TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (20ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (2ms), PolynomialLinearRange4iUR (988ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (4306ms), DependencyGraph (3ms), ReductionPairSAT (timeout)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| f#(s(s(s(s(s(s(s(s(x)))))))), y, y) | → | f#(id(s(s(s(s(s(s(s(s(x))))))))), y, y) |
Rewrite Rules
| f(s(s(s(s(s(s(s(s(x)))))))), y, y) | → | f(id(s(s(s(s(s(s(s(s(x))))))))), y, y) | | id(s(x)) | → | s(id(x)) |
| id(0) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: f, id, 0, s
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(s(s(s(s(s(s(s(x)))))))), y, y) | → | id#(s(s(s(s(s(s(s(s(x))))))))) | | id#(s(x)) | → | id#(x) |
| f#(s(s(s(s(s(s(s(s(x)))))))), y, y) | → | f#(id(s(s(s(s(s(s(s(s(x))))))))), y, y) |
Rewrite Rules
| f(s(s(s(s(s(s(s(s(x)))))))), y, y) | → | f(id(s(s(s(s(s(s(s(s(x))))))))), y, y) | | id(s(x)) | → | s(id(x)) |
| id(0) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: id, f, 0, s
Strategy
The following SCCs where found
| f#(s(s(s(s(s(s(s(s(x)))))))), y, y) → f#(id(s(s(s(s(s(s(s(s(x))))))))), y, y) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
Rewrite Rules
| f(s(s(s(s(s(s(s(s(x)))))))), y, y) | → | f(id(s(s(s(s(s(s(s(s(x))))))))), y, y) | | id(s(x)) | → | s(id(x)) |
| id(0) | → | 0 |
Original Signature
Termination of terms over the following signature is verified: id, f, 0, s
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed: