TIMEOUT

The TRS could not be proven terminating. The proof attempt took 60022 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (2581ms).
 | – Problem 2 was processed with processor SubtermCriterion (2ms).
 | – Problem 3 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 10 was processed with processor ReductionPairSAT (78ms).
 |    |    | – Problem 14 was processed with processor ReductionPairSAT (64ms).
 | – Problem 4 was processed with processor SubtermCriterion (1ms).
 | – Problem 5 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 11 was processed with processor ReductionPairSAT (91ms).
 |    |    | – Problem 15 was processed with processor ReductionPairSAT (25ms).
 | – Problem 6 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 12 was processed with processor ReductionPairSAT (79ms).
 |    |    | – Problem 16 was processed with processor ReductionPairSAT (24ms).
 | – Problem 7 was processed with processor SubtermCriterion (1ms).
 | – Problem 8 was processed with processor SubtermCriterion (3ms).
 | – Problem 9 was processed with processor ReductionPairSAT (6895ms).
 |    | – Problem 13 was processed with processor ReductionPairSAT (5219ms).
 |    |    | – Problem 17 was processed with processor ReductionPairSAT (6380ms).
 |    |    |    | – Problem 18 was processed with processor ReductionPairSAT (7303ms).
 |    |    |    |    | – Problem 19 remains open; application of the following processors failed [DependencyGraph (266ms), ReductionPairSAT (13988ms), DependencyGraph (266ms), SizeChangePrinciple (timeout)].

The following open problems remain:

Open Dependency Pair Problem 19

Dependency Pairs

mark#(head(X)) → active#(head(mark(X)))active#(2nd(cons(X, XS))) → mark#(head(XS))
mark#(take(X1, X2)) → mark#(X1)mark#(from(X)) → mark#(X)
mark#(cons(X1, X2)) → mark#(X1)active#(take(0, XS)) → mark#(nil)
mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))active#(from(X)) → mark#(cons(X, from(s(X))))
mark#(2nd(X)) → active#(2nd(mark(X)))mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
mark#(head(X)) → mark#(X)mark#(from(X)) → active#(from(mark(X)))
mark#(sel(X1, X2)) → mark#(X2)mark#(take(X1, X2)) → mark#(X2)
active#(head(cons(X, XS))) → mark#(X)active#(sel(0, cons(X, XS))) → mark#(X)
mark#(s(X)) → mark#(X)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
mark#(sel(X1, X2)) → mark#(X1)active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))
mark#(2nd(X)) → mark#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, nil, cons

Problem 1: DependencyGraph

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(take(X1, X2)) → mark#(X1)
mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))mark#(s(X)) → s#(mark(X))
active#(take(s(N), cons(X, XS))) → cons#(X, take(N, XS))mark#(s(X)) → mark#(X)
mark#(sel(X1, X2)) → mark#(X1)mark#(sel(X1, X2)) → sel#(mark(X1), mark(X2))
mark#(from(X)) → mark#(X)mark#(cons(X1, X2)) → mark#(X1)
active#(take(0, XS)) → mark#(nil)mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
take#(X1, mark(X2)) → take#(X1, X2)active#(from(X)) → mark#(cons(X, from(s(X))))
sel#(X1, mark(X2)) → sel#(X1, X2)active#(from(X)) → s#(X)
sel#(active(X1), X2) → sel#(X1, X2)cons#(X1, active(X2)) → cons#(X1, X2)
mark#(from(X)) → from#(mark(X))sel#(X1, active(X2)) → sel#(X1, X2)
mark#(from(X)) → active#(from(mark(X)))from#(active(X)) → from#(X)
head#(active(X)) → head#(X)active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))
active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))sel#(mark(X1), X2) → sel#(X1, X2)
mark#(head(X)) → active#(head(mark(X)))take#(mark(X1), X2) → take#(X1, X2)
active#(2nd(cons(X, XS))) → mark#(head(XS))cons#(mark(X1), X2) → cons#(X1, X2)
2nd#(mark(X)) → 2nd#(X)from#(mark(X)) → from#(X)
mark#(head(X)) → mark#(X)mark#(nil) → active#(nil)
2nd#(active(X)) → 2nd#(X)take#(X1, active(X2)) → take#(X1, X2)
active#(sel(s(N), cons(X, XS))) → sel#(N, XS)mark#(sel(X1, X2)) → mark#(X2)
active#(from(X)) → cons#(X, from(s(X)))active#(take(s(N), cons(X, XS))) → take#(N, XS)
active#(sel(0, cons(X, XS))) → mark#(X)cons#(X1, mark(X2)) → cons#(X1, X2)
mark#(cons(X1, X2)) → cons#(mark(X1), X2)mark#(2nd(X)) → 2nd#(mark(X))
active#(2nd(cons(X, XS))) → head#(XS)mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))head#(mark(X)) → head#(X)
cons#(active(X1), X2) → cons#(X1, X2)mark#(2nd(X)) → active#(2nd(mark(X)))
mark#(head(X)) → head#(mark(X))mark#(take(X1, X2)) → take#(mark(X1), mark(X2))
s#(mark(X)) → s#(X)mark#(take(X1, X2)) → mark#(X2)
s#(active(X)) → s#(X)active#(head(cons(X, XS))) → mark#(X)
mark#(2nd(X)) → mark#(X)active#(from(X)) → from#(s(X))
take#(active(X1), X2) → take#(X1, X2)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

The following SCCs where found

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(head(X)) → active#(head(mark(X)))
active#(2nd(cons(X, XS))) → mark#(head(XS))mark#(take(X1, X2)) → mark#(X1)
mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))mark#(head(X)) → mark#(X)
mark#(nil) → active#(nil)mark#(sel(X1, X2)) → mark#(X2)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(s(X)) → mark#(X)
mark#(sel(X1, X2)) → mark#(X1)mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))mark#(from(X)) → mark#(X)
mark#(cons(X1, X2)) → mark#(X1)active#(take(0, XS)) → mark#(nil)
active#(from(X)) → mark#(cons(X, from(s(X))))mark#(2nd(X)) → active#(2nd(mark(X)))
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))mark#(from(X)) → active#(from(mark(X)))
mark#(take(X1, X2)) → mark#(X2)active#(head(cons(X, XS))) → mark#(X)
active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))
mark#(2nd(X)) → mark#(X)
from#(active(X)) → from#(X)from#(mark(X)) → from#(X)
cons#(X1, active(X2)) → cons#(X1, X2)cons#(mark(X1), X2) → cons#(X1, X2)
cons#(X1, mark(X2)) → cons#(X1, X2)cons#(active(X1), X2) → cons#(X1, X2)
s#(mark(X)) → s#(X)s#(active(X)) → s#(X)
take#(mark(X1), X2) → take#(X1, X2)take#(X1, active(X2)) → take#(X1, X2)
take#(X1, mark(X2)) → take#(X1, X2)take#(active(X1), X2) → take#(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)
2nd#(active(X)) → 2nd#(X)2nd#(mark(X)) → 2nd#(X)
head#(mark(X)) → head#(X)head#(active(X)) → head#(X)

Problem 2: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 3: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 10: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, nil, cons

Strategy

Function Precedence

mark < take# < active < 2nd = 0 = s = take = from = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: 1
mark: collapses to 1
from: collapses to 1
take#: collapses to 2
head: 1
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

2nd: multiset
0: multiset
s: multiset
take: multiset
active: lexicographic with permutation 1 → 1
head: lexicographic with permutation 1 → 1
sel: 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.

take#(X1, active(X2)) → take#(X1, X2)

Problem 14: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

take#(X1, mark(X2)) → take#(X1, X2)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Function Precedence

mark < 2nd = 0 = s = take = active = from = take# = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: all arguments are removed from active
mark: 1
from: all arguments are removed from from
take#: collapses to 2
head: all arguments are removed from head
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

2nd: multiset
0: multiset
s: multiset
take: multiset
active: multiset
mark: multiset
from: multiset
head: multiset
sel: 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.

take#(X1, mark(X2)) → take#(X1, X2)

Problem 4: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

2nd#(active(X)) → 2nd#(X)2nd#(mark(X)) → 2nd#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

2nd#(active(X)) → 2nd#(X)2nd#(mark(X)) → 2nd#(X)

Problem 5: 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(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

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 11: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, nil, cons

Strategy

Function Precedence

mark < sel# < active < 2nd = 0 = s = take = from = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: 1
mark: collapses to 1
from: collapses to 1
sel#: collapses to 2
head: 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

2nd: multiset
0: multiset
s: multiset
take: multiset
active: lexicographic with permutation 1 → 1
sel: 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 15: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

sel#(X1, mark(X2)) → sel#(X1, X2)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Function Precedence

mark < 2nd = 0 = s = take = active = from = sel# = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: all arguments are removed from active
mark: 1
from: all arguments are removed from from
sel#: collapses to 2
head: all arguments are removed from head
sel: 1 2
cons: 1 2
nil: all arguments are removed from nil

Status

2nd: multiset
0: multiset
s: multiset
take: multiset
active: multiset
mark: multiset
from: multiset
head: multiset
sel: lexicographic with permutation 1 → 1 2 → 2
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.

sel#(X1, mark(X2)) → sel#(X1, X2)

Problem 6: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

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(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, nil, cons

Strategy

Function Precedence

active = mark < cons# < 2nd = 0 = s = take = from = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
cons#: 1 2
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: 1
mark: collapses to 1
from: all arguments are removed from from
head: all arguments are removed from head
sel: 1 2
cons: 1 2
nil: all arguments are removed from nil

Status

2nd: multiset
cons#: lexicographic with permutation 1 → 2 2 → 1
0: multiset
s: multiset
take: multiset
active: multiset
from: multiset
head: multiset
sel: lexicographic with permutation 1 → 2 2 → 1
cons: lexicographic with permutation 1 → 1 2 → 2
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(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Function Precedence

mark < 2nd = cons# = 0 = s = take = active = from = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
cons#: collapses to 2
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: all arguments are removed from active
mark: 1
from: 1
head: all arguments are removed from head
sel: all arguments are removed from sel
cons: 1 2
nil: all arguments are removed from nil

Status

2nd: multiset
0: multiset
s: multiset
take: multiset
active: multiset
mark: multiset
from: lexicographic with permutation 1 → 1
head: multiset
sel: multiset
cons: lexicographic with permutation 1 → 1 2 → 2
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 7: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

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

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

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 8: SubtermCriterion

Dependency Pair Problem

Dependency Pairs

head#(mark(X)) → head#(X)head#(active(X)) → head#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Projection

The following projection was used:

Thus, the following dependency pairs are removed:

head#(mark(X)) → head#(X)head#(active(X)) → head#(X)

Problem 9: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(head(X)) → active#(head(mark(X)))
active#(2nd(cons(X, XS))) → mark#(head(XS))mark#(take(X1, X2)) → mark#(X1)
mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))mark#(head(X)) → mark#(X)
mark#(nil) → active#(nil)mark#(sel(X1, X2)) → mark#(X2)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(s(X)) → mark#(X)
mark#(sel(X1, X2)) → mark#(X1)mark#(0) → active#(0)
mark#(s(X)) → active#(s(mark(X)))mark#(from(X)) → mark#(X)
active#(take(0, XS)) → mark#(nil)mark#(cons(X1, X2)) → mark#(X1)
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))mark#(2nd(X)) → active#(2nd(mark(X)))
active#(from(X)) → mark#(cons(X, from(s(X))))mark#(from(X)) → active#(from(mark(X)))
mark#(take(X1, X2)) → mark#(X2)active#(head(cons(X, XS))) → mark#(X)
active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))
mark#(2nd(X)) → mark#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Function Precedence

active# < active < 2nd = mark = from = mark# = 0 = s = take = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
mark: all arguments are removed from mark
from: all arguments are removed from from
mark#: all arguments are removed from mark#
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: collapses to 1
active#: collapses to 1
head: all arguments are removed from head
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

2nd: multiset
mark: multiset
from: multiset
mark#: multiset
0: multiset
s: multiset
take: multiset
head: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
head(active(X)) → head(X)from(mark(X)) → from(X)
mark(2nd(X)) → active(2nd(mark(X)))sel(X1, mark(X2)) → sel(X1, X2)
active(take(0, XS)) → mark(nil)mark(s(X)) → active(s(mark(X)))
active(2nd(cons(X, XS))) → mark(head(XS))cons(X1, mark(X2)) → cons(X1, X2)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))mark(from(X)) → active(from(mark(X)))
active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
mark(nil) → active(nil)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
from(active(X)) → from(X)cons(X1, active(X2)) → cons(X1, X2)
take(X1, active(X2)) → take(X1, X2)2nd(mark(X)) → 2nd(X)
mark(head(X)) → active(head(mark(X)))mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
sel(X1, active(X2)) → sel(X1, X2)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
take(mark(X1), X2) → take(X1, X2)head(mark(X)) → head(X)
take(X1, mark(X2)) → take(X1, X2)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))s(mark(X)) → s(X)
take(active(X1), X2) → take(X1, X2)sel(active(X1), X2) → sel(X1, X2)
2nd(active(X)) → 2nd(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 13: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(head(X)) → active#(head(mark(X)))
mark#(0) → active#(0)active#(2nd(cons(X, XS))) → mark#(head(XS))
mark#(take(X1, X2)) → mark#(X1)mark#(s(X)) → active#(s(mark(X)))
mark#(from(X)) → mark#(X)mark#(cons(X1, X2)) → mark#(X1)
active#(take(0, XS)) → mark#(nil)mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))
active#(from(X)) → mark#(cons(X, from(s(X))))mark#(2nd(X)) → active#(2nd(mark(X)))
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))mark#(head(X)) → mark#(X)
mark#(from(X)) → active#(from(mark(X)))mark#(sel(X1, X2)) → mark#(X2)
mark#(take(X1, X2)) → mark#(X2)active#(head(cons(X, XS))) → mark#(X)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(s(X)) → mark#(X)
active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))mark#(sel(X1, X2)) → mark#(X1)
active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))mark#(2nd(X)) → mark#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, nil, cons

Strategy

Function Precedence

active < 2nd = mark = from = mark# = 0 = s = take = active# = head = sel = cons = nil

Argument Filtering

2nd: all arguments are removed from 2nd
mark: all arguments are removed from mark
from: all arguments are removed from from
mark#: all arguments are removed from mark#
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: collapses to 1
active#: collapses to 1
head: all arguments are removed from head
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

2nd: multiset
mark: multiset
from: multiset
mark#: multiset
0: multiset
s: multiset
take: multiset
head: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
head(active(X)) → head(X)from(mark(X)) → from(X)
mark(2nd(X)) → active(2nd(mark(X)))sel(X1, mark(X2)) → sel(X1, X2)
active(take(0, XS)) → mark(nil)mark(s(X)) → active(s(mark(X)))
active(2nd(cons(X, XS))) → mark(head(XS))cons(X1, mark(X2)) → cons(X1, X2)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))mark(from(X)) → active(from(mark(X)))
active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
mark(nil) → active(nil)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
from(active(X)) → from(X)cons(X1, active(X2)) → cons(X1, X2)
take(X1, active(X2)) → take(X1, X2)2nd(mark(X)) → 2nd(X)
mark(head(X)) → active(head(mark(X)))mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
sel(X1, active(X2)) → sel(X1, X2)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
take(mark(X1), X2) → take(X1, X2)head(mark(X)) → head(X)
take(X1, mark(X2)) → take(X1, X2)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))s(mark(X)) → s(X)
take(active(X1), X2) → take(X1, X2)sel(active(X1), X2) → sel(X1, X2)
2nd(active(X)) → 2nd(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 17: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(head(X)) → active#(head(mark(X)))
active#(2nd(cons(X, XS))) → mark#(head(XS))mark#(take(X1, X2)) → mark#(X1)
mark#(s(X)) → active#(s(mark(X)))mark#(from(X)) → mark#(X)
mark#(cons(X1, X2)) → mark#(X1)active#(take(0, XS)) → mark#(nil)
mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))active#(from(X)) → mark#(cons(X, from(s(X))))
mark#(2nd(X)) → active#(2nd(mark(X)))mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))
mark#(head(X)) → mark#(X)mark#(from(X)) → active#(from(mark(X)))
mark#(sel(X1, X2)) → mark#(X2)mark#(take(X1, X2)) → mark#(X2)
mark#(s(X)) → mark#(X)active#(sel(0, cons(X, XS))) → mark#(X)
active#(head(cons(X, XS))) → mark#(X)mark#(sel(X1, X2)) → mark#(X1)
active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))
mark#(2nd(X)) → mark#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, cons, nil

Strategy

Function Precedence

active < 2nd = mark = from = mark# = take = active# = head = sel = cons < 0 = s = nil

Argument Filtering

2nd: all arguments are removed from 2nd
mark: all arguments are removed from mark
from: all arguments are removed from from
mark#: all arguments are removed from mark#
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: collapses to 1
active#: collapses to 1
head: all arguments are removed from head
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

2nd: multiset
mark: multiset
from: multiset
mark#: multiset
0: multiset
s: multiset
take: multiset
head: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
head(active(X)) → head(X)from(mark(X)) → from(X)
mark(2nd(X)) → active(2nd(mark(X)))sel(X1, mark(X2)) → sel(X1, X2)
active(take(0, XS)) → mark(nil)mark(s(X)) → active(s(mark(X)))
active(2nd(cons(X, XS))) → mark(head(XS))cons(X1, mark(X2)) → cons(X1, X2)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))mark(from(X)) → active(from(mark(X)))
active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
mark(nil) → active(nil)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
from(active(X)) → from(X)cons(X1, active(X2)) → cons(X1, X2)
take(X1, active(X2)) → take(X1, X2)2nd(mark(X)) → 2nd(X)
mark(head(X)) → active(head(mark(X)))mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
sel(X1, active(X2)) → sel(X1, X2)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
take(mark(X1), X2) → take(X1, X2)head(mark(X)) → head(X)
take(X1, mark(X2)) → take(X1, X2)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))s(mark(X)) → s(X)
take(active(X1), X2) → take(X1, X2)sel(active(X1), X2) → sel(X1, X2)
2nd(active(X)) → 2nd(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#(s(X)) → active#(s(mark(X)))

Problem 18: ReductionPairSAT

Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(head(X)) → active#(head(mark(X)))
active#(2nd(cons(X, XS))) → mark#(head(XS))mark#(take(X1, X2)) → mark#(X1)
mark#(from(X)) → mark#(X)mark#(cons(X1, X2)) → mark#(X1)
active#(take(0, XS)) → mark#(nil)mark#(take(X1, X2)) → active#(take(mark(X1), mark(X2)))
active#(from(X)) → mark#(cons(X, from(s(X))))mark#(2nd(X)) → active#(2nd(mark(X)))
mark#(sel(X1, X2)) → active#(sel(mark(X1), mark(X2)))mark#(head(X)) → mark#(X)
mark#(from(X)) → active#(from(mark(X)))mark#(sel(X1, X2)) → mark#(X2)
mark#(take(X1, X2)) → mark#(X2)active#(head(cons(X, XS))) → mark#(X)
active#(sel(0, cons(X, XS))) → mark#(X)mark#(s(X)) → mark#(X)
active#(sel(s(N), cons(X, XS))) → mark#(sel(N, XS))mark#(sel(X1, X2)) → mark#(X1)
active#(take(s(N), cons(X, XS))) → mark#(cons(X, take(N, XS)))mark#(2nd(X)) → mark#(X)

