 1
nil: all arguments are removed from nil
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
cons#(X1, active(X2)) → cons#(X1, X2) |
cons#(X1, mark(X2)) | → | cons#(X1, X2) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
cons#: collapses to 2
0: all arguments are removed from 0
s: all arguments are removed from s
indx: 2
active: all arguments are removed from active
mark: 1
dbl: all arguments are removed from dbl
from: 1
sel: 1 2
dbls: all arguments are removed from dbls
cons: 1 2
nil: all arguments are removed from nil
There are no usable rules.
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
cons#(X1, mark(X2)) → cons#(X1, X2) |
dbls#(active(X)) | → | dbls#(X) | dbls#(mark(X)) | → | dbls#(X) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
dbls#(active(X)) | → | dbls#(X) | dbls#(mark(X)) | → | dbls#(X) |
mark#(0) | → | active#(0) | active#(sel(0, cons(X, Y))) | → | mark#(X) | |
mark#(dbls(X)) | → | mark#(X) | mark#(dbls(X)) | → | active#(dbls(mark(X))) | |
mark#(dbl(X)) | → | active#(dbl(mark(X))) | mark#(from(X)) | → | active#(from(X)) | |
mark#(indx(X1, X2)) | → | mark#(X1) | mark#(sel(X1, X2)) | → | active#(sel(mark(X1), mark(X2))) | |
active#(from(X)) | → | mark#(cons(X, from(s(X)))) | active#(dbl(s(X))) | → | mark#(s(s(dbl(X)))) | |
mark#(dbl(X)) | → | mark#(X) | mark#(s(X)) | → | active#(s(X)) | |
mark#(cons(X1, X2)) | → | active#(cons(X1, X2)) | active#(sel(s(X), cons(Y, Z))) | → | mark#(sel(X, Z)) | |
mark#(nil) | → | active#(nil) | active#(dbls(cons(X, Y))) | → | mark#(cons(dbl(X), dbls(Y))) | |
active#(indx(nil, X)) | → | mark#(nil) | active#(dbl(0)) | → | mark#(0) | |
mark#(indx(X1, X2)) | → | active#(indx(mark(X1), X2)) | mark#(sel(X1, X2)) | → | mark#(X2) | |
active#(indx(cons(X, Y), Z)) | → | mark#(cons(sel(X, Z), indx(Y, Z))) | mark#(sel(X1, X2)) | → | mark#(X1) | |
active#(dbls(nil)) | → | mark#(nil) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
mark: all arguments are removed from mark
dbl: all arguments are removed from dbl
from: all arguments are removed from from
mark#: all arguments are removed from mark#
dbls: all arguments are removed from dbls
0: all arguments are removed from 0
s: all arguments are removed from s
indx: all arguments are removed from indx
active: collapses to 1
active#: collapses to 1
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil
active(dbls(nil)) → mark(nil) | cons(active(X1), X2) → cons(X1, X2) |
from(mark(X)) → from(X) | sel(X1, mark(X2)) → sel(X1, X2) |
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z)) | indx(mark(X1), X2) → indx(X1, X2) |
mark(cons(X1, X2)) → active(cons(X1, X2)) | mark(s(X)) → active(s(X)) |
cons(X1, mark(X2)) → cons(X1, X2) | active(indx(nil, X)) → mark(nil) |
active(dbl(0)) → mark(0) | active(dbl(s(X))) → mark(s(s(dbl(X)))) |
mark(indx(X1, X2)) → active(indx(mark(X1), X2)) | indx(X1, mark(X2)) → indx(X1, X2) |
active(sel(0, cons(X, Y))) → mark(X) | active(from(X)) → mark(cons(X, from(s(X)))) |
mark(nil) → active(nil) | s(active(X)) → s(X) |
dbl(mark(X)) → dbl(X) | mark(0) → active(0) |
from(active(X)) → from(X) | dbls(mark(X)) → dbls(X) |
cons(X1, active(X2)) → cons(X1, X2) | indx(active(X1), X2) → indx(X1, X2) |
indx(X1, active(X2)) → indx(X1, X2) | sel(X1, active(X2)) → sel(X1, X2) |
mark(from(X)) → active(from(X)) | mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2))) |
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z))) | cons(mark(X1), X2) → cons(X1, X2) |
dbls(active(X)) → dbls(X) | mark(dbls(X)) → active(dbls(mark(X))) |
s(mark(X)) → s(X) | sel(active(X1), X2) → sel(X1, X2) |
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y))) | mark(dbl(X)) → active(dbl(mark(X))) |
dbl(active(X)) → dbl(X) | sel(mark(X1), X2) → sel(X1, X2) |
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
mark#(0) → active#(0) |
active#(sel(0, cons(X, Y))) | → | mark#(X) | mark#(dbls(X)) | → | mark#(X) | |
mark#(dbls(X)) | → | active#(dbls(mark(X))) | mark#(dbl(X)) | → | active#(dbl(mark(X))) | |
mark#(from(X)) | → | active#(from(X)) | mark#(indx(X1, X2)) | → | mark#(X1) | |
mark#(sel(X1, X2)) | → | active#(sel(mark(X1), mark(X2))) | active#(from(X)) | → | mark#(cons(X, from(s(X)))) | |
active#(dbl(s(X))) | → | mark#(s(s(dbl(X)))) | mark#(dbl(X)) | → | mark#(X) | |
mark#(s(X)) | → | active#(s(X)) | mark#(cons(X1, X2)) | → | active#(cons(X1, X2)) | |
active#(sel(s(X), cons(Y, Z))) | → | mark#(sel(X, Z)) | mark#(nil) | → | active#(nil) | |
active#(dbls(cons(X, Y))) | → | mark#(cons(dbl(X), dbls(Y))) | active#(indx(nil, X)) | → | mark#(nil) | |
mark#(sel(X1, X2)) | → | mark#(X2) | mark#(indx(X1, X2)) | → | active#(indx(mark(X1), X2)) | |
active#(dbl(0)) | → | mark#(0) | mark#(sel(X1, X2)) | → | mark#(X1) | |
active#(indx(cons(X, Y), Z)) | → | mark#(cons(sel(X, Z), indx(Y, Z))) | active#(dbls(nil)) | → | mark#(nil) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil
mark: all arguments are removed from mark
dbl: all arguments are removed from dbl
from: all arguments are removed from from
mark#: all arguments are removed from mark#
dbls: all arguments are removed from dbls
0: all arguments are removed from 0
s: all arguments are removed from s
indx: all arguments are removed from indx
active: collapses to 1
active#: collapses to 1
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil
active(dbls(nil)) → mark(nil) | cons(active(X1), X2) → cons(X1, X2) |
from(mark(X)) → from(X) | sel(X1, mark(X2)) → sel(X1, X2) |
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z)) | indx(mark(X1), X2) → indx(X1, X2) |
mark(cons(X1, X2)) → active(cons(X1, X2)) | mark(s(X)) → active(s(X)) |
cons(X1, mark(X2)) → cons(X1, X2) | active(indx(nil, X)) → mark(nil) |
active(dbl(0)) → mark(0) | active(dbl(s(X))) → mark(s(s(dbl(X)))) |
mark(indx(X1, X2)) → active(indx(mark(X1), X2)) | indx(X1, mark(X2)) → indx(X1, X2) |
active(sel(0, cons(X, Y))) → mark(X) | active(from(X)) → mark(cons(X, from(s(X)))) |
mark(nil) → active(nil) | s(active(X)) → s(X) |
dbl(mark(X)) → dbl(X) | mark(0) → active(0) |
from(active(X)) → from(X) | dbls(mark(X)) → dbls(X) |
cons(X1, active(X2)) → cons(X1, X2) | indx(active(X1), X2) → indx(X1, X2) |
indx(X1, active(X2)) → indx(X1, X2) | sel(X1, active(X2)) → sel(X1, X2) |
mark(from(X)) → active(from(X)) | mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2))) |
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z))) | cons(mark(X1), X2) → cons(X1, X2) |
dbls(active(X)) → dbls(X) | mark(dbls(X)) → active(dbls(mark(X))) |
s(mark(X)) → s(X) | sel(active(X1), X2) → sel(X1, X2) |
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y))) | mark(dbl(X)) → active(dbl(mark(X))) |
dbl(active(X)) → dbl(X) | sel(mark(X1), X2) → sel(X1, X2) |
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
mark#(nil) → active#(nil) |
active#(sel(0, cons(X, Y))) | → | mark#(X) | mark#(dbls(X)) | → | mark#(X) | |
mark#(dbls(X)) | → | active#(dbls(mark(X))) | mark#(dbl(X)) | → | active#(dbl(mark(X))) | |
mark#(from(X)) | → | active#(from(X)) | mark#(indx(X1, X2)) | → | mark#(X1) | |
mark#(sel(X1, X2)) | → | active#(sel(mark(X1), mark(X2))) | active#(from(X)) | → | mark#(cons(X, from(s(X)))) | |
active#(dbl(s(X))) | → | mark#(s(s(dbl(X)))) | mark#(dbl(X)) | → | mark#(X) | |
mark#(s(X)) | → | active#(s(X)) | mark#(cons(X1, X2)) | → | active#(cons(X1, X2)) | |
active#(sel(s(X), cons(Y, Z))) | → | mark#(sel(X, Z)) | active#(dbls(cons(X, Y))) | → | mark#(cons(dbl(X), dbls(Y))) | |
active#(indx(nil, X)) | → | mark#(nil) | active#(dbl(0)) | → | mark#(0) | |
mark#(indx(X1, X2)) | → | active#(indx(mark(X1), X2)) | mark#(sel(X1, X2)) | → | mark#(X2) | |
active#(indx(cons(X, Y), Z)) | → | mark#(cons(sel(X, Z), indx(Y, Z))) | mark#(sel(X1, X2)) | → | mark#(X1) | |
active#(dbls(nil)) | → | mark#(nil) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
mark: all arguments are removed from mark
dbl: all arguments are removed from dbl
