YES
The TRS could be proven terminating. The proof took 809 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (71ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 was processed with processor PolynomialLinearRange4iUR (252ms).
| | – Problem 5 was processed with processor PolynomialLinearRange4iUR (109ms).
| | | – Problem 6 was processed with processor PolynomialLinearRange4iUR (141ms).
| | | | – Problem 7 was processed with processor DependencyGraph (6ms).
| – Problem 4 was processed with processor SubtermCriterion (0ms).
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| active#(g(X)) | → | mark#(h(X)) | | h#(mark(X)) | → | h#(X) |
| mark#(h(X)) | → | h#(X) | | mark#(g(X)) | → | g#(X) |
| g#(mark(X)) | → | g#(X) | | active#(c) | → | mark#(d) |
| active#(g(X)) | → | h#(X) | | mark#(h(X)) | → | active#(h(X)) |
| mark#(d) | → | active#(d) | | active#(h(d)) | → | mark#(g(c)) |
| mark#(g(X)) | → | active#(g(X)) | | mark#(c) | → | active#(c) |
| g#(active(X)) | → | g#(X) | | active#(h(d)) | → | g#(c) |
| h#(active(X)) | → | h#(X) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, active, mark, h
Strategy
The following SCCs where found
| active#(g(X)) → mark#(h(X)) | mark#(h(X)) → active#(h(X)) |
| active#(h(d)) → mark#(g(c)) | mark#(g(X)) → active#(g(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
| g#(active(X)) | → | g#(X) | | g#(mark(X)) | → | g#(X) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, 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) |
Problem 3: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(g(X)) | → | mark#(h(X)) | | mark#(h(X)) | → | active#(h(X)) |
| active#(h(d)) | → | mark#(g(c)) | | mark#(g(X)) | → | active#(g(X)) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): x
- active#(x): x + 1
- c: 0
- d: 1
- g(x): 2x
- h(x): x
- mark(x): 2x
- mark#(x): 2x + 1
Improved Usable rules
| g(active(X)) | → | g(X) | | h(mark(X)) | → | h(X) |
| g(mark(X)) | → | g(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:
| active#(h(d)) | → | mark#(g(c)) |
Problem 5: PolynomialLinearRange4iUR
Dependency Pair Problem
Dependency Pairs
| active#(g(X)) | → | mark#(h(X)) | | mark#(h(X)) | → | active#(h(X)) |
| mark#(g(X)) | → | active#(g(X)) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 3x + 3
- active#(x): 1
- c: 0
- d: 0
- g(x): 2
- h(x): 1
- mark(x): 2x
- mark#(x): x
Improved Usable rules
| h(mark(X)) | → | h(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
| active#(g(X)) | → | mark#(h(X)) | | mark#(h(X)) | → | active#(h(X)) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, active, mark, h
Strategy
Polynomial Interpretation
- active(x): 3x + 3
- active#(x): 2x
- c: 0
- d: 0
- g(x): 1
- h(x): 0
- mark(x): x + 3
- mark#(x): 0
Improved Usable rules
| h(mark(X)) | → | h(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:
| active#(g(X)) | → | mark#(h(X)) |
Problem 7: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| mark#(h(X)) | → | active#(h(X)) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, active, mark, h
Strategy
There are no SCCs!
Problem 4: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| h#(mark(X)) | → | h#(X) | | h#(active(X)) | → | h#(X) |
Rewrite Rules
| active(g(X)) | → | mark(h(X)) | | active(c) | → | mark(d) |
| active(h(d)) | → | mark(g(c)) | | mark(g(X)) | → | active(g(X)) |
| mark(h(X)) | → | active(h(X)) | | mark(c) | → | active(c) |
| mark(d) | → | active(d) | | 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: g, d, c, 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) |