Rewrite Rules

active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
active(2nd(cons(X, XS))) → mark(head(XS))active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))mark(from(X)) → active(from(mark(X)))
mark(cons(X1, X2)) → active(cons(mark(X1), X2))mark(s(X)) → active(s(mark(X)))
mark(head(X)) → active(head(mark(X)))mark(2nd(X)) → active(2nd(mark(X)))
mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))mark(0) → active(0)
mark(nil) → active(nil)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
from(mark(X)) → from(X)from(active(X)) → from(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)
s(mark(X)) → s(X)s(active(X)) → s(X)
head(mark(X)) → head(X)head(active(X)) → head(X)
2nd(mark(X)) → 2nd(X)2nd(active(X)) → 2nd(X)
take(mark(X1), X2) → take(X1, X2)take(X1, mark(X2)) → take(X1, X2)
take(active(X1), X2) → take(X1, X2)take(X1, active(X2)) → take(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)

Original Signature

Termination of terms over the following signature is verified: 2nd, 0, s, take, active, mark, from, head, sel, nil, cons

Strategy

Function Precedence

s < active# < 2nd = mark = from = mark# = take = active = head = sel = nil < 0 < cons

Argument Filtering

2nd: all arguments are removed from 2nd
mark: all arguments are removed from mark
from: all arguments are removed from from
mark#: all arguments are removed from mark#
0: all arguments are removed from 0
s: all arguments are removed from s
take: all arguments are removed from take
active: all arguments are removed from active
active#: collapses to 1
head: all arguments are removed from head
sel: all arguments are removed from sel
cons: all arguments are removed from cons
nil: all arguments are removed from nil

Status

2nd: multiset
mark: multiset
from: multiset
mark#: multiset
0: multiset
s: multiset
take: multiset
active: multiset
head: multiset
sel: multiset
cons: multiset
nil: multiset

Usable Rules

active(sel(0, cons(X, XS))) → mark(X)cons(active(X1), X2) → cons(X1, X2)
head(active(X)) → head(X)from(mark(X)) → from(X)
mark(2nd(X)) → active(2nd(mark(X)))sel(X1, mark(X2)) → sel(X1, X2)
active(take(0, XS)) → mark(nil)mark(s(X)) → active(s(mark(X)))
active(2nd(cons(X, XS))) → mark(head(XS))cons(X1, mark(X2)) → cons(X1, X2)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))mark(from(X)) → active(from(mark(X)))
active(from(X)) → mark(cons(X, from(s(X))))active(head(cons(X, XS))) → mark(X)
mark(nil) → active(nil)mark(0) → active(0)
s(active(X)) → s(X)active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
from(active(X)) → from(X)cons(X1, active(X2)) → cons(X1, X2)
take(X1, active(X2)) → take(X1, X2)2nd(mark(X)) → 2nd(X)
mark(head(X)) → active(head(mark(X)))mark(take(X1, X2)) → active(take(mark(X1), mark(X2)))
sel(X1, active(X2)) → sel(X1, X2)mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
take(mark(X1), X2) → take(X1, X2)head(mark(X)) → head(X)
take(X1, mark(X2)) → take(X1, X2)cons(mark(X1), X2) → cons(X1, X2)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))s(mark(X)) → s(X)
take(active(X1), X2) → take(X1, X2)sel(active(X1), X2) → sel(X1, X2)
2nd(active(X)) → 2nd(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(mark(X1), X2))