MAYBE
The TRS could not be proven terminating. The proof attempt took 5697 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 3 was processed with processor PolynomialLinearRange4iUR (0ms).
| | – Problem 5 was processed with processor PolynomialLinearRange4iUR (0ms).
| | | – Problem 7 was processed with processor PolynomialLinearRange4iUR (0ms).
| | | | – Problem 8 was processed with processor PolynomialLinearRange4iUR (0ms).
| | | | | – Problem 9 remains open; application of the following processors failed [DependencyGraph (2ms), PolynomialLinearRange4iUR (380ms), DependencyGraph (41ms), PolynomialLinearRange8NegiUR (2451ms), DependencyGraph (2ms), ReductionPairSAT (633ms), DependencyGraph (1ms), SizeChangePrinciple (37ms)].
The following open problems remain:
Open Dependency Pair Problem 9
Dependency Pairs
| mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) | | active#(f(X, X)) | → | mark#(f(a, b)) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(f(X1, X2)) | → | f#(mark(X1), X2) | | f#(X1, mark(X2)) | → | f#(X1, X2) |
| mark#(b) | → | active#(b) | | active#(b) | → | mark#(a) |
| mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) | | f#(X1, active(X2)) | → | f#(X1, X2) |
| mark#(a) | → | active#(a) | | f#(active(X1), X2) | → | f#(X1, X2) |
| active#(f(X, X)) | → | mark#(f(a, b)) | | active#(f(X, X)) | → | f#(a, b) |
| f#(mark(X1), X2) | → | f#(X1, X2) | | mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
The following SCCs where found
| f#(X1, mark(X2)) → f#(X1, X2) | f#(X1, active(X2)) → f#(X1, X2) |
| f#(active(X1), X2) → f#(X1, X2) | f#(mark(X1), X2) → f#(X1, X2) |
| mark#(b) → active#(b) | mark#(f(X1, X2)) → active#(f(mark(X1), X2)) |
| active#(b) → mark#(a) | mark#(a) → active#(a) |
| active#(f(X, X)) → mark#(f(a, b)) | mark#(f(X1, X2)) → mark#(X1) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(X1, mark(X2)) | → | f#(X1, X2) | | f#(X1, active(X2)) | → | f#(X1, X2) |
| f#(active(X1), X2) | → | f#(X1, X2) | | f#(mark(X1), X2) | → | f#(X1, X2) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(active(X1), X2) | → | f#(X1, X2) | | f#(mark(X1), X2) | → | f#(X1, X2) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(X1, mark(X2)) | → | f#(X1, X2) | | f#(X1, active(X2)) | → | f#(X1, X2) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): 3x
- b: 0
- f(x,y): 0
- f#(x,y): 3y
- mark(x): 2x + 1
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)) | → | f#(X1, X2) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| f#(X1, active(X2)) | → | f#(X1, X2) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): x + 1
- b: 0
- f(x,y): 0
- f#(x,y): y + x + 2
- mark(x): 0
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)) | → | f#(X1, X2) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(b) | → | active#(b) | | mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) |
| active#(b) | → | mark#(a) | | mark#(a) | → | active#(a) |
| active#(f(X, X)) | → | mark#(f(a, b)) | | mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): x
- active#(x): x
- b: 0
- f(x,y): 2x + 2
- mark(x): x
- mark#(x): x
Improved Usable rules
| f(active(X1), X2) | → | f(X1, X2) | | f(X1, mark(X2)) | → | f(X1, X2) |
| active(b) | → | mark(a) | | active(f(X, X)) | → | mark(f(a, b)) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| mark(a) | → | active(a) | | f(X1, active(X2)) | → | f(X1, X2) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(f(X1, X2)) | → | mark#(X1) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(b) | → | active#(b) | | active#(b) | → | mark#(a) |
| mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) | | mark#(a) | → | active#(a) |
| active#(f(X, X)) | → | mark#(f(a, b)) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Polynomial Interpretation
- a: 1
- active(x): x + 1
- active#(x): 1
- b: 2
- f(x,y): 1
- mark(x): 2
- mark#(x): x
Improved Usable rules
| f(active(X1), X2) | → | f(X1, X2) | | f(X1, mark(X2)) | → | f(X1, X2) |
| f(mark(X1), X2) | → | f(X1, X2) | | f(X1, active(X2)) | → | f(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 7: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) | | active#(b) | → | mark#(a) |
| mark#(a) | → | active#(a) | | active#(f(X, X)) | → | mark#(f(a, b)) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): 3
- active#(x): x
- b: 1
- f(x,y): 1
- mark(x): 3
- mark#(x): 1
Improved Usable rules
| f(active(X1), X2) | → | f(X1, X2) | | f(X1, mark(X2)) | → | f(X1, X2) |
| f(mark(X1), X2) | → | f(X1, X2) | | f(X1, active(X2)) | → | f(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(b) | → | mark#(a) | | mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) |
| active#(f(X, X)) | → | mark#(f(a, b)) |
Rewrite Rules
| active(f(X, X)) | → | mark(f(a, b)) | | active(b) | → | mark(a) |
| mark(f(X1, X2)) | → | active(f(mark(X1), X2)) | | mark(a) | → | active(a) |
| mark(b) | → | active(b) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, mark(X2)) | → | f(X1, X2) | | f(active(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) |
Original Signature
Termination of terms over the following signature is verified: f, b, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): 0
- active#(x): x
- b: 1
- f(x,y): 0
- mark(x): 1
- mark#(x): 0
Improved Usable rules
| f(active(X1), X2) | → | f(X1, X2) | | f(X1, mark(X2)) | → | f(X1, X2) |
| f(mark(X1), X2) | → | f(X1, X2) | | f(X1, active(X2)) | → | f(X1, X2) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed: