YES

The TRS could be proven terminating. The proof took 15596 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (429ms).
 | – Problem 2 was processed with processor SubtermCriterion (0ms).
 | – Problem 3 was processed with processor SubtermCriterion (33ms).
 | – Problem 4 was processed with processor SubtermCriterion (0ms).
 | – Problem 5 was processed with processor PolynomialLinearRange4iUR (7018ms).
 |    | – Problem 9 was processed with processor PolynomialLinearRange4iUR (3542ms).
 |    |    | – Problem 10 was processed with processor PolynomialLinearRange4iUR (1966ms).
 |    |    |    | – Problem 11 was processed with processor PolynomialLinearRange4iUR (2036ms).
 |    |    |    |    | – Problem 12 was processed with processor PolynomialLinearRange4iUR (67ms).
 |    |    |    |    |    | – Problem 13 was processed with processor PolynomialLinearRange4iUR (89ms).
 |    |    |    |    |    |    | – Problem 14 was processed with processor PolynomialLinearRange4iUR (67ms).
 |    |    |    |    |    |    |    | – Problem 15 was processed with processor PolynomialLinearRange4iUR (126ms).
 |    |    |    |    |    |    |    |    | – Problem 16 was processed with processor PolynomialLinearRange4iUR (44ms).
 |    |    |    |    |    |    |    |    |    | – Problem 17 was processed with processor DependencyGraph (0ms).
 | – Problem 6 was processed with processor SubtermCriterion (1ms).
 |    | – Problem 8 was processed with processor SubtermCriterion (1ms).
 | – Problem 7 was processed with processor SubtermCriterion (0ms).

Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2))active#(cons(mark(X1), X2))cons#(mark(X1), X2)cons#(X1, X2)
active#(length(nil))mark#(0)mark#(length1(X))active#(length1(X))
from#(mark(X))from#(X)mark#(s(X))s#(mark(X))
mark#(nil)active#(nil)length#(active(X))length#(X)
mark#(length(X))active#(length(X))active#(from(X))cons#(X, from(s(X)))
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(s(X))mark#(X)
cons#(X1, mark(X2))cons#(X1, X2)mark#(cons(X1, X2))cons#(mark(X1), X2)
length#(mark(X))length#(X)mark#(length(X))length#(X)
mark#(0)active#(0)mark#(s(X))active#(s(mark(X)))
mark#(from(X))mark#(X)mark#(cons(X1, X2))mark#(X1)
cons#(active(X1), X2)cons#(X1, X2)active#(length(cons(X, Y)))length1#(Y)
active#(from(X))mark#(cons(X, from(s(X))))length1#(mark(X))length1#(X)
mark#(length1(X))length1#(X)active#(from(X))s#(X)
s#(mark(X))s#(X)active#(length(cons(X, Y)))s#(length1(Y))
cons#(X1, active(X2))cons#(X1, X2)mark#(from(X))from#(mark(X))
active#(length1(X))mark#(length(X))mark#(from(X))active#(from(mark(X)))
from#(active(X))from#(X)s#(active(X))s#(X)
active#(length1(X))length#(X)active#(from(X))from#(s(X))
length1#(active(X))length1#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


The following SCCs where found

length#(mark(X)) → length#(X)length#(active(X)) → length#(X)

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

length1#(mark(X)) → length1#(X)length1#(active(X)) → length1#(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)

mark#(cons(X1, X2)) → active#(cons(mark(X1), X2))mark#(s(X)) → active#(s(mark(X)))
active#(length1(X)) → mark#(length(X))mark#(from(X)) → active#(from(mark(X)))
mark#(from(X)) → mark#(X)mark#(length(X)) → active#(length(X))
active#(length(cons(X, Y))) → mark#(s(length1(Y)))mark#(cons(X1, X2)) → mark#(X1)
mark#(s(X)) → mark#(X)mark#(length1(X)) → active#(length1(X))
active#(from(X)) → mark#(cons(X, from(s(X))))

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

length1#(mark(X))length1#(X)length1#(active(X))length1#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

length1#(mark(X))length1#(X)length1#(active(X))length1#(X)

Problem 3: 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(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, 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 4: 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(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, 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 5: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(cons(X1, X2))active#(cons(mark(X1), X2))mark#(s(X))active#(s(mark(X)))
active#(length1(X))mark#(length(X))mark#(from(X))active#(from(mark(X)))
mark#(from(X))mark#(X)mark#(length(X))active#(length(X))
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(cons(X1, X2))mark#(X1)
mark#(s(X))mark#(X)mark#(length1(X))active#(length1(X))
active#(from(X))mark#(cons(X, from(s(X))))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Polynomial Interpretation

Improved Usable rules

length1(mark(X))length1(X)cons(active(X1), X2)cons(X1, X2)
cons(X1, mark(X2))cons(X1, X2)from(mark(X))from(X)
length1(active(X))length1(X)cons(mark(X1), X2)cons(X1, X2)
length(active(X))length(X)s(mark(X))s(X)
length(mark(X))length(X)s(active(X))s(X)
cons(X1, active(X2))cons(X1, X2)from(active(X))from(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

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

Problem 9: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(s(X))active#(s(mark(X)))mark#(from(X))active#(from(mark(X)))
active#(length1(X))mark#(length(X))mark#(length(X))active#(length(X))
mark#(from(X))mark#(X)mark#(cons(X1, X2))mark#(X1)
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(s(X))mark#(X)
active#(from(X))mark#(cons(X, from(s(X))))mark#(length1(X))active#(length1(X))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, length1, from, nil, cons

Strategy


Polynomial Interpretation

Improved Usable rules

length1(mark(X))length1(X)from(mark(X))from(X)
length1(active(X))length1(X)s(mark(X))s(X)
length(active(X))length(X)length(mark(X))length(X)
s(active(X))s(X)from(active(X))from(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

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

Problem 10: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

active#(length1(X))mark#(length(X))mark#(from(X))active#(from(mark(X)))
mark#(from(X))mark#(X)mark#(length(X))active#(length(X))
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(cons(X1, X2))mark#(X1)
mark#(s(X))mark#(X)mark#(length1(X))active#(length1(X))
active#(from(X))mark#(cons(X, from(s(X))))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Polynomial Interpretation

Improved Usable rules

cons(active(X1), X2)cons(X1, X2)from(mark(X))from(X)
mark(length(X))active(length(X))length1(active(X))length1(X)
mark(s(X))active(s(mark(X)))active(length1(X))mark(length(X))
length(mark(X))length(X)active(length(nil))mark(0)
length1(mark(X))length1(X)cons(X1, mark(X2))cons(X1, X2)
active(length(cons(X, Y)))mark(s(length1(Y)))cons(mark(X1), X2)cons(X1, X2)
mark(length1(X))active(length1(X))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(from(X))active(from(mark(X)))length(active(X))length(X)
s(mark(X))s(X)active(from(X))mark(cons(X, from(s(X))))
mark(nil)active(nil)mark(0)active(0)
s(active(X))s(X)from(active(X))from(X)
cons(X1, active(X2))cons(X1, X2)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

mark#(from(X))mark#(X)active#(from(X))mark#(cons(X, from(s(X))))

Problem 11: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(from(X))active#(from(mark(X)))active#(length1(X))mark#(length(X))
mark#(length(X))active#(length(X))mark#(cons(X1, X2))mark#(X1)
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(s(X))mark#(X)
mark#(length1(X))active#(length1(X))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, length1, from, nil, cons

Strategy


Polynomial Interpretation

Improved Usable rules

length1(mark(X))length1(X)length1(active(X))length1(X)
length(active(X))length(X)s(mark(X))s(X)
length(mark(X))length(X)s(active(X))s(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

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

Problem 12: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

active#(length1(X))mark#(length(X))mark#(length(X))active#(length(X))
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(cons(X1, X2))mark#(X1)
mark#(s(X))mark#(X)mark#(length1(X))active#(length1(X))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Polynomial Interpretation

Improved Usable rules

length1(mark(X))length1(X)length1(active(X))length1(X)
length(active(X))length(X)s(mark(X))s(X)
length(mark(X))length(X)s(active(X))s(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

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

Problem 13: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

active#(length1(X))mark#(length(X))mark#(length(X))active#(length(X))
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(s(X))mark#(X)
mark#(length1(X))active#(length1(X))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, length1, from, nil, cons

Strategy


Polynomial Interpretation

Improved Usable rules

length1(mark(X))length1(X)length1(active(X))length1(X)
length(active(X))length(X)s(mark(X))s(X)
length(mark(X))length(X)s(active(X))s(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

mark#(s(X))mark#(X)

Problem 14: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

active#(length1(X))mark#(length(X))mark#(length(X))active#(length(X))
active#(length(cons(X, Y)))mark#(s(length1(Y)))mark#(length1(X))active#(length1(X))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Polynomial Interpretation

Improved Usable rules

length(active(X))length(X)s(mark(X))s(X)
length(mark(X))length(X)s(active(X))s(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

mark#(length1(X))active#(length1(X))

Problem 15: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

active#(length1(X))mark#(length(X))mark#(length(X))active#(length(X))
active#(length(cons(X, Y)))mark#(s(length1(Y)))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, length1, from, nil, cons

Strategy


Polynomial Interpretation

Improved Usable rules

length(active(X))length(X)length(mark(X))length(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

active#(length1(X))mark#(length(X))

Problem 16: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(length(X))active#(length(X))active#(length(cons(X, Y)))mark#(s(length1(Y)))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Polynomial Interpretation

Improved Usable rules

s(mark(X))s(X)s(active(X))s(X)

The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:

mark#(length(X))active#(length(X))

Problem 17: DependencyGraph



Dependency Pair Problem

Dependency Pairs

active#(length(cons(X, Y)))mark#(s(length1(Y)))

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, length1, from, nil, cons

Strategy


There are no SCCs!

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(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, 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 8: SubtermCriterion



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(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, length1, from, nil, cons

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

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

Problem 7: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

length#(mark(X))length#(X)length#(active(X))length#(X)

Rewrite Rules

active(from(X))mark(cons(X, from(s(X))))active(length(nil))mark(0)
active(length(cons(X, Y)))mark(s(length1(Y)))active(length1(X))mark(length(X))
mark(from(X))active(from(mark(X)))mark(cons(X1, X2))active(cons(mark(X1), X2))
mark(s(X))active(s(mark(X)))mark(length(X))active(length(X))
mark(nil)active(nil)mark(0)active(0)
mark(length1(X))active(length1(X))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)length(mark(X))length(X)
length(active(X))length(X)length1(mark(X))length1(X)
length1(active(X))length1(X)

Original Signature

Termination of terms over the following signature is verified: 0, s, length, active, mark, from, length1, cons, nil

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

length#(mark(X))length#(X)length#(active(X))length#(X)