MAYBE
The TRS could not be proven terminating. The proof attempt took 1014 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (0ms).
| – Problem 2 was processed with processor SubtermCriterion (0ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (1ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (210ms), DependencyGraph (2ms), PolynomialLinearRange8NegiUR (301ms), DependencyGraph (1ms), ReductionPairSAT (355ms), DependencyGraph (1ms), SizeChangePrinciple (9ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| f#(s(x), x) | → | f#(s(x), round(s(x))) |
Rewrite Rules
| f(s(x), x) | → | f(s(x), round(s(x))) | | round(0) | → | 0 |
| round(0) | → | s(0) | | round(s(0)) | → | s(0) |
| round(s(s(x))) | → | s(s(round(x))) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, round
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| f#(s(x), x) | → | f#(s(x), round(s(x))) | | f#(s(x), x) | → | round#(s(x)) |
| round#(s(s(x))) | → | round#(x) |
Rewrite Rules
| f(s(x), x) | → | f(s(x), round(s(x))) | | round(0) | → | 0 |
| round(0) | → | s(0) | | round(s(0)) | → | s(0) |
| round(s(s(x))) | → | s(s(round(x))) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, round
Strategy
The following SCCs where found
| f#(s(x), x) → f#(s(x), round(s(x))) |
| round#(s(s(x))) → round#(x) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| round#(s(s(x))) | → | round#(x) |
Rewrite Rules
| f(s(x), x) | → | f(s(x), round(s(x))) | | round(0) | → | 0 |
| round(0) | → | s(0) | | round(s(0)) | → | s(0) |
| round(s(s(x))) | → | s(s(round(x))) |
Original Signature
Termination of terms over the following signature is verified: f, 0, s, round
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| round#(s(s(x))) | → | round#(x) |