YES
The TRS could be proven terminating. The proof took 3345 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (120ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (0ms).
| – Problem 4 was processed with processor PolynomialLinearRange4iUR (1691ms).
| | – Problem 6 was processed with processor PolynomialLinearRange4iUR (731ms).
| | | – Problem 7 was processed with processor PolynomialLinearRange4iUR (676ms).
| | | | – Problem 8 was processed with processor DependencyGraph (1ms).
| – Problem 5 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| c#(active(X)) | → | c#(X) | | active#(f(f(a))) | → | mark#(c(f(g(f(a))))) |
| g#(mark(X)) | → | g#(X) | | mark#(g(X)) | → | mark#(X) |
| mark#(g(X)) | → | g#(mark(X)) | | mark#(c(X)) | → | active#(c(X)) |
| mark#(a) | → | active#(a) | | f#(active(X)) | → | f#(X) |
| mark#(f(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
| g#(active(X)) | → | g#(X) | | active#(f(f(a))) | → | c#(f(g(f(a)))) |
| active#(f(f(a))) | → | f#(a) | | active#(f(f(a))) | → | f#(g(f(a))) |
| active#(f(f(a))) | → | g#(f(a)) | | mark#(g(X)) | → | active#(g(mark(X))) |
| mark#(c(X)) | → | c#(X) | | c#(mark(X)) | → | c#(X) |
| mark#(f(X)) | → | f#(mark(X)) | | f#(mark(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, a, active, mark
Strategy
The following SCCs where found
| f#(active(X)) → f#(X) | f#(mark(X)) → f#(X) |
| g#(active(X)) → g#(X) | g#(mark(X)) → g#(X) |
| active#(f(f(a))) → mark#(c(f(g(f(a))))) | mark#(g(X)) → mark#(X) |
| mark#(c(X)) → active#(c(X)) | mark#(g(X)) → active#(g(mark(X))) |
| mark#(f(X)) → mark#(X) | mark#(f(X)) → active#(f(mark(X))) |
| c#(active(X)) → c#(X) | c#(mark(X)) → c#(X) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| c#(active(X)) | → | c#(X) | | c#(mark(X)) | → | c#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, a, active, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| c#(active(X)) | → | c#(X) | | c#(mark(X)) | → | c#(X) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, 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 4: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(a))) | → | mark#(c(f(g(f(a))))) | | mark#(g(X)) | → | mark#(X) |
| mark#(c(X)) | → | active#(c(X)) | | mark#(g(X)) | → | active#(g(mark(X))) |
| mark#(f(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): 0
- active#(x): 0
- c(x): 0
- f(x): 2x
- g(x): x + 1
- mark(x): 0
- mark#(x): 2x
Improved Usable rules
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(g(X)) | → | mark#(X) | | mark#(g(X)) | → | active#(g(mark(X))) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(a))) | → | mark#(c(f(g(f(a))))) | | mark#(c(X)) | → | active#(c(X)) |
| mark#(f(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): 0
- active#(x): x
- c(x): 0
- f(x): 2
- g(x): 3
- mark(x): x
- mark#(x): 2
Improved Usable rules
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
| f(active(X)) | → | f(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:
| mark#(c(X)) | → | active#(c(X)) |
Problem 7: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(f(a))) | → | mark#(c(f(g(f(a))))) | | mark#(f(X)) | → | mark#(X) |
| mark#(f(X)) | → | active#(f(mark(X))) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, a, active, mark
Strategy
Polynomial Interpretation
- a: 0
- active(x): 0
- active#(x): 0
- c(x): 0
- f(x): 2x + 1
- g(x): 0
- mark(x): 0
- mark#(x): 2x
Improved Usable rules
| c(mark(X)) | → | c(X) | | c(active(X)) | → | c(X) |
The following dependency pairs are strictly oriented by an ordering on the given polynomial interpretation, thus they are removed:
| mark#(f(X)) | → | mark#(X) | | mark#(f(X)) | → | active#(f(mark(X))) |
Problem 8: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(f(f(a))) | → | mark#(c(f(g(f(a))))) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, a, active, mark
Strategy
There are no SCCs!
Problem 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(active(X)) | → | f#(X) | | f#(mark(X)) | → | f#(X) |
Rewrite Rules
| active(f(f(a))) | → | mark(c(f(g(f(a))))) | | mark(f(X)) | → | active(f(mark(X))) |
| mark(a) | → | active(a) | | mark(c(X)) | → | active(c(X)) |
| mark(g(X)) | → | active(g(mark(X))) | | f(mark(X)) | → | f(X) |
| f(active(X)) | → | f(X) | | c(mark(X)) | → | c(X) |
| c(active(X)) | → | c(X) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, c, 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) |