YES
The TRS could be proven terminating. The proof took 1946 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (82ms).
| – Problem 2 was processed with processor SubtermCriterion (1ms).
| | – Problem 5 was processed with processor SubtermCriterion (1ms).
| – Problem 3 was processed with processor SubtermCriterion (1ms).
| – Problem 4 was processed with processor PolynomialLinearRange4iUR (1259ms).
| | – Problem 6 was processed with processor PolynomialLinearRange4iUR (466ms).
| | | – Problem 7 was processed with processor DependencyGraph (0ms).
| | | | – Problem 8 was processed with processor PolynomialLinearRange4iUR (11ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(f(g(X), Y)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
| active#(f(g(X), Y)) | → | f#(X, f(g(X), Y)) | | mark#(g(X)) | → | g#(mark(X)) |
| mark#(g(X)) | → | mark#(X) | | f#(X1, active(X2)) | → | f#(X1, X2) |
| g#(active(X)) | → | g#(X) | | mark#(f(X1, X2)) | → | f#(mark(X1), X2) |
| f#(X1, mark(X2)) | → | f#(X1, X2) | | active#(f(g(X), Y)) | → | mark#(f(X, f(g(X), Y))) |
| mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) | | f#(active(X1), X2) | → | f#(X1, X2) |
| mark#(g(X)) | → | active#(g(mark(X))) | | f#(mark(X1), X2) | → | f#(X1, X2) |
| active#(f(g(X), Y)) | → | f#(g(X), Y) | | mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, 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#(f(X1, X2)) → active#(f(mark(X1), X2)) | active#(f(g(X), Y)) → mark#(f(X, f(g(X), Y))) |
| mark#(g(X)) → mark#(X) | mark#(g(X)) → active#(g(mark(X))) |
| mark#(f(X1, X2)) → mark#(X1) |
| g#(active(X)) → g#(X) | g#(mark(X)) → g#(X) |
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(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, 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 5: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| f#(X1, mark(X2)) | → | f#(X1, X2) | | f#(X1, active(X2)) | → | f#(X1, X2) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| f#(X1, mark(X2)) | → | f#(X1, X2) | | f#(X1, active(X2)) | → | f#(X1, X2) |
Problem 3: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, 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
| mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) | | active#(f(g(X), Y)) | → | mark#(f(X, f(g(X), Y))) |
| mark#(g(X)) | → | mark#(X) | | mark#(g(X)) | → | active#(g(mark(X))) |
| mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark
Strategy
Polynomial Interpretation
- active(x): 1
- active#(x): x
- f(x,y): 1
- g(x): 0
- mark(x): 1
- mark#(x): 1
Improved Usable rules
| f(active(X1), X2) | → | f(X1, X2) | | f(X1, mark(X2)) | → | f(X1, X2) |
| g(active(X)) | → | g(X) | | f(mark(X1), X2) | → | f(X1, X2) |
| f(X1, active(X2)) | → | f(X1, X2) | | 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#(g(X)) | → | active#(g(mark(X))) |
Problem 6: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(f(g(X), Y)) | → | mark#(f(X, f(g(X), Y))) | | mark#(f(X1, X2)) | → | active#(f(mark(X1), X2)) |
| mark#(g(X)) | → | mark#(X) | | mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): x
- f(x,y): 2x
- g(x): 2x + 1
- mark(x): x
- mark#(x): x + 2
Improved Usable rules
| f(active(X1), X2) | → | f(X1, X2) | | f(X1, mark(X2)) | → | f(X1, X2) |
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | g(active(X)) | → | g(X) |
| f(mark(X1), X2) | → | f(X1, X2) | | f(X1, active(X2)) | → | f(X1, X2) |
| mark(g(X)) | → | active(g(mark(X))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| 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(X1, X2)) | → | active#(f(mark(X1), X2)) | | mark#(g(X)) | → | mark#(X) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(f(g(X), Y)) | → | mark#(f(X, f(g(X), Y))) | | mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark
Strategy
The following SCCs where found
| mark#(f(X1, X2)) → mark#(X1) |
Problem 8: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| mark#(f(X1, X2)) | → | mark#(X1) |
Rewrite Rules
| active(f(g(X), Y)) | → | mark(f(X, f(g(X), Y))) | | mark(f(X1, X2)) | → | active(f(mark(X1), X2)) |
| mark(g(X)) | → | active(g(mark(X))) | | 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) | | g(mark(X)) | → | g(X) |
| g(active(X)) | → | g(X) |
Original Signature
Termination of terms over the following signature is verified: f, g, active, mark
Strategy
Polynomial Interpretation
- active(x): 0
- f(x,y): x + 1
- g(x): 0
- mark(x): 0
- 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:
| mark#(f(X1, X2)) | → | mark#(X1) |