TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60012 ms.

The following DP Processors were used


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).

The following open problems remain:

Open Dependency Pair Problem 18

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

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))

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

The following SCCs where found

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)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

s#(mark(X)) → s#(X)s#(active(X)) → s#(X)

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

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)

Problem 10: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

sel#(X1, active(X2)) → sel#(X1, X2)sel#(X1, mark(X2)) → sel#(X1, X2)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil

Strategy

Function Precedence

active < 0 = s = indx = mark = dbl = from = sel# = sel = dbls = cons = nil

Argument Filtering

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

Status

0: multiset
indx: multiset
mark: multiset
dbl: multiset
from: multiset
sel#: lexicographic with permutation 2 → 1
sel: lexicographic with permutation 1 → 1
nil: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

sel#(X1, mark(X2)) → sel#(X1, X2)

Problem 14: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

sel#(X1, active(X2)) → sel#(X1, X2)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Function Precedence

sel# < active < 0 = s = indx = mark = dbl = from = sel = dbls = cons = nil

Argument Filtering

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

Status

0: multiset
s: multiset
indx: multiset
active: multiset
mark: multiset
dbl: multiset
from: multiset
sel: lexicographic with permutation 1 → 1 2 → 2
dbls: multiset
cons: multiset
nil: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

sel#(X1, active(X2)) → sel#(X1, X2)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

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)

Problem 11: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

indx#(X1, active(X2)) → indx#(X1, X2)indx#(X1, mark(X2)) → indx#(X1, X2)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil

Strategy

Function Precedence

mark < active < 0 = s = indx = dbl = from = sel = dbls = indx# = cons = nil

Argument Filtering

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

Status

0: multiset
s: multiset
indx: multiset
active: multiset
dbl: multiset
from: multiset
sel: multiset
dbls: multiset
indx#: lexicographic with permutation 1 → 1 2 → 2
cons: multiset
nil: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

indx#(X1, active(X2)) → indx#(X1, X2)

Problem 15: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

indx#(X1, mark(X2)) → indx#(X1, X2)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Function Precedence

0 = s = indx = active = mark = dbl = from = sel = dbls = indx# = cons = nil

Argument Filtering

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

Status

0: multiset
s: multiset
indx: multiset
mark: multiset
from: multiset
sel: multiset
dbls: multiset
cons: lexicographic with permutation 1 → 2 2 → 1
nil: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

indx#(X1, mark(X2)) → indx#(X1, X2)

Problem 5: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

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)

Problem 12: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

cons#(X1, active(X2)) → cons#(X1, X2)cons#(X1, mark(X2)) → cons#(X1, X2)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil

Strategy

Function Precedence

cons# < active = mark < 0 = s = indx = dbl = from = sel = dbls = cons = nil

Argument Filtering

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

Status

cons#: lexicographic with permutation 2 → 1
0: multiset
s: multiset
indx: lexicographic with permutation 1 → 1 2 → 2
active: multiset
dbl: multiset
from: multiset
sel: multiset
dbls: lexicographic with permutation 1 → 1
nil: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

cons#(X1, active(X2)) → cons#(X1, X2)

Problem 16: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

cons#(X1, mark(X2)) → cons#(X1, X2)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Function Precedence

mark < cons# < 0 = s = indx = active = dbl = from = sel = dbls = cons = nil

Argument Filtering

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

Status

0: multiset
s: multiset
indx: lexicographic with permutation 2 → 1
active: multiset
mark: multiset
dbl: multiset
from: lexicographic with permutation 1 → 1
sel: lexicographic with permutation 1 → 2 2 → 1
dbls: multiset
cons: lexicographic with permutation 1 → 2 2 → 1
nil: multiset

Usable Rules

There are no usable rules.

The dependency pairs and usable rules are stronlgy conservative!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

cons#(X1, mark(X2)) → cons#(X1, X2)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

dbls#(active(X)) → dbls#(X)dbls#(mark(X)) → dbls#(X)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

dbls#(active(X)) → dbls#(X)dbls#(mark(X)) → dbls#(X)

Problem 7: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Function Precedence

mark = dbl = from = mark# = dbls = 0 = s = indx = active = active# = sel = cons = nil

Argument Filtering

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

Status

mark: multiset
dbl: multiset
from: multiset
mark#: multiset
dbls: multiset
0: multiset
s: multiset
indx: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

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!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

mark#(0) → active#(0)

Problem 13: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, indx, s, active, dbl, mark, from, sel, dbls, cons, nil

Strategy

Function Precedence

active# < active < mark = dbl = from = mark# = dbls = 0 = s = indx = sel = cons = nil

Argument Filtering

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

Status

mark: multiset
dbl: multiset
from: multiset
mark#: multiset
dbls: multiset
0: multiset
s: multiset
indx: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

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!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

mark#(nil) → active#(nil)

Problem 17: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Function Precedence

active# < mark = dbl = from = mark# = dbls = s = indx = active = sel < 0 = cons = nil

Argument Filtering

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

Status

mark: multiset
dbl: multiset
from: multiset
mark#: multiset
dbls: multiset
0: multiset
s: multiset
indx: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

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!

Eliminated dependency pairs

The following dependency pairs (at least) can be eliminated according to the given precedence.

mark#(cons(X1, X2)) → active#(cons(X1, X2))

Problem 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

dbl#(mark(X)) → dbl#(X)dbl#(active(X)) → dbl#(X)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

dbl#(mark(X)) → dbl#(X)dbl#(active(X)) → dbl#(X)

Problem 9: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

from#(active(X)) → from#(X)from#(mark(X)) → from#(X)

Rewrite Rules

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)

Original Signature

Termination of terms over the following signature is verified: 0, s, indx, active, mark, dbl, from, sel, dbls, nil, cons

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

from#(active(X)) → from#(X)from#(mark(X)) → from#(X)