MAYBE

The TRS could not be proven terminating. The proof attempt took 19904 ms.

The following DP Processors were used


Problem 1 was processed with processor DependencyGraph (0ms).
 | – Problem 2 was processed with processor SubtermCriterion (0ms).
 |    | – Problem 4 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    |    | – Problem 6 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    |    |    | – Problem 8 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    |    |    |    | – Problem 10 was processed with processor PolynomialLinearRange4iUR (0ms).
 | – Problem 3 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    | – Problem 5 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    |    | – Problem 7 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    |    |    | – Problem 9 was processed with processor PolynomialLinearRange4iUR (0ms).
 |    |    |    |    | – Problem 11 remains open; application of the following processors failed [DependencyGraph (2ms), PolynomialLinearRange4iUR (881ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (8485ms), DependencyGraph (1ms), ReductionPairSAT (7424ms), DependencyGraph (1ms), SizeChangePrinciple (47ms)].

The following open problems remain:



Open Dependency Pair Problem 11

Dependency Pairs

mark#(f(X1, X2, X3))active#(f(X1, X2, mark(X3)))active#(f(a, b, X))mark#(f(X, X, X))

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark


Problem 1: DependencyGraph



Dependency Pair Problem

Dependency Pairs

active#(f(a, b, X))f#(X, X, X)mark#(a)active#(a)
f#(X1, mark(X2), X3)f#(X1, X2, X3)mark#(f(X1, X2, X3))mark#(X3)
active#(c)mark#(a)mark#(c)active#(c)
f#(X1, X2, mark(X3))f#(X1, X2, X3)mark#(b)active#(b)
f#(mark(X1), X2, X3)f#(X1, X2, X3)active#(f(a, b, X))mark#(f(X, X, X))
mark#(f(X1, X2, X3))active#(f(X1, X2, mark(X3)))mark#(f(X1, X2, X3))f#(X1, X2, mark(X3))
active#(c)mark#(b)f#(active(X1), X2, X3)f#(X1, X2, X3)
f#(X1, active(X2), X3)f#(X1, X2, X3)f#(X1, X2, active(X3))f#(X1, X2, X3)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


The following SCCs where found

f#(X1, X2, mark(X3)) → f#(X1, X2, X3)f#(mark(X1), X2, X3) → f#(X1, X2, X3)
f#(X1, mark(X2), X3) → f#(X1, X2, X3)f#(X1, active(X2), X3) → f#(X1, X2, X3)
f#(active(X1), X2, X3) → f#(X1, X2, X3)f#(X1, X2, active(X3)) → f#(X1, X2, X3)

mark#(b) → active#(b)active#(f(a, b, X)) → mark#(f(X, X, X))
mark#(f(X1, X2, X3)) → active#(f(X1, X2, mark(X3)))mark#(a) → active#(a)
mark#(f(X1, X2, X3)) → mark#(X3)active#(c) → mark#(b)
mark#(c) → active#(c)active#(c) → mark#(a)

Problem 2: SubtermCriterion



Dependency Pair Problem

Dependency Pairs

f#(X1, X2, mark(X3))f#(X1, X2, X3)f#(mark(X1), X2, X3)f#(X1, X2, X3)
f#(X1, mark(X2), X3)f#(X1, X2, X3)f#(X1, active(X2), X3)f#(X1, X2, X3)
f#(active(X1), X2, X3)f#(X1, X2, X3)f#(X1, X2, active(X3))f#(X1, X2, X3)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Projection

The following projection was used:

Thus, the following dependency pairs are removed:

f#(mark(X1), X2, X3)f#(X1, X2, X3)f#(active(X1), X2, X3)f#(X1, X2, X3)

Problem 4: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

f#(X1, X2, mark(X3))f#(X1, X2, X3)f#(X1, mark(X2), X3)f#(X1, X2, X3)
f#(X1, active(X2), X3)f#(X1, X2, X3)f#(X1, X2, active(X3))f#(X1, X2, X3)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

There are no usable rules

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

f#(X1, X2, active(X3))f#(X1, X2, X3)

Problem 6: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

f#(X1, X2, mark(X3))f#(X1, X2, X3)f#(X1, mark(X2), X3)f#(X1, X2, X3)
f#(X1, active(X2), X3)f#(X1, X2, X3)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

There are no usable rules

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

f#(X1, X2, mark(X3))f#(X1, X2, X3)

Problem 8: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

f#(X1, mark(X2), X3)f#(X1, X2, X3)f#(X1, active(X2), X3)f#(X1, X2, X3)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

There are no usable rules

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

f#(X1, active(X2), X3)f#(X1, X2, X3)

Problem 10: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

f#(X1, mark(X2), X3)f#(X1, X2, X3)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

There are no usable rules

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

f#(X1, mark(X2), X3)f#(X1, X2, X3)

Problem 3: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(b)active#(b)active#(f(a, b, X))mark#(f(X, X, X))
mark#(f(X1, X2, X3))active#(f(X1, X2, mark(X3)))mark#(a)active#(a)
mark#(f(X1, X2, X3))mark#(X3)active#(c)mark#(b)
mark#(c)active#(c)active#(c)mark#(a)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

Improved Usable rules

f(X1, X2, mark(X3))f(X1, X2, X3)active(f(a, b, X))mark(f(X, X, X))
f(mark(X1), X2, X3)f(X1, X2, X3)f(active(X1), X2, X3)f(X1, X2, X3)
active(c)mark(b)f(X1, X2, active(X3))f(X1, X2, X3)
mark(c)active(c)f(X1, active(X2), X3)f(X1, X2, X3)
active(c)mark(a)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
f(X1, mark(X2), X3)f(X1, X2, X3)mark(b)active(b)
mark(a)active(a)

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

mark#(f(X1, X2, X3))mark#(X3)

Problem 5: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(b)active#(b)mark#(a)active#(a)
mark#(f(X1, X2, X3))active#(f(X1, X2, mark(X3)))active#(f(a, b, X))mark#(f(X, X, X))
active#(c)mark#(b)active#(c)mark#(a)
mark#(c)active#(c)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

Improved Usable rules

f(X1, X2, mark(X3))f(X1, X2, X3)f(mark(X1), X2, X3)f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, mark(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)

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

active#(c)mark#(b)mark#(c)active#(c)
active#(c)mark#(a)

Problem 7: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(b)active#(b)active#(f(a, b, X))mark#(f(X, X, X))
mark#(f(X1, X2, X3))active#(f(X1, X2, mark(X3)))mark#(a)active#(a)

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

Improved Usable rules

f(X1, X2, mark(X3))f(X1, X2, X3)f(mark(X1), X2, X3)f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, mark(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)

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

mark#(b)active#(b)

Problem 9: PolynomialLinearRange4iUR



Dependency Pair Problem

Dependency Pairs

mark#(a)active#(a)mark#(f(X1, X2, X3))active#(f(X1, X2, mark(X3)))
active#(f(a, b, X))mark#(f(X, X, X))

Rewrite Rules

active(f(a, b, X))mark(f(X, X, X))active(c)mark(a)
active(c)mark(b)mark(f(X1, X2, X3))active(f(X1, X2, mark(X3)))
mark(a)active(a)mark(b)active(b)
mark(c)active(c)f(mark(X1), X2, X3)f(X1, X2, X3)
f(X1, mark(X2), X3)f(X1, X2, X3)f(X1, X2, mark(X3))f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)

Original Signature

Termination of terms over the following signature is verified: f, b, c, a, active, mark

Strategy


Polynomial Interpretation

Improved Usable rules

f(X1, X2, mark(X3))f(X1, X2, X3)f(mark(X1), X2, X3)f(X1, X2, X3)
f(active(X1), X2, X3)f(X1, X2, X3)f(X1, mark(X2), X3)f(X1, X2, X3)
f(X1, X2, active(X3))f(X1, X2, X3)f(X1, active(X2), X3)f(X1, X2, X3)

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

mark#(a)active#(a)