from: all arguments are removed from from
mark#: all arguments are removed from mark#
dbls: all arguments are removed from dbls
0: all arguments are removed from 0
s: all arguments are removed from s
indx: all arguments are removed from indx
active: collapses to 1
active#: collapses to 1
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil
active(dbls(nil)) → mark(nil) | cons(active(X1), X2) → cons(X1, X2) |
from(mark(X)) → from(X) | sel(X1, mark(X2)) → sel(X1, X2) |
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z)) | indx(mark(X1), X2) → indx(X1, X2) |
mark(cons(X1, X2)) → active(cons(X1, X2)) | mark(s(X)) → active(s(X)) |
cons(X1, mark(X2)) → cons(X1, X2) | active(indx(nil, X)) → mark(nil) |
active(dbl(0)) → mark(0) | active(dbl(s(X))) → mark(s(s(dbl(X)))) |
mark(indx(X1, X2)) → active(indx(mark(X1), X2)) | indx(X1, mark(X2)) → indx(X1, X2) |
active(sel(0, cons(X, Y))) → mark(X) | active(from(X)) → mark(cons(X, from(s(X)))) |
mark(nil) → active(nil) | s(active(X)) → s(X) |
dbl(mark(X)) → dbl(X) | mark(0) → active(0) |
from(active(X)) → from(X) | dbls(mark(X)) → dbls(X) |
cons(X1, active(X2)) → cons(X1, X2) | indx(active(X1), X2) → indx(X1, X2) |
indx(X1, active(X2)) → indx(X1, X2) | sel(X1, active(X2)) → sel(X1, X2) |
mark(from(X)) → active(from(X)) | mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2))) |
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z))) | cons(mark(X1), X2) → cons(X1, X2) |
dbls(active(X)) → dbls(X) | mark(dbls(X)) → active(dbls(mark(X))) |
s(mark(X)) → s(X) | sel(active(X1), X2) → sel(X1, X2) |
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y))) | mark(dbl(X)) → active(dbl(mark(X))) |
dbl(active(X)) → dbl(X) | sel(mark(X1), X2) → sel(X1, X2) |
The dependency pairs and usable rules are stronlgy conservative!
The following dependency pairs (at least) can be eliminated according to the given precedence.
mark#(cons(X1, X2)) → active#(cons(X1, X2)) |
dbl#(mark(X)) | → | dbl#(X) | dbl#(active(X)) | → | dbl#(X) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
dbl#(mark(X)) | → | dbl#(X) | dbl#(active(X)) | → | dbl#(X) |
from#(active(X)) | → | from#(X) | from#(mark(X)) | → | from#(X) |
active(dbl(0)) | → | mark(0) | active(dbl(s(X))) | → | mark(s(s(dbl(X)))) | |
active(dbls(nil)) | → | mark(nil) | active(dbls(cons(X, Y))) | → | mark(cons(dbl(X), dbls(Y))) | |
active(sel(0, cons(X, Y))) | → | mark(X) | active(sel(s(X), cons(Y, Z))) | → | mark(sel(X, Z)) | |
active(indx(nil, X)) | → | mark(nil) | active(indx(cons(X, Y), Z)) | → | mark(cons(sel(X, Z), indx(Y, Z))) | |
active(from(X)) | → | mark(cons(X, from(s(X)))) | mark(dbl(X)) | → | active(dbl(mark(X))) | |
mark(0) | → | active(0) | mark(s(X)) | → | active(s(X)) | |
mark(dbls(X)) | → | active(dbls(mark(X))) | mark(nil) | → | active(nil) | |
mark(cons(X1, X2)) | → | active(cons(X1, X2)) | mark(sel(X1, X2)) | → | active(sel(mark(X1), mark(X2))) | |
mark(indx(X1, X2)) | → | active(indx(mark(X1), X2)) | mark(from(X)) | → | active(from(X)) | |
dbl(mark(X)) | → | dbl(X) | dbl(active(X)) | → | dbl(X) | |
s(mark(X)) | → | s(X) | s(active(X)) | → | s(X) | |
dbls(mark(X)) | → | dbls(X) | dbls(active(X)) | → | dbls(X) | |
cons(mark(X1), X2) | → | cons(X1, X2) | cons(X1, mark(X2)) | → | cons(X1, X2) | |
cons(active(X1), X2) | → | cons(X1, X2) | cons(X1, active(X2)) | → | cons(X1, X2) | |
sel(mark(X1), X2) | → | sel(X1, X2) | sel(X1, mark(X2)) | → | sel(X1, X2) | |
sel(active(X1), X2) | → | sel(X1, X2) | sel(X1, active(X2)) | → | sel(X1, X2) | |
indx(mark(X1), X2) | → | indx(X1, X2) | indx(X1, mark(X2)) | → | indx(X1, X2) | |
indx(active(X1), X2) | → | indx(X1, X2) | indx(X1, active(X2)) | → | indx(X1, X2) | |
from(mark(X)) | → | from(X) | from(active(X)) | → | from(X) |
Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons
The following projection was used:
Thus, the following dependency pairs are removed:
from#(active(X)) | → | from#(X) | from#(mark(X)) | → | from#(X) |