YES
The TRS could be proven terminating. The proof took 3644 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (96ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor PolynomialLinearRange4iUR (1484ms).
| | – Problem 6 was processed with processor PolynomialLinearRange4iUR (1062ms).
| | | – Problem 7 was processed with processor PolynomialLinearRange4iUR (469ms).
| | | | – Problem 8 was processed with processor PolynomialLinearRange4iUR (476ms).
| | | | | – Problem 9 was processed with processor DependencyGraph (0ms).
| – Problem 5 was processed with processor SubtermCriterion (1ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | g#(mark(X)) | → | g#(X) |
| mark#(g(X)) | → | g#(X) | | active#(f(X)) | → | h#(f(X)) |
| f#(active(X)) | → | f#(X) | | active#(f(X)) | → | f#(X) |
| mark#(f(X)) | → | mark#(X) | | active#(f(X)) | → | mark#(g(h(f(X)))) |
| mark#(f(X)) | → | active#(f(mark(X))) | | mark#(g(X)) | → | active#(g(X)) |
| g#(active(X)) | → | g#(X) | | mark#(h(X)) | → | active#(h(mark(X))) |
| h#(active(X)) | → | h#(X) | | f#(mark(X)) | → | f#(X) |
| mark#(f(X)) | → | f#(mark(X)) | | active#(f(X)) | → | g#(h(f(X))) |
| mark#(h(X)) | → | mark#(X) | | mark#(h(X)) | → | h#(mark(X)) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
The following SCCs where found
| mark#(h(X)) → active#(h(mark(X))) | mark#(f(X)) → mark#(X) |
| mark#(h(X)) → mark#(X) | active#(f(X)) → mark#(g(h(f(X)))) |
| mark#(f(X)) → active#(f(mark(X))) | mark#(g(X)) → active#(g(X)) |
| f#(active(X)) → f#(X) | f#(mark(X)) → f#(X) |
| g#(active(X)) → g#(X) | g#(mark(X)) → g#(X) |
| h#(mark(X)) → h#(X) | h#(active(X)) → h#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | h#(active(X)) | → | h#(X) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| h#(mark(X)) | → | h#(X) | | h#(active(X)) | → | h#(X) |
Problem 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(h(X)) | → | active#(h(mark(X))) | | mark#(f(X)) | → | mark#(X) |
| mark#(h(X)) | → | mark#(X) | | active#(f(X)) | → | mark#(g(h(f(X)))) |
| mark#(f(X)) | → | active#(f(mark(X))) | | mark#(g(X)) | → | active#(g(X)) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 0
- active#(x): x
- f(x): 1
- g(x): 0
- h(x): 1
- mark(x): 2
- mark#(x): 1
Improved Usable rules
| g(active(X)) | → | g(X) | | h(mark(X)) | → | h(X) |
| f(active(X)) | → | f(X) | | g(mark(X)) | → | g(X) |
| f(mark(X)) | → | f(X) | | h(active(X)) | → | h(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(g(X)) | → | active#(g(X)) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(h(X)) | → | active#(h(mark(X))) | | mark#(f(X)) | → | mark#(X) |
| mark#(h(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
| active#(f(X)) | → | mark#(g(h(f(X)))) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 0
- active#(x): 0
- f(x): x
- g(x): 0
- h(x): x + 1
- mark(x): 2
- mark#(x): 2x
Improved Usable rules
| g(active(X)) | → | g(X) | | g(mark(X)) | → | g(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(h(X)) | → | active#(h(mark(X))) | | mark#(h(X)) | → | mark#(X) |
Problem 7: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(f(X)) | → | mark#(X) | | active#(f(X)) | → | mark#(g(h(f(X)))) |
| mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 0
- active#(x): 2
- f(x): 2x + 1
- g(x): 1
- h(x): 3
- mark(x): 0
- mark#(x): 2x
Improved Usable rules
| g(active(X)) | → | g(X) | | g(mark(X)) | → | g(X) |
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
| mark#(f(X)) | → | active#(f(mark(X))) | | active#(f(X)) | → | mark#(g(h(f(X)))) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): 2
- f(x): 3
- g(x): 2
- h(x): 0
- mark(x): 3
- mark#(x): x
Improved Usable rules
| g(active(X)) | → | g(X) | | g(mark(X)) | → | g(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(f(X)) | → | active#(f(mark(X))) |
Problem 9: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(f(X)) | → | mark#(g(h(f(X)))) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
There are no SCCs!
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Rewrite Rules
| active(f(X)) | → | mark(g(h(f(X)))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(g(X)) | → | active(g(X)) | | mark(h(X)) | → | active(h(mark(X))) |
| f(mark(X)) | → | f(X) | | f(active(X)) | → | f(X) |
| g(mark(X)) | → | g(X) | | g(active(X)) | → | g(X) |
| h(mark(X)) | → | h(X) | | h(active(X)) | → | h(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark, h
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |