YES
The TRS could be proven terminating. The proof took 1196 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (58ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (593ms).
| | – Problem 5 was processed with processor PolynomialLinearRange4iUR (501ms).
| | | – Problem 6 was processed with processor DependencyGraph (0ms).
| – Problem 4 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| g#(mark(X)) | → | g#(X) | | mark#(g(X)) | → | g#(mark(X)) |
| mark#(g(X)) | → | mark#(X) | | mark#(a) | → | active#(a) |
| f#(active(X)) | → | f#(X) | | mark#(f(X)) | → | active#(f(X)) |
| g#(active(X)) | → | g#(X) | | active#(f(f(a))) | → | mark#(f(g(f(a)))) |
| active#(f(f(a))) | → | f#(g(f(a))) | | active#(f(f(a))) | → | f#(a) |
| active#(f(f(a))) | → | g#(f(a)) | | mark#(g(X)) | → | active#(g(mark(X))) |
| f#(mark(X)) | → | f#(X) | | mark#(f(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(f(g(f(a)))) | | mark(f(X)) | → | active(f(X)) |
| mark(a) | → | active(a) | | mark(g(X)) | → | active(g(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, a, active, mark
Strategy
The following SCCs where found
| active#(f(f(a))) → mark#(f(g(f(a)))) | mark#(g(X)) → mark#(X) |
| mark#(g(X)) → active#(g(mark(X))) | mark#(f(X)) → active#(f(X)) |
| f#(active(X)) → f#(X) | f#(mark(X)) → f#(X) |
| g#(active(X)) → g#(X) | g#(mark(X)) → g#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(f(g(f(a)))) | | mark(f(X)) | → | active(f(X)) |
| mark(a) | → | active(a) | | mark(g(X)) | → | active(g(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, a, active, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(a))) | → | mark#(f(g(f(a)))) | | mark#(g(X)) | → | mark#(X) |
| mark#(g(X)) | → | active#(g(mark(X))) | | mark#(f(X)) | → | active#(f(X)) |
Rewrite Rules
| active(f(f(a))) | → | mark(f(g(f(a)))) | | mark(f(X)) | → | active(f(X)) |
| mark(a) | → | active(a) | | mark(g(X)) | → | active(g(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): x
- active#(x): x
- f(x): 0
- g(x): x + 1
- mark(x): x
- mark#(x): x
Improved Usable rules
| g(active(X)) | → | g(X) | | mark(a) | → | active(a) |
| active(f(f(a))) | → | mark(f(g(f(a)))) | | f(active(X)) | → | f(X) |
| mark(g(X)) | → | active(g(mark(X))) | | g(mark(X)) | → | g(X) |
| f(mark(X)) | → | f(X) | | mark(f(X)) | → | active(f(X)) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(a))) | → | mark#(f(g(f(a)))) | | mark#(g(X)) | → | active#(g(mark(X))) |
| mark#(f(X)) | → | active#(f(X)) |
Rewrite Rules
| active(f(f(a))) | → | mark(f(g(f(a)))) | | mark(f(X)) | → | active(f(X)) |
| mark(a) | → | active(a) | | mark(g(X)) | → | active(g(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, a, active, mark
Strategy
Polynomial Interpretation
- a: 1
- active(x): x
- active#(x): 2x
- f(x): x
- g(x): 0
- mark(x): 2x
- mark#(x): 2x
Improved Usable rules
| g(active(X)) | → | g(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | f(mark(X)) | → | f(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| active#(f(f(a))) | → | mark#(f(g(f(a)))) |
Problem 6: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(g(X)) | → | active#(g(mark(X))) | | mark#(f(X)) | → | active#(f(X)) |
Rewrite Rules
| active(f(f(a))) | → | mark(f(g(f(a)))) | | mark(f(X)) | → | active(f(X)) |
| mark(a) | → | active(a) | | mark(g(X)) | → | active(g(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, a, active, mark
Strategy
There are no SCCs!
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(f(g(f(a)))) | | mark(f(X)) | → | active(f(X)) |
| mark(a) | → | active(a) | | mark(g(X)) | → | active(g(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, a, active, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |